The Impact of Company Leverage on the CAPM and Parametric … · CAPM Alpha and Beta coe cients for...

The Impact of Company Leverage on the

CAPM and Parametric Portfolio Construction

by

Stefano Dova

EDHEC Business School

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy (PhD)

Dissertation Committee

Abraham Lioui, PhD

Raman Uppal, PhD

Michael W. Brandt, PhD

mailto:[email protected]

http://www.something.net

c©Copyright by Stefano Dova, 2018

All Rights reserved

Abstract

This thesis is divided in two chapters, in which I analyze the impact of

company leverage on stock returns and optimal portfolios. In the first

chapter, I derive a CAPM for levered equity from the unlevered one-

factor CAPM (or asset CAPM), correcting for the presence of debt at

both the individual company and market level. I show that the levered

representation of the one-factor asset CAPM contains at least three

factors conditioning its beta to the market excess return: leverage,

debt maturity, and asset volatility of the firm relative to the market.

The conditioning factors are correlated to the Fama-French SML and

HML factors and give a theoretical explanation of why certain factor

models work. The second chapter uses the conditioning factors of

the first paper as the characteristics to run the parametric portfolio

approach by Brandt, Santa-Clara and Valkanov and extend it to a

continuous time setting. I 1) extract characteristics from a portfolio of

credit-risky single stocks, 2) solve the dynamic programming problem

for portfolio weights, and 3) show that accounting for credit risk in

a multi-period framework achieves better Sharpe ratios than naive

strategies such as EW and VW as well as results that are comparable

to those of portfolios built on size, book-to-market, momentum and

gross profitability.

Contents

List of Figures ix

List of Tables x

1 From the CAPM to Fama-French: A road paved with leverage 1

1.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Risky Asset and Firm Value . . . . . . . . . . . . . . . . . 7

1.1.2 Equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.3 The Market . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.4 The Capital Structure CAPM . . . . . . . . . . . . . . . . 17

1.2 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2.2 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . 24

1.2.3 Cross-Sectional Analysis: levered and unlevered Betas . . . 26

1.2.4 Correlation Analysis . . . . . . . . . . . . . . . . . . . . . 40

1.2.5 Statistical Properties of Quarterly Excess Returns . . . . . 42

1.2.6 Fama-Macbeth Regression Analysis . . . . . . . . . . . . . 44

1.2.7 Empirical Results: Testing the CS-CAPM Model . . . . . 48

1.2.7.1 Equity Beta: A function of Asset Beta, Leverage,

and Credit Risk . . . . . . . . . . . . . . . . . . . 49

1.2.7.2 Testing Excess Returns with the CS-CAPM . . . 53

1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Bibliography 60

iv

CONTENTS

Appendices 66

.1 Proof of Propositions . . . . . . . . . . . . . . . . . . . . . . . . . 67

2 Dynamic Parametric Portfolio Policies 69

2.1 The Model for Returns . . . . . . . . . . . . . . . . . . . . . . . . 73

2.1.1 Single Stocks . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.1.2 Stock Portfolio . . . . . . . . . . . . . . . . . . . . . . . . 75

2.1.3 The Market . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.2 PPP Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.2.1 PPP implementation . . . . . . . . . . . . . . . . . . . . . 78

2.2.2 The Characteristics . . . . . . . . . . . . . . . . . . . . . . 80

2.2.3 The State Variables . . . . . . . . . . . . . . . . . . . . . . 82

2.3 Portfolio Choice Model . . . . . . . . . . . . . . . . . . . . . . . . 85

2.3.1 The Utility Function . . . . . . . . . . . . . . . . . . . . . 85

2.3.2 The Dynamic Budget Constraint . . . . . . . . . . . . . . 86

2.3.3 The Bellman Equation . . . . . . . . . . . . . . . . . . . . 88

2.3.4 Solving for the Value Function . . . . . . . . . . . . . . . . 90

2.3.4.1 A guess for the function . . . . . . . . . . . . . . 90

2.3.4.2 Implicit Portfolio Weights . . . . . . . . . . . . . 90

2.3.4.3 Solving for G(t) . . . . . . . . . . . . . . . . . . . 91

2.3.4.4 Explicit Portfolio Weights . . . . . . . . . . . . . 96

2.3.5 Comparative Statics . . . . . . . . . . . . . . . . . . . . . 99

2.4 Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . 101

2.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.4.2 Timeline and Variables . . . . . . . . . . . . . . . . . . . . 102

2.4.3 Performance Statistics . . . . . . . . . . . . . . . . . . . . 104

2.4.4 One-Period Optimisation . . . . . . . . . . . . . . . . . . . 106

2.4.5 Intertemporal Optimization . . . . . . . . . . . . . . . . . 111

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Bibliography 120

v

CONTENTS

Appendices 124

.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

.2 Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

.3 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

.4 Results including financial stocks . . . . . . . . . . . . . . . . . . 130

.5 Empirical evidence of market structure . . . . . . . . . . . . . . . 135

vi

To my Family

Acknowledgements

I would like to thank my advisor, Prof. Raman Uppal, and the entire

faculty at EDHEC, for the help and encouragement to achieve the best

possible results. I am grateful to my employer, Mediobanca SpA, for

supporting me along the way. I thank my friend Dr. David Mascio for

his tips around dealing with the emotional impact of writing a thesis

(while he was writing one himself!). I thank my parents for teaching

me the value of self-sacrifice to learn new things. Finally, I am forever

grateful to my family that gave me the time and emotional strength

to write this thesis.

List of Figures

1.1 Equity Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Equtiy return as a function of debt and recovery . . . . . . . . . . 14

1.3 Equity return as a function of debt and maturity . . . . . . . . . 15

1.4 Debt Drivers: Asset Beta at 0.8 . . . . . . . . . . . . . . . . . . . 20

1.5 Debt Drivers: Asset Beta at -0.5 . . . . . . . . . . . . . . . . . . . 20

1.6 Risk free rate and debt duration . . . . . . . . . . . . . . . . . . . 26

1.7 Excess return for asset and equity . . . . . . . . . . . . . . . . . . 26

1.8 Rolling asset and equity CAPM p-values . . . . . . . . . . . . . . 28

1.9 Normality test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.10 Autocorrelation test . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.11 Rolling Fama-MacBeth . . . . . . . . . . . . . . . . . . . . . . . . 48

1.12 Regressor p-values . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.1 Theta weight of a characteristic as a function of γ. . . . . . . . . . 100

2.2 Theta weight and volatility . . . . . . . . . . . . . . . . . . . . . . 100

2.3 Theta weight and market . . . . . . . . . . . . . . . . . . . . . . . 100

2.4 Theta weight and maturity . . . . . . . . . . . . . . . . . . . . . . 100

ix

List of Tables

1.1 Asset Excess Return Statistics . . . . . . . . . . . . . . . . . . . 25

1.2 Equity Excess Return Statistics . . . . . . . . . . . . . . . . . . . 25

1.3 Summary Asset and Equity CAPM Estimates . . . . . . . . . . . 27

1.4 Equally Weighted - Beta Univariate Portfolio Analysis . . . . . . 30

1.5 Value-Weighted - Beta Univariate Portfolio Analysis . . . . . . . . 31

1.6 Beta and Size Sorts — Assets . . . . . . . . . . . . . . . . . . . . 33

1.7 Beta and Size Sorts — Equity . . . . . . . . . . . . . . . . . . . . 34

1.8 Beta and Book-to-Market Sorts — Assets . . . . . . . . . . . . . . 34

1.9 Beta and Book-to-Market Sorts — Equity . . . . . . . . . . . . . 35

1.10 Beta and Leverage Sorts — Assets . . . . . . . . . . . . . . . . . 35

1.11 Beta and Leverage Sorts — Equity . . . . . . . . . . . . . . . . . 36

1.12 Beta and Debt Duration Sorts — Assets . . . . . . . . . . . . . . 36

1.13 Beta and Debt Duration Sorts — Equity . . . . . . . . . . . . . . 37

1.14 Beta and Volatility Sorts — Assets . . . . . . . . . . . . . . . . . 37

1.15 Beta and Volatility Sorts — Equity . . . . . . . . . . . . . . . . . 38

1.16 Book-to-Mkt and Leverage Sorts — Equity . . . . . . . . . . . . . 39

1.17 Size and Debt Duration Sorts — Equity . . . . . . . . . . . . . . 40

1.18 Factor Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.19 Factor Correlation P-value . . . . . . . . . . . . . . . . . . . . . . 42

1.20 Cross-Sectional average Fama-MacBeth Regression . . . . . . . . 46

1.21 Equity Beta Regressions — Individual Stocks . . . . . . . . . . . 50

1.22 Equity Beta Regressions — Industry Portfolios . . . . . . . . . . . 52

1.23 CS-CAPM Regressions — Single Stocks . . . . . . . . . . . . . . . 55

1.24 CS-CAPM Regressions — Industry Portfolios . . . . . . . . . . . 58

x

LIST OF TABLES

2.1 Portfolio statistics for myopic individuals . . . . . . . . . . . . . . 107

2.2 Certainty equivalent . . . . . . . . . . . . . . . . . . . . . . . . . 111

2.3 Portfolio statistics for intertemporal optimisation . . . . . . . . . 113

2.4 Certainty equivalent for low risk aversion . . . . . . . . . . . . . . 116

5 Portfolio statistics for myopic individuals including financial stocks 131

6 Portfolio statistics for intertemporal optimization including finan-

cial stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7 State Variables Correlation . . . . . . . . . . . . . . . . . . . . . 135

8 P-Value of State Variables Correlation . . . . . . . . . . . . . . . 135

9 Characteristic Ptf. Corr. . . . . . . . . . . . . . . . . . . . . . . . 136

10 P-Value of Characteristic Ptf. Corr. . . . . . . . . . . . . . . . . 136

xi

1

From the CAPM to

Fama-French: A road paved with

leverage

This paper develops a conditional CAPM model for levered equity starting from

a one-factor unlevered equity CAPM for asset returns. The need to start mod-

eling from asset (or unlevered equity) returns is driven by the existence of a

number of capital structure characteristics that affect the distribution of levered

equity returns making them non linear. The CAPM relies on asset payoffs having

an elliptical distribution or agents having mean-variance preferences over those

payoffs, Berk (1997), and, most importantly, on identifying the equilibrium, tan-

gency portfolio as the market. Now, because traded securities are not normally

distributed, the CAPM fails when tested on levered equity because it does not

measure the right underlying priced risk. The idea is that every firm consists of

real assets whose payoffs are uncertain and lognormal, the sum of these assets,

not the securities written on them, should be taken as the market portfolio in the

CAPM. Securities are just, as pointed out by Merton (1974), non linear payoffs

1

on those assets: stocks a long call, and bonds are a short put on company asset

returns.

Leverage drives the in-the-moneyness of stock returns with respect to the

underlying firm asset returns. In other words, given a certain capital structure,

the higher the leverage, the lower the delta of the option: when a company has a

higher leverage (and credit risk), it will have to return a higher premium for its

levered equity. As credit risk approaches distress, stock values tend towards zero,

and erratic return scenarios are possible, while the actual return on the real assets

is best described by debt or default claims. This idea is old in finance, dating

back to Modigliani and Miller (1958), and applied to both individual firms and

the market. Correcting for the impact of leverage, helps see the CAPM in a new

light.

I adopt a Merton model framework to describe the dynamics of equity and

debt at the firm and at the aggregate level. My model is consistent with much of

the literature on structural credit risk and adjusts returns based on the capital

structure of the underlying company. Examples of recent empirical literature on

the importance of financial leverage and distress to capture anomalies in the cross-

section of stock returns are by Vassalou and Xing (2004), Garlappi and Yan (2011)

and Avramov et al. (2013). Further studies develop theoretical models capturing

the relevance of firm asset dynamics when describing expected equity returns:

Berk et al. (1999) develop a model that induces anomalies on security returns,

like momentum and book-to-market by looking at firm’s investment options. Berk

et al. (2004) expand on their earlier work by proving that investment risk at the

asset level is partly non-diversifiable and requires a risk premium. Carlson et al.

(2004) further develop this idea by introducing operating leverage and frictions

2

and Bhamra et al. (2009), with their structural-equilibrium model, suggest that

leverage is directly related to the risk premium of levered stocks. Finally, Choi

(2013) shows empirically how leverage and asset risk affect equity risk. Most of

these studies focus on the firm and do not recognize the interactions of the firm

with the aggregate market.

Empirically, multi-factor models improve on the CAPM by including addi-

tional relevant variables to explain the cross-section of returns but fail to rec-

ognize that using a market equity index as the proxy for the market factor is

inappropriate because it does not capture the measure of credit risk intrinsic in

the overall economy. Measuring the CAPM as a single stock versus the market

equity index excess returns, I would be making two mistakes at once on both

the dependent and independent variable: unless market levered equity has the

same delta on the underlying production assets as the single stock, the Betas

I measure are bound to be biased and an alpha or unexplained systematic re-

turn component will emerge in the regression. My conditional model proposes a

theoretical explanation of the conditioning factors affecting the market portfolio

representation.

Notable examples of papers on the Beta bias due to the omission of market

leverage are the works by Ferguson and Shockley (2003) and Aretz and Shackleton

(2011), where most of the attention is focused on the omission of debt dynamics

from the market factor (but not from the individual firm). The main argument

of these papers is that, by omitting debt in the market return specification, the

researcher is biasing Beta downward so they correct by unlevering market equity.

The uncertainty on the direction of the bias and challenges faced in empirical

testing by these papers are due, in my view, to the omission of debt dynamics

3

from the individual firm.

The empirical literature on asset pricing, from the Fama and French (1993)

three-factor model to the recent Hou et al. (2015) or Fama and French (2016)

five-factor models, has well recognized the issue of a failing single factor or market

Beta model. One line of research in the last two decades has been to try and

identify the anomalies of the classical CAPM escaping the market Beta factor and

to capture them in a broadened multi-factor framework or through firm charac-

teristics. Accruals, net share issues, momentum, and volatility are a few of a long

list of eighty anomalies identified and summarized in a recent paper by Chordia

et al. (2015). Their paper also represents a first attempt to bridge the literature

on empirical factors with the growing focus on firms’ individual characteristics.

Keim and Stambaugh (1986) already saw that bond as well as stock prices played

a role in predicting equity returns. Jagannathan and Wang (1996) saw that credit

risk was a good proxy to drive the conditional market risk premium. Avramov

et al. (2013) study anomalies in the context of financial distress and find that

many anomalies are strongly related to distress. Few theoretical contributions

have been made to analytically account for credit risk as an important factor

influencing equity and market returns. One contribution is by Eisdorfer et al.

(2012), who build the option to default by shareholders as an endogenous de-

terminant of equity price behavior and the paper by Vassalou and Xing (2004)

that applies the Merton (1974) option framework to build an analytical measure

of credit risk to be used as a factor. I add to this literature deriving a simple

levered CAPM representation that reconciles the general equilibrium and APT

theories while relating the capital structure of the firm to that of the market.

4

1.1 The Model

1.1 The Model

I start by applying the CAPM framework to production assets and then derive

from it equity security dynamics consistent with credit risk as a state variable.

My initial CAPM representation defines the excess return on a generic individual

risky unlevered equity (or production asset1) as proportional to the excess return

on the market risky unlevered equity:

Et(ri,t+dt)− rf t+dt = β[Et(rm,t+dt)− rf t+dt]. (1.1)

where ri indicates gross returns, β = Cov(ri, rm)/V ar(rm), and everything is

based on information at time t.2 The CAPM was conceived to represent the

relationship of proportionality between the entire economy and one specific firm

or industry sector as an aggregate of firms. It relates firm production processes,

not non linear securities: it is about one real asset’s (log)normal returns in relation

to the market’s (log)normal returns.

Ferguson and Shockley (2003) show that the Beta of individual equity should

be calculated using the market asset (and not the equity index), but their model

should also be amended to take into account credit risk of the individual firm

because the behavior of firm leverage changes through time, hence the need to

take it into account. Other important elements consider here are the recovery

factor and the term structure of debt, crucial also to understanding why certain

industries allow for more leverage than others.

1I use unlevered equity, production asset, real asset and simply asset as synonyms.2I assume everything occurs at time t and I will drop the time subscript. In addition, I will

assume r rates are instantaneous and will not use dt unless in the context of a larger expressionthat requires it for sake of clarity.

5

1.1 The Model

In this section, I model both the firm and market asset dynamics indepen-

dently, starting from the unlevered equity level to derive the true asset Betas.

I then use the asset Betas to describe the non linear behavior of levered equity

returns as a function of the asset Beta, leverage, recovery and debt maturity.

This specification reconciles the one-factor CAPM model for real assets with a

conditional CAPM model for securities (equity, in this case).

Assumptions

1. Firms consist of equity and debt only: I simplify data to accommodate this

definition.

2. Debt maturity is allowed to change deterministically through time.

3. I assume that Debt consists of coupon paying instruments with a fixed

coupon rate.

4. Shareholders who manage the company can make credible threats to debt

holders and achieve debt forgiveness.

5. Liquidation occurs when the value of assets is lower than a critical value,

which is, in general, lower than the total liabilities: equity often loses control

when debt is already highly risky.

6. When a company is liquidated, it incurs loss of productivity and other costs

are idiosyncratic to the firm or industry: these costs are incurred even if

the company is sold in distress to a third party.

6

1.1 The Model

1.1.1 Risky Asset and Firm Value

Companies’ unlevered equity returns are what should feed, under my assump-

tions, the CAPM equation (1.1). I assume assets have a lognormal distribution:

dAiAi

= (µai − δai)dt+ σaidBi, (1.2)

= (rf − δai)dt+ σaidB∗i . (1.3)

where µ indicates the physical drift (gross of any cash payout δ)1 for the assets of

firm i, r is the net real risk-free interest rate and σ the volatility of the production

asset process. B is a Wiener-process, and ∗ indicates the risk-neutral probability

measure.

Following Leland (1998), I model the firm value at time T , when debt expires

and either refinancing or distress occur, as the value of assets less the cost of liq-

uidation contingent on the actual liquidation probability.2 By default I mean the

moment when the firm fails to pay any of its existing indebtedness but production

continues, whereas bankruptcy or liquidation occur when the firm is liquidated

and production stops or the firm is sold in distress and production is impaired. K

is the fraction of total debt that indicates the liquidation threshold of a company

or industry: the value of K represents a proxy of how much debt a company can

take depending on its asset composition and business dynamics. In the KMV

model for instance, see Crosbie (2003), K is assumed to be the sum of all short-

term debt plus some long-term debt, depending on the industry. ρ is the cost

of liquidation, i.e., the fraction of assets lost when the company is liquidated: it

1This is typically the average coupon on outstanding debt plus the net share sales or repur-chases. To simplify formulas, I omit the net cash payout δ in formulas.

2I use liquidation and bankruptcy as synonyms.

7

1.1 The Model

is a function of recovery for the firm or industry sector. The firm’s value can

therefore be represented in terms of asset values before and after bankruptcy:

VT = (1− ρ)AT1{AT<K} + AT1{AT>K}. (1.4)

Expression (1.4) tells us that the firm value is equal to the asset value only when

debt is minimal and the firm is far from bankruptcy. So to appropriately study

the dynamics of a levered firm, it is crucial to derive the determinants of K that

lead creditors to liquidate a firm or renegotiate its debt away from the natural

strike level that should be the debt face value at maturity, AT = D. Finding the

value of K is fundamental to deriving the asset value under our framework.

