Reducing Bias Problems When Estimating Cost of Capital in Company Valuations

Dr. Klaus MarkVice President, Credit Analysis, Power & Renewables KfW IPEX-Bank

Global Association of Risk ProfessionalsMay 2016

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The views expressed in the following material are the

author’s and do not necessarily represent the views of

the Global Association of Risk Professionals (GARP),

its Membership or its Management.

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Reducing bias problems when estimating cost of capital in company valuations – simple can be better

Dr. Klaus Mark

Vice President

Credit Analysis, Power & Renewables

KfW IPEX-Bank

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References

Breuer, W., K. Kohn, and K. Mark (2016). A Note on Corporate Valuation Using Imprecise Cost of Capital (March 23, 2016). Available at SSRN: http://ssrn.com/abstract=2657025.

Breuer, W., D. Fuchs, and K. Mark (2014). Estimating cost of capital in firm valuations with arithmetic or geometric mean – or better use the Cooper estimator? The European Journal of Finance 20: 568-594.

Cooper, I. (1996). Arithmetic versus geometric mean estimators: Setting discount rates for capital budgeting. European Financial Management 2: 157-167.

Elsner, S., and H.-C. Krumholz (2013). Corporate valuation using imprecise cost of capital. Journal of Business Economics 83: 985-1014.

Jacquier, E., A. Kane, and A.J. Marcus (2005). Optimal estimation of the risk premium for the long run and asset allocation. Journal of Financial Econometrics 3: 37-55.

Disclaimer

Views expressed in this presentation are the personal opinion of the author and do not necessarily reflect the position of KfW IPEX-Bank.

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Biases in discount factor estimations in a DCF / company valuation framework – let’s start with a simple example!

Expected value of cost of capital – risk is given by three states of nature:

�(�) =1

3∗ 0.05 +

1

3∗ 0.1 ∗

1

3∗ 0.15 = 0.1

Corresponding one-period present value:

1

1 + �(�)= 0.9091

But actually, we estimate discount factors:

� �1

1 + �� =

1

3∗

1

1 + 0.05+1

3∗

1

1 + 0.1+1

3∗

1

1 + 0.15= 0.9103 >

1

1 + �(�)

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Does this really matter with respect to company valuations? – Yes, it does, particularly regarding the Terminal Value (TV)

� �1

�� =

1

3∗

1

0.05+1

3∗1

0.1+1

3∗

1

0.15= 12.22 >

1

0.1= 10

r2r1

���

�(�, �)�

�

�(�(�, �))

Some obvious results:

- Bias increases with the convexity of the function of discount factor estimates, and thus with the length of the forecasting horizon

- Bias is also larger the higherstandard error of (average) returns used for estimation

- TV is a key driver of estimation

distortions

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Formal derivation of the bias problem when estimating cost of capital in company valuations using historical returns in a CAPM-world

Consider an all equity-financed firm with a cash flow of € 1 at time n = 1 that is growing at a constant rate from n = 1 until infinity:

n 1NN

0 n Nn 1 0,n 0,N

E(V )(1 g)V

(1 i RP) (1 i RP)

0,ni : rate of riskless zero bonds with maturity n

RP risk premium computed as the product of the market risk premium and the firm specific beta value. For simplicity reasons, we assume that beta = 1 and that the (market) risk premium is constant over time.

g: per-period cash flow growth rate

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…Whilst the riskless interest rate can be assumed to be known, the market risk premium has to be estimated

To estimate RP we use historical realizations of excess return of a stock market index serving as a proxy for the CAPM market portfolio over a corresponding risk free rate (e.g.: DAX/CDAX over REXP or S&P 500 over T-Bonds).

At first glance: The arithmetic mean of historical one-period excess returns (which are i.i.d) seemsappropriate. – It is the unbiased estimator of the one-period future expected return:

T

T tt 1

1A (1 r )

T

.

- Note that the arithmetic mean estimator – as a sample result – is stochastic from an ex-ante perspective.

- Of course, we can also interpret historical returns as total returns which can be used – on an average basis – to estimate “full” cost of equity capital.

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The arithmetic mean: a biased estimator of the discount rate!

With r as expected (excess) return that serves as the relevant (true) discount rate we search for:

n n

1ˆE D(1 r )

Since n

Tˆ1/ A is a convex function for all n = 1, …, N, Jensen’s inequality implies:

T n n

T

nn

T

1 1ˆE A 1 r E(1 r )A

1r : 1 r.

1E

A

As the geometric mean is always smaller than the arithmetic mean (under uncertainty), corresponding estimators for the discount rate are even more biased.

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Cooper (1996) derived an estimator that can reduce bias effects considerably using Taylor-Approximation

Utilizing the i.i.d.-property of historical returns, Cooper reduces the second order Taylor-Approxi-mation of the “true” discount rate to a simple term that only entails the arithmetic and the geometric mean weighted by the length of the forecast horizon and the historical time series:

n, T n,T n,Tn n

T T

1 1C : c (1 c )

ˆ ˆA G

with: n,Tc : (T n) /(T 1)

Key characteristics:

- Reduces bias, when standard errors of historical returns and forecast horizons are “moderate”

- However: Cooper estimation results can completely deteriorate when standard errors and/orforecast horizons get large - even possible: negative discount factor estimates.

