Finance Summaries

8/12/2019 Finance Summaries

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Utility function for various Coefficients of Risk Aversion

Differing utility for a given level of risk of aversion

Notes to the utility function

Risk lover will accept lower returns, even as risk increases (insanity) Risk neutral investor doesnt care for the variability of the returns, he only cares about the

expected return, i.e. E(r) = U. Depending on his/her coefficient of risk aversion, the risk averse investor will demand higher

returns for an increase in risk, the more risk averse the investor, the higher the return

he/she demand for a given increase in risk. For the risk free assets, the utility any investor derives from it, will be E(r) as = 0.


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CERR

The utility derived from a risky asset = Certainty equivalent Rate of return , i.e. the rate thata risk free asset must meet or beat if it is to be preferred.

If the utility of an any risky investment is less than the expected return of the risk free asset,then the risk free asset will be preferred.

For the risk-neutral investor, as risk is irrelevant, a risky asset is preferred so long as its E(r) >rf .

AIM: FIND THE RISKY ASSET WHICH PROVIDES THE INVESTOR WITH THE HIGHEST UTILITY (and makesure that this utility is > r f . * [r f is normally the T-bill rate]

THE MEAN VARIANCE CRITERION

Basically, the portfolio with the highest ratio is preferred.

Complete portfolio

Construction of a portfolio of risky and risk-free assets (we separate this choice from whichportfolio of risky asset to choose from).

The risky-portfolio is chosen, from available risky assets, so as to maximise the utility of aninvestor given their level of risk aversion.

AFTER this is chosen, we then decide on the mix between risky and risk-free assets. y = percentage of funds in the RISKY asset (1- y) = percentage of funds in the RISK-FREE asset .

() ()


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If borrowing at some other rate = >

o () o

()

In general, the CAL, generated by the SD and E(r) of a given portfolio gives us the CAL for agiven security or a selection of securities.

If the risky asset chosen is itself a market portfolio consisting of all risky assets we call theline the CML (capital market line).

Chapter 7 Risk Aversion and Capital Allocation to risky assets

General

Previously the risky portfolio was taken as a given, we now derive the optimal such portfolio.

Short term perspective (Returns are roughly normally distributed over the short-term).Diversification

Two-security portfolio

We now look at the returns and variability of a two security portfolio. Two RISKY assets.

Corporate bonds (debt) are nonetheless risky assets.

Returns

= return of the combined portfolio = return of debt/ corporate bonds (still risky) = return of stock/equity = weight of debt (if < 1 we are shorting equity) = weight of equity (if > 1 we are shorting debt) Where


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Variance/ Risk

It should be clear, that a negative covariance of two shares, reduces the overall variability of

the portfolio. Implies perfectly positive correlation

o o

o Implies perfectly negative correlationo o o o if i.e.

These elementary relationships lead to the following:

If we extend this to a multi-asset portfolio (potentially consisting of a market portfolio withall assets in proportion so as to minimise the risk, we can construct a frontier.

However, now, we take the matrix as given, and vary only the weights.


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The relevant portion of the frontier is however smaller, because the red points on the

frontier offer lower levels of return than other points on the efficient frontier which havethe same level of risk, and are thus less efficient.

Therefore the EFFICIENT FRONTIER:

PORTFOLIOS ON THE EFFICIENT FRONTIER

The Optimal risky portfolio the portfolio with the highest Sharpe ratio (the most north-easterly portfolio).

The global-min variance portfolio , the efficient portfolio with the lowest standarddeviation/ risk [does not maximise the Sharpe ratio].

The optimal complete portfolio , the portfolio chosen as a mix between the risk free andoptimal risky portfolio, given ones l evel of risk aversion.

Our metric for the evaluation of these portfolios is the Sharpe ratio

() Achieving portfolios on the efficient frontier

We will analyse this from a two risky security perspective. Global Min Variance portfolio

Optimal Risky portfolio (all investors should want this)


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Optimal Complete Portfolio (Heres where their risk aversion is factored in)

1.

2.

3. 4. 5. ()

Chapter 8 Single Factor Index ModelGeneral

o Previously we looked at the variance of a portfolio or asset as a function of all the individualvariances and covariances, we then established the efficient frontier portfolios by examining

all possible weights of the various assets available to the market place.o We established the optimal risky portfolio by maximising the Sharpe ratio on the efficient

frontier.o However, the establishment of this portfolio is computationally intensive, requiring many

estimates (the covariance matrix for example). () where n is the number of assets in the market

o We now attempt to avoid this computational excess by implementing a simpler model,where the risk of a portfolio/share is a function of the market (systematic risk) and of firmspecific characteristics (non-systematic risk).

o This is done by breaking up the residual/ errors/ unexpected return into a:1. Systematic component (as result of variation in a common macro-economic factor)2. Non-systematic component (firm specific variation).

Building the model

o Initially, given any model for E(r) the following will always hold. o Where: is the return of security i is the expected return of the share as given by some model.

is the error term or residual (the return not explained by the model) i.e. theunexpected return

~ N(0, )

o If we then further break up into the unexpected return that comes from someunexpected movement in a common macro-economic variable and all other unexpectedreturns we have:

Where ~ N(0, ) & ~ N(0, )


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Active vs. Passive Portfolio

2 choices:1. Invest in the market through the index2. Invest in the market but add specific securities to your portfolio based on alpha

analysis. Complication:

1. If we identify positive alphas and decide to add them to our portfolio, we will alsotake on extra risk, i.e. we lose some of the benefits of diversification.