Mella-Barral and Perraudin (1997) analyze the negotiation process between

equity and debt holders that leads to the determination of the optimal liquidation

point. Their conclusion is that, when shareholders can make credible threats to

debt holders, the optimal liquidation point coincides with the recovery value of

the assets. I am assuming that selling the company as a going concern achieves

the same result as liquidation, i.e., a cost of ρAT proportional to the current asset

value. So the optimal liquidation point K is when the amount of debt forgiveness

necessary to continuing operations in the hands of current shareholders is higher

than the proportional expected cost of liquidation: Et(DT − AT ) > Et(ρAT ).1

Solving this inequality for A, I obtain the level of assets K at which debt-holders

liquidate the company and shareholders lose control as a function of the level and

1E indicates the expectation operator.

8

1.1 The Model

maturity of debt, recovery, and asset growth rate:

K = h(D, ρ, µ) =DT

(1 + ρ)eµa(T−t) . (1.5)

The optimal level of liquidation is directly proportional to the level of debt and

inversely proportional to the cost of liquidation. Naturally, when the time to

maturity of debt moves, the strike level decreases. I use this in my model to

calculate levered equity dynamics as the strike of equity option on the company’s

assets.

1.1.2 Equity

The literature provides a number of structural and reduced-form alternatives

to modeling credit risk: I have chosen to run my analysis following Merton (1974).

I model equity as a call option on the firm’s value and take liquidation costs into

account by adopting K as the strike of the equity option:

Ei = max(Ai,T −Ki, 0),

= Aie−d(T−t)N(d1)−Kie

−rf (T−t)N(d2), (1.6)

9

1.1 The Model

with T being the maturity of debt instruments and d being the dividend rate.1

The terms d1 and d2, under the risk-neutral measure, are equivalent to:

d1 =lnAi

Ki+(rf + 1

2σ2a

)(T − t)

σa√T − t

, (1.7)

d2 = d1− σa√T − t. (1.8)

I call d2 the Risk-Neutral Distance to Liquidation (RNDL) and, provided I knew

how to map the physical and risk-neutral asset return drift, I could rewrite it

under the physical probability measure to obtain the Physical Distance to Liqui-

dation (DL) :

DL =ln(Ai)− ln(Ki) +

(µa − 1

2σ2a

)(T − t)

σa√T − t

(1.9)

is an expression that indicates the distance between the asset value and the

liquidation value taking asset growth into account and standardizing by the asset’s

volatility: it is a measure of how far the firm is from liquidation. The cumulative

density of this number yields the general probability of default. I can define

equity as a function of the Distance to Liquidation or, implicitly, its credit risk,

where RNDL is itself a function of the asset, nominal debt, and recovery:

Ei = f(RNDL) = f(Ai, rf , σa, Ki, T ). (1.10)

1To simplify formulas, I assume that the dividend rate is zero, but I do account for it inthe empirical work.

10

1.1 The Model

Following Jones et al. (1984), I also define the instantaneous equity volatility as

a function of the asset’s volatility, both assumed to be constant at this time:

σe = g(σa, Ai, Ei, N(d1)) =AiEiN(d1)σa. (1.11)

Combining equations (1.10) and (1.11), I could solve for two unknowns iteratively

to find values of Ai and then calculate µ from the time series of Ai’s. Nonetheless,

I will follow the alternative approach suggested in Eom et al. (2004), working out

the value of assets simply by adding equity and nominal debt and then determin-

ing the asset volatility inverting the g function:

σa = g−1(σe, Ai, Ei, N(d1)) (1.12)

Applying Ito’s lemma to (2.2), I obtain the individual stock dynamics, an

expression of the instantaneous expected return on equity:

re ∼dEiEi

=1

Ei

[∂Ei∂t

dt+∂Ei∂Ai

dAi +1

2

∂2Ei∂2Ai

(dAi)2

]=

1

Ei

[Θidt+ ∆iAi [rfdt+ σadB

∗] +1

2Γiσ

2aA

2i dt

]=

Θi

Eidt+

1

Ei

[∆iAirf +

1

2Γiσ

2aA

2i

]dt+

∆i

EiσaAidB

∗, (1.13)

and taking expectations, under the risk neutral measure, for every individual firm

i,

E∗(rei,t+dt) ∼ E∗(rei) =Θi

Eidt+

[∆iAiEirf +

1

2Γiσ

2a,i

A2i

Ei

]dt. (1.14)

11

1.1 The Model

This representation of the equity dynamics shows that, when the capital struc-

ture of the company changes (equity delta or leverage amount), the equity return,

traditionally used in CAPM tests on levered equity, will move away from the true

asset return, thereby distorting the measurement of Beta.

Figure (1.1) gives a 3D representation of the relationship between equity value,

equity return, and debt. Fixing the initial values of assets, recovery, time to

maturity, and risk-free return, I calculate equity value and return as a function

of debt. In this example, time to maturity is five years, the risk-free rate is 1.2%,

and the initial value of assets is standardized at 100. Equity value is convex in

debt while equity returns are concave, and as debt increases to 100% they move

toward negative.

12

1.1 The Model

Figure 1.1: Equity return on the left axis and equity value on the right one. Inthe absence of a stochastic element, the five year equity return in the case of nodebt is equivalent to the cumulative risk-free rate.

Figure (1.2) plots the return on equity under the same assumptions as in

figure (1.1) while allowing for recovery to change. The higher the recovery is the

higher is the return, given a level of debt. This has a massive impact on certain

industries that have a solid and valuable hard asset base and, therefore, a high

recovery rate.

13

1.1 The Model

Figure 1.2: Firm equity return as a function of both debt and recovery.When recovery is equal to zero we have the concave slope of figure (1.1).

Figure (1.3) plots the return on equity under the same assumptions as in

figure (1.1) while allowing for the time to maturity of debt to change. Time to

maturity of debt impacts returns by increasing them when the drift of the asset

process is positive. At the same time, though, it also affects overall volatility

which increases with the square of time to maturity, affecting the probability of

liquidation. As firm debt increases, this effect bends the linearity of the return

drift. This explains why the surface twists for high debt levels.

14

1.1 The Model

Figure 1.3: Firm equity return as a function of both debt and debt maturity.

1.1.3 The Market

A similar model applies to the market. The market is the aggregate of all

return producing assets, and, if I assume, simplistically, that it is just a conglom-

erate of individual firms, then I can say its asset follows a lognormal distribution:

dM

M= µmdt+ σmdBi, (1.15)

= rfdt+ σmdB∗i . (1.16)

Again µ and σ indicate the drift and volatility of the market asset process, and

they will be derived from observed market equity data: this way I do not need to

specify how the aggregation of the stochastic processes of individual firms occurs.

15

1.1 The Model

Following the same reasoning of the previous paragraph,1 I conclude that there

is a representation of the market equity dynamics similar to (1.14):

E∗(rem,t+dt) ∼ E∗(rem) =Θm

Emdt+

[∆m

M

Emrf +

1

2Γmσ

2m

M2

Em

]dt. (1.17)

If my assumptions are correct, by using the levered market equity index return as

a proxy for the one-factor CAPM market return, I misrepresent the factor and the

mistake gets larger as the credit risk of the market increases. By regressing indi-

vidual asset returns against the equity market proxy, when the market is highly

levered, I will find little correlation because the right end side of the equation is

not properly representing the market asset. I need to find the unlevered market

dynamics similarly to what I do for the individual companies. Once I have a time

series for the market and individual unlevered returns, I can run a regression on

the basis of equation (1.1) and find the true unlevered Betas.

To find the value of Ki for the market, I have to not only specify the market

level of debt (as an aggregate of the debt of all market firms) but also indicate the

appropriate recovery value and debt maturity. While for the individual firm this

is a relatively easy task, ρ takes a very unintuitive meaning for the market assets;

it becomes much more complex to understand how to aggregate the liquidation

costs of all market firms. As the entire market approaches distress it is expected

that liquidation costs will increase at a systemic level for at least two reasons:

the buyers of distressed assets are more likely to also be affected by distress, and

the cost of liquidating an asset when many others are also being sold is likely

to be greater. Market recovery is very complex, and I choose to take a practical

1I am only changing notation on formulas, so for proofs, please refer to section B, on theequity, above

16

1.1 The Model

approach by assuming that the base case recovery is similar to what is used in

practice for government recovery rates (approximately 20%). In reality, market

recovery will be a function of individual marginal recovery rates, according to a

correlation structure that increases when a systemic default event occurs.

1.1.4 The Capital Structure CAPM

Using accounting and financial data, I can calculate the greeks and leverage

for both the individual firm and the market and use the CAPM to explain equity

price movements conditional on those greeks. The greeks, in turn, are functions

of credit risk and recovery. The first step to explain equity returns is to write the

CAPM for assets and express individual unlevered equity returns as their Beta

to the unlevered market:

E(rai)− rf = βai [E(ram)− rf ],

E(dAi) = Aiβai [E(ram)− rf ] + Airf . (1.18)

Once I have worked out production asset dynamics and Betas, I can use them to

determine the levered equity dynamics, thanks to expression (1.14):

E(rei) = E

(1

Ei

[∂Ei∂t

dt+∂Ei∂Ai

dAi +1

2

∂2Ei∂2Ai

(dAi)2

]),

=Θi

Eidt+ ∆i

AiEi

[βai [E(ram)− rf ] + rf ] +

+ E

(1

2

ΓiEi

(Ai [βai [(ram)− rf ] + rf ])2

),

=Θi

Eidt+ ∆i

AiEi

[βai [E(ram)− rf ] + rf ] +1

2

ΓiEiAi

2β2aiσ2am . (1.19)

17

1.1 The Model

This expression is more articulated than the standard CAPM but conveys a

simple message: equity returns are a function of the firm production asset Betas

adjusted for the firm’s credit risk, as determined by the firm’s leverage, debt

maturity, and volatility. This can be better seen by rearranging the expression

as the traditional CAPM:

E(rei)− rf︸︷︷︸Equity Excess Ret.

=

Di

Eirf︸︷︷︸

Leverage

+Θi

Ei︸︷︷︸Debt Maturity

+ΓiAi

2σ2ai

2Ei︸︷︷︸Volatility

dt︸︷︷︸

Additional Factors

+

+ ∆AiEiβai︸︷︷︸

MarketAssetBeta

[E(ram)− rf ]︸︷︷︸Market Factor

, (1.20)

The above representation also highlights that the Beta of the single stock to the

market asset returns is a function of the in-the-moneyness of the equity call option

on the company assets as well as the level of leverage of the company.

The model nests the linear CAPM as a special case: when a firm is entirely

capitalized with equity and the market is not levered, its theta is zero, its delta is

one, and its gamma is zero, so the relationship between its production assets and

equity returns becomes linear. The model also nests the Ferguson and Shockley

(2003) CAPM, when the market is levered and the firm is not. In general though,

this is a quadratic relationship that relates equity to market assets and market

assets volatility. In fact, the quadratic term can be demonstrated to be the market

asset volatility by construction.

Expression (1.19) links observable firm-levered equity with unobservable mar-

ket assets: I can say something about the impact of the firm leverage on its

levered equity return but still nothing about how it relates to the market. To un-

18

1.1 The Model

derstand the real dynamics of firm and market leverage together, I need to reduce

the expression to obtain observable variables on both sides of the equation.

PROPOSITION 1. Capital-Structure CAPM (CS-CAPM) — I can write ex-

pression (1.19) in a format similar to the CAPM and using only observable vari-

ables:

E(rei)−AiEirf

Θi

Eidt+

∆i

EiAiβai [E(ram)− rf ] +

ΓiAi2σ2

ai

2Eidt,

=

[Θi

Ei+

ΓiA2iσ

2ai

2Ei−B

(Θm

Em+

ΓmM2σ2

am

2Em

)]dt+

+B

[E(rem)− ∆mM

Emrf

], (1.21)

with B = βai∆i

∆m

AiEi

EmM

. Notably, asset returns, although unobservable and un-

der certain conditions, are invariable irrespective of the capital structure (see

Modigliani and Miller (1958)). When the CAPM is estimated using levered eq-

uity as regression variable, the equity Beta will be a function of the asset Beta,

the relative leverage of the firm, and the market and the ratio of their probability

of default. In particular, all else being equal:

• For positive asset Betas: Market leverage higher than firm leverage induces

lower equity Betas and vice-versa: when both leverage measures are similar,

they offset each other and Beta estimates are stable across leverage levels.

• For negative asset Betas: Market leverage higher than firm leverage induces

higher equity Betas and vice-versa: when both leverage measures are similar,

they offset and Beta estimates are stable across leverage levels.

• Equity excess returns on both sides of the equation are adjusted for leverage

19

1.1 The Model

with respect to the risk-free rate as well.

Interpreting the overall directional impacts of (1.21) is not trivial because of

the interaction of leverage ratios and deltas of both the dependent and indepen-

dent variables. To illustrate the relative effect of market and firm debt on the

estimates of equity Beta (here called B), I plot, in figure (1.4) and (1.5), its value

as a function of market and firm leverage. There is an interesting dynamic when

the market and the firm have similar debt levels (i.e., it moves on the diagonal

line through zero), the value of B hovers around its true asset Beta value even if

we are looking at levered equity.

Figure 1.4: Starting Asset Beta= 0.8: market and firm debt asdrivers of the equity return Beta.

Figure 1.5: Starting Asset Beta= -0.5: market and firm debt asdrivers of the equity return Beta.

Proposition 1 can be rearranged to obtain a conditional model directly compa-

rable to the classical Sharpe and Lintner CAPM with the conditioning variables

appearing as Additional Factors:

20

1.1 The Model

COROLLARY 1. A representation of 1.21 as a conditional model:

E(rei |I)− AiEirf =

Θi

Eidt+

∆i

EiAiβai [E(ram)− rf ] +

ΓiAi2σ2

ai

2Eidt (1.22)

can be written as:

E(rei)− rf︸︷︷︸Equity Excess Ret.

=

=

Di

Eirf −

B(∆mM − Em)

Emrf︸︷︷︸

Leverage

+Θi

Ei− BΘm

Em︸︷︷︸Debt Maturity

+ΓiAi

2σ2ai

2Ei−BΓmM

2σ2am

2Em︸︷︷︸Volatility

dt︸︷︷︸

Additional Factors

+

+ B︸︷︷︸MarketEquityBeta

[E(rem)− rf ]︸︷︷︸Market Factor

, (1.23)

with B = βai∆i

∆m

AiEi

EmM

. The main characteristics of this model are:

1. The CS-CAPM has additional factors with respect to a CAPM and resem-

bles an APT. These factors are expected to capture most of the systematic

effects left unexplained by the CAPM (or one-factor models). The additional

factors are:

(a) Leverage: this term pays a premium proportional to the risk-free rate

adjusted by the difference between leverage of the firm and the market.

(b) Debt Maturity: the difference between the debt maturity of the firm

and the market weighted by equity market values.

(c) Volatility: the difference between the firm and market volatility weighted

by a combination of Γ, asset values and equity values.

21

1.1 The Model

2. The higher the firm leverage is relative to market, the higher the expected

excess return is.1

3. A longer debt maturity at the firm rather than in the market predicates a

lower excess return (the reason being that Θ is negative).

4. All else being equal, a higher asset volatility at the firm, rather than in the

market drives a higher excess return.

5. B encompasses the asset β as well as a weighting based on the relative

leverage of firm and market and the relative probability of default expressed

implicitly by the delta of the equity options.

6. Finally, the presence of B in the additional factors gives some insight into

traditional estimation issues for Betas, as they often seem to be structurally

correlated to the intercept element of one-factor models.

Equation 1.23 has been derived from the CAPM model applied to unlevered

equity of equation 1.18 by introducing leverage and capital structure character-

istics in the firm asset dynamics. This way I have established a link between

the classical CAPM and conditional CAPM models. The presence of condition-

ing factors allows a comparison of this model with multi-factor models although

admittedly here, I still have only one single source of priced risk: the market,

albeit time-varying. I am giving an analytical explanation as to why condition-

ing factors are there and why they are better at explaining excess returns than

the simple static Beta on the market. The next section focuses on empirically

1Interestingly, if risk-free rates are negative, they could have a negative contribution toexcess return.

22

1.2 Empirical Evidence

proofing my conclusions and comparing my factors with those in the existing

literature, in particular with the classical Fama-French factors.


1.2.1 Data

My database contains 1,065 stocks that are, or have been, constituents of the

S&P 500 index from March 1989 to December 2013. I have eliminated all financial

stocks because, in their case, leverage takes on a different meaning than in the case

of corporates. In order to keep the market beta unbiased, I have also eliminated

financial stocks when reconstructing the S&P 500 index.1 The data comes from

Compustat and CRSP for financial and accounting variables, Barclays, SIFMA,

and Bloomberg for debt analytics such as debt duration. The sample is quarterly

with the first observation on March 31, 1989: quarterly is the frequency adopted

by Compustat for reporting balance sheet items such as accounting liabilities, ac-

counting assets, shares outstanding, and dividends per share. Interest expenses

are inferred from the annual interest expenses and evenly apportioned on a quar-

terly basis. The duration of firm liabilities has been calculated assuming that

short-term liabilities have a duration of one year and long-term liabilities have

the same average duration as the one reported in the Barclays U.S. High Grade

Duration Index. Recovery rates are assigned according to the current Moody’s

industry specific recovery tables. The model poses a series of challenges with

respect to input data: the main one is finding both the firm and the market asset

1In general, results are very similar to the ones one would obtain when including financialstocks, only slightly less conclusive at the level of the return regressions.

23


values as well as their respective volatilities. To establish asset values, I add debt

values from Compustat and equity values from CRSP to obtain a time series of

asset values as suggested in Eom et al. (2004). The gross asset return drift µ

is calculated from the asset value time series, while asset volatility is estimated

as a function of equity volatility adjusting for the leverage effect as in expres-

sion (1.11). Equity return volatility is built by taking the absolute value of the

quarterly price range. Industry-specific recoveries reported are taken from rating

agencies’ datasets and assumed static throughout the sample. Finally, market

recovery is assumed constant at the level of Government CDS recovery (currently

approximately 20% for the United States). This choice of market recovery sug-

gests that when a systemic default event occurs, the recovery ratio is expected to

be much lower than the average industry sector recoveries.

1.2.2 Descriptive Statistics

Tables 1.1 and 1.2 report arithmetic average statistics on the distribution of

asset and equity excess returns for individual firms. The tables also include the

same statistics for the S&P index built as a value-weighted average of the firms

in my sample.1

1The S&P index I build here is intended to be calculated as the real one, but in someinstances, it may show some differences due to the elimination of those stocks that do not haveenough observations or have irreparably incorrect balance sheet data.

24


Table 1.1 Asset Excess Return Statis-tics

Firms S&P

Mean 0.073 0.047

Volatility 0.659 0.219

Skewness 3.156 -0.648

Kurtosis 57.267 3.760

Table 1.2 Equity Excess ReturnStatistics

Firms S&P

Mean 0.083 0.068

Volatility 0.785 0.282

Skewness 1.695 -0.690

Kurtosis 25.389 3.698

As expected, average stock returns are higher than average asset returns be-

cause firms use a leverage that is, on average, higher than zero. The same ob-

servation is valid for the aggregate perspective, so the S&P shows higher equity

than asset returns.