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Bias of discount factors in a simple binomial model1)

Bias of respective discount factor estimations (n: forecast horizon):

Arithmetic mean Geometric mean CooperSD n = 10 n = 30 n = 50 n = 10 n = 30 n = 50 n = 10 n = 30 n = 500.15 0.0061 0.0053 0.0016 0.0350 0.0157 0.0040 -0.0004 -0.0013 -0.00090.25 0.0174 0.0176 0.0076 0.1095 0.0695 0.0321 -0.0033 -0.0152 -0.01800.35 0.0354 0.0466 0.0384 0.2602 0.3103 0.4469 -0.0151 -0.1202 -0.38681) Stochastic returns by a binomial model, expected return: 0.125; standard deviation: 0.15, 0.25, 0.35; sample length 50.

Arithmetic mean

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 10 20 30 40

Forecast horizon

Bia

s

0.15 0.25 0.35

Geometric mean

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 10 20 30 40

Forecast horizon

Bia

s

0.15 0.25 0.35

Cooper

-0.45

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0 10 20 30 40

Forecast horizon

Bia

s

0.15 0.25 0.35

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Breuer/Fuchs/Mark (2014) found that a “truncated” arithmetic mean estimator outperforms Cooper using real-world capital market data

BFM (2014) tested Cooper versus the arithmetic mean and the geometric mean in simulated firm valuations using our above CAPM framework based on real-world capital market data.

n 1NN

0 n Nn 1 0,n 0,N

E(V )(1 g)V

(1 i RP) (1 i RP)

- Bootstrap simulations with 10,000 runs

- Market risk premium (“time series”) data of 13 developed countries from Morningstar

- “Typical” term structure of interest (US term structure for all countries)

- Stylized cash flow growth rates of 0%, 2%, 4%, 6%.

- Correcting for unrealistic estimation results (e.g. no Cooper discount rate estimates < 0)

- In particular: Necessarily limiting the denominator of the TV to: r – g > 0.005, (otherwise: basic TV is simply set to 1 / 0.005 = 200)

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Some key results of Breuer/Fuchs/Mark (2014)

- Cooper clearly outperforms the arithmetic mean as well as the geometric mean.

- Bias problems are massively driven by the TV (thus, also by the cash flow growth rate).

- The simple “BFM estimator” – which truncates the NPVs of future cash flows after 30 years(i.e. neglects the TV) and uses the simple arithmetic mean – outperforms Cooper.

(in percent) Arithmetic mean Geometric mean Cooper estimatorTruncated

Arithmetic mean (after 30 years)("BFM estimator")

g = 0% 13.0 48.1 6.5 -4.4

g = 2% 30.0 90.0 20.8 -9.9

g = 4% 128.8 333.3 109.3 -21.3

g = 6% 107 265.2 77.3 -43.71)

Australia, Austria, Belgium, Canada, France, Germany, Ireland, Italy, Japan, Netherlands, Switzerland, UK, US

Source: BFM (2014)

Average relative bias of firm valuations (13 countries)1)

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Assuming continuous-time lognormal returns Elsner/Krumholz (2013) derived a discount factor estimator which is in fact unbiased

Building on the fundamentals of Jacquier et al. (2005) – who derived an unbiased estimator for long-term compounding rates – EK found an unbiased estimator of the discount factor:

2n 2

n TˆD (A ) exp 0,5 n

T

depending on:

- the arithmetic mean of historical returns – which are assumed to be continuous time

lognormally distributed with parameters µ and 2

- the length of the time series T

- the length of the forecast horizon n

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Particularly: EK (2013) derived a closed form estimator for thediscount rate to be put into the TV-formula

With respect to the TV, as an infinite sum of discounted cash flows, we face the problem that, for every future period, we have to use different corrective adjustments to the discount factor.

Solution:

- In a continuous-time frame-work, the TV is an integral over discount rates from today until infinity.

- EK (2013) demonstrate that they can derive an unbiased estimator of the discount rate to be put into the TV formula by calculating a corresponding infinite integral using their estimation function for the single-period discount factor.

Picture source: EK (2013)

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Breuer/Kohn/Mark (2016) then tested the “BFM estimator” vs. the “EK estimator” within the same framework and a broader data base

Same valuation equation:

n 1NN

0 n Nn 1 0,n 0,N

E(V )(1 g)V

(1 i RP) (1 i RP)

- Bootstrap simulations with 10,000 runs

- Market risk premium data of 19 countries from Morningstar

- Valuations as of 9 points in time (June 2008 to June 2012), thus: 19 x 9 = 171 “cases”.

- Country and time specific term structures of riskless interest rates (1 - 10 years) derived from government bond zero rates (data: Bloomberg), from 10 years onward: flat term structure.

