portfolio theory and risk

8/12/2019 Portfolio Theory and Risk

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CAS Exam 8 Notes - Parts A&B

Portfolio Theory and Equilibrium in Capital Markets

Fixed Income Securities


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Part I

Table of Contents

A Portfolio Theory and Equilibrium in Capital Markets 1

BKM - Ch. 6: Risk aversion and capital allocation to risky assets . . . . . . . . . . . . . . . . . . . 3

BKM - Ch. 7: Optimal risky portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

BKM - Ch. 8: Index models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

BKM - Ch. 9: The Capital Asset Pricing Model (CAPM) . . . . . . . . . . . . . . . . . . . . . . . 27BKM - Ch. 10: Arbitrage pricing theory and multifactor models of risk and return . . . . . . . . . 37

BKM - Ch. 11: The efficient market hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

BKM - Ch. 12: Behavioral finance and technical analysis . . . . . . . . . . . . . . . . . . . . . . . 51

BKM - Ch. 13: Empirical evidence on security returns . . . . . . . . . . . . . . . . . . . . . . . . . 57

B Fixed Income Securities 69

BKM - Ch. 14: Bond prices and yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Hull - Ch. 4: Interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

BKM - Ch. 15: The term structure of interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Hull - Ch. 6.1: Day count and quotation conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Hull - Ch. 22 - Part 1: Credit risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Altman: Measuring corporate bond mortality and performance . . . . . . . . . . . . . . . . . . . . 101

Cummins: CAT Bonds and other risk-linked securities . . . . . . . . . . . . . . . . . . . . . . . . . 105

Additional Notes 113


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A Portfolio Theory and Equilibrium in Capital Markets


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BKM - Ch. 6: Risk aversion and capital allocation to risky assets

Introduction

The process of constructing an investor portfolio can be viewed as a sequence of two steps:1. Selecting the composition of ones portfolio of risky assets such as stocks and long-term bonds2. Deciding how much to invest in that risky portfolio vs. safe assets such as short-term T-bills

Fundamental part of asset allocation problem: Characterize risky portfolio risk-return trade-off Then, the fundamental decision is capital allocation between the risk-free and the risky portfolio Two themes in portfolio theory:1. Investors will avoid risk unless they can anticipate a reward for engaging in risky investments

2. Utility model allows to quantify investors trade-offs between portfolio risk/expected return

Risk and risk aversion

Risk, speculation, and gambling Speculation: The assumption of considerable investment risk to obtain commensurate gain

Considerable risk: The risk is sufficient to affect the decision Commensurate gain: Positive risk premium, i.e., expected profit > risk-free alternative

Gamble: To bet or wager on an uncertain outcome The central difference with speculation is the lack of commensurate gain To turn a gamble into a speculative prospect requires an adequate risk premium to compensate

risk-averse investors for the risks they bear A risky investment with a risk premium of zero, akafair game, amounts to a gamble

A risk-averse investor will reject it In some cases a gamble may appear to the participants as speculation:

Economists call this case of differing beliefs heterogeneous expectations Risk aversion and utility values

Risky assets command a risk premium in the marketplaceMost investors are risk averse Investors who are risk averse reject investment portfolios that are fair games or worse A risk-averse investor penalizes the expected rate of return of a risky portfolio by a certain

% (or penalizes the expected profit by a dollar amount) to account for the risk involved The greater the risk, the larger the penalty

We will assume that each investor can assign a welfare, or utility, score to competing investmentportfolios based on the expected return and risk of those portfolios Higher utility values are assigned to portfolios with more attractive risk-return profiles Portfolios have higher utility scores for higher expected returns/lower scores for higher volatility

E.g.,utility score for portfolio with expected return E(r) and variance of returns 2: utility score:

U=E(r) 12A2 (1)

WhereUis the utility value and Ais an index of the investors risk aversion How variance of risky portfolios lowers utility depends on A, the investors degree of risk aversion

More risk-averse investors (larger values ofA) penalize risky investments more severely

Investors select the investment portfolio providing the highest utility level

Risk-free portfolios utility score = their (known) rate of return (no penalty for risk) We can interpret the utility score of risky portfolios as a certainty equivalent rate of return

Certainty equivalent rate: The rate that risk-free investments would need to offer to providethe same utility score as the risky portfolio

Natural way to compare the utility values of competing portfolios A portfolio is desirable only if its certainty equivalent return > risk-free alternative

Risk-neutral investors (A= 0) judge risky prospects solely by their expected rates of return The level of risk is irrelevant to the risk-neutral investor: There is no penalty for risk For this investor, a portfolios certainty equivalent rate is simply its expected rate of return

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p

Q

P)pr

(

E

)r

(

E Indifference curve

(preferred direction)

Northwest

II

IV

I

III

Figure 1: The trade-off between risk and return of a potential investment portfolio P

A risk lover (A < 0) is willing to engage in fair games and gambles: This investor adjusts theexpected return upward to take into account the fun of confronting the prospects risk

PortfolioP (expected return E(rp), standard deviationp) is preferred by risk-averse investors toany portfolio in quadrant IVbecause it has an expected return any portfolio in that quadrantand a standard deviationany portfolio in that quadrant

Conversely, any portfolio in quadrant Iis preferable to portfolioP Mean-variance (M-V) criterion: PortfolioAdominates B if:

E(rA) E(rB) and A B In theE-plane in Fig. 1, the preferred direction is northwest, because we simultaneously increase

the expected return/decrease the variance of the rate of return Indifference curve: Equally preferred portfolios will lie in the mean-standard deviation plane on

a curve called the indifference curve that connects all portfolio points with the same utility value Estimating risk aversion

One way is to observe individuals decisions when confronted with risk Consider an investor with risk aversion Awhose entire wealth is in a piece of real estate Suppose that in any given year there is a probability p of a disaster that will wipe out the

investors entire wealth. Such an event would amount to a rate of return of100% With probability 1 p, real estate remains intact, and rate of return is zero The expected rate of return of this prospect is:

E(r) =p (1) + (1 p) 0 = p The variance of the rate of return equals the expectation of the squared deviation:

2(r) =p (p 1)2 + (1 p) p2 =p(1 p) Utility score:

U=E(r) 12A2(r) = p 12Ap(1 p) (2) We can relate the risk-aversion parameter to the amount that an individual would be willing

to pay for insurance against the potential loss. Suppose an insurance company offers to coverany loss over the year for a fee ofdollars per dollar of insured property

Such a policy amounts to a sure negative rate of return of, with a utility score: U= Maximum value ofthe investor is willing to pay? Equate the utility score of the uninsuredproperty to that of the insured property, and solve for

= p[1 + 12A(1 p)] (3) Square brackets in Eq. 3 = multiple of expected loss p the investor is willing to pay

Economists estimate that investors exhibit degrees of risk aversion in the range of 2 to 4 More support for the hypothesis that A is somewhere in the range of 2 to 4 can be obtained from

estimates of the expected rate of return and risk on a broad stock-index portfolio

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Capital allocation across risky and risk-free portfolios The most straightforward way to control the risk of the portfolio is through the fraction of the portfolio

invested in Treasury bills and other safe money market securities versus risky assets This capital allocation decision is an example of an asset allocation choice - a choice among broad

investment classes, rather than among the specific securities within each asset class Most investment professionals: Asset allocation = most important part of portfolio construction

Take composition of risky portfolio as given and focus on allocation between it/risk-free securities When we shift wealth from the risky portfolio to the risk-free asset, we do not change the relative

proportions of the various risky assets within the risky portfolio Rather, we reduce the relative weight of the risky portfolio as a whole in favor of risk-free assets

As long as we do not alter the weights of each security within the risky portfolio, the probabilitydistribution of the rate of return on the risky portfolio remains unchanged by the asset reallocation

What will change is the probability distribution of the rate of return on the complete portfolio thatconsists of the risky asset and the risk-free asset

The risk-free asset There are no true risk-free assets

Only the government can issue default-free bonds Even the default-free guarantee by itself is not sufficient to make the bonds risk-free in real terms The only risk-free asset in real terms would be a perfectly price-indexed bond Moreover, a default-free perfectly indexed bond offers a guaranteed real rate to an investor only i

the maturity of the bond is identical to the investors desired holding period Even indexed bonds are subject to interest rate risk: Real interest rates change unpredictably

Nevertheless, it is common practice to view Treasury bills as the risk-free asset Their short term nature makes their values insensitive to interest rate fluctuations An investor can lock in a short-term nominal return by buying a bill and holding it to maturity Inflation uncertainty over a few weeks/monthsuncertainty of stock market returns