Figure 1.6 shows the relationship between risk-free rates and the average debt

duration of the S&P index stocks. Risk-free rates are reported quarterly, and

debt duration is in years. Duration here has the financial mathematics meaning

of average point of repayment of all liabilities and does not reflect the weighted

average life that is expected to be longer because of the presence of numerous

coupon-paying assets.

Figure 1.7 shows the impact of leverage on returns by plotting two leverage

measures on the same time scale as the cumulative return series: whenever there is

a period of high leverage, there seems to be a trough in the excess returns affecting

more the levered equity performance than the unlevered one. This is clear for

the leverage measure of total assets (calculated as the sum of total nominal debt

25


and market value of equity) to market value of equity, less so for the total assets

to book-value ratio. This suggests that I should expect my measure of leverage

to perform better than the Fama-French HML factor.

Figure 1.6: On the left axis absolutequarterly risk-free rate and on the rightaxis debt duration of S&P stocks. Debtduration has almost always monotoni-cally increased.

Figure 1.7: On the left axis cumula-tive excess returns for S&P asset andequity, and on the right axis two lever-age measures versus market value andversus book value.

1.2.3 Cross-Sectional Analysis: levered and unlevered Be-

tas

In this part I estimate the traditional CAPM by rolling windows and use

the asset and equity Beta estimates to run uni- and bi-variate cross-sectional

analyzes. To facilitate comparison with the existing CAPM and APT literature,

I construct decile portfolios following the procedure outlined in Fama and French

(1993). My cross sectional analysis will include both the factors emerging from

my model and the Fama-French Size and Book-to-Market factors.

The first step is to estimate Betas for unlevered equities and compare them

26


to those for levered equities. I use the simple CAPM specifications:

For assets: E (rai,t)− rf = αai + βai [E(ram,t)− rf ] + εi,t (1.24)

For stocks: E (rei,t)− rf = αei + βei [E(rem,t)− rf ] + εi,t (1.25)

with subscripts a and e indicating assets and equity respectively, t indicating time

and i the cross-sectional dimension of the individual firm.

I run regressions (1.24) and (1.25) with a quarterly rolling observation window.

I estimate Alphas and Betas for both the asset and equity specification on excess

returns1 calculated versus the 3m risk-free.

Table 1.3 Summary Asset and Equity CAPM Estimates: Column headers corre-spond to Alpha and Beta estimates, p-values based on a Normal Distribution andAdjusted R-squared. Rows report average values of moments on a quarterly basis.

Alpha Beta P-value Alpha P-value Beta R2

Asset Mean 0.007 1.010 0.497 0.24 0.149

Volatility (0.036) (0.973)

Skewness 0.871 0.890

Kurtosis 7.594 6.425

Equity Mean 0.009 0.966 0.504 0.199 0.193

Volatility (0.039) (0.823)

Skewness 0.419 0.906

Kurtosis 5.619 7.051

1For asset excess returns, I use the gross return measure calculated before subtractinginterest costs.

27


Figure 1.8: The chart shows rolling estimates and p-values of 20 quarterseach averaged across the cross section of firms included in the sample. Yabindicates assets and Yeb equity.

Table 1.3 reports summary estimates and figure (1.8) the corresponding quar-

terly results. The values are means calculated across both the time and cross-

sectional dimension. Estimates are quarterly, so, while Beta is not expected to

change significantly on an annual basis and its average estimate is close to one,

the level of Alpha should be multiplied by four to obtain the annual intercept

value. The average annual level of Alpha for levered equity is around 3.6%, while

the one based on assets is around 2.7%. This phenomenon suggests that there

is a larger systematic excess return component that remains unexplained by the

CAPM Beta at the levered equity level as opposed to the unlevered equity CAPM

Beta.

28


Tables 1.4 and 1.5 present univariate Beta analysis with estimates of the

CAPM Alpha and Beta coefficients for assets and levered equities according to

equations 1.24 and 1.25. Buckets are built according to their average Beta, and

I additionally indicate their corresponding average Alpha. The rightmost Beta

bucket corresponds to the difference between the tenth and first bucket. Focusing

on the value-weighted table in particular, I observe that, when looking at levered

equity, excess returns for high Beta stocks in the 10th bucket are lower than the

ones in the first bucket but this effect is not there when looking at unlevered

equity. This effect is usually one of the points raised against the CAPM, and,

because I do not observe the phenomenon in the unlevered equity specification,

I conclude that, under the bivariate analysis, unlevering equity produces results

that are more consistent with the CAPM theory.

29

1.2

Em

pirica

lE

vid

ence

Table 1.4 Equally Weighted - Beta Univariate Portfolio Analysis: Column headers correspond to 10buckets created by taking 10 percentile equally weighted Beta portfolios. The rightmost column is thedifference between the 10th and first bucket. Avg. Beta stands for the average Beta of the referencebucket. Avg. Alpha stands for the average Alpha of the reference bucket. All numbers are quarterly.

Beta Buckets b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b10-1

Avg. Beta -0.328 0.200 0.429 0.621 0.805 1.007 1.225 1.469 1.793 2.770 3.098Avg. Alpha 0.026 0.010 0.009 0.006 0.005 0.003 0.003 0.001 0.003 0.005 -0.020Asset Ret. 0.020 0.013 0.015 0.014 0.010 0.014 0.015 0.018 0.022 0.027 0.007Equity Ret 0.022 0.013 0.017 0.017 0.010 0.019 0.017 0.019 0.029 0.027 0.006

30

1.2

Em

pirica

lE

vid

ence

Table 1.5 Value-Weighted - Beta Univariate Portfolio Analysis: Column headers correspond to 10buckets created by taking 10 percentile equally-weighted Beta portfolios. The rightmost column is thedifference between the 10th and first bucket. Avg. Beta stands for the average Beta of the referencebucket. Avg. Alpha stands for the average Alpha of the reference bucket. All numbers are quarterly.

Beta Buckets b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b10-1

Avg. Beta -0.308 0.215 0.457 0.621 0.801 1.018 1.235 1.479 1.777 2.670 2.978

Avg. Alpha 0.038 0.015 0.015 0.011 0.013 0.010 0.010 0.010 0.014 0.023 -0.015

Asset Ret. 0.017 0.015 0.020 0.005 0.005 0.009 0.010 0.021 0.019 0.021 0.005

Equity Ret. 0.018 0.013 0.020 0.005 0.005 0.011 0.011 0.019 0.021 0.022 0.004

31


Hereafter, I extend my analysis to bivariate cross-sectional sorting for quar-

terly stock returns. I run a series of sorting routines based on the factors of my

model in equation 1.23: Leverage, Debt Duration, and Volatility, in addition to

Beta. Beta sorts are based on estimates for assets and levered equity respec-

tively. Portfolios are sorted into five buckets by Beta and the second variable of

interest. Portfolios of stocks are built using value weights. In general, I find that

the intensity of anomalies are much lower when observed in the unlevered equity

dimension than the levered equity one. The Fama-French Size and BM effects are

more intense for levered equity and, in particular, they are much more consistent

across equity Beta buckets than in the case of the asset Beta buckets.

The tables, including my factors as drivers of excess returns (Leverage, Debt

Duration, and Volatility), should be looked at in the context of the theoretical

model:

• Equity Beta sorts here refer to the B coefficient of model 1.23 which is also

impacting the other three factors, suggesting mutual interaction between

equity Beta and the second sorting variable;

• The results on the effects of Leverage, Debt Duration, and Volatility do

not depend on their absolute value, but rather on the relative value, with

respect to the same measure in the S&P 500 benchmark.

Notwithstanding the above, I can still detect a few common trends: high Leverage

and Beta buckets appear to have higher excess returns, both in the asset case and

the equity case. Higher Debt duration is inducing higher returns, which grow

with the Beta buckets in the equity case. Stock volatility relates to higher excess

returns mostly in the case of low Beta stocks. Overall, these effects emerging

32


from the cross-sectional analysis are coherent with the conclusions of the model.

The directional influence of certain factors on the excess returns is confirmed, and

the impact of the variables is more intense in the equity specification. Ideally, we

would expect to see no impact at all in the asset specification.

Table 1.6 Beta and Size Sorts — Assets: The row Beta buckets arebased on estimates of an asset CAPM 20 quarter rolling regression. Thecolumn Size buckets are average company sizes. Table contents are averagequarterly excess returns.

Size ($bn) b1 b2 b3 b4 b5 b5-b1

Beta 2.444 5.394 9.115 15.791 66.252 63.808

b1 -0.117 0.018 0.017 0.019 0.021 0.022 0.004

b2 0.526 0.023 0.015 0.010 0.014 0.009 -0.014

b3 0.909 0.011 0.011 0.017 0.008 0.003 -0.008

b4 1.345 0.034 0.015 0.012 0.023 0.008 -0.026

b5 2.351 0.037 0.020 0.018 0.027 -0.008 -0.045

33


Table 1.7 Beta and Size Sorts — Equity: The row Beta buckets arebased on estimates of an equity CAPM 20 quarter rolling regression. Thecolumn Size buckets are average company sizes. Table contents are averagequarterly excess returns.

Size ($bn) b1 b2 b3 b4 b5 b5-b1

Beta 2.444 5.394 9.115 15.791 66.252 63.808

b1 -0.012 0.026 0.024 0.021 0.026 0.025 -0.001

b2 0.572 0.018 0.023 0.014 0.013 0.011 -0.007

b3 0.985 0.031 0.021 0.013 0.013 -0.001 -0.032

b4 1.416 0.017 0.014 0.018 0.025 0.004 -0.013

b5 2.176 0.047 0.014 0.013 0.017 -0.016 -0.063

Table 1.8 Beta and Book-to-Market Sorts — Assets: The row Beta buck-ets are based on estimates of an asset CAPM 20 quarter rolling regression.The column Book-to-Market buckets are average company BMs. Tablecontents are average quarterly excess returns.

Book-to-Mkt b1 b2 b3 b4 b5 b5-b1

Beta 00.062 0.266 0.378 0.515 0.833 0.771

b1 -0.117 0.035 0.004 0.021 0.013 0.025 -0.010

b2 0.526 0.013 0.002 0.009 0.014 0.018 0.005

b3 0.909 -0.005 0.011 0.008 0.004 0.018 0.023

b4 1.345 0.001 0.012 0.013 0.020 0.025 0.023

b5 2.351 -0.018 0.006 0.014 0.009 0.049 0.068

34


Table 1.9 Beta and Book-to-Market Sorts — Equity: The row Beta buck-ets are based on estimates of an equity CAPM 20 quarter rolling regres-sion. The column Book-to-Market buckets are average company BMs.Table contents are average quarterly excess returns.

Book-to-Mkt b1 b2 b3 b4 b5 b5-b1

Beta 0.062 0.266 0.378 0.515 0.833 0.771

b1 -0.012 0.035 0.005 0.027 0.020 0.036 0.002

b2 0.572 0.009 0.007 0.012 0.017 0.026 0.017

b3 0.985 -0.012 0.004 0.004 0.017 0.032 0.044

b4 1.416 0.001 0.008 0.006 0.010 0.034 0.033

b5 2.176 -0.025 -0.005 0.007 -0.003 0.048 0.073

Table 1.10 Beta and Leverage Sorts — Assets: The row Beta bucketsare based on estimates of an asset CAPM 20 quarter rolling regression.The column Leverage Ratio buckets are average company leverage. Tablecontents are average quarterly excess returns.

Leverage b1 b2 b3 b4 b5 b5-b1

Beta 1.035 1.135 1.250 1.449 2.319 1.283

b1 -0.117 0.050 0.005 0.016 0.025 0.007 -0.043

b2 0.526 0.011 0.016 0.007 0.009 0.003 -0.008

b3 0.909 -0.004 0.013 0.016 0.001 0.001 0.006

b4 1.345 0.010 0.012 0.006 0.010 0.020 0.010

b5 2.351 -0.007 -0.001 0.021 0.014 0.017 0.024

35


Table 1.11 Beta and Leverage Sorts — Equity: The row Beta bucketsare based on estimates of an equity CAPM 20 quarter rolling regression.The column Leverage Ratio buckets are average company leverage. Tablecontents are average quarterly excess returns.

Leverage b1 b2 b3 b4 b5 b5-b1

Beta 1.035 1.135 1.250 1.449 2.319 1.283

b1 -0.012 0.047 0.006 0.013 0.037 0.021 -0.027

b2 0.572 0.009 0.012 0.014 0.014 0.015 0.006

b3 0.985 -0.004 0.007 0.014 0.007 0.002 0.007

b4 1.416 0.007 0.007 0.013 0.009 0.009 0.002

b5 2.176 -0.018 -0.010 0.006 0.010 0.029 0.046

Table 1.12 Beta and Debt Duration Sorts — Assets: The row Beta buck-ets are based on estimates of an asset CAPM 20 quarter rolling regression.The column Debt Duration buckets are average company debt duration.Table contents are average quarterly excess returns

Debt Dur b1 b2 b3 b4 b5 b5-b1

Beta 2.887 4.939 5.364 5.651 6.156 3.269

b1 -0.117 0.017 0.016 0.027 0.023 0.024 0.007

b2 0.526 0.011 0.008 -0.005 0.010 0.025 0.014

b3 0.909 -0.003 -0.000 0.007 0.011 0.017 0.020

b4 1.345 0.007 -0.006 0.018 0.001 0.028 0.021

b5 2.351 -0.008 0.003 0.001 0.006 0.020 0.028

36


Table 1.13 Beta and Debt Duration Sorts — Equity: The row Betabuckets are based on estimates of an equity CAPM 20 quarter rollingregression. The column Debt Duration buckets are average company debtduration. Table contents are average quarterly excess returns.

Debt Dur b1 b2 b3 b4 b5 b5-b1

Beta 2.887 4.939 5.364 5.651 6.156 3.269

b1 -0.012 0.032 0.025 0.009 0.037 0.018 -0.013

b2 0.572 0.002 0.012 0.008 0.010 0.030 0.028

b3 0.985 0.006 -0.021 0.008 -0.000 0.020 0.015

b4 1.416 0.003 -0.002 0.014 0.011 0.022 0.020

b5 2.176 -0.032 0.006 -0.001 0.011 0.018 0.051

Table 1.14 Beta and Volatility Sorts — Assets: The row Beta buckets arebased on estimates of an asset CAPM 20 quarter rolling regression. Thecolumn Volatility buckets are average company volatility. Table contentsare average quarterly excess returns.

Stock Vol b1 b2 b3 b4 b5 b5-b1

Beta 0.124 0.167 0.194 0.236 0.398 0.274

b1 -0.117 0.002 0.030 0.018 0.016 0.051 0.049

b2 0.526 0.008 -0.000 0.023 0.002 0.027 0.019

b3 0.909 0.011 0.012 -0.000 -0.003 -0.006 -0.017

b4 1.345 0.010 0.009 0.027 0.002 0.007 -0.002

b5 2.351 0.007 -0.006 0.001 0.016 -0.006 -0.013

37


Table 1.15 Beta and Volatility Sorts — Equity: The row Beta buckets arebased on estimates of an equity CAPM 20 quarter rolling regression. Thecolumn Volatility buckets are average company volatility. Table contentsare average quarterly excess returns.

Stock Vol b1 b2 b3 b4 b5 b5-b1

Beta 0.124 0.167 0.194 0.236 0.398 0.274

b1 -0.012 0.006 0.015 0.016 0.024 0.069 0.062

b2 0.572 0.011 0.010 0.020 0.005 0.015 0.003

b3 0.985 0.004 0.016 0.009 -0.002 -0.008 -0.011

b4 1.416 0.013 -0.006 0.014 0.007 0.010 -0.003

b5 2.176 0.007 0.002 0.002 -0.000 -0.017 -0.024

A further step in my analysis is a table to compare the FF Book-to-Market

factor with my Leverage factor (directly from the model). FF is defined as

Book Value of EquityMarket Value of Equity

and my Leverage is defined as Market Value of Equity + Book Value of DebtMarket Value of Equity

.

The results show a mix of intuitions: high-leverage companies with low BM have

negative average excess returns, which is coherent with most of the empirical

literature. At the same time, an increase in leverage paired with an increase in

BM somewhat reduces the average excess return for the highest-leverage buckets.

In other words, the relationship between leverage and BM does not appear to

be monotonic. This is expected because, as discussed in the model section, my

factors are always to be analyzed as ratios to the market equivalent measure:

so it is possible that idiosyncratic leverage both increases or decreases the risk

premium depending on the sign of Beta and the market leverage.

38


Table 1.16 Book-to-Mkt and Leverage Sorts — Equity: Therow Book-to-Market buckets are based on the FF factor.The column Leverage Ratio buckets are averages on com-pany leverage. Table contents are average quarterly excessreturns.

Leverage b1 b2 b3 b4 b5

BM 1.035 1.135 1.250 1.449 2.319

b1 0.062 -0.002 -0.001 0.005 0.016 0.007

b2 0.266 0.008 0.008 0.006 -0.004 -0.023

b3 0.378 0.010 0.009 0.012 0.016 0.003

b4 0.515 0.021 0.015 0.019 0.004 0.007

b5 0.833 0.029 0.040 0.042 0.032 0.033

Finally, I am investigating the relationship between the FF Size factor and my

factors. In particular, I look at the relationship between Debt Duration and the

Size of companies. I expect that, in the levered equity domain, there should be a

strong correlation between the ability to raise long-term debt and the dimension of

a company. Table 1.17 indirectly confirms my hypothesis: smaller companies pay

a risk premium that is increasing in their debt duration, while large companies

pay almost no premium if not in the highest debt duration bucket.

39


Table 1.17 Size and Debt Duration Sorts — Equity: The rowSize buckets are based on the FF factor. The column Debt Du-ration buckets are averages on company leverage. Table contentsare average quarterly excess returns.

Debt Duration b1 b2 b3 b4 b5

Size 2.887 4.939 5.364 5.651 6.156

b1 2.444 0.021 0.028 0.024 0.033 0.050

b2 5.394 0.010 0.022 0.014 0.026 0.027

b3 9.115 0.010 0.004 0.015 0.028 0.029

b4 15.791 0.012 0.013 0.018 0.021 0.026

b5 66.252 -0.000 -0.003 0.005 0.005 0.020

1.2.4 Correlation Analysis

To provide more background on the relationships among variables to be used

in the empirical analysis, I have conducted a correlation study of the company

characteristics used by my model as well as the Fama-French Size and Book-to-

Market characteristics.

40

1.2

Em

pirica

lE

vid

ence

Table 1.18 Factor Correlation: Pairwise Pearson correlations of the company characteristicsused as an input for the model factors. Correlations are computed on the time series ofindividual company observations and then averaged across companies. Numbers with an *indicate a p-value lower than 0.1.

Lev. Debt Dur. Vol. A.Beta E.Beta Size BM

Leverage 1.000 0.028 -0.007 -0.015 -0.021 -0.488* 0.441*

Debt Duration 0.028 1.000 -0.060 0.025 0.010 0.090 0.050

Volatility -0.007 -0.060 1.000 0.010 0.009 -0.011 -0.006

Asset Beta -0.015 0.025 0.010 1.000 0.802* 0.002 0.007

Equity Beta -0.021 0.010 0.009 0.802* 1.000 0.001 -0.005

Size -0.488* 0.090 -0.011 0.002 0.001 1.000 -0.501*

BM 0.441* 0.050 -0.006 0.007 -0.005 -0.501* 1.000

41


Asset and Equity Beta are obviously highly and significantly correlated as

are Size and BM. Furthermore, Leverage is positively and significantly correlated

to BM and negatively and significantly correlated to Size. Other correlations

that are not below 0.1 of p-value, but are still relatively low, are the positive

correlation between Size and Debt Duration, as well as BM and Debt Duration.