2. The question then is: how much weight do we give to the active portfolio, given thisadditional risk

Active portfolio weight / Optimal risky portfolio

Assume a beta = 1 for the entire active portfolio.

Contribution to risk/reward ratio of the active portfolio o Where = the alpha of the entire portfolioo Where = the entire risk of portfolio A

Contribution to the risk/reward ratio of the passive portfolio =

If we now relax the assumption that and let

we then have:

1. 2. 3. The more something moves with the market, the less we need the market portfolio.

Returns and risk

Recall that:1. 2. () Let = the weight in the active portfolio

Let = the weight in the market portfolio


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Information Ratio

Where is the information ratio, how much extra return we can leverage from

security analysis, relative to the risk incurred when we over/under-weight securities relativeto the market index.

Chapter 9 CAPM

Although this section is related to the index model section, its point of departure isentirely different.

The derivation follows directly from the Markowitz efficient frontier and the OPTIMALRISKY PORTFOLIO

This model is an EQUILBRIUM model. By equilibrium, we mean that this model pricesgives the fair return for an asset when all information pertaining to that asset is constantand there are no unexpected returns.

Assumptions

1. Individual investors are price takers (their actions cannot affect the price)2. Single Period Investment horizon allows for uniform risky asset and risk measure3. Financial Assets stocks, no human capital, non -traded assets4. No taxes, No transaction costs (different investor, different tax bracket, different

preferences)5. INVESTORS ARE RATIONAL MEAN-VARIANCE OPTIMIZERS

Investors will choose the ORP as their risky asset.

6. There are homogenous expectations Identical PDFS for rates of return Same efficient frontier Same ORP7. { } { } 8. Borrowing and lending at 9. Capital markets are in equilibrium all assets give appropriate return for their risk

Market Portfolio

The market portfolio is the ORP on the efficient frontier of all risky assets As such it forms a CML (capital market line).

Though some people borrow (Y


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() ()

() () Thus, in equilibrium, the market premium of the market is determined by the average level

of risk aversion in the market. From Markowitz CAPM we now describe the risk return relationship with instead of

SD.

Expected-Return- Beta relationship

We now argue that a stocks contribution to the variance of the market portfolio is the onlyrelevant information.

The risk of stock is the risk it brings to the market portfolio. We seek now to find the contribution of the stock to the variance of the portfolio. This is the variance of the stock and its covariance with all other stocks multiplied by the

weight it holds in .

We now follow the same approach with As contribution to the market portfolios risk

premium:

IN equilibrium, all assets must offer the same reward-risk ratio

[ ]


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Chapter 10 Multi-factor index models (MFIM)

Where:

{} {} NOTE: The Fi are the unexpected returns, or the deviation of the actual returns from their

expected value, and on average are = 0 . is the Factor sensitivity, or the loading for F k To answer questions about the results we need values for which are often obtained

from a CAPM like multifactor model.

Multi-factor SML MODELS

There is no clear derivation, or rigorous treatment of this model, it is simply a crudeextension of the CAPM model, those this time in n dimension.

is the Factor sensitivity

This gives us a means of calculating the expected return for the MFIM

Chapter 11 Arbitrage Pricing Theory

Key Propositions

1. Security returns can be described by a factor model 2. There are SUFFICIENT securities to diversify away company specific risk.3. Well-functioning security markets don t allow for arbitrage opportunities .

Background

Law of one price if 2 assets are identical in all relevant economic ways must have thesame price.

o LOP is enforced by arbitrageurso If an arbitrage opportunity exists arbitrageurs engage in arbitrage activitieso Arbitrage activities eliminate the arbitrage opportunity

If an arbitrage opp. Exists Investors will want an infinite position in it Opp. Disappearsquickly

o This is in contrast to (+)/(-) alpha positions, which require (+) investors to interact to

restore the relevant risk-return relationship.


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Model

() Where:

() Because () () Note: this holds for any well-diversified portfolio

Implications

The following situation cannot hold

Here, the expected returns for the same beta are different yielding the possibility forarbitrage.

This arbitrage position will disappear over time (quickly)


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Any well diversified portfolios must lie on this line if the no-arbitrage condition is to hold. If we now let the factor be the market premium we have that:

We have now derived the CAPM relationship under arbitrage conditions.

Benefits of APT

Does not require - only a well-diversified portfolio = (+) flexible Says that index models are sufficient approximations. Applies well to WELL-DIVERSIFIED portfolios (not necessarily stocks). Allows individual stocks to be mispriced. Has Multi-factor capability.

CAPM vs. APT

APT CAPMEqu. = no arbitrage Equ. = fairly priced securitiesWell-diversified portfolio Requires unobservable Equ. Quickly restored by only a few investors Equ. Requires + investors to restore mean-

variance efficiencyExpected Return Beta relationship forportfolios

Describes equ. For all assets

Chapter 12 Behavioural Finance & Technical analysis

Behavioural Finance

Intro

In conventional finance: prices are correct, people are rational, allocative efficiency, EMH Behavioural Finance seeks to explain unrationality


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