In practice, most investors use a broader range of money market instruments as a risk-free asset All the money market instruments are virtually free of interest rate risk because of their short

maturities and are fairly safe in terms of default or credit risk Most money market funds hold three types of securities: (i)T-bills,(ii)Bank certificates of deposit

(CDs), and (iii) Commercial paper (CP), differing slightly in their default risk

Portfolios of one risky asset and a risk-free asset The concern is with the proportion of the investment budget y to be allocated to the risky portfolioP

The remaining proportion 1 y is to be invested in the risk-free asset F Denote the risky rate of return ofP byrp, its expected rate of return by E(rp) and its standard

deviation by p. The rate of return on the risk-free asset is denoted as rf The risk premium on the risky asset is: E(rp) rf The rate of return on the complete portfolio C is rc = yrp+ (1 y)rf

E(rc) =rf+ y[E(rp) rf] Base rate of return = risk-free rate. The portfolio is also expected to earn a risk premium that

depends on risk premium of risky portfolio E(rp) rf and investors position y in P When combining risky and risk-free assets, standard deviation c of complete portfolio = standard

deviation p of risky asset multiplied by weight y of risky asset:

c = yp (4)

Investment opportunity set with risky/risk-free asset in the E- plane Equation for the straight line between F and P:

E(rc) =rf+ y[E(rp) rf] =rf+ cp

[E(rp) rf] (5)

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)r

(

E

)pr

(

E

f

Br

frF

Capital Allocation Line (CAL)

P

fr){pr(E1)y(S

Capital Allocation Line (CAL)

f

Brwith borrowing rate

p

=SSlopefr) {pr(E

p

Figure 2: The investment opportunity set in the expected return-standard deviation plane

Investment opportunity set The set of feasible expected return and standard deviation pairs of portfolios resulting from

different values ofy The Capital Allocation Line (CAL) and the Sharpe ratio

The CAL depicts all the risk-return combinations available to investors The slope Sof the CAL equals the increase in the expected return of the complete portfolio

per unit of additional standard deviation, i.e. incremental return per incremental risk

The slope is called the reward-to-volatility ratio or the Sharpe ratioS=

E(rp) rfp

(6)

If investors can borrow at rf, they can construct portfolios to the right ofPon the CAL However, non-government investors cannot borrow at the risk-free rate Then in the borrowing range, the reward-to-volatility ratio (i.e. the slope of the CAL) will be lower The CAL will therefore be kinked at pointP

Risk tolerance and asset allocation Investor confronting the CAL must choose one optimal portfolio Cfrom set of feasible choices

This choice entails a trade-off between risk and return

Differences in risk aversionDifferent investors choose different positions in risky asset Investors attempt to maximize utility by choosing the best allocation to the risky asset y As allocation to risky asset increases (y), expected return increases, but so does volatility

Solving the utility maximization problem:max

yU=E(rc) 12A2c =rf+ y[E(rp) rf] 12Ay22p

Setting the derivative of this expression to zero and solving for y yields the optimal position:

y =E(rp) rf

A2p(7)

The optimal position in the risky asset is inversely proportional to the level of risk aversion and

the level of risk (variance) and directly proportional to the risk premium offered by the risky asset Indifference curve analysis

First calculate the utility value of a risk-free portfolio yielding rf Then, find the expected return the investor would require to maintain the same level of utility

when holding a risky portfolio for a given This yields all combinations of expected return/volatility with a given constant utility level Any investor prefers a portfolio on higher indifference curve (higher certainty equivalent)

Portfolios on higher indifference curves offer a higher return for any given level of risk Higher indifference curves correspond to higher levels of utility

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CAL

P

C

fr=U

UComplete portfolio maximizing

UHigher

)r

(

E

)pr

(

E

)cr

(

E

fr

pc

Figure 3: Indifference curve analysis

More risk-averse investors have steeper indifference curves than less risk-averse investors Steeper curves: Investors require greater increase in expected return for increase in risk

Investors thus attempt to find the complete portfolio on the highest possible indifference curve Superimpose indifference curves on the investment opportunity set represented by the CAL as

in Fig. 3, and identify highest possible indifference curve that still touches CAL That indifference curve is tangent to the CAL, and the tangency point corresponds to the

standard deviation and expected return of the optimal complete portfolio

The choice for y the fraction of overall investment funds to place in the risky portfolio versus the saferbut lower expected-return risk-free asset, is in large part a matter of risk aversion

Passive strategies: The capital market line Passive strategy

Describes a portfolio decision that avoids any direct or indirect security analysis Natural candidate for passively held risky asset is a well-diversified portfolio of common stocks Because a passive strategy requires that we devote no resources to acquiring information on any

individual stock or group of stocks, we must follow a neutral diversification strategy Select diversified stock portfolio that mirrors the value of the US corporate sector

This results in a portfolio in which,e.g., the proportion invested in Microsoft stock will be the ratioof Microsofts total market value to the market value of all listed stocks

The Capital Market Line (CML) Defined as the CAL provided by l-month T-bills and a broad index of common stocks A passive strategy generates an investment opportunity set that is represented by the CML

How reasonable is it for an investor to pursue a passive strategy?1. The alternative active strategy is not free, Whether you choose to invest the time and cost to

acquire the information needed to generate an optimal active portfolio of risky assets, or whetheryou delegate the task to a professional who will charge a fee

2. Free-rider benefit: Another reason to pursue a passive strategy If there are many active investors who quickly bid up prices of undervalued assets and force

down prices of overvalued assets, then at any time most assets will be fairly priced Therefore, a well-diversified portfolio of common stock will be a reasonably fair buy, and the

passive strategy may not be inferior to that of the average active investor To summarize, a passive strategy involves investment in two passive portfolios: (i) Virtually risk-freeshort-term T-bills (or a money market fund) and (ii) A fund of common stocks that mimics a broadmarket index. The capital allocation line representing such a strategy is called the capital market line

Criticisms of index funds dont hold up They are undiversified: The same complaint could be leveled at actively managed funds They are top-heavy: True, but S&P 500 not so narrow focused (77.2% of US stock-market value) They are chasing performance: This is what all investors do You can do better: As a group, investors in cant outperform market (collectively, they = market)

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BKM - Ch. 7: Optimal risky portfolios

Introduction The investment decision can be viewed as a top-down process:

1. Capital allocation between the risky portfolio and risk-free assets2. Asset allocation across broad classes (US stocks, international stocks, long-term bonds)3. Security selection of individual assets within each asset class

The optimal capital allocation is determined by risk aversion as well as expectations for the risk-returntrade-off of the optimal risky portfolio

In principle, asset allocation and security selection are technically identical: Both aim at identifying that optimal risky portfolio, namely, the combination of risky assets that

provides the best risk-return trade-off In practice, however, asset allocation and security selection are typically separated into two steps:

1. The broad outlines of the portfolio are established first (asset allocation)2. Details concerning specific securities are filled in later (security selection)

Diversification and portfolio risk When all risk is firm-specific, diversification can reduce risk to arbitrarily low levels

With all risk sources independent, exposure to any specific source of risk reduced to negligible level The insurance principle: Reduction of risk to very low levels for independent risk sources Risk eliminated by diversification: Unique/firm-specific/nonsystematic/diversifiable risk

When common sources of risk affect all firms, even extensive diversification cannot eliminate risk The risk that remains even after extensive diversification is called market risk, risk that is at-

tributable to marketwide risk sources (aka systematic risk, or nondiversifiable risk)

Portfolio of two risky assets Efficient diversification: Constructing risky portfolios to provide the lowest possible risk for any given

level of expected return Consider two mutual funds: A bond portfolio D , and a stock fund E

wD is invested in the bond fund, and the remainder 1 wD =wEis invested in the stock fund Denoting rD, rEthe rate of return on debt/equity funds, the rate of return rp on portfolio is:

rp= wDrD+ wErE (1)

The expected return on the portfolio is a weighted average of expected returns on the componentsecurities with portfolio proportions as weights:

E(rp) =wDE(rD) + wEE(rE) (2)

The variance of the two-asset portfolio is:

2p =w2D

2D+ w

2E

2E+ 2wDwECov(rD, rE) (3)

Variance is reduced if the covariance term is negative Even if covariance term

0, the portfolio standard deviation is less than weighted average of

individual security s, unless the two securities are perfectly positively correlated The covariance can be computed from the correlation coefficient DE:

Cov(rD, rE) =DEDE 2p =w2D2D+ w2E2E+ 2wDwEDEDE (4)

A hedge asset has negative correlation with the other assets in the portfolio Such assets will be particularly effective in reducing total risk Expected return is unaffected by correlation between returns

Always prefer to add to portfolio assets with low or negative correlation with existing position