In addition there appears to be positive correlation between Debt Duration and

Leverage. Correlations of variables with Equity Betas seem weak overall, and the

possible reason is that Pearson correlation only measures linear correlations while

we know, from the model, that Equity Beta is non linearly related to a product

of variables.

Table 1.19 Factor Correlation P-values: P-values for the Pearson correlation matrixcalculated as above.

Lev. Debt Dur. Vol. A.Beta E.Beta Size BM

Leverage 1.000 0.196 0.363 0.507 0.499 0.105 0.115

Debt Duration 0.196 1.000 0.368 0.513 0.491 0.221 0.263

Volatility 0.363 0.368 1.000 0.522 0.489 0.336 0.360

Asset Beta 0.507 0.513 0.522 1.000 0.020 0.502 0.497

Equity Beta 0.499 0.491 0.489 0.020 1.000 0.493 0.496

Size 0.105 0.221 0.336 0.502 0.493 1.000 0.091

BM 0.115 0.263 0.360 0.497 0.496 0.091 1.000

1.2.5 Statistical Properties of Quarterly Excess Returns

Before moving on to regression analysis, I have tested the main statistical as-

sumptions of my model implementation. My empirical work is based on assuming

42


Figure 1.9: S&P 500 standardized quarterly stock return cumulativedistribution compared to a normal cumulative distribution: KS testoutput is null.

that returns are Normally distributed and that autocorrelation is not detectable,

given the quarterly observation windows. To make sure these assumptions are

robust, I have run the Kolmogorov-Smirnov test on the 100 sample observations

of the S&P 500 aggregate index built with the stocks in my sample and found

that the test has failed to reject normality.

Also, I have extensively tested autocorrelation of returns at both the overall

S&P 500 and at the individual stocks level. With quarterly observations, au-

tocorrelation (even of level 1) is not significant, please refer to the confidence

bounds in figure 1.10. As a consequence of the lack of autocorrelation at this

sampling frequency, I have chosen not to calculate Newey-West standard errors

for my regression coefficients and instead utilize the OLS ones.

43


Figure 1.10: Autocorrelation structure, up to 20 period lag, of S&P500 quarterly stock returns with confidence intervals.

1.2.6 Fama-Macbeth Regression Analysis

Following Fama and MacBeth (1973), I use the estimates of Beta from the

previous sections to derive the market factor and check the consistency of the

CAPM predictions with market evidence. I run a cross sectional regression under

the specification:

E (ri,t+1)− rf,t+1 = ν0,i + νiβ1,i + εi,t, (1.26)

where βi are the estimated Beta coefficients obtained by the 20 quarter rolling

window regression of the previous section. I wish to check, both with the asset and

the equity specification of equation 1.26, if it is confirmed that higher Betas induce

higher returns and that the intercept term is not significant. My expectation is

44


that the CAPM theory will hold better in the asset than in the levered equity

domain.

I estimate 3-, 4-, and 5-year quarterly rolling window regressions that produce

results that are encouraging with respect to my model relationship between Asset

Beta and Equity Beta: in the asset return framework, the relationship between

the market and the firm has a higher coefficient in all estimation windows. The

overall average coefficients for Beta, i.e., the market factor, are estimated to be

positive for the shorter windows and negative for the longer ones.

• ν0 estimates have higher p-values for the Assets than for the Equity.

• ν1 estimates are much less negative for the Assets than for the Equity.

• Estimates for the excess returns measured against the duration adjusted

risk-free rate are just slightly better in terms of coefficient expectations and

slightly worst in terms of R-squared.

45

1.2

Em

pirica

lE

vid

ence

Table 1.20 Cross-Sectional average Fama-MacBeth Regression: The table presents time-series av-erages estimates of the coefficients ν from 1.26. The ν1’s are estimated by rolling 3-, 4-, and5-year quarterly regressions of one quarter ahead excess returns against an intercept and the betacoefficients estimated in 1.24 and 1.25.

Asset 3yr Stock 3yr Asset 4yr Stock 4yr Asset 5yr Stock 5yr

ν0 Mean 0.002 -0.002 0.013 -0.002 0.021 0.029

ν1 Mean 0.007 0.012 -0.005 -0.015 -0.015 -0.028

Adj.R2 0.019 0.027 0.015 0.023 0.012 0.020

46


The average coefficient measures presented in this section are to be read with

caution. Because of the time-averaging, they may give an indication of the rela-

tionship between the firms and the market that is biased at times or for certain

firms. To complete the picture of this analysis and provide more color on the

nature of the levered and unlevered Beta factors, in figure (1.11) I present the

rolling estimates in the case of a 20 quarter window for the underlying asset and

equity CAPMs. The pattern followed by the recursive estimates of ν0 and ν1

shows that persistence is much higher for the Asset specification than for the

Equity one. The intercept (ν0) p-value pattern hints at a higher likelihood of the

Asset intercept being zero (especially over the recent economic crisis, when the

impact of leverage on equities was at all time high).

47


Figure 1.11: Rolling estimates and statistics for the 20 quarter regressionspecification.

1.2.7 Empirical Results: Testing the CS-CAPM Model

In this final part I test the main implication of my model: levered equity

returns are better explained by a conditional model whose factors derive from the

capital structure of the firms and the market and are consistent with a CAPM

representation at the unlevered equity level.

In order to substantiate my conclusion, I prove two concepts: (i) that the

classical Market factor, the levered Equity Beta, is explained by Asset Beta,

relative leverage and credit risk so it cannot be constant, and (ii) that given the

previous point, Equity Excess Returns depend on Asset Beta plus another three

48


conditioning factors mimicking firm characteristics: Leverage, Debt Duration,

and the Volatility of the individual firm, all relative to the Market.

1.2.7.1 Equity Beta: A function of Asset Beta, Leverage, and Credit

Risk

I test the prediction of the model that Stock Betas can be explained by Asset

Betas the ratio of Firm and Market Leverage and the ratio of Firm and Market

equity deltas; these measure the in-the-moneyness of the option in the hands of

the equity holders with respect to the return of the assets of the firm: the lower

the delta is the higher is the credit risk of the firm. The equation I test is non

linear:

B = βei = βai∆i

∆m

AiEi

EmM

, (1.27)

and I propose the test its linearized version. I also plan to check the potential

impact of the Fama-French factors on levered equity Beta:

Bt+1 ∼ βei,t+1 =

(φ0 + φ1βai + φ2

∆i

∆m

+ φ3AiEi

EmM

+ φ4SMB + φ5HML+ εi

)t

.

(1.28)

I run a 20 quarter rolling window regressions of equation 1.28 using one quarter

ahead equity Betas regressed on current quarter asset Betas and other factors.

The regression is run under various specifications, from the basic one factor to

the comprehensive one testing also for the relevance of Fama-French factors. I

calculate coefficients and R-square reported in table 1.21 by averaging across the

49


time-series and the individual firms.

Table 1.21 Equity Beta Regressions — Individual Stocks: Time-series averageestimates of the coefficients φ from 1.28. Estimates are obtained by rolling5-year quarterly sectoral regressions and taking averages over the 62 periodsand the cross-section of 563 companies that alternatively constitute the S&P500.

Version 1 2 3 4 5 6

φ0: Intercept Average 0.48 1.01 0.52 0.40 0.56

P-value 0.15 0.09 0.32 0.36 0.32

φ1: Asset Beta Average 0.51 0.47 0.58 0.46

P-value 0.09 0.11 0.06 0.12

φ2: Delta Ratio Average -0.05 -0.03 -0.03 -0.06

P-value 0.59 0.49 0.47 0.50

φ3: Leverage Ratio Average 0.03 0.09 0.16 -0.03

P-value 0.50 0.49 0.24 0.48

φ4: SMB Average 0.01

P-value 0.49

φ5: HML Average -0.01

P-value 0.57

Adj. R2 0.42 0.05 0.20 0.51 0.43 0.51

The results confirm that equity Betas depend on asset Betas and that other

variables contribute to explaining their variance. The ratio of deltas and leverage

factors appear to contribute to explaining the out-of-sample variance of the lev-

ered equity Betas, but their significance and sign are less stable than the outright

asset Beta. Also some variables like the Leverage Ratio appear to change sign

50


depending on the specification, suggesting that there may be issues with the or-

thogonality of regressors. This is due to both the non linear model specification

and the fact that the measures of delta and leverage for the firm and the market

are only relevant to explain equity Betas when their ratio is different than one,

if those ratios are close to one the impact on Betas is low generating a high p-

value. The specification without intercept has a lower R-square, suggesting that

there may be some systematic effects that are not perfectly captured by the linear

regression. Finally, the Fama-French size and value factors neither improve on

the R-square nor affect the stability of the other coefficients in the regression.

I conclude that they are insignificant when explaining equity Beta variations in

combination with factors from my model.

I complete this section by introducing the same as table 1.21, computed by

aggregating firms into value-weighted industry groups. I aggregate companies

into the 10 basic GIC industry groups to average out any individual firm noise.

Portfolio aggregation by value, though, may also be considered a bias as it has the

effect of giving larger companies a higher relevance in the portfolio. The industry

groups I consider are: Energy, Materials, Industrials, Consumer, Staples, Health,

Financials, Technology, Telecom, and Utility. I treat these as a single stock and

run the above analysis obtaining:

51


Table 1.22 Equity Beta Regressions — Industry Portfolios: Time-series av-erage estimates of the coefficients φ from 1.28. Estimates are obtained byrolling 5-year quarterly sectoral regressions and taking averages over the 62periods and the cross-section of 10 industries.

Version 1 2 3 4 5 6

φ0: Intercept Average 0.40 0.92 0.98 0.45 0.50

P-value 0.14 0.03 0.10 0.22 0.21

φ1: Asset Beta Average 0.56 0.49 0.72 0.47

P-value 0.08 0.11 0.03 0.12

φ2: Delta Ratio Average 0.01 -0.01 0.03 -0.01

P-value 0.41 0.47 0.39 0.50

φ3: Leverage Ratio Average -0.03 0.01 0.08 -0.01

P-value 0.18 0.34 0.26 0.33

φ4: SMB Average 0.01

P-value 0.52

φ5: HML Average 0.01

P-value 0.48

Adj. R2 0.47 0.03 0.27 0.55 0.58 0.54

The outcome of the industry-level analysis is similar to that of the single com-

pany. The Fama-French factors do not make any positive marginal contribution

to the regression. In addition, some of the systematic effects relating to the in-

tercept at the level of the single stock analysis seem to have been filtered out

by means of the industry aggregation. There is still a mixed effect between the

Leverage Ratio and Delta Ratio that impacts the sign of those coefficients and is

due to the correlation between them: debt is an input of the calculation of the

52


equity delta and an input for the calculation of leverage.

1.2.7.2 Testing Excess Returns with the CS-CAPM

In this final part, I focus on the central implication of the model to show how

much of future quarterly stock returns can be explained by adding my condi-

tioning factors to the classical CAPM market factor, thereby correcting for the

leverage effect on both the market and the single stock. I test equation 1.23 by

regressing one quarter ahead Equity Excess Returns on Beta and the additional

factors of my model according to the following formula:

[E(rei)− rf ]t+1︸︷︷︸Equity Excess Return

=

=

Di

Eirf −

B(∆mM − Em)

Emrf︸︷︷︸

Leverage

+Θi

Ei− BΘm

Em︸︷︷︸Debt Maturity

+ΓiAi

2σ2ai

2Ei−BΓmM

2σ2am

2Em︸︷︷︸Volatility

t

dt+

+ Bt [E(rem)− rf ]t︸︷︷︸Market Factor

+εt, (1.29)

with B as the Beta market factor established by regressing excess returns on mar-

ket excess returns, ε indicating a random error orthogonal to the other variables

and the subscripts t+ 1, and t indicating the point in time at which the variables

are taken.

I run rolling regressions using a number of specifications, including Fama-

French factors. The equation to be tested is going to require some simplifications

53


with respect to the non linear elements of the formula. I linearize it by adopting

the following proxy factors:

1. Leverage factor (LEV): the difference of firm minus market leverage ratios

multiplied by the risk-free rate, disregarding the Beta and Delta terms.

2. Debt Maturity factor (DM): the difference in debt duration divided by the

standardized market value, disregarding the Beta.

3. Volatility factor (VOL): the difference of variances, disregarding the gamma,

asset value, equity value, and Beta components.

In order to ensure orthogonality, I choose to replace B with its component βa,

as the other components, namely leverage ratios and delta ratios, are proxied by

the Leverage factor. The final expression looks as follows:

(E(rei)− rf )t+1 =

=(µ0 + µ1LEV + µ2DM + µ3V OL+ µ4βa,i + µ5SMB + µ6HML+ εi

)t,

(1.30)

with all variables on the left-hand side forward looking by one quarter with respect

to those on the right-hand side.

54


Table 1.23 CS-CAPM Regressions — Single Stocks: The table presents time-seriesaverage estimates of the coefficients µ from 1.30 under 8 specifications. They areestimated by rolling 5-year quarterly sectoral regressions and taking averages over the62 periods and the 563 companies.

Version 1 2 3 4 5 6 7 8

µ0: Intercept Average 0.03 -0.01 0.02 0.02 0.02 0.02 -0.02 -0.02

P-value 0.38 0.51 0.38 0.41 0.36 0.38 0.51 0.51

µ1: LEV Average 5.17 -0.61 -2.5

P-value 0.37 0.46 0.46

µ2: DM Average 0.02 0.02 0.02

P-value 0.33 0.37 0.37

µ3: VOL Average 0.16 0.20 0.21

P-value 0.36 0.35 0.36

µ4: Beta Average -0.01 -0.01 -0.01

P-value 0.50 0.51 0.51

µ5: SMB Average -0.01 0.01

P-value 0.50 0.52

µ6: HML Average 0.01 0.01

P-value 0.44 0.43

Adj. R2 0.02 0.03 0.04 0.01 -0.01 0.01 0.08 0.08

Looking at the R-square coefficients in table 1.23, it appears that including

the Fama-French factors does not contribute to improving the Adjusted R-square

of the regression. This suggests that the marginal contribution of those factors

is null and already captured by the other factors that, according to my model,

complement the market factor of the classical CAPM. In particular, I also find

55


the desirable outcome that, with the exception of the intercept, all coefficients

maintain a rather constant pattern across specifications both in terms of sign and

magnitude.

The average p-values for every specification are highly insignificant, but look-

ing at their frequency distribution across the observations, it is possible to observe

that some of them experience periods of significance more than others. In partic-

ular, note that in figure 1.12, the three predictive factors (LEV, DM, VOL) of my

model, all have a high percentage of significant p-value observations (those close

to zero), while only Beta seems to have a majority of insignificant observations.

The Fama-French HML factor appears similar to my factors, but we have seen

that its effect is counterbalanced by the presence of my factors in the regression.

The Fama-French SMB factor, instead, has a distribution that resembles that of

the more insignificant variables such as the intercept and the market Beta.

56


Figure 1.12: Frequency distribution of p-values of the individualregressors. On the x-axis are bars of 10% probability each, and onthe vertical y-axis the actual number of observations.

To complete the analysis, I aggregate stocks according to the 10 major in-

dustry groups of the previous section and rerun the regressions to check that the

coefficients are stable while filtering out the noise by means of averaging. The

portfolios are, as usual, value-weighted and treated as 10 individual stocks.

57


Table 1.24 CS-CAPM Regressions — Industry Portfolios: The table presents time-series average estimates of the coefficients µ from 1.30 under 8 specifications. They areestimated by rolling 5 year quarterly sectoral regressions and taking averages over the62 periods and the 10 sectors, for a total of 610 observations.

Version 1 2 3 4 5 6 7 8

µ0: Intercept Average 0.01 0.00 0.01 -0.00 0.01 0.01 0.04 0.03

P-value 0.40 0.45 0.33 0.48 0.35 0.38 0.50 0.52

µ1: LEV Average 2.11 3.97 2.50

P-value 0.41 0.43 0.48

µ2: DM Average -0.01 -0.01 -0.01

P-value 0.46 0.48 0.49

µ3: VOL Average 0.29 0.32 0.39

P-value 0.29 0.29 0.30

µ4: Beta Average 0.02 -0.01 -0.02

P-value 0.48 0.50 0.49

µ5: SMB Average -0.00 -0.00

P-value 0.50 0.48

µ6: HML Average 0.01 0.01

P-value 0.43 0.44

Adj. R2 0.02 0.05 0.04 0.03 -0.01 0.02 0.13 0.11

The outcome of the single stock analysis is largely confirmed at the industry

level. The Fama-French factors’ marginal contribution to explaining future stock

returns is fully captured by the factors that my analytical model uses to correct

for the levered nature of the firm and the market equity returns.

58

1.3 Conclusion

1.3 Conclusion

The Capital Structure CAPM provides a theoretical explanation for some of

the puzzles that have affected empirical tests of the Sharpe and Lintner CAPM for

many decades. In conditional CAPM seems to capture most of the explanatory

power of the Fama-French SMB and HML and, in so doing, provide a theoretical

background as to why those exist.

The assumptions of the model suggest that, when looking at equities, we

look at non linear payoffs that have to be adjusted by a number of terms to be

effectively explained by a linear asset-pricing model. The main assumption is that

the equity world is a call option on the total asset payoffs. By modeling the call

option on firms and market, I achieve a new CAPM representation that, from the

general equilibrium asset-level formulation, produces conditional representation

of returns in the levered equity world where the conditional Beta is affected by

three factors. The model also shows that I can expect correlation between equity

Beta and Alpha to be mostly driven by the relative firm-to-market leverage ratio.

The overall conclusion of the empirical analysis is that the risk premium of a

stock or industry is driven by its asset Beta as well as Volatility, Debt Duration,

and Leverage as conditioning variables for the levered Beta. The Fama-French

factors tested on the same sample and with the same frequency of observation

have negative marginal contribution to explaining the variability of the equity

premia.

59

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65

Appendices

66

.1 Proof of Propositions

Appendix for chapter 1


Proof. Proof of Proposition (1) I begin with the expressions that approximate equity

returns:

E(rei) =1

Ei

[∂Ei∂t

dt+∂Ei∂Ai

dAi +1

2

∂2Ei∂2Ai

(dAi)2

], (31)

=1

Ei

[Θidt+ ∆idAi +

Γi2

(dAi)2

]. (32)

then I use the asset-CAPM in 1.18 to obtain the following:

E(rei)−AiEirf =

1

Ei

[Θidt+ ∆iAiβai [E(ram)− rf ] +

Γi2A2iσ

2aidt

]. (33)

I then express the market asset return as a function of its equity return:

E(rem) ∼ E(dEmEm

)=

1

Em

[∂Em∂t

dt+∂Em∂M

dM +1

2

∂2Em∂2M

(dM)2

], (34)

=1

Em

[Θmdt+ ∆mdM +

Γm2M2σ2

amdt

], (35)

E(ram) ∼ E(dM

M

)' Em

∆mM

[E(rem)− Θm

Emdt− Γm

2EmM2σ2

amdt

]. (36)

67


then I use the market asset return expression (70) to finalize (67):

E(rei)−AiEirf =

[Θi

Ei+

Γi2Ei

A2iσ

2ai

]dt+

∆i

EiAi [βai [E(ram)− rf ]] , (37)

=

[Θi

Ei+

Γi2Ei

A2iσ

2ai

]dt+

∆i

EiAiβai∗

∗[E

(Em

∆mM

[E(rem)− Θm

Emdt− Γm

2EmM2σ2

amdt

])− rf

], (38)

=

[Θi

Ei+

Γi2Ei

A2iσ

2ai

]dt− βai

∆i

∆m

AiEi

EmM

[Θm

Em+

Γm2Em

M2σ2am

]dt+

+ βai∆i

∆m

AiEi

EmM

[E(rem)− ∆mM

Emrf

], (39)

=

[Θi

Ei+

ΓiA2iσ

2ai

2Ei−B

(Θm

Em+

ΓmM2σ2am

2Em

)]dt+

+B

[E(rem)− ∆mM

Emrf

], (40)

with B = βai∆i∆m

AiEi

EmM . Q.E.D.