- Country and time specific cash flow growth derived from IMF average long-term forecasts regarding real GDP growth (respective “current” World Economic Outlook-forecasts).

- Looking at different detailed planning horizons encompassing 5, 10, 30 or 50 years.

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Data basis – study by Breuer/Kohn/Mark (2016)

*ADF tests (Dickey and Fuller, 1979) for random walks without drift. Specifications of differenced equations include one lagged difference, respectively. Corresponding panel tests of Levin et al. (2002) and Im et al. (2003) also indicate stationarity. # skewness and kurtosis-based tests in the tradition of D’Agostino et al. (1990) and Royston (1991). Data source: Morningstar (2012).

(A) (B) (C) (D) (E) (F) (G) (H)

Length of

time series

Arithmetic

mean

Geometric

mean

Standard

deviation

Skewness of

log (MRP)

Standard

errorStationarity

Normality of

log (MRP)

Years Percent Percent Percentage points Percentage points Percentage pointsp-value

ADF Tests *

p-value

NV Tests #

Australia 42 3.4 0.7 23.8 -35.7 3.6 0.000 0.530Austria 40 3.5 -1.1 31.5 -96.9 4.9 0.000 0.001Belgium 42 5.6 2.1 25.6 -173.8 3.9 0.000 0.000Canada 76 5.0 3.6 17.2 -59.8 2.0 0.000 0.087Denmark 42 6.7 2.3 31.9 -6.4 4.8 0.000 0.968Finland 24 11.6 1.1 55.1 22.6 11.0 0.025 0.687France 42 5.7 2.5 25.7 -62.4 3.9 0.000 0.195Germany 42 4.0 0.6 26.4 -64.0 4.0 0.000 0.140Italy 42 1.8 -2.7 31.9 -210.8 4.9 0.000 0.912Japan 60 8.0 4.5 29.2 6.9 4.5 0.000 0.281Malaysia 19 8.7 2.9 36.7 19.0 8.1 0.000 0.142Netherlands 42 5.8 3.2 23.2 -64.2 3.5 0.000 0.045Norway 42 9.7 2.9 41.9 -83.5 6.4 0.000 0.503Spain 33 9.5 5.9 29.0 3.5 5.0 0.049 0.565Sweden 42 10.3 6.2 30.3 -33.6 4.6 0.000 0.413Switzerland 42 5.7 3.2 23.3 -46.0 3.6 0.000 0.411Thailand 24 11.8 0.8 50.6 -46.1 10.2 0.004 0.657UK 80 4.2 1.8 23.3 -34.3 2.6 0.000 0.000US 80 7.2 5.5 19.1 -67.1 2.1 0.000 0.021Average 45 6.7 2.4 30.3 -54.4 4.9

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Country-averages of the Relative Biases of tested estimators (in percent)

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Average Relative Bias results over 171 valuation cases and estimation results assuming “optimal” detail planning horizons (in percent)

TV Trunc TV Trunc

Average Relativ Bias (171 cases):

N = 5 114.9 -72.2 44.0 -71.3N = 10 98.6 -46.7 53.5 -47.4N = 30 114.9 0.4 91.5 -8.8N = 50 134.9 24.3 193.3 10.6

Optimal planning horizons:

Period 10 33 26 43

Value Relative Bias 90.3 -0.6 25.8 -4.5

Arithmetic mean EK

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Random effects panel regression of Relative Biases (RB) – key results

RB is explained as a function of the above variables. In addition, time dummies (D_Dec_2008 to D_June_2012) and random country effects were included (not shown in the table). Standard errors in brackets. */**/*** significant at 10 %/5 % /1 %. 171 observations.

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Determinants of performance differences: BFM versus EK in a random effects panel regression

Detailed planning horizon N = 30. Differences of absolute values of RB, when applying the BFM vs. the EK approach without (panel (A)) or with (panel (B)) truncation. Estimations including time dummies and random country effects as above (not shown). 171 Observations. Furthermore, differences in Relative Root Mean Squared Errors (RRMSE) of the respective estimatiors.

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To sum up

Indeed, simple can be better when estimating cost of capital in firm valuations:

- Although bias effects strongly depend on the data, in a CAPM world we receive overall robust results that recommend: ‘stick to the simple arithmetic mean estimator without terminal value computation, consider future cash flows up to (about) 30 years’

- This overall result is strongly confirmed if we also look at other error measures such as (relative) root mean squared errors.

- Still, the Cooper estimator and the estimator by Elsner/Krumholz can considerably reduce bias, and, could particularly be recommended for “smooth”/ low risk data environments.

Some caveats:

- The approaches by Cooper and Elsner/Krumholz were “not allowed” to perform in thoseenvironments they are made for: full equity returns (not CAPM!), zero cash flow growth, no TV-calculation (Cooper) – however: we tested “real-life” valuations…

- If we assume full equity returns or if we use a different error measure (log-scaled errors), the EK-approach gains ground (see robustness checks in Breuer/ Kohn/Mark, 2016)

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Thank you for your attention!

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