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How low can portfolio standard deviation be? With perfect negative correlation, = 1:

2p = (wDD wEE)2 (5)

When =1, a perfectly hedged position can be obtained by choosing the portfolio proportionsto solve wDD wEE= 0. Then:

wD = E

D

+ E

and wE= D

D

+ E

= 1 wD (6)

What happens when wD >1 and wE


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16141210

8

9

Standard Deviation (%)

ExpectedReturn(%)

E

D

3:== 0

= 1

= {1

10

Figure 2: Portfolio expected return as a function of standard deviation

For = 1, the portfolio opportunity set is linear, but now it offers a perfect hedging oppor-tunity and the maximum advantage from diversification

The lower the correlation, the greater the potential benefit from diversification Suppose now an investor wishes to select the optimal portfolio from the opportunity set

The best portfolio will depend on risk aversion Portfolios to the northeast in Fig. 2 provide higher rates of return but impose greater risk The best trade-off among these choices is a matter or personal preference

Investors with greater risk aversion prefer southwest portfolios (lower expected return, lower risk) Given a level of risk aversion, determine the portfolio that provides highest level of utility

UsingU=E(rp) 12A2p and the portfolio mean/variance determined by the portfolio weightsin the two funds wE and wD, the optimal investment proportions in the two funds are:

wD =E(rD) E(rE) + A(2E DEDE)

A(2D+ 2E 2DEDE)

and wE= 1 wD

Asset allocation with stocks, bonds, and bills

The optimal risky portfolio with two risky assets and a risk-free asset Graphical solution

Ratchet the CAL upward until it reaches point of tangency with investment opportunity set This must yield the CAL with the highest feasible reward-to-volatility ratio Thus, the tangency portfolio (P in Fig. 3) is the optimal risky portfolio to mix with T-bills We can read the expected return and standard deviation of portfolio Pfrom the graph

PortfolioCompleteOptimal

PortfolioGlobal M-V%= 5fr

CAL(E)

Minimum-Variance Frontier

Opportunity Set of Risky Assets

Efficient FrontierIndifference Curve

C

E

G

P

CAL(P)

20151050

14

12

108

6

4

2

Standard Deviation (%)

ExpectedReturn(%)

Figure 3: The opportunity set of the debt and equity funds with the CAL

Portfolio construction with only two risky assets and a risk-free asset The objective is to find the weights wD and wEthat result in the highest slope of the CAL

(i.e., the weights that result in the risky portfolio with the highest reward-to-volatility ratio)

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Thus our objective function is the slope (equivalently, the Sharpe ratio) Sp:

Sp =E(rp) rf

p

For portfolio with two risky assets, expected return and standard deviation of portfolioP are:

E(rp) =wDE(rD) + wEE(rE)

p

= [w2D

2D

+ w2E

2E

+ 2wD

wE

Cov(rD

, rE

)]1/2

Therefore, we solve an optimization problem formally written as (subject towi= 1):maxwi

Sp =E(rp) rf

p

In the case of two risky assets, the solution for the weights of the optimal risky portfolio P,using excess rates of return R rather than total returns r, is:

wD = E(RD)

2E E(RE)Cov(RD, RE)

E(RD)2E+ E(RE)

2D [E(RD) + E(RE)]Cov(RD, RE)

and wE= 1 wD (7)

The steps to arrive at the complete portfolio are:1. Specify the return characteristics of all securities (expected returns, variances, covariances)2. Establish the risky portfolio:

(a) Calculate the optimal risky portfolioP [Eq. (7)](b) Calculate properties ofPusing weights determined in step (a) and Eqs. (2) and (3)

3. Allocate funds between the risky portfolio and the risk-free asset:(a) Calculate the fractiony of the complete portfolio allocated to portfolio P(the risky port-

folio) and to T -bills (the risk-free asset)

y=E(rp) rf

A2p(8)

(b) Calculate the share of the complete portfolio invested in each asset and in T-bills Our two risky assets, the bond and stock mutual funds, are already diversified portfolios. The

diversification within each of these portfolios must be credited for a good deal of the risk reductioncompared to undiversified single securities

Optimizing the asset allocation between bonds and stocks contributed incrementally to the im-provement in the reward-to-volatility ratio of the complete portfolio

The CAL with stocks, bonds, and bills shows that the standard deviation of the complete portfoliocan be further reduced while maintaining the same expected return as the stock portfolio

The Markowitz portfolio selection model

Generalizing the portfolio construction problem to the case of many risky securities and a risk-free asset

1. Identify the risk-return combinations available from the set of risky assets2. Identify optimal portfolio of risky assets by finding portfolio weights resulting in steepest CAL3. Choose appropriate complete portfolio by mixing risk-free asset with optimal risky portfolio

Security selection In the risk-return analysis, the portfolio manager needs as inputs a set of estimates for the expected

returns of each security and a set of estimates for the covariance matrix Hence, we have n estimates ofE(ri) and the n n estimates of the covariance matrix in which

the n diagonal elements are estimates of the variances 2i and the n2 n= n(n 1) off-diagonal

elements are the estimates of the covariances between each pair of asset returns

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The expected return/variance of any risky portfolio with weights in each security wi is:

E(rp) =

ni=1

wiE(ri) (9)

2p =n

i=1

nj=1

wiwjCov(ri, rj) (10)

Markowitz model is precisely step one of portfolio management: The identification of the efficientset of portfolios, or the efficient frontier of risky assets

M-V frontier: Graph of the lowest possible variance for a given portfolio expected return All individual assets lie to the right inside the frontier, at least when short sales are allowed

When short sales are prohibited, single securities may lie on the frontier The part of the frontier above the global M-V portfolio is the efficient frontier The principal idea behind the frontier set of risky portfolios is that, for any risk level, we are

interested only in that portfolio with the highest expected return The frontier is the set of portfolios that minimizes variance for any target expected return

Some clients may be subject to additional constraints. E.g., prohibited from taking short positions For these clients the portfolio manager will add to the optimization program constraints that

rule out negative (short) positions in the search for efficient portfolios

In this special case, single assets may be, in and of themselves, efficient risky portfolios

E.g., asset with highest expected return is a frontier portfolio because, without short sales, theonly way to obtain that rate of return is to hold the asset as ones entire risky portfolio

Some may want to ensure minimal level of expected dividend yield from optimal portfolio In this case the input list will be expanded to include a set of expected dividend yields d1, , dn

and the optimization program will include an additional constraint that ensures that the ex-pected dividend yield of the portfolio will equal or exceed the desired leveld

Any constraint carries a price tag in the sense that an efficient frontier constructed subject to extraconstraints will offer a reward-to-volatility ratio inferior to that of a less constrained one

Another type of constraint is aimed at ruling out investments in industries or countries consideredethically or politically undesirable. This is referred to as socially responsible investing

Capital allocation and the separation property We ratchet up the CAL by selecting different portfolios until we reach portfolio P which is the

tangency point of a line from Fto the efficient frontier Portfolio Pmaximizes the reward-to-volatility ratio and is the optimal risky portfolio

The most striking conclusion is that a portfolio manager will offer the same risky portfolio Pto alclients regardless of their degree of risk aversion

The degree of risk aversion comes into play only in the selection of desired point along CAL More risk-averse clients invest more in risk-free asset and less in optimal risky portfolio thanless risk-averse clients. However, both use portfolioPas their optimal risky investment vehicle

Separation property: Portfolio choice problem may be separated into 2 independent tasks:1. Determination of the optimal risky portfolio (purely technical)2. Allocation of the complete portfolio to T-bills vs. the risky portfolio (personal preference)

In practice, different managers estimate different input lists, thus deriving different efficient fron-tiers, and offer different optimal portfoliosSource of disparity lies in security analysis This analysis suggests that a limited number of portfolios may be sufficient to serve the demands

of a wide range of investorsTheoretical basis of the mutual fund industry The power of diversification

Consider the naive diversification strategy in which an equally weighted portfolio is constructedwi = 1/nfor each security. In this case Eq. (10) becomes:

2p = 1

n

ni=1

1

n2i +

nj=1,j=i

ni=1

1

n2Cov(ri, rj) (11)

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Define the average variance and average covariance of the securities as:

2 = 1

n

ni=1

2i and Cov = 1

n(n 1)n

j=1,j=i

ni=1

Cov(ri, rj) (12)

Then, the portfolio variance is:

2p = 1

n2 +

n 1n

Cov (13)

Effect of diversification When the average covariance among security returns is zero, as it is when all risk is firm-specific,

portfolio variance can be driven to zero Hence when security returns are uncorrelated, the power of diversification is unlimited However, usually, economy-wide risk factors impart positive correlation among stocks