68

2

Dynamic Parametric Portfolio

Policies

Modern investment theory has traditionally been based on the mean-variance

approach by Markowitz (1952). That approach presents two main challenges

to the investor: 1) practical implementation with a large number of assets be-

comes computationally heavy with unstable results and 2) the model is essentially

static. That is, it does not enable the investor to hedge its investment decisions

intertemporally.

Merton (1969) suggests a solution for the latter challenge above; he extends

portfolio theory into the intertemporal dimension, introducing a model that al-

lows the investor to select her portfolio allocation continuously while taking into

account her consumption decisions through time.

The first challenge of portfolio theory has remained open for a longer pe-

riod, and many authors have introduced modifications of the Markowitz (1952)

setup to accommodate the need to reflect, in portfolio investment decisions, a

factor structure of the underlying assets. In essence, the literature has attempted

to aggregate assets according to common factors in order to shrink the multi-

69

dimensionality problem relating to the variance covariance matrix of portfolio

returns. In parallel, a multifactor model has been introduced to explain return

dynamics in the pioneering work by Fama and French (1993). In the paper by

Daniel and Titman (1997), the factor approach is then challenged in favor of a

characteristics approach, whereby returns are not a function of stocks’ covariance

with common factors (such as size or book-to-market) but depend on the individ-

ual firms’ characteristics (which, also, for instance, could be measuring the size

and book-to-market relevant to a firm and not as its covariance to a common size

or book-to-market factor).

Brandt et al. (2009) propose a Parametric Portfolio Policy (PPP) to model

directly portfolio weights as a function of stock characteristics and not the indi-

vidual stocks themselves, allowing for a dramatic simplification of the Markowitz

(1952) problem, thanks to the parametrization of the portfolio policy. Their

approach provides a solution to the first challenge, mentioned above, of the tra-

ditional portfolio theory but does not tackle the intertemporal issue, solved sep-

arately by Merton (1969). This paper combines the PPP developed by Brandt

with the dynamic portfolio choice theory introduced by Merton.

The PPP model requires a choice of characteristics to drive the portfolio

choice. A number of studies, such as the recent Green et al. (2017), have investi-

gated which characteristics are relevant to capturing stock return dynamics. My

choice of relevant characteristics is based on those that emerge when modeling

stocks as call options on the underlying firm assets, as in Dova (2016), where rele-

vant return drivers depend on the options’ pricing inputs: the strike is an amount

close to the overall company indebtedness, the delta is an amount close to the

company’s leverage, and, finally, the volatility is the underlying asset volatility,

70

that is, equity de-levered volatility.

In keeping with Merton, my portfolio choice problem is in continuous time for

a multi-period investor that has a CRRA utility function. I assume that stocks’

returns are explained by a market portfolio and three characteristic long-short

return portfolios. The long-short portfolios of my choice are mostly uncorrelated,

and the market portfolio is, by construction, an affine function of the long-short

portfolios.1 I solve the dynamic programming problem in closed form by adopting

a linear approximation of the characteristics’ returns.

This paper is related to three lines of research. The first is the modeling of

stock returns through structural credit risk models initiated by Merton (1974)

and continued in a number of more recent studies by Vassalou and Xing (2004)

and Choi and Richardson (2008). In these studies, stocks are represented as a,

generally in-the-money, option on the corporate asset-generated cashflows. The

optional payout makes the stocks a function of the underlying asset volatility as

well as the debt characteristics, including the amount of leverage and the time

to maturity of debt. In addition, individual assets have a beta to the aggregate

production assets, making asset beta a driver of returns. In essence, in line with

Dova (2016), describing stocks in this way leads to a characterization of returns as

depending on a number of factors, including volatility and other capital structure

variables as well as the fundamental asset beta.

The second line of research that this paper relates to is portfolio optimization

in the presence of a multitude of assets. Brandt et al. (2009) introduce the PPP

method that represents a solution to portfolio choice problems in which the sheer

1Please see the appendix for characteristics correlation structure. The market portfoliostructure as an affine function of the individual characteristics for stocks is a consequence ofthe linear aggregation of stocks.

71

multitude of assets makes it computationally difficult and statistically unreliable

to allocate an optimal weight to each of them. By aggregating assets according to

their characteristics, Brandt et al. (2009) simplify the allocation and estimation

problem to a handful of variables for a myopic investor. DeMiguel et al. (2017)

apply this technique to the mean-variance framework. I adopt the parametric

portfolio technique using, as characteristics, the stocks option-like features and

extend the parametric approach to a multiperiod setting.

My work is also related to the continuous time dynamic programming liter-

ature for portfolio choice, pioneered by Merton (1971) through to Liu (2007),

which identifies those problem setups that allow for a closed-form solution. I

extend the work by Liu (2007) to the PPP representation and find a closed-form

solution for parametric weights.

The paper is divided into four sections. The first section presents a model for

individual and portfolios of stocks following a structural credit risk model. The

second section extracts the relevant characteristics for building a PPP from the

return dynamics of the portfolio of credit-risky stocks modeled as options. The

third section introduces and solves the dynamic programming problem, and the

fourth section estimates the dynamic PPP model and compares it to classical

portfolio allocations such as equal-weight or value-weight.

72

2.1 The Model for Returns


I adopt the Merton (1974) structural credit risk model and describe equity as

an option on the underlying company assets. The asset dynamics are

dAiAi

= (µai − δai)dt+ σaidBi, (2.1)

where µ indicates the physical drift (gross of any cash payout δ)1 for the assets

a of firm i and σ the volatility of the production asset process. B is a Wiener

process.

2.1.1 Single Stocks

Following Merton, I model equity as an option on the underlying firm’s asset

value:

Ei = max(Ai,T −Ki, 0)

= Aie−d(T−t)N(d1)−Kie

−r(T−t)N(d2), (2.2)

with T the maturity of debt instruments, r the return of a zero coupon bond,

and d the dividend rate.2 The terms d1 and d2, under the risk-neutral measure,

1This is typically the average coupon on outstanding debt plus the net share sales or repur-chases. Hereafter, to simplify formulas, I omit the net cash payout δ.

2To simplify formulas, I assume that the dividend rate is zero, but I do account for it inthe empirical work.

73


are equivalent to

d1 =lnAi

Ki+(r + 1

2σ2ai

)(T − t)

σai√T − t

and (2.3)

d2 = d1− σai√T − t. (2.4)

In line with Dova (2016), I define the strike level for equity to be the level at

which the debt at maturity minus the recovery amount in liquidation equals the

value of assets at that point in time. Inherently, this means that debt holders

will take over the company when assets fall below the level of debt minus what

can be recovered in liquidation:

K = h(DT , χ, µ) =Decpn(T−t)

(1 + χ)eµai (T−t), (2.5)

where D is the level of debt at maturity (assuming that it compounds at the

average coupon rate) and χ is the recovery rate. The optimal level at which the

firm is liquidated or debt holders take over the company is directly proportional

to the level of debt and inversely proportional to the cost of liquidation.

Applying Ito’s lemma to (2.2), I obtain the continuous-time individual stock

dynamics, an expression of the instantaneous expected return on equity under

the following physical measure:1

reEi ∼ dEi =

[∂Ei∂t

dt+∂Ei∂Ai

dAi +1

2

∂2Ei∂2Ai

(dAi)2

]=

[Θi + ∆iAiµai +

ΓiA2i

2σ2ai

]dt+ ∆iAiσadBi, (2.6)

1Note that from now on I will indicate with r indexed the instantaneous return on any assetprice dP/P .

74


and taking expectations,

E(rei,t+dt) ∼ E(rei) =1

Ei

[Θi + ∆iAiµai +

ΓiA2i

2σ2ai

]dt. (2.7)

I conclude that equity return is a function of the maturity of the firm’s leverage,

the underlying asset levered return, and its volatility. It depends on the capital

structure of the company (equity theta, delta, and gamma change, with leverage

and volatility). Because under the Merton model, equity and asset volatility are

proportional

σei = g(σa, Ai, Ei,∆i) =AiEi

∆iσai . (2.8)

I can say that equity returns are affected by their own second moment or, vice

versa, use equity volatility to compute asset volatility.

2.1.2 Stock Portfolio

Given that there is a multitude of stocks in the market, the main issue is to

define the relations between idiosyncratic and systemic volatility. In my model,

the investor allocates her entire wealth to risky assets 1 and a stock portfolio is

the weighted sum of individual stock positions. I assume I have N stocks, P =∑Ni=1wiEi (with

∑Ni=1wi = 1), and I can derive the portfolio return dynamics

1The inclusion of the risk-free asset in a portfolio impacts the portfolio leverage per se, but,here, I am already considering leverage at the individual firm level, and I prefer not to allowfor the additional layer of leverage. This choice is in line with Brandt et al. (2009).

75


applying Ito’s Lemma as above:

rp ∼dP

P=

1

P

N∑i=1

wi

[∂Ei∂t

dt+∂Ei∂Ai

dAi +1

2

∂2Ei∂2Ai

(dAi)2

]

=1

P

N∑i=1

wi

[Θi + ∆iAiµai +

ΓiA2i

2σ2ai

]dt+

1

P

N∑i=1

wi∆iAiσaidBi. (2.9)

The portfolio expected return is

E(rp) ∼dP

P=

1

P

N∑i=1

wi

[Θi + ∆iAiµai +

ΓiA2i

2σ2ai

]dt. (2.10)

The unconditional expected return is the sum of the individual expected returns.

At this point, I do not say anything about the correlation structure of the in-

dividual shocks. I will deal with them later, while describing the nature of the

portfolio return characteristics.

2.1.3 The Market

I can model the market asset as the weighted average of the individual firm

asset processes. It is distributed as a weighted sum of lognormal variables, and

I approximate it with a lognormal according to Dufresne (2004). I also assume

that the market shocks are related to the structure of the characteristics driving

returns, but I will describe this relation later.

dM

M= µamdt+ σmdBm, (2.11)

with∑

iwi,mµai = µam and limN→∞∑N

i,j=1,i 6=j[w2i,mσ

2ai

+2wi,mwj,mρi,jσaiσaj ] = σ2m

(here wi,m indicates the value-weight of the individual stock in the market). I can

76

2.2 PPP Methodology

then derive expression (2.6) for the aggregate market, as follows:

rm ∼dEmEm

=1

Em

[∂Em∂t

dt+∂Em∂M

dM +1

2

∂2Em∂2M

(dM)2

]=

1

Em

[Θm + ∆mMµam +

ΓmM2

2σ2am

]dt+

M

Em∆mσmdBm, (2.12)

where M represents the assets and Em is the price of the equity market. I will

later use the market return expression to set up the PPP.

2.2 PPP Methodology

The PPP approach, developed by Brandt et al. (2009), introduces an effective

technique to find optimal portfolios by parameterizing stocks’ portfolio weights

according to their characteristics and by greatly reducing the degrees of freedom

of the optimization problem. The technique is based on representing the portfolio

return as the sum of a benchmark portfolio return and the returns of long-short

portfolios whose weights are parameterized according to the deviations of certain

characteristics from the ones of the benchmark portfolio itself:

rp,t+dt(θ) = rb,t+dt +1

Nt

θ>X>t rt+dt, (2.13)

where θ is a vector of weights for the characteristics, X represents a vector of

standardized characteristics, and subscripts indicate the time at which variables

are observed. In short, the investor allocates by default to a benchmark portfolio

and then optimizes her portfolio, finding the weights that maximize her utility

function (note that the weights are assumed to be time-invariant). I choose the

77

2.2 PPP Methodology

market portfolio of value-weighted stocks as a benchmark and then derive an

expression for the continuous time dynamics of my basket of stocks modeled

according to the structural credit risk model using the option return drivers as

characteristics.

2.2.1 PPP implementation

To move from the expressions for the portfolio (2.10) and market returns (2.12)

to the PPP framework, I start by subtracting one from the other, as follows:

rp − rm =1

P

N∑i=1

wi

[Θi + ∆iAiµai +

ΓiA2i

2σ2ai

]dt+

− 1

Em

[Θm + ∆mMµam +

ΓmM2

2σ2am

]dt

+1

P

N∑i=1

wi∆iAiσaidBi −M

Em∆mσmdBm, (2.14)

I then observe that, given an initial endowment sufficient to purchase the market

portfolio and in the absence of borrowing and short selling restrictions, I can

assume that Em = P , the price of the chosen portfolio. Also, to every firm

corresponds a weight in the market index that I define wi,m. I can transform the

previous expression as follows:

rp − rm =1

P

(N∑i=1

(wi − wi,m)

[Θi + ∆iAiµai +

ΓiA2i

2σ2ai

])dt

+1

P

(N∑i=1

(wi − wi,m)∆iAiσaidBi

), (2.15)

78

2.2 PPP Methodology

I note that the sum of weights wi−wi,m is equal to zero because there is no risk-

free rate. I can normalize the return explaining factors to excess returns with

respect to the market. The excess return drivers are in the square parentheses:

theta, delta, asset return, gamma, and volatility.1 I split those characteristics

into a weight and a return associated to that, so the terms including theta, delta,

and gamma become characteristics. I have a normalized weight xΘ,i, x∆,i ... xk,i

and a return attribution E(re) = 1P

(∑Ni=1 wi

[Θi + ∆iAiµai +

ΓiA2i

2σ2ai

])dt.

Now, θk is defined as the loading of the characteristics (in my case three: time

value, levered asset returns, and volatility) that contribute to the stock return

and it is constant across stocks. I can write

rp − rm =

(K∑k=1

θk

N∑i=1

(wi − wi,m)rei

)=

(K∑k=1

θk

N∑i=1

1

Nxk,irei

)

=1

Nθ>X>re = θ>rk

= θ>µkdt+ θ>σkdBk, (2.16)

where xk,i represents standardized2 characteristic weights and rk,i is the long-short

characteristic return. θ is the Kx1 vector of weights for the characteristics, X is

an NxK matrix of characteristic long-short standardized weights for the entire

universe of stocks, and re is the Nx1 vector of the individual stock returns. µk

and σk represent the mean and volatility of the long-short characteristic return

vectors. In essence, I have obtained an expression in which, to get an optimal

1The absolute asset value and its square Ai and A2i , as well as the market equity price P ,

are scaling factors.2They are standardized because they are by definition a difference of weights with the same

mean, and I can divide by their standard deviation and multiply the loading by the sameamount.

79

2.2 PPP Methodology

portfolio, it is sufficient to maximize expected utility with respect to the weights

θ of characteristics derived from the underlying single stock return drivers.

Note that the assumption that characteristic loadings are constant through

the cross section is very reasonable in my framework in which stocks’ dynamics

are driven by the components (or characteristics) of the stochastic differential

equation for a call option. Therefore, the individual loading of the components

depend on the Ito’s lemma application and stay the same across firms.

2.2.2 The Characteristics

It is important to analyze the characteristic long-short return portfolios driv-

ing equity excess returns and discuss their nature before moving on with the

optimization. There are three distinct aggregates of variables that I will call

characteristics, k1, k2, k3:

Θi

P︸︷︷︸k1=Time Decay

, ∆iAiPµai︸︷︷︸

k2=Levered Asset Returns

, ΓiA2i

2Pσ2ai︸︷︷︸

k3=Scaled Volatility

︸︷︷︸

Characteristic Return Components

. (2.17)

The elements of the three non linear characteristics are as follows:

1. Θ(ttomat): measures the time decay of the call option and is related to the

average debt duration of the company.

2. ∆(pdef): a measure of the stock risk-neutral in-the-moneyness with respect

to the underlying assets. It is closely related to the physical probability of

default of the company.

80

2.2 PPP Methodology

3. Ai/P(lev): measures the company leverage, and I define it as the sum of

company debt plus market value of equity.

4. µai(agr): measures the company physical asset growth rate.

5. Γ(gamma): measures the sensitivity of the Delta with respect to movements

in the asset value.

6. A2i /2P(sqrlev): a scaled measure of company leverage.

7. σai(mvol): the volatility of assets.

Interestingly, this list contains three of the six characteristics that, DeMiguel

et al. (2017) in their recent study, find to be significant for achieving effective

portfolio construction before transaction costs, in particular, return volatility,

asset growth and beta. Overall, returns are explained by nonlinear combinations

of seven variables. These combinations correspond to the elements of the Ito

lemma for the instantaneous change of stock price: time decay, delta times asset

return drift, and gamma times volatility. I can then write my characteristic

returns in PPP notation as follows:

θT rk = θ1rK1 + θ2rk2 + θ3rk3 . (2.18)

In essence, my portfolio choice problem has four assets: the market and three

characteristic long-short portfolios corresponding to the Greeks expansion of the

optional stock returns. Because the weight of the market is always one, under

the PPP setup, I only have to decide the weights for the remaining three long-

short characteristics: normalized theta, normalized delta times asset returns and

normalized gamma times volatility.

81

2.2 PPP Methodology

2.2.3 The State Variables

The three characteristics are driven by common factors:

• debt maturity, which I measure as the average time to maturity of corporate

debt,

• levered asset returns, which I measure as the annual expected asset returns

weighted by the delta of the equity option and the leverage ratio of the

company, and

• asset volatility, which I measure as the square of corporate unlevered asset

returns.

I assume that debt maturity, levered asset returns, and volatility change stochas-

tically through time and drive the returns of both the market portfolio and the

long-short portfolios (or characteristics portfolios). They are my state variables.

I indicate them with D for debt maturity, L for levered asset returns, and V for

volatility. They can be aggregated into a 3-by-1 state variables vector S.

The three state variables are assumed to be linearly independent of each other

so that debt maturity, levered asset returns, and asset volatility have zero cor-

relation.1 The distribution of the state variables vector is assumed to be of the

1Please see empirical evidence in the appendix.

82

2.2 PPP Methodology

square-root type1

dS = (ks −KsS)dt+ Σs

√SdBs =

= d

D

L

V

=

kD −KDD

kL −KLL

kv −KvV

dt+

σD√D 0 0

0 σL√L 0

0 0 σV√V

dBD

dBL

dBV

. (2.19)

Therefore,

µs = (ks −KsS) =

µD

µL

µV

and (2.20)

ΣsΣTs = h0 + h1S =

σ2DD 0 0

0 σ2LL 0

0 0 σ2V V

. (2.21)

Each characteristic return portfolio is assumed to be an affine function of the

state variables. In particular, the characteristic portfolios are

rk = µk(S)dt+σk(S)dBk =

rk1 = µk1dt+ σk1dBk1 = (b0 + b1D)dt+ b2

√DdBD

rk2 = µk2dt+ σk2dBk2 = (c0 + c1L)dt+ c2

√LdBL

rk3 = µk3dt+ σk3dBk3 = (e0 + e1V )dt+ e2

√V dBV

,

(2.22)

1This assumption reflects a modeling choice by which de-trended debt duration, leveredreturns, and volatilities follow mean-reverting heteroskedastic distributions. There is significantliterature around the mean-reversion of asset prices as well as asset volatilities. I will assumethis holds also for the de-trended debt duration series.