The irreducible risk of a diversified portfolio depends on covariance of the returns of com-ponent securities, which is a function of the importance of systematic economic factors

Suppose that all securities have a common and all security pairs have a common The covariance between all pairs of securities is 2 and Eq. (13) becomes:

2

p

= 1

n2 +

n 1n

2 (14)

When = 0, we again obtain the insurance principle: p 0 as n For >0, however, portfolio variance remains positive For = 1, portfolio variance equals 2 regardless ofnDiversification is of no benefit

In the case of perfect correlation, all risk is systematic More generally, as n becomes greater, Eq. (14) shows that systematic risk becomes 2

For diversified portfolios, the contribution to portfolio risk of a particular security depends on thecovariance of that securitys return with other securities, and not on the securitys variance

Asset allocation and security selection Why distinguish between asset allocation and security selection? 3 reasons:

1. As a result of greater need and ability to save (for college, recreation, longer life, health care

needs, etc.), the demand for sophisticated investment management has increased enormously2. The widening spectrum of financial markets and financial instruments has put sophisticated

investment beyond the capacity of many amateur investors3. There are strong economies of scale in investment analysis

The end result is that the size of a competitive investment company has grown with the industry,and efficiency in organization has become an important issue

The practice is therefore to optimize the security selection of each asset-class portfolio independently At the same time, top management continually updates the asset allocation of the organization,

adjusting the investment budget allotted to each asset-class portfolio

Risk pooling, risk sharing, and risk in the long run

Risk pooling and the insurance principle

The insurance principle Suppose an insurer sells 10,000 uncorrelated policies, each with a standard deviation Because the covariance between any two insurance policies is zero and is the same for each

policy, the standard deviation of the rate of return on the 10,000-policy portfolio is:

2p = 1

n2 p=

n (15)

p could be further decreased by selling even more policiesThis is the insurance principle

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It seems that as the film sells more policies, its risk continues to fall. Flaw in this argument: Probability of loss = inadequate measure of risk: Does not account for the magnitude of loss If 10,000 policies are sold, maximum possible loss is 10,000 bigger and comparison with a

one-policy portfolio cannot be made based on means/standard deviations of rates of return Similar flaw as the argument that investing in stocks for the long run reduces risk

Increasing the size of the bundle of policies does not make for diversification! Diversifying a portfoliomeans dividing a fixed investment budget across more assets

When we combinenuncorrelated insurance policies, each with an expected profit $, both expectedtotal profit and standard deviation (SD) grow in direct proportion to n:

E(n) =nE()

V ar(n) =n2V ar() =n22

SD(n) =n

The ratio of mean to standard deviation does not change when nincreases The risk-return trade-off therefore does not improve with the assumption of additional policies

Risk sharing If risk pooling (sale of additional independent policies) does not explain insurance, what does?

The answer is risk sharing, the distribution of a fixed amount of risk among many investors An underwriter will contact other underwriters who each will take a piece of the action: Each will

choose to insure a fraction of the project risk Each underwriter has a fixed amount of equity capital. Underwriters engage in risk sharing. They

limit their exposure to any single source of risk by sharing that risk with other underwriters Underwriter diversifies its risk by allocating its budget across many projects that are not perfectly

correlatedOne underwriter will decline to u/w too large a fraction of any single project This is the proper use of risk pooling: Pooling many sources of risk in a portfolio of given size

The bottom line is that portfolio risk management is about the allocation of a fixed investment budgetto assets that are not perfectly correlated

In this environment, rate of return statistics (expected returns, variances, and covariances) aresufficient to optimize the investment portfolio

Choices among alternative investments of a different magnitude require that we abandon rates of

return in favor of dollar profits This applies as well to investments for the long run

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BKM - Ch. 8: Index models

Introduction

The Markowitz procedure suffers from two drawbacks:1. The model requires a huge number of estimates to fill the covariance matrix2. The model does not provide any guideline to the forecasting of the security risk premiums

Index models simplify estimation of covariance matrix and enhance analysis of risk premiums By allowing to explicitly decompose risk into systematic and firm-specific components, these models

also shed considerable light on both the power and limits of diversification Further, they allow to measure these components of risk for particular securities and portfolios

Despite simplification, index models remain true to concepts of efficient frontier/portfolio optimization Empirically, index models are as valid as the assumption of normality of rates of return on securities

A single-factor security market

The input list of the Markowitz model The success of a portfolio selection rule depends on the quality of the input list The Markowitz model necessitates n expected returns, n variances, and n(n 1)/2 covariances Markowitz model: Errors in estimation of correlation coefficients can lead to nonsensical results

This can happen because some sets of correlation coefficients are mutually inconsistent Introducing a model that simplifies the way we describe the sources of security risk allows us to

use a smaller, consistent set of estimates of risk parameters and risk premiums The simplification emerges because positive covariances among security returns arise fromcommon economic forces that affect the fortunes of most firms

Examples of common economic factors: Business cycles, interest rates, cost of natural resources The unexpected changes in these variables cause, simultaneously, unexpected changes in the

rates of return on the entire stock market By decomposing uncertainty into these system-wide versus firm-specific sources, we vastly simplify

the problem of estimating covariance and correlation Normality of returns and systematic risk

Decompose rate of return on security i into its expected plus unanticipated components:

ri= E(ri) + ei (1)

Where the unexpected return ei has a mean of zero and a standard deviation ofi When security returns can be well approximated by normal distributions that are correlated across

securities, we say that they are joint normally distributed At any time, security returns are driven by one or more common variables

Suppose the common factor m that drives innovations in security returns is some macroeconomicvariable that affects all firms Decompose sources of uncertainty into uncertainty about economyas a whole (captured by m) and uncertainty about firm in particular (captured by ei)

ri= E(ri) + m + ei (2)

The macroeconomic factor mmeasures unanticipated macro surprises m has a mean of zero (over time, surprises average out) with standard deviation ofm mand ei are uncorrelated, because ei is firm-specific Independent of shocks to the common

factor that affect the entire economy The variance ofri thus arises from two uncorrelated sources, systematic and firm specific

2i =2m+

2(ei) (3)

The common factor m generates correlation across securities Because all securities will respond to the same macroeconomic news

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Sincem is uncorrelated with any firm-specific surprises, the covariance between any i and j is:Cov(ri, rj) =C ov(m + ei, m + ej) =

2m (4)

We recognize that some securities are more sensitive than others to macroeconomic shocks: Capture this refinement by assigning each firm a sensitivity coefficient to macro conditions

This leads to the single-factor model:

ri= E(ri) + im + ei (5)

The systematic risk of security i is 2i

2m, and its total risk is:

2i =2i

2m+

2(ei) (6)

The covariance between any pair of securities also is determined by their betas:Cov(ri, rj) =C ov(im + ei, jm + ej) =ij

2m (7)

Normality of security returns alone guarantees that portfolio returns are also normal and that thereis a linear relationship between security returns and the common factor

Seek a variable that can proxy for common factor. To be useful, variable must be observable, sowe can estimate its volatility/sensitivity of individual securities returns to variation in its value

The single-index model

A reasonable approach is to assert that the rate of return on a broad index of securities such as the S&P

500 is a valid proxy for the common macroeconomic factor Single-index modelsingle-factor model because market index = proxy for common factor

The regression equation of the single-index model Denote the market index by M, with excess return ofRM=rMrfand standard deviation ofM We regress the excess return of a security Ri= ri rfon the excess return of the index RM We collect a historical sample of paired observations Ri(t) and RM(t) where t denotes the date of

each pair of observations. The regression equation is:

Ri(t) =i+ iRM(t) + ei(t) (8)

The intercept i is the securitys expected excess return when the market excess return is zero The slope coefficienti is the securitys sensitivity to the index

ei is the zero-mean, firm-specific surprise in the security return in time t, aka the residual The expected return-beta relationship Taking expected values, we obtain the expected return-beta relationship of the single-index model:

E(Ri) =i+ iE(RM) (9)

Part of a securitys risk premium is due to the risk premium of the index The market risk premium is multiplied by the relative sensitivity of the individual security This is the systematic risk premium because it derives from the risk premium that characterizes

the entire market, which proxies for the condition of the full economy or economic system is the non-market premium

Risk and covariance in the single-index model Both variances/covariances are determined by security betas/properties of market index:

Total risk = Systematic risk + Firm-specific risk

2i =2i

2M+

2(ei) (10)

Covariance = Product of betas Market index riskCov(ri, rj) =ij

2M (11)

Correlation = Product of correlations with the market index

Corr(ri, rj) =ij

2M

ij=Corr(ri, rM) Corr(rj, rM) (12)

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The set of parameter estimates needed for the single-index model consists of only , , and (e)for the individual securities, plus the risk premium and variance of the market index

The set of estimates needed for the single-index model1. i: Stocks expected return if the market is neutral, i.e. rM rf= 02. i(rM rf): The component of return due to movements in the overall market3. ei: The unexpected component of return due to unexpected firm specific events4. 2i

2M: The variance attributable to the uncertainty of the common macroeconomic factor

5. 2(ei): The variance attributable to firm-specific uncertainty Advantages of the model

The index model is a very useful abstraction because for large universes of securities, thenumber of estimates is only a small fraction of what otherwise would be needed

The index model abstraction is crucial for specialization of effort in security analysis: If acovariance term had to be calculated for each security pair, then no specialization by industry

Disadvantages Cost of the model: Restrictions it places on structure of asset return uncertainty Classification of uncertainty into dichotomy - macro vs. micro risk - oversimplifies sources of

real-world uncertainty and misses important sources of dependence in stock returns E.g., dichotomy rules out industry events (affect an industry but not the broad macroeconomy)

The optimal portfolio derived from the single-index model therefore can be significantly inferiorto that of the full-covariance (Markowitz) model when stocks with correlated residuals have large

alpha values and account for a large fraction of the Portfolio If many pairs of the covered stocks exhibit residual correlation, it is possible that a multi-

index model, which includes additional factors to capture those extra sources of cross-securitycorrelation, would be better suited for portfolio analysis and construction

The index model and diversification Suppose that we choose an equally weighted portfolio ofn securities. The excess rate of return on

each security is given by: Ri = i+ iRM+ ei Similarly, we can write the excess return on the portfolio of stocks as: Rp= p+ pRM+ ep As the number of stocks included in this portfolio increases, the part of the portfolio risk attributable

to nonmarket factors becomes ever smaller: This part of the risk is diversified away In contrast, market risk remains, regardless of the number of firms combined into the portfolio

Rp=n

i=1

wiRi= 1

n

ni=1

(i+ iRM+ ei) = 1

n

ni=1

i+

1

n

ni=1

i

RM+

1

n

ni=1

ei (13)

Portfolios sensitivity to market, nonmarket return and avg. of firm-specific components:

p = 1

n

ni=1

i p= 1

n

ni=1

i ep = 1

n

ni=1

ei (14)

Hence the portfolios variance is:

2p =2p

2M+

2(ep) (15)

The systematic risk component of the portfolio variance, which depends on marketwide movements

is 2p2M and depends on the sensitivity coefficients of the individual securities. This part of therisk will persist regardless of the extent of portfolio diversification

The nonsystematic component of the variance is 2(ep) and is attributable to firm-specific ei Because eis are independent and have zero expected value, as more stocks are added to the

portfolio, firm-specific components cancel out, resulting in ever-smaller nonmarket risk Such risk is thus termed diversifiable: Because the eis are uncorrelated,

2(ep) =n

i=1

1

n

22(ei) =

1

n2(e) (16)

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Estimating the single-index model The security characteristic line for stock i

Regression: Line with intercept i, slope i = security characteristic line (SCL) for stock i:

Ri(t) =i+ iRS&P500(t) + ei(t)

The explanatory power of the SCL for stock i The R-square tells us the percentage of the variation in the stock i series that is explained by the

variation in the S&P 500 excess returns

The adjustedR2 (slightly smaller) corrects for upward bias in R2 that arises because we use fittedvalues of parameters (slope and intercept ) rather than true, but unobservable values

In general, the adjusted R-square (R2A) is derived from the unadjusted by:R2A= 1 (1 R2) (n1)(nk1) where k is the number of independent variables

An additional degree of freedom is lost to the estimate of the intercept Analysis of variance

The sum of squares (SS) is the portion of the variance of the dependent variable (stock is return)that is explained by the independent variable (S&P 500 return); it is equal to 2i

2S&P500

The MS column for the residual shows the variance of the unexplained portion of stock is return,i.e. the portion of return that is independent of the market index. The square root of this value isthe standard error (SE) of the regression

Dividing the total SS of the regression by the number of dof (59 for 60 observations), we obtainthe estimate of the variance of the dependent variable (stock i), per month

R2 (ratio of explained/total variance) = [explained (regression) SS]/[total SS]

R2 = 2i

2S&P500

2i 2S&P500+

2(ei)= 1

2(ei)

2i 2S&P500+

2(ei)

The estimate of alpha The intercept is the estimate of stock is alpha for the sample period (per month) The standard error of the estimate is a measure of the imprecision of the estimate. If the standard

error is large, the range of likely estimation error is correspondingly large

We can relate the standard error of to the standard error of the residuals as:

SE(i) =(ei)

1

n+

(Avg. S&P500)2

V ar(S&P500) (n 1)

The t-statistic is the ratio of the regression parameter to its standard error This statistic equals the number of standard errors by which our estimate exceeds zero, and

therefore can be used to assess the likelihood that the true but unobserved value might actuallyequal zero rather than the estimate derived from the data

For , we are interested in avg. value of stock is return net of market movements Define the nonmarket component of return as actual return minus the return attributable to

market movements. Call this stock is firm-specific returnRfs = Ri iRS&P500 IfRfs were normally distributed with a mean of zero, the ratio of its estimate to its standarderror would have a t-distribution From a table of the t-distribution, we can find the probability that the true is actually zero This is called the level of significance or the probability or p-value. Conventional cut-off for

statistical significance is a probability of less than 5%, which requires t-statistic2.0 Even ifwas both economically and statistically significant within the sample, we still would not

use that as a forecast for a future period Overwhelming empirical evidence shows that 5-year alpha values do not persist over time Virtually no correlation between estimates from one sample period to the next

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The estimate of beta When the value of beta and its SE produce a large t-statistic and a p-value of practically zero, we

can confidently reject the hypothesis that stock is true beta is zero

SE(i) = (ei)

i

n 1 A more interestingt-statistic might test a null hypothesis that i is greater than the market-wide

average beta of 1. Thist-statistic would measure how many standard errors separate the estimated

beta from a hypothesized value of 1:

t-statistics =Estimated value Hypothesized value

Standard error

Firm-specific risk The annual standard deviation of stock is residual is

12 times its monthly standard deviation

The standard deviation of systematic risk is (S&P500) It is common for individual stocks to have a firm-specific risk as large as its systematic risk

Correlation and covariance matrix The variance estimates for the individual stocks equal2i

2M+

2(ei) The off-diagonal terms are covariance values and equal ij

2M

Portfolio construction and the single-index model Alpha and security analysis Most important advantage of single-index model: Framework it provides for macroeconomic/securit

analysis in preparing input list critical to efficiency of the optimal portfolio The single-index model separates macroeconomic and individual-firm sources of return variation

and makes it easier to ensure consistency across analysts Hierarchy of the preparation of the input list using the single-index model:

1. Macroeconomic analysis is used to estimate the risk premium and risk of the market index2. Statistical analysis used to estimates of all securities and their residual variances 2(ei)3. The portfolio manager uses the estimates for the market-index risk premium and the beta

coefficient of a security to establish the expected return of that security absent any contribution

from security analysis. This market-driven expected return can be used as a benchmark4. Security-specific expected return forecasts (s) are derived from security-valuation models The alpha value distills the incremental risk premium attributable to private informationdeveloped from security analysis

is more than just one of the components of expected return: It is the key variable that tells uswhether a security is a good or a bad buy

The index portfolio as an investment asset To avoid inadequate diversification, include the S&P 500 portfolio as one of the assets S&P 500passive portfolio that the manager would select in the absence of security analysis

The single-index model input list If the portfolio manager plans to compile a portfolio from a list ofn actively researched firms and

a passive market index portfolio, the input list will include the following estimates:1. Risk premium on the S&P 500 portfolio2. Estimate of the standard deviation of the S&P 500 portfolio3. n sets of estimates of(i) coefficients, (ii) Stock residual variances, and (iii) values

The optimal risky portfolio of the single-index model With the estimates of and coefficients, plus the risk premium of the index portfolio, we can

generate the n + 1 expected returns using Eq. (9) With the estimates ofcoefficients and residual variances, together with the variance of the index

portfolio, we can construct the covariance matrix using Eq. (10) Given risk premiums and the covariance matrix, we can conduct the optimization program

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The,, and residual variance of a weighted portfolio are the simple averages of those parametersacross component securities (where the indexn +1 corresponds to the market index: n+1= M =0, n+1= M= 1, and (en+1) =(eM) = 0):

p =

n+1i=1

wii p =

n+1i=1

wii 2(ep) =

n+1i=1

w2i 2(ei) (17)