83

2.2 PPP Methodology

where µk and σk indicate the generic characteristic 3-by-1 drift and volatility

vectors. In general, b0 = c0 = e0 = 0 because the portfolios are long-short

portfolios and are expected to have no constant drift. The first characteristic

portfolio expresses the scaled time decay of the equity option that is driven by

the maturity of company debt. The second characteristic portfolio return is

a scaled version of the levered asset return state variable. Finally, the third

characteristic portfolio return tracks volatility scaled by the stocks’ gamma: the

volatility variable is, itself, stochastic.

By definition, the market portfolio is a weighted average of all stocks in the

market, and, as such, it can be described as an affine function of the same state

variables that drive returns at the individual stock level, as follows:

rm = µmdt+ ΣmdBm

= (a0 + a1D + a2L+ a3V )dt+ a4

√DdBD + a5

√LdBL + a6

√V dBV (2.23)

with ΣmdBm =

[a4

√D a5

√L a6

√V

] [dBD dBL dBV

]T.

I expect the market portfolio to have a drift that depends on the weighted

average of individual stock drifts. Also, empirically, it will mostly depend on

levered asset drifts, while its shocks will depend on all state variables. In fact,

at the aggregate level, equity duration is not expected to change dramatically

over time. Also, with respect to the impact of volatility, the literature offers

broad evidence that individual stock expected returns depend on volatility, but

this effect almost disappears at the aggregate level. Overall, I expect to have

a1 = a3 ∼ 0.

84

2.3 Portfolio Choice Model


In my setup, markets are incomplete, and I have the following problem:

• Maximize utility to a finite horizon for a multiperiod investor in continuous

time with no intermediate consumption.

• In addition to investing in the market, I have to choose weights to assign to

long-short return portfolios corresponding to the stock characteristics, one

corresponding to debt maturity, one to levered returns, and one to volatility.

• The market and the two characteristic portfolios are affine functions of the

state variables, levered asset returns and asset volatility.

• Debt duration is assumed to be deterministic over the optimisation period.

• The investor utility is of the power type.

• All wealth is allocated to risky assets.

I refer back to Merton (1971) and Liu (2007) for the continuous time dynamic

programming portfolio choice.

2.3.1 The Utility Function

I assume that the investor has a CRRA utility function. The utility at time

t of wealth in t+ dt is represented by

U(W, t) = e−βdtEt(W 1−γt+dt

)1− γ

, (2.24)

85


where β is the subjective discount factor and γ the coefficient of risk aversion.

I assume the investor choice does not involve intermediate consumption and, in

keeping with Brandt et al. (2009), I assume she has to invest all her wealth in

risky assets.

2.3.2 The Dynamic Budget Constraint

Following the PPP approach, the default allocation is to the market portfolio,

and the investor has to choose the weights to allocate to each of the long-short

portfolios that represent the universe of investable stocks. Wealth gets allocated

to a market tracking portfolio and a series of other excess return or overlay

strategies. The budget constraint for this problem is:

dW = Wt

[rm + θT rk

]dt

= Wt

[µm + θTµk

]dt+Wt

[ΣmdBm + θTΣkdBk

], (2.25)

where rk indicates a 3-by-1 vector of instantaneous long-short returns and rm

the instantaneous return on the market. W is wealth and the variables µ in-

dicate drifts and Σ volatilities. In particular, Σk is the matrix of characteristic

volatilities, as follows:

Σk =

b2

√D 0 0

0 c2

√L 0

0 0 e2

√V

. (2.26)

The correlation matrix of the disturbances to the characteristic portfolios and

the ones to the state variables is, in absolute value, a diagonal unit matrix because,

86


as discussed, the characteristic portfolios are affine functions of the state variables

and there are no significant cross correlations. The empirical evidence in appendix

shows correlations and p-values among debt duration, volatility, and levered asset

returns. The only correlation that appears to be significantly different from zero

is the one between debt duration and volatility. Hereafter, I will assume it is zero

for computational ease:

|ρ| =

1 0 0

0 1 0

0 0 1

. (2.27)

In addition, I define the correlation matrix of the market portfolio disturbance

with the disturbances of the characteristic portfolios. In my setup, this matrix is

also, in absolute value,1 an identity matrix because the market is the weighted

average of the individual stocks and, by construction, an affine function of the

characteristics themselves:

|ρm| =

1 0 0

0 1 0

0 0 1

. (2.28)

In essence, the market disturbance consists of the combination of three Brownian

motions that represent the characteristics and, in turn, the state variables.

Because they are both diagonal unit matrices, I could avoid using both ρ and

ρm. Nonetheless, I will solve the optimization problem in the general case and I

1Correlations can be both positive or negative but are assumed to be perfect so that theabsolute value will be one.

87


will then find the particular solution to my problem.

2.3.3 The Bellman Equation

Using the Bellman principle, I can write the portfolio choice problem of max-

imizing utility at maturity, subject to the budget constraint, as the solution to

the following:

maxθEt

[1

dtdJ(W,S, t)

]= 0. (2.29)

Final wealth will be the result of a succession of dynamic investment decisions

with J , the indirect utility function, depending on wealth, the state variables,

time, and, most importantly, the choice of the weight vector. Using Ito to expand

the differential, I get

maxθEt

[Jt + JWdW +

1

2JWWdW

2 + JSdS +1

2JSSdSdS + JWSdWdS

]= 0.

(2.30)

I then insert the budget constraint into (2.30) and take expectations to get:

maxθ

(Jt + JWWt[µm + θTµk] +

1

2JWWW

2t

[ΣmΣT

m + 2θTΣkρTmΣT

m + θTΣkΣTk θ]

+

+ µTs JS +1

2Tr(ΣsΣ

Ts JSST

)+Wt

[Σmρ

TmΣT

s + θTΣkρTΣT

s

]JWS

)= 0. (2.31)

88


Differentiating with respect to θ and simplifying, yields the first-order condition

for optimality1

JWµk + JWWWt

[Σkρ

TmΣT

m + ΣkΣTk θ]

+ ΣkρTΣT

s JWS = 0, (2.32)

and solving for the 3-by-1 characteristic weights vector θ, I get

θ = − JWWtJWW

(ΣkΣTk )−1µk︸︷︷︸

Characteristic Hedging

− (ΣkΣTk )−1Σkρ

TmΣT

m︸︷︷︸Benchmark Correlation Hedging

+

− JWS

WtJWW

(ΣkΣTk )−1Σkρ

TΣTs︸︷︷︸

Intertemporal Hedging

. (2.33)

I can identify three components to the optimal weight vector. The Characteris-

tic Hedging component takes care of the links between characteristics optimiz-

ing allocation among them. The Benchmark Correlation Hedging contributes to

the weights, taking into account the correlation of the characteristics with the

benchmark portfolio because the investor is assumed to be always invested in

it. Finally, the Intertemporal Hedging component takes care of the correlation

between the changing opportunity set represented by the state variables and the

characteristics.

1The utility function is concave, so this is both a necessary and sufficient condition foroptimality.

89


2.3.4 Solving for the Value Function

2.3.4.1 A guess for the function

My investor has a CRRA utility function and, in line with Liu (2007), I

conjecture the value function to be of the following type:

J(W,S, t) = e−βtW 1−γ

1− γ[f(S, t)]γ , (2.34)

with β as the subjective discount factor, γ the coefficient of risk aversion and f

a function of the state variable defined as follows:

f(S, t) = eC(t)+G(t)TS+ 12ST ηTQ(t)ηS, (2.35)

with C(t) a scalar and G(t), Q(t) vector and matrix functions of time, η a vector

of scalars.

2.3.4.2 Implicit Portfolio Weights

Given the guess for the value function, I can substitute it into (2.33) and get

the expression for the characteristic weights as a function of the state variable:

θ = (γΣkΣTk )−1µk︸︷︷︸



TmΣm︸︷︷︸

Benchmark Correlation Hedging

+ (ΣkΣTk )−1Σkρ

TΣTs

∂lnf

∂S︸︷︷︸Intertemporal Hedging

,

(2.36)

90


where the log of the function f measures the elasticity of weights to changes in

the state variable. Calculating this derivative I obtain

θ = (γΣkΣTk )−1µk︸︷︷︸



TmΣT

m︸︷︷︸Benchmark Correlation Hedging

+

+ (ΣkΣTk )−1Σkρ

TΣTs [G(t) +Q(t)ηS]︸︷︷︸

Intertemporal Hedging

. (2.37)

In order to fully characterize the portfolio weights I have to find a solution for the

G(t) and Q(t). Because I have no intermediate consumption, the weights do not

depend on C(t). In addition, since the processes that describe assets are affine in

the state variables, I also conclude that Q(t) = 0. So the problem simplifies to

finding G(t) and (2.37) simplifies to

θ =(γΣkΣ

Tk

)−1 (µk − γΣkρ

TmΣT

m + γΣkρTΣT

sG(t)). (2.38)

2.3.4.3 Solving for G(t)

The form of G(t) follows from Liu (2007) with a few modifications to accom-

modate for a portfolio allocation where there is no risk-free rate, and the weights

are measured as long-short characteristic deviations from the market portfolio.

I should replace the solution for θ and the conjecture for the J function into

(2.31). Before I do that, I simplify the portfolio weight expression by defining

a drift measure that is the combination of the characteristics drifts and their

static correlation to the market portfolio. My new measure of drift becomes

µ = µk − γΣkρTmΣT

m, which translates into the following 3-by-1 portfolio weight

91


vector for the characteristics:

θ =(γΣkΣ

Tk

)−1 (µ+ γΣkρ

TΣTsG(t)

). (2.39)

I proceed and replace (2.39) and (2.34) into (2.31) to obtain the following

Lemma 1.

(−βγ

+1− γγ

µm

)f+

+(1− γ

2γ2(ΣkΣ

Tk )−1µTµ− 1− γ

2ΣmΣT

m −1− γγ

(ΣkΣTk )−1µTΣkρ

TmΣT

m

)f+

+ ft +1

2Tr(ΣsΣ

Ts fSST

)+γ − 1

2ffTS

(ΣsΣ

Ts − Σsρρ

TΣTs

)fS+

+

(µTs + (1− γ)Σmρ

TmΣT

s +1− γγ


TΣTs

)fS = 0. (2.40)

The expression consists of seven stochastic components. By construction, they

are affine functions of the state variables. Hereafter, I describe them one-by-one,

I assign them generic affine coefficients, and I show their mapping to my model

and the reason why they are affine in the state variables.

1. The market expected return is as follows:

µm = δ0 + δ1S

= a0 + a1D + a2L+ a3V, (2.41)

linear in S by construction. a0 represents a drift that can be decomposed in

three individual drifts assigned to the three state variables.

92


2. The squared sharpe ratio is as follows:

µT (ΣkΣTk )−1µ = H0 +H1S

= µT(D,L,V )

1b22D

0 0

0 1c22L

0

0 0 1e22V

µ(D,L,V ). (2.42)

It is linear in S and not quadratic because the inverse of the variance nor-

malizes the drifts, which are linear functions of the state variables. This

can be verified by looking at the explicit form of µ:

µ = µk − γΣkρTmΣT

m

=

b1D

c1L

e1V

− γb2a4D

c2a5L

e2a6V

. (2.43)

3. The market variance is as follows:

ΣmΣTm = q0 + q1S

= a24D + a2

5L+ a26V. (2.44)

93


4. The characteristic-market hedging covariance is as follows:

(µTΣkρTmΣT

m)(ΣkΣTk )−1 = z0 + z1S

= µT(D,L,V )

a4b2

a5c2

a6e2

. (2.45)

It is linear in S because the quadratic terms in the volatility elements of the

numerator are normalized by the denominator.

5. The unspanned state variables covariance matrix is as follows:

−(ΣsΣTs − Σsρρ

TΣTs ) = l0 + l1S

=

0 0 0

0 0 0

0 0 0

. (2.46)

It is a matrix of zeros because the correlation matrix ρ is an identity matrix.

6. The state-market hedging covariance vector is as follows:

(ΣmρTmΣT

s )T = p0 + p1S

=

[a4σDD a5σLL a6σV V

]. (2.47)

94


7. The state-characteristic hedging covariance vector is as follows:

((µTΣkρTΣT

s )(ΣkΣTk )−1)T = g0 + g1S

= µT(D,L,V )

σDb2

0 0

0 σLc2

0

0 0 σVe2

. (2.48)

It is linear in S because quadratic terms at the numerator and denominator

simplify.

I am using the simplified expression in the generic coefficients above to find a

solution to the stochastic PDE. In order for the conjecture for J to be the solu-

tion of (2.40), I can replace these seven components and expression (2.36) for the

portfolio weights in expression (2.31) to obtain an expression in C(t), G(t), Q(t).

I know that the solution here does not depend on either C(t) or Q(t) because I

have no consumption and no quadratic terms. As a consequence, once I replace

expressions 1. to 7. above into (2.40), I have to make sure that all terms contain-

ing G(t), that is those linear in S, are equal to zero in order to have a solution

to the PDE. I conclude that the value of G(t) that solves equation (2.40) is

Lemma 2.

G(t) =

− 2[exp(ξτ)−1]

(κ+ξ)[exp(ξτ)−1]+2ξψ, if ξ2 ≥ 0

− 2

κ+ζcos(ζτ/2)sin(ζτ/2)

ψ, if ζ2 ≥ 0,

(2.49)

with τ being the time difference between the finite horizon of the investor and the

date the portfolio allocation choice is made. The solution above is what solves the

95


ODE that makes all coefficients linear in S equal to zero, as follows:

∂

∂td−

(K − (1− γ)p1 −

1− γγ

g1

)Td+

1

2dT [h1 + l1(1− γ)]d+

+1− γ2γ2

H1 +1− γγ

δ1 −1− γ

2q1 −

1− γγ

z1 = 0, (2.50)

with initial condition d(T ) = 0 and given the following variable definitions:

ψ = −(

1− γ2γ2

H1 +1− γγ

δ1 −1− γ

2q1 −

1− γγ

z1

)(2.51)

κ = K − (1− γ)p1 −1− γγ

g1 (2.52)

ξ =√κ2 + 2ψ[h1 + l1(1− γ)] (2.53)

ζ = −iξ. (2.54)

2.3.4.4 Explicit Portfolio Weights

To find the explicit portfolio weights, I apply a separation result that allows

me to represent the overall solution as a linear combination of individual solutions

for the three characteristics. This approach is subject to a number of conditions

that Liu (2007) states in his work and can be summarized as

1. The state variable drifts can be partitioned.

2. The state variables volatility matrix is diagonal.

3. The correlation matrix between assets and state variables is diagonal.

4. The assets volatility matrix is lower or upper triangular.

96


5. The sharpe ratio can be partitioned with each element depending only on

one state variable.

6. The risk-free rate (constant or stochastic) can be partitioned.

I can easily check that these apply to my setup1 by mapping out the main com-

ponents of my problem in matrix format. The state variables are

µs =

µD

µL

µV

(2.55)

Σs =

σD√D 0 0

0 σL√L 0

0 0 σV√V

(2.56)

and the characteristic variables are

µk =

b1D

c1L

e1V

(2.57)

Σk =

b2

√D 0 0

0 c2

√L 0

0 0 e2

√V

. (2.58)

1In the appendix, I report a series of statistics that support this modeling choice for thecharacteristic variables.

97


The Sharpe ratio takes the following form:

(Σk)−1µ =

b1−γb2a4

b2

√D

c1−γc2a5c2

√L

e1−γe2a6e2

√V

, (2.59)

and, finally, the correlation matrix of characteristics and state variables is

|ρ| =

1 0 0

0 1 0

0 0 1

. (2.60)

Lemma 3. My variables verify the conditions for the separation theorem pre-

sented in Liu (2007). I hereby extend that theorem to express the weights of each

characteristic portfolio as a linear function of the individual solutions to three

separate equations in (2.40),

(−βγ

+1− γγ

µm,i +1− γ2γ2

(Σk,iΣTk,i)−1µi

TµiΣk,iΣTk,i −

1− γ2

Σm,iΣTm,ii+

− 1− γγ


TΣk,iρTm,iΣ

Tm,i

)fi+

+ ft,i +1

2Tr(Σs,iΣ

Ts,ifSST ,i

)+γ − 1

2ffTS,i

(Σs,iΣ

Ts,i − Σs,iρiρ

Ti ΣT

s,i

)fS,i+

+

(µTs,i + (1− γ)Σm,iρ

Tm,iΣ

Ts,i +

1− γγ


TΣk,iρTi ΣT

s,i

)fS,i = 0,

(2.61)

where i indicates which characteristic I am solving for. In particular, the implicit

function f takes the form f(S, t) = f1(S1, t)f2(S2, t)f3(S3, t). Solving for the three

independent f functions above allows to express the portfolio weights as linear

98


combinations of the portfolio weights for the individual characteristics. In this

specific case, because the characteristics volatility matrix is diagonal, the weight

vector will have three distinct and separate components:

θ = (γΣkΣTk )−1µk − (ΣkΣ

Tk )−1Σkρ

TmΣT

m + (ΣkΣTk )−1Σkρ

TΣTsG(t) = (2.62)

θ1

θ2

θ3

=

b1γb22

c1γc22

e1γe22

−a4b2

a5c2

a6e2

+

σDb2

0 0

0 σLc2

0

0 0 σVe2

G(t)D

G(t)L

G(t)V

. (2.63)

In a more general setup, it is sufficient that the volatility matrix of character-

istic portfolios is either lower or upper triangular to apply the separation theorem

and find recursive solutions. I would first solve the equation that has only one

non-zero element in the variance-covariance matrix and then proceed with solving

the one immediately below and so on to complete all individual variable solutions.

Finally, I would express weights as a linear combination of the individual solu-

tions weighted by the correlation elements in the variance-covariance matrix of

characteristic return portfolios.

2.3.5 Comparative Statics

It is important to understand how variations in the underlying variables im-

pact the weights for the three long-short portfolios resulting from the solution

above (2.63). In order to do that, I run a simple simulation, varying some of

the key underlying model variables and observing the impact of that variation on

the overall portfolio weights vector Θ. A high-risk aversion parameter impacts

the Characteristic Hedging component, reducing its weight in favor of the Bench-

99


mark Correlation Hedging component and, partially, the Intertemporal Hedging

Component. A growing characteristic variance reduces the weight of that charac-

teristic in the portfolio allocation. A characteristic having a positive covariance

with the market, has a negative weight contribution to Θ. Finally, by construc-

tion, the contribution of the intertemporal hedging component to the portfolio

weights is zero when the horizon is just one period and grows up to an asymptotic

value when the time to maturity increases above 10 years.

Figure 2.1: Theta weight of a charac-teristic as a function of γ.

Figure 2.2: Theta weight of a charac-teristic as a function of the characteris-tic volatility.

Figure 2.3: Theta weight of a charac-teristic as a function of its covariancewith the market.

Figure 2.4: Contribution to Θ of in-tertemporal hedging component as afunction of time to maturity.

100

2.4 Empirical Application


2.4.1 Data

My database contains a total of 1,078 stocks that have been constituents of

the S&P 500 index from March 1989 to June 2017. At every quarterly date,

my investible universe consists of only those stocks that belong to the S&P 500

index. Those same stocks are also the ones I use to calculate my benchmark.1

Here, I exclude the financial stocks from both the benchmark and the portfolios

(see appendix B for results that include financial stocks).

The data comes from Compustat and CRSP for financial and accounting

variables and Barclays, SIFMA, and Bloomberg for debt analytics, such as debt

duration. The sample is quarterly with the first observation on March 31, 1989:

quarterly is the frequency adopted by Compustat for reporting balance sheet

items, such as accounting liabilities, accounting assets, shares outstanding, and

dividends per share. Interest expenses are inferred from the annual interest ex-

penses and evenly apportioned on a quarterly basis. The duration of firm lia-

bilities has been calculated assuming that short-term liabilities have a duration

of one year and long-term liabilities have the same average duration as the one

reported in the Barclays U.S. High Grade Duration Index. Recovery rates are

assigned according to the current Moody’s industry specific recovery tables.

Some of the input data for my model are not directly observable in the mar-

ket and have to be calculated or inferred from observable variables: these are

mainly the firm asset values and volatilities. To establish asset values, I add debt

1Nonetheless, my benchmark will be slightly different from the S&P 500 because, at times,some of the financials required for my model will not be available, and I will exclude thosecompanies from the database.

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values from Compustat and equity values from CRSP to obtain a time series of

asset values as suggested in Eom et al. (2004). The gross asset return drift µ

is calculated from the asset value time series, while asset volatility is estimated

as a function of equity volatility adjusting for the leverage effect. Equity return

volatility is built by taking the absolute value of the price range within the last

month of the quarter.

In order to deal with outliers at the level of balance sheet variables, I win-

sorized data series using the method suggested by Green et al. (2017) at the first

and third quartile of the cross-section at every quarterly observation. I eliminated

all stocks that had missing data at the level of input variables for my model, such

as total debt, market cap, debt maturity, and at least two valid consecutive stock

return observations within the sample. In keeping with Brandt et al. (2009), I

also eliminated the stocks with a negative book-to-market ratio. Finally, I win-

sorized characteristics at the first and third quartile, on a quarterly basis, setting

the extremes at the quartile level this time.1

2.4.2 Timeline and Variables

The portfolio optimization is run out of sample: at date t, returns reflect the

change in prices between date t and date t+ 1. All balance sheet data, reported

quarterly and indexed at time t, are collected between time t − 2 and t − 1 and

made available at time t to ensure that there is, at least, one quarter between the

data publication and the investment decision. In the case of semi-annual data,

the input data is the one relating to at least one reporting period prior to the

1I chose to keep the winsorized characteristics at the quartile level as opposed to what I havedone for the raw variables in order to reduce the potential noise on strictly positive variablesinduced by the Green procedure.

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current quarter.

Financial market data is observed on a spot basis. In particular, with respect

to the state variables in my model, I have adopted the following conventions:

• Debt Duration state variable is observed at quarter end and it is the nor-

malized average duration of the company debt at the end of the previous

quarter multiplied by the leverage ratio at the end of the previous quarter.1

So, for example, if the maturity of the individual company’s debt is six

years, I will pick the six-year interpolated rate on the treasury curve. The

treasury rate is a valid instrument from a statistical point of view because it

has a strong negative correlation to corporate debt with a p-value of 0.000

over the more than 50,000 observations in my sample.

• Levered Asset Returns are measured as the value of assets at the end of the

previous quarter divided by the value of assets at the end of two quarters

prior to the current, all multiplied by the leverage ratio as measured at the

end of the previous quarter.

• Asset Volatility is measured as the realized equity price range for the month

previous to quarter end scaled by the leverage ratio at the end of the pre-

vious quarter.

These state variables, all measured at t − 1, drive returns for the period going

from time t to timet+ 1. This ensures that my portfolio optimisation is truly for-

ward looking. In addition, I will report the performance of alternative portfolios

based on three characteristics taken from the recent literature. In particular, I

1In the model, the time decay element is measured by the option Θ divided by the value ofequity; here I use the leverage ratio, as it is stationary.

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will run portfolios based on the size, book-to-market and momentum character-

istics as well as size, book-to-market, and gross productivity. These variables are

calculated as follows:1

• Size is the log of shares outstanding times the share price, all lagged by one

quarter.

• Book-to-Market, (BTM) is the log of one plus assets minus liabilities divided

by market value of equity, all lagged by one period.

• Momentum, (MOM) at time t corresponds to quarterly momentum and is

the ratio between share price at time t−1 plus pro-rata quarterly dividends

divided by share price at time t− 2.

• Gross Profitability, (GMA) is measured as the ratio between revenues minus

costs and total assets. All lagged by one quarter.

2.4.3 Performance Statistics

All out-of-sample simulations are compared with value-weighted and equal-

weighted portfolios built with S&P 500 stocks. In order to compare the active

portfolios built according to my model and the benchmarks, I will calculate a

series of statistics: mean return, median return, maximum drawdown, standard

deviation of returns, Sharpe ratio, Certainty Equivalent, turnover and contribu-

tion to the performance from the dynamic optimisation component. All statistics

are calculated on a quarterly basis. I report returns, standard deviations, con-

1I choose to adopt variable definitions taken from Brandt et al. (2009) and DeMiguel et al.(2017)

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tribution of the dynamic component, and Sharpe ratios on an annualised basis,

while maximum drawdown, Certainty Equivalent and turnover are quarterly.

In particular, turnover is calculated according to the method utilized by

Brandt et al. (2009) as follows:

Tt =Nt∑i=1

|wi,t − wi,t−1|. (2.64)

In particular, for every quarterly date I calculate the change between a stock’s

weight in the portfolio and its weight in the previous period. Whenever a stock

disappears from the portfolio it is assigned a weight of zero. I disregard the

change in weight occurring naturally because of the natural evolution of company

characteristics, and I report the turnover required to adjust to the portfolio weight

net of the natural change in weight at the individual company account level.

Finally, the turnover is reported as the average quarterly turnover over the entire

sample.

The certainty-equivalent (CE) return, defined as the risk-free rate that an

investor is willing to accept rather than adopting a particular risky portfolio

strategy, is computed by taking a third order Taylor expansion of the power

utility function:

CE = µr − γσ2r

2+ γ(1 + γ)

σ3r

3(2.65)

in which µr, σ2r and σ3

r are the first, second and third moment of the out-of-sample

excess returns for a generic strategy r, and γ is the risk aversion.

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2.4.4 One-Period Optimisation

In this part, I present portfolio optimisation results for an investor that opti-

mizes returns across a single period. In order to estimate the model parameters,

I use a sample window of 10 years rolling it forward by one quarter at a time. Es-

timation starts with an initial window running from March 31, 1989 to March 31,

1999. The investor, whose preferences are expressed by a power utility function,

uses the estimated variables to allocate her portfolio, assuming the investment

horizon is one quarter. The investor’s portfolio is built using the value-weighted

market portfolio as a base, and individual stock transactions are put in place

to reallocate stock characteristic weights according to the output of the model.

To measure the validity of my model (dubbed the Capital Structure Portfolio,

(CSP), I benchmark the Parametric Portfolio returns against both the value-

weighted (VW) and the equally-weighted (EW) return on S&P 500 stocks. In

addition, I also present the performance of portfolios built using the size, book-

to-market, and momentum characteristics as well as using size, book-to-market,

and gross profitability. Next to the characteristic-based portfolios’ Sharpe ratios

I have added two ** when they pass a one-sided t-test to detect if their measure

is higher than that of both the VW and EW benchmarks with a 97.5% level of

confidence. I have added only one * if the t-test is passed only with respect to the

VW benchmark. No asterisk indicates that the t-test is not passed with respect

to any benchmark.1 Results are presented for risk aversion parameters of γ = 5,

γ = 10, γ = 15, and γ = 30.

1The t-test is built by bootstrapping 1,000 samples with replacement and assuming i.i.d.data.

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Table 2.1 Return statistics for portfolios built by individuals with an investment horizon of one periodonly. I report annualized mean return, annualized median return, maximum quarterly drawdown, an-nualized standard deviation of portfolio return, annualized Sharpe ratio, certainty equivalent and meanquarterly turnover. Sharpe ratio is significantly higher than VW’s and EW’s when accompanied by **symbol or only higher than VW’s when accompanied by * symbol. The first three portfolios are built onthree characteristics: 1) CSP portfolio: Debt Duration, Asset Levered Returns and Asset Volatility; 2)SBM portfolio: size, book-to-market, and momentum and 3) SBG portfolio: size, book-to-market, andgross profitability. The last two portfolios represent the value-weighted and the equal-weighted portfoliosconstructed using all S&P 500 stocks included in my investment universe. The three panels correspond tothe three different risk aversion parameters used: γ = 5, γ = 10, γ = 15, and γ = 30.

Panel A: γ = 5 Mean Ret Median Ret Max D.down Std SR CE Turnover

CSP 0.334 0.409 -0.625 0.397 0.841** -0.006 4.994

SBM 0.105 0.111 -0.527 0.256 0.410* -0.010 1.604

SBG 0.109 0.096 -0.486 0.202 0.538* 0.003 1.141

VW B.mark 0.046 0.095 -0.449 0.149 0.308 -0.004 0.085

EW B.mark 0.092 0.115 -0.474 0.178 0.515 0.002 0.167

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Panel B: γ = 10 Mean Ret Median Ret Max D.down Std SR CE Turnover

CSP 0.186 0.217 -0.516 0.247 0.753** -0.036 2.453

SBM 0.074 0.110 -0.462 0.188 0.394* -0.034 0.693

SBG 0.076 0.102 -0.445 0.156 0.483* -0.020 0.641

VW B.mark 0.046 0.095 -0.449 0.149 0.308 -0.024 0.086

EW B.mark 0.092 0.115 -0.474 0.178 0.515 -0.025 0.166

Panel C: γ = 15 Mean Ret Median Ret Max D.down Std SR CE Turnover

CSP 0.137 0.137 -0.480 0.203 0.677** -0.065 1.608

SBM 0.064 0.096 -0.440 0.169 0.378 -0.062 0.422

SBG 0.064 0.085 -0.432 0.145 0.443* -0.041 0.497

VW B.mark 0.046 0.095 -0.449 0.149 0.308 -0.049 0.086

EW B.mark 0.092 0.115 -0.474 0.178 0.515 -0.056 0.166

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Panel D: γ = 30 Mean Ret Median Ret Max D.down Std SR CE Turnover

CSP 0.088 0.111 -0.444 0.165 0.536* -0.172 0.771

SBM 0.054 0.091 -0.423 0.153 0.350 -0.173 0.236

SBG 0.053 0.071 -0.419 0.137 0.388 -0.136 0.384

VW B.mark 0.046 0.075 -0.410 0.144 0.321 -0.158 0.085

EW B.mark 0.092 0.110 -0.448 0.172 0.538 -0.182 0.167

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The output of the model is highly sensitive with respect to the individual’s risk

aversion parameter. Portfolios that do well across most statistics with low risk

aversion lose to other portfolios when the risk aversion parameter is increased. My

CSP portfolio has significantly higher mean returns across risk aversion scenarios,

but its turnover changes significantly when risk aversion changes. With low high

risk aversion, the CSP portfolio has high return, high Sharpe ratio and relatively

low turnover. As expected, the EW portfolio outperforms the VW portfolio in

terms of Sharpe ratio and turnover. Also the EW portfolio outperforms the

SBM and the SBG portfolios in terms of Sharpe ratio in all cases when the

risk aversion is medium-high (γ > 5). The CE performance measures depicts a

slightly different scenario where, the CSP, SBM and SBG strategies outperform

the passive indices in case of very low and very high risk aversion. In addition,

on a CE basis the SBG strategy seems to be doing better than the others. This

is due to the strong impact of the risk aversion coefficient adjustment in the CE

formula when strategies have high volatility, like the CSP. When the coefficient

of risk aversion tends to 1, then the rankings displayed in the Sharpe Ratio

performances are observable also in the CE indicator with a clear dominance of

the active investment strategies:

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Table 2.2 I report, for all strategies and benchmarks, the certainty equivalentreturn for levels of relative risk aversion below 5, ranging from the log utilityfunction to a γ = 4. Investment horizon is one quarter ahead.

CE γ = 1 γ = 2 γ = 3 γ = 4

CSP 0.094 0.033 0.014 0.003

SBM 0.034 0.010 0.001 -0.005

SBG 0.056 0.026 0.015 0.009

VW B.mark 0.008 0.006 0.003 -0.001

EW B.mark 0.019 0.015 0.011 0.006

2.4.5 Intertemporal Optimization

In this part, I present portfolio optimization results for an investor that op-

timizes returns across an investment period of five years. The investor takes

decisions about her portfolio today, aware that she will have to reallocate on a

quarterly basis with the intent to maximize her wealth in five years’ time. In or-

der to estimate the model parameters, I use a sample window of 10 years, rolling

it forward by one quarter at a time. Estimation starts with an initial window run-

ning from March 31, 1989 to March 31, 1999. The investor, whose preferences are

expressed by a power utility function, uses the estimated variables to allocate her

portfolio, assuming the investment horizon is long term. The investor’s portfolio

is built using the value-weighted market portfolio as a base, and individual stock

transactions are put in place to reallocate stock characteristic weights according

to the output of the model. To measure the validity of my CSP model, I bench-

mark the Parametric Portfolio returns against both the value-weighted (VW) and

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the equally-weighted (EW) return on S&P500 stocks. In addition, I also present

the performance of portfolios built using the size, book-to-market, and momen-

tum characteristics as well as using size, book-to-market, and gross profitability.

Next to the characteristic-based portfolios’ Sharpe ratios I have added two **

when they pass a one-sided t-test to detect if their measure is higher than that

of both the VW and EW benchmarks with a 97.5% level of confidence. I have

added only one * if the t-test is passed only with respect to the VW benchmark.

No asterisk indicates that the t-test is not passed with respect to any benchmark.

Results are presented for risk aversion parameters of γ = 5, γ = 10, γ = 15, and

γ = 30.

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Table 2.3 Return statistics for portfolios built by individuals with an investment horizon of five years(intertemporal optimization). I report annualized mean return, annualized median return, maximumquarterly drawdown, annualized standard deviation of portfolio return, annualized Sharpe ratio, certaintyequivalent and mean quarterly turnover, as well as the average annual contribution to return by theintertemporal hedging component Dynamic Contribution. Sharpe ratio is significantly higher than VW’sand EW’s when accompanied by ** symbol or only higher than VW’s when accompanied by * symbol.The first three portfolios are built on three characteristics: 1) CSP portfolio: Debt Duration, AssetLevered Returns and Asset Volatility; 2) SBM portfolio: size, book-to-market, and momentum, and 3)SBG portfolio: size, book-to-market, and gross productivity. The last two portfolios represent the value-weighted and the equal-weighted portfolios constructed using all S&P 500 stocks included in my investmentuniverse. The three panels correspond to the three different risk aversion parameters used: γ = 5, γ = 10,γ = 15, and γ = 30.

Panel A: γ = 5 Mean Ret Median D.down Std SR CE T.over Dyn.Con.

CSP 0.338 0.411 -0.586 0.398 0.848** 0.010 4.853 0.004

SBM 0.210 0.139 -0.482 0.325 0.645** 0.019 2.408 0.105

SBG 0.166 0.143 -0.555 0.247 0.672** 0.010 1.446 0.057

VW B.mark 0.046 0.075 -0.410 0.144 0.321 -0.004 0.085 0.000

EW B.mark 0.092 0.110 -0.448 0.172 0.538 0.002 0.167 0.000

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Panel B: γ = 10 Mean Ret Median D.down Std SR CE T.over Dyn.Con.

CSP 0.198 0.222 -0.479 0.257 0.767** -0.020 2.447 0.011

SBM 0.130 0.095 -0.654 0.248 0.523* -0.031 2.514 0.056

SBG 0.129 0.119 -0.409 0.193 0.671** 0.001 1.379 0.054

VW B.mark 0.046 0.075 -0.410 0.144 0.321 -0.024 0.085 0.000

EW B.mark 0.092 0.110 -0.448 0.172 0.538 -0.025 0.167 0.000

Panel C: γ = 15 Mean Ret Median D.down Std SR CE T.over Dyn.Con.

CSP 0.151 0.153 -0.444 0.216 0.701** -0.044 1.663 0.014

SBM 0.121 0.091 -0.498 0.223 0.545* 0.008 2.004 0.058

SBG 0.129 0.137 -0.525 0.200 0.645** -0.065 1.554 0.065

VW B.mark 0.046 0.075 -0.410 0.144 0.321 -0.049 0.085 0.000

EW B.mark 0.092 0.110 -0.448 0.172 0.538 -0.056 0.167 0.000

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Panel D: γ = 30 Mean Ret Median D.down Std SR CE T.over Dyn.Con.

CSP 0.104 0.123 -0.408 0.178 0.583* -0.134 0.911 0.016

SBM 0.062 0.051 -0.532 0.209 0.296 0.013 2.390 0.008

SBG 0.121 0.140 -0.376 0.186 0.649** -0.087 1.281 0.068

VW B.mark 0.046 0.075 -0.410 0.144 0.321 -0.150 0.085 0.000

EW B.mark 0.092 0.110 -0.448 0.172 0.538 -0.182 0.167 0.000

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The performance of characteristic-driven portfolios is, again, very much de-

pendent on the parameter of risk aversion of the investor. The CSP portfolio

shows a moderate turnover (in line with the SBG portfolio) in case of high risk

aversion and a much higher turnover in case of lower risk aversion. The higher

turnover is compensated by a higher Sharpe ratio. All three characteristic portfo-

lios outperform the VW benchmarks in terms of Sharpe ratio in all risk aversion

scenarios. The EW portfolio displays a much better performance than the VW

one, and, as the risk aversion parameter of the individual increases, it becomes

a very compelling alternative to the dynamic portfolios built on characteristics.

The CE performance of active strategies is highly impacted by the degree of risk

aversion. Similarly to the case where investment horizon is one period ahead, the

performance rankings of active strategies are closer to the Sharpe Ratio rankings

for low relative risk aversion parameters. In particular, my CSP model, being the

most volatile, performs best of all when risk aversion is at 5 or below.

Table 2.4 I report, for all strategies and benchmarks, the certainty equivalentreturn for levels of relative risk aversion below 5, ranging from the log utilityfunction to a γ = 4. Investment horizon is intertemporal.

CE γ = 1 γ = 2 γ = 3 γ = 4

CSP 0.094 0.050 0.030 0.018

SBM 0.036 0.017 0.008 0.028

SBG 0.058 0.037 0.021 0.018

VW B.mark 0.008 0.006 0.003 -0.001

EW B.mark 0.019 0.015 0.011 0.006

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2.5 Conclusion

2.5 Conclusion

This paper has three main objectives: 1) to derive parametric portfolio char-

acteristics in the case of credit risky stocks extracting them from the analytic

stocks’ representation as an option on the firm’s assets, 2) to solve the intertem-

poral parametric portfolio allocation problem for those characteristics, and 3) to

empirically test chacteristic-driven optimal portfolios and compare them to the

SBM, SBG, VW, and EW portfolios.