The objective is to maximize the Sharpe ratio of the portfolio by using portfolio weights w1, , wn+1.With this set of weights, the portfolio expected return, standard deviation, and Sharpe ratio are:

E(Rp) =p+ E(RM)p =

n+1i=1

wii+ E(RM)

n+1i=1

wii (18)

p = [2p

2M+

2(ep)]1/2 =

2M

n+1i=1

wii

2+

n+1i=1

w2i 2(ei)

1/2

(19)

Sp =E(Rp)

p(20)

The optimal risky portfolio trades off search for against departure from efficient diversification

The optimal risky portfolio turns out to be a combination of two component portfolios:1. An active portfolio A comprised of the nanalyzed securities2. The market-index passive portfolio is the (n + l)-th asset included to aid in diversification

Assume first that the active portfolio has a beta of 1: In that case, the optimal weight in active portfolio is proportional to ratio A/2(eA) This ratio balances the contribution of the active portfolio (its alpha) against its contribution

to the portfolio variance (residual variance) The analogous ratio for the index portfolio is E(RM)/2Mand hence the initial position in the

active portfolio (i.e., if its beta were 1) is:

w0A= A/

2(eA)

E(RM)/2M(21)

Next, we amend this position to account for the actual beta of the active portfolio For any level of2A, the correlation between the active and passive portfolios is greater when

the beta of the active portfolio is higher This implies less diversification benefit from the passive portfolio and a lower position in it.

Correspondingly, the position in the active portfolio increases:

wA= w0A

1 + (1 A)w0A(22)

The information ratio Sharpe ratio of optimally constructed risky portfolio > index portfolio (passive strategy):

S2p =S2M+

A(eA)

2(23)

The information ratio The contribution of the active portfolio (when held in its optimal weight wA) to the Sharpe

ratio of the overall risky portfolio is determined by the ratio of its alpha to its residual standarddeviation, akathe information ratio

Ratio measures extra return we can obtain from security analysis compared to firm-specificrisk we incur when we over-/underweight securities relative to passive market index

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To maximize the overall Sharpe ratio, maximize the information ratio of active portfolio Information ratio maximized if we invest in each security its ratio ofi/2(ei) Rescaling so that total position in active portfolio = wA, weight in security i is:

wi =wA

i2(ei)

ni=1

i2(ei)

(24)

With this set of weights, the contribution of each security to the information ratio of the activeportfolio depends on its own information ratio:

A(eA)

2=

ni=1

i(ei)

2(25)

In contrast to alpha, the market (systematic) component of the risk premiumiE(RM) is offset bythe securitys nondiversifiable (market) risk 2i

2Mand both are driven by the same beta

The beta of a security is neither vice nor virtue: It is a property that simultaneously affects the risk and risk premium of a security Only the aggregate of the active portfolio, rather than each individual securitys matters

The index portfolio is an efficient portfolio only if all alpha values are zero Unless a security has = 0, including it in active portfolio makes portfolio less attractive

In addition to the securitys systematic risk, which is compensated for by the market riskpremium (through beta), the security would add its firm-specific risk to portfolio variance If all securities have zero alphas, the optimal weight in the active portfolio will be zero, and

the weight in the index portfolio will be 1 However, when security analysis uncovers securities with nonmarket risk premiums (nonzero

alphas), the index portfolio is no longer efficient Summary of the optimization procedure

1. Compute the initial position of each security in the active portfolio as w0i =i/2(ei)

2. Scale initial positions to force

weights = 1 by dividing by their sum: wi= w0i /

w0i3. Compute the alpha of the active portfolio: A=

wii

4. Compute the residual variance of the active portfolio: 2(eA) =

w2i

2(ei)5. Compute the initial position in the active portfolio: w0A = [A/

2(eA)]/[E(RM)/2M]

6. Compute the beta of the active portfolio: A=

wii7. Adjust the initial position in the active portfolio: wA = w

0A/[1 + (1 A)w0A]

8. Note: The optimal risky portfolio now has weights: wM= 1 wA and wi =wAwi9. Calculate the risk premium of the optimal risky portfolio from the risk premium of the index

portfolio and the alpha of the active portfolio: E(Rp) = (wM+ w

AA)E(RM) + w

AA

10. Compute the variance of the optimal risky portfolio from the variance of the index portfolio andthe residual variance of the active portfolio: 2p = (w

M+ w

AA)

22M+ [wA(eA)]

2

Practical aspects of portfolio management with the index model Is the index model inferior to the full-covariance model?

A parsimonious model that is stingy about inclusion of independent variables is often superior:

Predicting the value of the dependent variable depends on two factors: (i) The precision othe coefficient estimates and (ii) The precision of the forecasts of the independent variables

When we add variables, we introduce errors on both counts Markowitz model more flexible in for asset covariance structure compared to single-index model

Advantage illusory if we cant estimate covariances with any degree of confidence Using full-covariance matrix invokes estimation of thousands of termsPossible that cumu-

lative effect of so many estimation errors results in an inferior than the single-index model Advantages of the single-index framework:

Clear practical advantage Decentralizes macro and security analysis

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The industry version of the index model Merrill Lynch publishes a monthly Security Risk Evaluation book (aka beta book) which uses

the S&P 500 as the proxy for the market portfolio It relies on the 60 most recent monthly observations to calculate regression parameters Uses total returns, rather than excess returns, in the regressionsVariant of index-model:

r= a + brM+ e instead of r rf = + (rM rf) + e (26)

Ifrf is constant, both equations have same independent variable rm and residual e

The slope coefficient will be the same in the two regressions However, the intercept that Merrill Lynch calls alpha is really an estimate of + rf(1 ) Merrill Lynch departs from the index model is in its use of percentage changes in price instead

of total rates of returnIgnores the dividend component of stock returns For most firms, R2 0.5, indicating that stocks have far more firm-specific than systematic risk

Highlights the practical importance of diversification Adjusted beta

Motivation for adjusting estimates: On average, s of stocks move toward 1 over time As it grows, a firm diversifies, expanding to similar products and later to more diverse opera-

tions. As firm becomes more conventional, it resembles rest of economy even more Its beta coefficient will tend to change in the direction of 1

Another explanation for this phenomenon is statistical: When we estimate this beta coefficient over a particular sample period, we sustain someunknown sampling error of the estimated beta

The greater the difference between our estimate and 1, the greater the chance that wehad a large estimation error and that in a later sample period will be closer to 1

Merrill Lynch adjusts beta estimates in a simple way:Adjusted beta = 23Sample beta +

13

estimates are ex post (after the fact) measures. They do not mean that anyone could haveforecast these alpha values ex ante (before the fact)

Given the R2 of the regression and the residual standard deviation of the stock (ei), we can solvefor the total standard deviation i of stocki:

i=

2(ei)

1 R2

i = stock is monthly std. dev. for sample periodAnnualized std. dev. is

12i Predicting betas

Simple approach: Collect data on in different periods and then estimate a regression:

Current beta =a + b(Past beta) (27)

Given estimates ofa and b, we would then forecast future betas using:

Forecast beta =a + b(Current beta) (28)

If belief that firm size/debt ratios are two determinants of, an expanded version of Eq. (27) is:

Current beta =a + b1(Past beta) + b2(Firm size) + b3(Debt ratio)

Variables to help predict betas:1. Variance of earnings 3. Growth in earnings per share 5. Dividend yield2. Variance of cash flow 4. Market capitalization (firm size) 6. Debt-to-asset ratio

Even after controlling for a firms financial characteristics, industry group helps to predict

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Index models and tracking portfolios Suppose a portfolio manager believes she has identified an underpriced portfolio. The index model

equation for this portfolio is:

Rp= .04 + 1.4RS&P500+ ep (29)

Manager confident in quality of security analysis but wary about near term performance of broadmarket. Wants a position that takes advantage of teams analysis but is independent of overalmarket performance

To this end, a tracking portfolio (T) can be constructed

Tracking portfolio A tracking portfolio for Pis designed to match the systematic component ofPs return The idea is for the portfolio to track the market-sensitive component ofPs return This means the tracking portfolio must have the same beta on the index portfolio as Pand as

little nonsystematic risk as possible. This procedure is also called beta capture A tracking portfolio for Pwill have a levered position in the S&P 500 to achieve a beta of 1.4

Therefore,T includes positions of 1.4 in the S&P 500 and -0.4 in T-bills Because T is constructed from the index and bills, it has an alpha value of zero

Now buy Pbut offset systematic risk by assuming a short position in tracking portfolio The short position in Tcancels out the systematic exposure of the long position in P The overall combined position is thus market neutral Therefore, even if the market does poorly, the combined position should not be affected But the alpha on portfolio P will remain intact