In the first part of the paper I have analytically extracted the three main

drivers of credit-risky stock returns and identified them as company debt duration,

company levered asset return, and company asset volatility (where by asset I mean

the production process).

In the second part of the paper, I have applied the characteristics found to

be significant for credit risky stocks to the parametric portfolio approach first

introduced by Brandt et al. (2009).

In the third part of the paper, I have solved, in closed form following Liu

(2007), my optimization problem using the three characteristics identified in sec-

tion one for a dynamic parametric portfolio policy. I have benchmarked my CSP

optimal portfolio to the value-weighted S&P 500 index, and I built the paramet-

ric portfolio as deviations from the benchmark characteristics. I have obtained

a vector of weights for three long-short portfolios: debt duration, levered asset

returns, and volatility whose performances I have added to the benchmark to

calculate my portfolio.

In the final empirical section, I have taken the model to the data. I have used

10 year quarterly rolling windows to estimate the model parameters and presented

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2.5 Conclusion

optimal portfolio results for individuals with different investment horizons: those

that have a one-period horizon and those that have long-run, dynamic investment

objectives. The performance comparison has been conducted on a strictly out-of-

sample basis for three dynamic portfolios built with the CSP, the SBM, and the

SBG characteristics, as well as for two passive portfolios: the VW and the EW

portfolios. To measure performance, I have calculated a number of statistics for

two different risk aversion parameters.

Given my choice of variables, utility function, and model setup, I conclude

that my CSP active, characteristic-driven portfolio is superior, in terms of Sharpe

ratio, to passive investments both in case of a one-period horizon and in case of

inter-temporal investing. When considering the performance in terms of CE, the

CSP portfolio performs best in low risk aversion scenarios, and with very high

risk aversion. This is due to the non linear impact of volatility weighted by the

risk aversion coefficient in the CE calculation. At high risk aversion scenarios the

convexity effect takes over, given the shape of my utility function, and reverts

the negative CE contribution of the more volatile models.

The actual performance of the CSP strategy, post-trading costs, depends on

the amount of costs assumed for the implementation of the strategies. I have also

considered alternative characteristic-driven portfolios, the SBM and SBG, which

perform better than the passive strategies in case of intertemporal optimization

and worse than the passive strategies in case of one-period optimisation. All ac-

tive strategies have worse turnovers than the passive ones and whether this affects

their Sharpe ratio or CE superiority depends on the filters utilised for implement-

ing the portfolio decisions and the associated costs. Among the characteristic-

driven portfolios, the CSP has a better performance in case of low risk aversion.

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2.5 Conclusion

In addition, its Sharpe ratio is quite stable across the two investment horizon

scenarios, while its turnover improves when implementing the dynamic strategy.

The other two characteristic-driven portfolios show a significant improvement in

Sharpe ratio from the one-period horizon to the dynamic scenarios, but their

turnover statistic dramatically worsens suggesting that some of the Sharpe ratio

improvement may be offset by higher costs. The CSP strategy is the one that

displays the best CE performance at low risk aversion scenarios. When risk aver-

sion increases, the active portfolios and the passive ones alternate in terms of CE

dominance.

All in all, the characteristic-driven portfolios appear to contain information

that improves performance, in Sharpe ratio terms, with respect to the VW and

EW portfolios. The performance improvement increases when moving from the

one-period to the intertemporal horizon. Turnover costs are crucial in assess-

ing net performance, and the active portfolios considered here display a higher

turnover than the naive investment strategies.

A series of filters to portfolio re-weighting signals could be considered to reduce

the cost impact of the many transactions implied by the active strategies. Finally,

this paper presents a strategy for a fully invested individual, while it would be

interesting to test the dynamic parametric portfolio policy allowing the individual

to have an allocation that goes from one-time long to one-time short.1

1These will be the subject of a separate study.

119

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Finance, 59(2):831–868.

123

Appendices

124

.1 Proof of Lemma 1

Appendix for chapter 2

.1 Proof of Lemma 1

Proof. I plan to prove that plugging (2.34) and (2.38) into (2.31) results in Lemma 1.

First, I plug the guess for the value function into the objective function and get

− βf + γft + (1− γ)[µm + θTµk]f+

− γ(1− γ)

2

[ΣmΣT

m + 2θTΣkρTmΣT

m + θTΣkΣTk θ]f+

+ γµTs fS + γ1

2Tr(ΣsΣ

Ts fSST

)+γ(γ − 1)

2ffTS ΣsΣ

Ts fS+

+ γ(1− γ)[Σmρ

TmΣT

s + θTΣkρTΣT

s

]fS = 0. (66)

I expand expression (66) using the new definition of drift (2.39) to obtain the PDE that

I will have to solve, finding the appropriate value for G(t):

(−βγ

+1− γγ

[µm + (γΣkΣTk )−1

(µ+ γΣkρ

TΣTs G(t)

)T (µ+ γΣkρ

TmΣT

m

)]+

− 1− γ2

[ΣmΣT

m + 2(γΣkΣTk )−1

(µ+ γΣkρ

TΣTs G(t)

)TΣkρ

TmΣT

m+

+ (γΣkΣTk )−1

(µ+ γΣkρ

TΣTs G(t)

)TΣkΣ

Tk

(µ+ γΣkρ

TΣTs G(t)

)(γΣkΣ

Tk )−1

])f+

+ ft + µTs fS +1

2Tr(ΣsΣ

Ts fSST

)+

(γ − 1)

2ffTS ΣsΣ

Ts fS+

+ (1− γ)[Σmρ

TmΣT

s + (γΣkΣTk )−1

(µ+ γΣkρ

TΣTs G(t)

)TΣkρ

TΣTs

]fS = 0, (67)

125

.1 Proof of Lemma 1

expanding

(−βγ

+1− γγ

[µm + (γΣkΣ

Tk )−1µTµ+ γG(t)TΣsρρ

TmΣT

m + (ΣkΣTk )−1µTΣkρ

TmΣT

m

+ (ΣkΣTk )−1G(t)TΣsρΣT

k µ]+

− 1− γ2

[ΣmΣT

m + 2(γΣkΣTk )−1µTΣkρ

TmΣT

m + 2G(t)TΣsρρTmΣT

m+

+ (ΣkΣTk )−1µ

Tµ

γ2+G(t)TΣsρρ

TΣTs G(t) + (γΣkΣ

Tk )−1µTΣkρ

TΣTs G(t)+

+ (γΣkΣTk )−1G(t)TΣsρΣT

k µ])f+

+ ft + µTs fS +1

2Tr(ΣsΣ

Ts fSST

)+

(γ − 1)

2ffTS ΣsΣ

Ts fS+

+ (1− γ)[Σmρ

TmΣT

s + (γΣkΣTk )−1µΣkρ

TΣTs +G(t)TΣsρρ

TΣTs

]fS =

= 0. (68)

I recognize that (γΣkΣTk )−1µTΣkρ

TΣTs G(t) = (γΣkΣ

Tk )−1G(t)TΣsρΣT

k µ and I can sim-

plify (68):

(−βγ

+1− γγ

µm +1− γ2γ2

(γΣkΣTk )−1µTµ− 1− γ

2ΣmΣT

m+

− 1− γ2

G(t)TΣsρρTΣT

s G(t))f + ft + µTs fS+

+1

2Tr(ΣsΣ

Ts fSST

)+

(γ − 1)

2ffTS ΣsΣ

Ts fS+

+[(1− γ)Σmρ

TmΣT

s + (1− γ)(γΣkΣTk )−1µTΣkρ

TΣTs + (1− γ)G(t)TΣsρρ

TΣTs

]fS =

= 0. (69)

126

.2 Proof of Lemma 2

Now, I note that in my setup G(t) = fsf , and I replace terms and simplify further:

(−βγ

+1− γγ

µm

)f+

+1− γ2γ2

(ΣkΣTk )−1µTµ− 1− γ

2ΣmΣT

m −1− γγ


TmΣT

m

)f+

+ ft +1

2Tr(ΣsΣ

Ts fSST

)+γ − 1

2ffTS

(ΣsΣ

Ts − Σsρρ

TΣTs

)fS+

+


TmΣT

s +1− γγ


TΣTs

)fS = 0. (70)

.2 Proof of Lemma 2

Proof. I plan to prove expression (2.50). I substitute the functional forms of the seven

components of the expression into the following function:

(−βγ

+1− γγ

µm

)f+

+1− γ2γ2

(ΣkΣTk )−1µTµ− 1− γ

2ΣmΣT

m −1− γγ


TmΣT

m

)f+

+ ft +1

2Tr(ΣsΣ

Ts fSST

)+γ − 1

2ffTS

(ΣsΣ

Ts − Σsρρ

TΣTs

)fS+

+


TmΣT

s +1− γγ


TΣTs

)fS =(−β

γ+

1− γγ

(δ0 + δ1S))f+

+1− γ2γ2

(H0 +H1S)− 1− γ2

(q0 + q1S)− 1− γγ

(z0 + z1S))f+

+ ft +1

2Tr(ΣsΣ

Ts fSST

)+

1− γ2f

fTS

(l0 + l1S

)TfS+

+

(µTs + (1− γ)(p0 + p1S) +

1− γγ

(g0 + g1S)T)fS = 0. (71)

127

.3 Proof of Lemma 3

The only term left out is the expression ΣsΣTs fSS , for which I am using a result in Liu

(2007) to express it as dT (h0 + h1S)df . I now obtain

(−βγ

+1− γγ

(δ0 + δ1S))f+

+1− γ2γ2

(H0 +H1S)− 1− γ2

(q0 + q1S)− 1− γγ

(z0 + z1S))f+

+ ft +1

2fS(h0 + h1S)

fSf

+1− γ

2ffTS

(l0 + l1S

)TfS+

+

(µTs + (1− γ)(p0 + p1S) +

1− γγ

(g0 + g1S)T)fS = 0. (72)

Now, to find the solution for G(t), I set the sum of all terms linear in S equal to zero,

and I obtain (2.50).

.3 Proof of Lemma 3

Proof. I note that the portfolio of characteristics has a Sharpe ratio that can be par-

titioned into two distinct components with each one depending only on one, state

variable as outlined in equation (2.59). The intertemporal hedging component can also

be partitioned as follows:

(ΣkΣTk )−1Σkρ

TΣTs G(t) =

σDb2

0 0

0 σLc2

0

0 0 σVe2

G(t)D

G(t)L

G(t)V

. (73)

Finally, to prove that f = f1f2f3, I therefore can partition G(t) as above, and it is

sufficient to note that f is exponential in the state variables. Since the solution is, by

assumption, additive in the state variables, the f function will be the product of the

individual fi because of the properties of exponentials.

128

.3 Proof of Lemma 3

Certainty Equivalent Approximation

Proof. I note that the utility of the CE is the expected value of the utility of future

wealth (1 + r)W under a portfolio strategy return r. I can take a generic Taylor series

expansion to the third order to capture non-linearities:

E(U((1 + r)W )) = U(W ) + U ′(W )Wµr +U ′′(W )W 2

2σ2r+

+U ′′′(W )W 3

3σ3r + o(U ′′′), (74)

where µr, σ2r , σ

3r have the usual meaning of first, second and third moment of the dis-

tribution of the random return r. Now, I can write the CE as a Taylor expansion of

the first order as follows:

U(CE) = U(W ) + U ′(W )(CE −W ) + o(U ′). (75)

Following the definition of certainty equivalent I can write E(U((1 + r)W )) = U(CE)

and replacing the generic utility function with the power utility I obtain the expression

for the CE.

129

.4 Results including financial stocks

.4 Results including financial stocks

This section presents the same evidence as in Tables I and II with the inclusion,

in the data sample, of financial companies. While the level of magnitude of outperfor-

mance by the characteristic-driven portfolios is lower than in the case without financial

companies, most of the conclusions in terms of directionality of the impact of risk

aversion and average returns remain the same as in the base case.

130

.4R

esu

ltsin

cludin

gfinan

cial

stock

sTable 5 Return statistics for portfolios built by individuals with an investment horizon of oneperiod only. I report annualized mean return, annualized median return, maximum quarterlydrawdown, annualized standard deviation of portfolio return, annualized Sharpe ratio and meanquarterly turnover. Sharpe ratio is significantly higher than VW’s and EW’s when accompaniedby ** symbol or only higher than VW’s when accompanied by * symbol. The first three portfoliosare built on three characteristics: 1) CSP portfolio: Debt Duration, Asset Levered Returns andAsset Volatility; 2) SBM portfolio: size, book-to-market, and momentum and 3) SBG portfolio:size, book-to-market, and gross profitability. The last two portfolios represent the value-weightedand the equal-weighted portfolios constructed using all S&P 500 stocks included in my investmentuniverse. The three panels correspond to the three different risk aversion parameters used: γ = 5,γ = 10, and γ = 15.

Panel A: γ = 5 Mean Ret Median Ret Max D.down Std SR Turnover

CSP 0.276 0.392 -0.763 0.418 0.661** 3.991

SBM 0.102 0.119 -0.533 0.261 0.390* 1.763

SBG 0.094 0.116 -0.475 0.172 0.545 0.885

VW B.mark 0.046 0.095 -0.449 0.149 0.308 0.086

EW B.mark 0.092 0.115 -0.474 0.178 0.515 0.166

131

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cludin

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cial

stock

s

Panel B: γ = 10 Mean Ret Median Ret Max D.down Std SR Turnover

CSP 0.157 0.216 -0.602 0.258 0.610** 1.941

SBM 0.072 0.096 -0.484 0.189 0.383* 0.775

SBG 0.067 0.097 -0.460 0.146 0.458* 0.525

VW B.mark 0.046 0.095 -0.449 0.149 0.308 0.086

EW B.mark 0.092 0.115 -0.474 0.178 0.515 0.166

Panel C: γ = 15 Mean Ret Median Ret Max D.down Std SR Turnover

CSP 0.118 0.194 -0.548 0.210 0.560* 1.262

SBM 0.063 0.104 -0.468 0.169 0.370* 0.484

SBG 0.058 0.085 -0.456 0.140 0.413* 0.421

VW B.mark 0.046 0.095 -0.449 0.149 0.308 0.086

EW B.mark 0.092 0.115 -0.474 0.178 0.515 0.166

132

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esu

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stock

sTable 6 Return statistics for portfolios built by individuals with an investment horizon of five years(intertemporal optimization). I report annualized mean return, annualized median return, maximumquarterly drawdown, annualized standard deviation of portfolio return, annualized Sharpe ratio andmean quarterly turnover, as well as the average annual contribution to return by the intertemporalhedging component Dynamic Contribution. Sharpe ratio is significantly higher than VW’s and EW’swhen accompanied by ** symbol or only higher than VW’s when accompanied by * symbol. Thefirst three portfolios are built on three characteristics: 1) CSP portfolio: Debt Duration, Asset Lev-ered Returns and Asset Volatility; 2) SBM portfolio: size, book-to-market, and momentum, and 3)SBG portfolio: size, book-to-market, and gross productivity. The last two portfolios represent thevalue-weighted and the equal-weighted portfolios constructed using all S&P 500 stocks included in myinvestment universe. The three panels correspond to the three different risk aversion parameters used:γ = 5, γ = 10, and γ = 15.

Panel A: γ = 5 Mean Ret Median Ret D.down Std SR T.over Dyn.Con.

CSP 0.280 0.340 -0.741 0.419 0.670** 3.865 0.004

SBM 0.146 0.128 -0.440 0.278 0.526* 2.638 0.044

SBG 0.135 0.122 -0.549 0.224 0.601** 1.197 0.041

VW B.mark 0.046 0.095 -0.449 0.149 0.308 0.086 0.000

EW B.mark 0.092 0.115 -0.474 0.178 0.515 0.166 0.000

133

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s

Panel B: γ = 10 Mean Ret Median Ret D.down Std SR T.over Dyn.Con.

CSP 0.168 0.209 -0.587 0.270 0.631** 1.937 0.010

SBM 0.138 0.136 -0.430 0.216 0.639** 2.099 0.065

SBG 0.125 0.119 -0.639 0.237 0.525* 1.863 0.058

VW B.mark 0.046 0.095 -0.449 0.149 0.308 0.086 0.000

EW B.mark 0.092 0.115 -0.474 0.178 0.515 0.166 0.000

Panel C: γ = 15 Mean Ret Median Ret D.down Std SR T.over Dyn.Con.

CSP 0.130 0.176 -0.536 0.225 0.580** 1.312 0.012

SBM 0.114 0.092 -0.516 0.228 0.499* 2.335 0.051

SBG 0.115 0.114 -0.349 0.182 0.635** 1.233 0.057

VW B.mark 0.046 0.095 -0.449 0.149 0.308 0.086 0.000

EW B.mark 0.092 0.115 -0.474 0.178 0.515 0.166 0.000

134

.5 Empirical evidence of market structure


This section presents evidence to support the choices made in terms of format of

the market structure.

The State Variables

First, I address the state variables: debt duration (D), levered asset returns (L) and

asset variance (V) and the market returns. Here, I present the correlation structure of

the cross-sectional averages of the percentage return of the three state variables among

themselves and with the market levered equity returns and the corresponding p-values

as measured on the data sample. Albeit the correlations are non-zero, the p-values

indicate that the coefficients for two of the three crosses are not significant within a

5% confidence interval. The cross between volatility and debt duration appears to be

significant within a 99% confidence interval, but for computational ease I will assume it

to be insignificant. Table IV is the standardized version of matrix Σs. The correlation

of state variables with the market is significant for all state variables of the CSP model

indicating they are good predictors of the dynamics of market returns.

Table 7 State Variables Correlation

Mkt sD sL sV

Mkt 1.000 0.334 0.207 -0.545

sD 0.334 1.000 0.074 -0.247

sL 0.207 0.074 1.000 0.059

sV -0.545 -0.247 0.059 1.000

Table 8 P-Value of State VariablesCorrelation

Mkt sD sL sV

Mkt 1.000 0.001 0.028 0.000

sD 0.001 1.000 0.439 0.009

sL 0.028 0.439 1.000 0.535

sV 0.000 0.009 0.535 1.000

135


Market and Characteristic Portfolios

I run a correlation analysis among the characteristic portfolio and the market port-

folio returns. Numbers represent the correlation and p-values estimated on the observa-

tions of the data sample.1 The four-by-four matrices relate each characteristic portfolio

to the market portfolio along the first row and column. The remaining three-by-three

sub-matrix is the characteristic portfolios correlation matrix.

In the characteristic portfolios sub-matrix, correlations are not significant within a

95% confidence interval for two out of three crosses. The cross between volatility and

levered return is significant within a 99% confidence interval. I choose to ignore this

and assume it is zero for computational ease.

In the case of correlation between the market portfolio and the characteristic port-

folio returns, the significance levels are all within a 95% interval.

The first row or column of Table VI is the diagonal of matrix Σm, while the sub-

matrix of the characteristic portfolios, excluding the first row and column, is the stan-

dardized equivalent of Σk.

Table 9 Characteristic Ptf. Corr.

Mkt D L V

Mkt 1.000 -0.220 -0.233 0.541

D -0.220 1.000 0.158 -0.043

L -0.233 0.158 1.000 -0.257

V 0.541 -0.043 -0.257 1.000

Table 10 P-Value of CharacteristicPtf. Corr.

Mkt D L V

Mkt 1.000 0.019 0.013 0.000

D 0.019 1.000 0.095 0.649

L 0.013 0.095 1.000 0.006

V 0.000 0.649 0.006 1.000

1I have 117 actual observations.

136

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