The combined portfolio Cprovides an excess return per dollar of:

RC=Rp RT = (.04 + 1.4RS&P500+ ep) 1.4RS&P500 = .04 + ep (30)

While this portfolio is still risky (residual risk ep), the systematic risk has been eliminated, and ifPis reasonably well-diversified, the remaining nonsystematic risk will be small Manager can take advantage of the 4% without inadvertently taking on market exposure

Process of separating the search for from the choice of market exposure is called transport This long-short strategy is characteristic of the activity of many hedge funds

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BKM - Ch. 9: The Capital Asset Pricing Model (CAPM)

Introduction CAPM gives precise prediction of relationship between risk of an asset and its expected return CAPM serves two vital functions:

1. It provides a benchmark rate of return for evaluating possible investments2. Helps make a guess as to the expected return on assets not yet traded in the marketplace

Although the CAPM does not fully withstand empirical tests, it is widely used because of the insight itoffers and because its accuracy is deemed acceptable for important applications

The Capital Asset Pricing Model Simplifying assumptions that lead to the basic version of the CAPM:

1. There are many investors, each with a wealth that is small compared to the total endowment oall investors. Investors act as though security prices are unaffected by their own trades

2. All investors plan for one identical holding period. This behavior is myopic: It ignores everythingthat might happen after the end of single-period horizon. Myopic behavior is suboptimal

3. Investments limited to publicly traded financial assets and to risk-free borrowing/lending Rules out investment in nontraded assets such as human capital, private enterprises, and gov-ernmentally funded assets. Assumes that investors may borrow/lend any amount at risk-free rate

4. Investors pay no taxes on returns and no transaction costs on trades in securities5. All investors are rational M-V optimizers, they all use the Markowitz model6. All investors analyze securities in the same way and share the same economic view of the world

I.e., for any set of security prices, they all derive the same input list to feed into the Markowitzmodel. This assumption is often referred to as homogeneous expectations

Summary of the equilibrium that will prevail in this hypothetical world of securities and investors:1. All investors will choose to hold a portfolio of risky assets in proportions that duplicate represen

tation of the assets in the market portfolio M, which includes all traded assets. The proportion oeach stock in the market portfolio equals the market value of the stock (price per sharenumberof shares outstanding) divided by the total market value of all stocks

2. Not only will the market portfolio be on the efficient frontier, but it also will be the tangencyportfolio to the optimal capital allocation line (CAL) derived by each and every investor

The capital market line (CML), the line from the risk-free rate through the market portfolio

M is also the best attainable capital allocation line. All investors hold M as their optimal riskyportfolio, differing only in the amount invested in it versus in the risk-free asset

3. The risk premium on the market portfolio will be proportional to its risk and the degree of riskaversion of the representative investor:

E(rM) rf= A2MWhere2M= variance of market portfolio and

A= avg. degree of risk aversion across investors4. The risk premium on individual assets will be proportional to the risk premium on the market

portfolio Mand the beta coefficient of the security relative to the market portfolio, where:

i=

Cov(ri, rM)

2M

And the risk premium on individual securities is:

E(ri) rf=i[E(rM) rf]

Why do all investors hold the market portfolio? When we aggregate the portfolios of all individual investors, lending/borrowing cancel out, and the

value of the aggregate risky portfolio will equal the entire wealth of the economy This is the market portfolio M

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The CAPM implies that as individuals attempt to optimize their personal portfolios, they eacharrive at the same portfolio, with weights on each asset equal to those of the market portfolio

If all investors use identical Markowitz analysis (Assumption 5) applied to the same universe ofsecurities (Assumption 3) for the same time horizon (Assumption 2) and use the same input list(Assumption 6), they all must arrive at the same composition of the optimal risky portfolio, theportfolio on the efficient frontier identified by the tangency line from T-bills to that frontier

As a result, the optimal risky portfolio of all investors is simply a share of the market portfolio All assets have to be included in the market portfolio

When all investors avoid a stock i, the demand is zero, and is price takes a free fall

Ultimately, stock i reaches a price attractive enough to be included in the optimal portfolio Such price adjustment process guarantees that all stocks are included in optimal portfolio

The passive strategy is efficient In the simple world of the CAPM, Mis the optimal tangency portfolio on the efficient frontier Thus the passive strategy of investing in a market index portfolio is efficient

We sometimes call this result a mutual fund theorem Another incarnation of separation property: Separate portfolio selection into 2 components:

1. A technical problem: Creation of mutual funds by professional managers2. A personal problem that depends on an investors risk aversion: Allocation of the complete

portfolio between the mutual fund and risk-free assets

The risk premium of the market portfolio

Each individual investor chooses a proportion y allocated to the optimal portfolio M such that:

y=E(rM) rf

A2M(1)

Net borrowing and lending across all investors must be zero Substituting the representativeinvestors risk aversion Afor A, the average position in the risky portfolio is 100%, or y= 1

Risk premium on market portfolio related to its variance by avg. degree of risk aversion:

E(rM) rf = A2M (2)

Expected returns on individual securities

The contribution of one stockito portfolio variance can be expressed as the sum of all the covarianceterms corresponding to the stockContribution of stock i to variance of market portfolio:

wi[w1Cov(r1, ri) + + wiCov(ri, ri) + + wnCov(rn, ri)] (3)

When there are many stocks, there will be many more covariance terms than variance termsCovariance of a stock with all others dominates that stocks contribution to total portfolio risk Measure stock is contribution to market portfolio risk by its covariance with that portfolio:

Stocki contribution to variance =wiCov(ri, rM)

Since rM= wkrk, the covariance of return on stock i with market portfolio is:Cov(ri, rM) =C ov

ri,

nk=1

wkrk

=

nk=1

wkCov(ri, rk) (4)

The contribution of our holding of stock i to the risk premium of the market portfolio is wi[E(ri)rf] Therefore, the reward-to-risk ratio for investments in i is:

Stocki contribution to risk premium

Stock i contribution to variance =

wi[E(ri) rf]wiCov(ri, rM)

= E(ri) rfCov(ri, rM)

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The market portfolio is the tangency (efficient M-V) portfolio with reward-to-risk ratio:

Market risk premium

Market variance =

E(rM) rf2M

(5)

Market price of risk: Ratio in Eq. (5), quantifies extra return demanded to bear portfolio risk A basic principle of equilibrium is that all investments should offer the same reward-to-risk ratio

The reward-to-risk ratios ofi and the market portfolio should be equal:E(r

i)

rf

Cov(ri, rM)=

E(rM

)

rf

2M(6)

Hence, the fair risk premium of stock i is:

E(ri) rf= Cov(ri, rM)2M

[E(rM) rf] (7)

Expected return-beta relationship Ratio Cov(ri, rM)/2M measures contribution of stock i to variance of market portfolio as

fraction of total variance of market portfolio. This ratio is called and:

E(ri) =rf+ [E(rM)

rf] (8)

Assumptions that made individuals act similarly are very useful: If everyone holds an identical risky portfolio, then everyone will find that the beta of each asset

with the market portfolio equals the assets beta with his or her own risky portfolio Hence everyone will agree on the appropriate risk premium for each asset Even if one does not hold precise market portfolio, a well-diversified portfolio is so highly

correlated with market that a stocks relative to market is a useful risk measure Modified CAPMs hold true even if differences among individuals lead them to different portfolios If CAPM relationship holds for any individual asset, it must hold for any combination of assets:

w1E(r1) = w1rf+ w11[E(rM) rf]+w2E(r2) = w2rf+ w22[E(rM) rf]+w2E(r2) = w2rf+ w22[E(rM)

rf]

= +wnE(rn) = wnrf+ wnn[E(rM) rf]

E(rp) = rf+ p[E(rM) rf] whereE(rp) =

wkE(rk) and p=

wkk

This result has to be true for market portfolio itself: E(rM) =rf+ M[E(rM) rf] M= 1 This also establishes 1 as the weighted-average value of beta across all assets s greater than 1 are considered aggressive, s below 1 can be described as defensive

The security market line CAPM relationship as a reward-risk equation: of a security = appropriate measure of its risk

because is proportional to the risk that security contributes to optimal risky portfolio CAPM states that the securitys risk premium is directly proportional to both: (i) The beta and

(ii) The risk premium of the market portfolio

The security market line (SML) graphs individual asset risk premiums as a function of asset risk Relevant measure of risk for individual assets part of diversified portfolios is not assets std

dev./variance. It is the asset contribution of the to portfolio variance (measured by assets ) The SML is valid for both efficient portfolios and individual assets Fairly priced assets plot exactly on the SML If stock perceived to be a good buy (underpriced) it will provide an expected return in excess

of fair return stipulated by SML Underpriced stocks plot above SML: Given their s, theirexpected returns are greater than implied by CAPM. Overpriced stocks plot below the SML

Difference between fair/actually expected rates of return on a stock is called the stocks

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Market

0 1.5

Stock

Slope of SML=fr) {Mr(E

Security Market Line (SML)

0:= 1M

)

Mr(

E

)r(

E

fr

Figure 1: The security market line

The security market line provides a benchmark for the evaluation of investment performance: Giventhe risk of an investment, as measured by its beta, the SML provides the required rate of returnnecessary to compensate investors for both risk as well as the time value of money

In contrast, the capital market line (CML) graphs the risk premiums of efficient portfolios ( i.e.portfolios composed of the market and the risk-free asset) as a function of portfolio

Analysis suggests: Starting point of portfolio management = passive market-index portfolio The portfolio manager will then increase the weights of securities with positive alphas and

decrease the weights of securities with negative alphas The CAPM is also useful in capital budgeting decisions

For a firm considering a new project, the CAPM can provide the required rate of return thatthe project needs to yield, based on its beta, to be acceptable to investors

Yet another use of the CAPM is in utility ratemaking cases In this case the issue is the rate of return that a regulated utility should be allowed to earn

on its investment in plant and equipment. The firm would be allowed to set prices at a levelexpected to generate rate of returns indicated by the CAPM

The CAPM and the index model Actual returns versus expected returns

One central prediction of CAPM: The market portfolio is a M-V efficient portfolio

However, testing its efficiency has not been feasible CAPM involves expected returns. We can only observe actual/realized HPRs The second central set of CAPM predictions is the expected return-beta relationship

Problem of measuring expectations as well We must make additional assumptions to make CAPM implementable and testable

The index model and realized returns To go from expected to realized returns, use the index model in excess return form:

Ri= i+ iRM+ ei (9)

Covariance between the returns on stock i and the market index Firm-specific component independent of market wide component Cov(RM, ei) = 0 Hence, the covariance of the excess rate of return on security i with the market index is:

Cov(Ri, RM) =C ov(iRM+ ei, RM) =i2M

Sensitivity coefficient i in Eq. (9), which is the slope of the index model regression, equals:

i=Cov(Ri, RM)

2M

The index model = in CAPM relationship, except that we replace the CAPM (theoretical)market portfolio with the well-specified/observable market index

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The index model and the expected return-beta relationship If the index Min Eq. (9) represents the true market portfolio, we can take the expectation of each

side of the equation to show that the index model specification is: E(ri) rf=i[E(rm) rf] A comparison of the index model relationship to the CAPM expected return-beta relationship Eq

(8) shows that the CAPM predicts that i should be zero for all assets i is the expected return in excess of (or below) fair expected return predicted by CAPM If the stock is fairly priced, its alpha must be zero

Therefore, if we estimate the index model for several firms, using Eq. (9) as a regression, we shouldfind that ex post/realized s (regression intercepts) center around zero

CAPM states thatE[] = 0 for all securities, whereas index model representation of CAPM holdsthat realized value of should average out to zero for sample of historical observed returns

Indirect evidence on the efficiency of the market portfolio can be found in a study that estimatesvalues for a large sample of equity mutual funds: The distribution ofs is roughly bell shapedwith a mean that is slightly negative but statistically indistinguishable from zero

The market model Other applicable variation on the intuition of the index model, which divides returns into firm-

specific and systematic components somewhat differently from the index model The market model states that the return surprise of any security is proportional to the return

surprise of the market, plus a firm-specific surprise:

ri E(ri) =i[rM E(rM)] + ei If CAPM is valid, substituting for E(ri) from Eq. (8), the market model equation is identical to

the index modelIndex model and market model are used interchangeablyIs the CAPM practical?

If CAPM was valid, single-index model in which index includes all traded securities also valid All alpha values in security risk premiums would be identically zero

Alls= 0 feasible in principle, but cannot be expected to emerge in real markets Such an equilibrium may be one that the real economy can approach, but not necessarily reach

Actions of security analysts drive security prices to proper levels at which = 0 But if all s 0, there would be no incentive to engage in such security analysis

A more reasonable standard, that the CAPM is the best available model to explain rates of return onrisky assets, means that in the absence of security analysis, one should take security alphas as zero

Is the CAPM testable? A model consists of(i) A set of assumptions,(ii) Logical/mathematical development of the model

and (iii) a set of predictions. We can test a model in two ways: Normative and positive Normative tests examine the assumptions of the model Positive tests examine the predictions

Model robust wrt. assumption if predictions not highly sensitive to assumption violation If we use only assumptions to which the model is robust, the models predictions will be

reasonably accurate despite its shortcomings Tests of models are almost always positive: Judge model on success of empirical predictions

Because the nonrealism of CAPM assumptions precludes a normative test, the positive test is reallya test of the robustness of the model to its assumptions The CAPM implications are embedded in two predictions:

1. The market portfolio is efficient2. Security market line (CAPM relationship) accurately describes risk-return trade-off (s= 0)

Central problem in testing predictions: The hypothesized market portfolio is unobservable It is difficult to test the efficiency of an observable portfolio, let alone an unobservable one These problems alone make adequate testing of the model infeasible

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The CAPM fails empirical tests Tests use proxies such as the S&P 500 index to stand in for the true market portfolio. Assumption

that the market proxy is sufficiently close to the true, unobservable market portfolio CAPM fails these tests: Data reject hypothesis that s 0 at acceptable levels of significance E.g., on average, low-beta securities have >0 and high-beta securities have


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2. The expected return of any asset can be expressed as an exact linear function of the expectedreturn on any two efficient-frontier portfolios P and Q:

E(ri) E(rQ) = [E(rP) E(rQ)] Cov(ri, rP) Cov(rP, rQ)2P Cov(rP, rQ)

(10)

3. Every portfolio on the efficient frontier, except for the global M-V portfolio, has a companionportfolio on bottom (inefficient) half of the frontier with which it is uncorrelated

UncorrelatedCompanion portfolio called zero-beta portfolio of efficient portfolio If we choose the market portfolio M and its zero-beta companion portfolio Z, then Eq

(10) simplifies to the CAPM-like equation:

E(ri) E(rZ) = [E(rM) E(rZ)] Cov(ri, rM)2M

=i[E(rM) E(rZ)] (11)

Eq. (11) resembles the SML of the CAPM, except that the risk-free rate is replaced with theexpected return on the zero-beta companion of the market index portfolio

Eq. (11): CAPM equation for investors restricted when borrowing/investing in risk-free asset Because average returns on the zero-beta portfolio are greater than observed T-bill rates, the zero-

beta model can explain why average estimates of alpha values are positive for low-beta securitiesand negative for high-beta securities, contrary to the prediction of the CAPM

Labor income and nontraded assets An important departure from realism is the CAPM assumption that all risky assets are traded.

Two important asset classes that are not traded are:1. Human capital2. Privately held businesses

Privately held business may be the lesser of the two sources of departures from the CAPM Nontraded firms can be incorporated or sold at will, save for liquidity considerations Owners of private business also can borrow against their value

Suppose that privately held business have similar risk characteristics as those of traded assets Individuals can partially offset the diversification problems posed by their nontraded en

trepreneurial assets by reducing their portfolio demand for securities of similar, traded assets

CAPM equation may not be greatly disrupted by presence of entrepreneurial income To the extent that risk characteristics of private enterprises differ from those of traded securities,a portfolio of traded assets that best hedges the risk of typical private business would enjoy excessdemand from the population of private business owners

The price of assets in this portfolio will be bid up relative to the CAPM considerations, andthe expected returns on these securities will be lower in relation to their systematic risk

Adding proprietary income to a standard asset-pricing model improves its predictive performance The size of labor income and its special nature is of greater concern for validity of CAPM

Despite individuals borrowing against labor income (via a home mortgage) and reducing someuncertainty about future labor income via life insurance, human capital is less portable acrosstime and more difficult to hedge using traded securities than nontraded business

This induces pressure on security prices and results in departures from CAPM equation

Equilibrium expected return-beta equation for an economy in which individuals are endowed withlabor income of varying size relative to their nonlabor capital: The resultant SML equation is:

E(Ri) =E(RM)Cov(Ri, RM) +

PHPM

Cov(Ri, RH)

2M+ PHPM

Cov(RM, RH)(12)

Where PH = Value of aggregate human capitalPM = Market value of traded assets (market portfolio)PH = Excess rate of return on aggre

portfolio theory and risk

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