Equilibrium Pricing Utility function · Equilibrium Pricing ... expected value (mean) and variance, is its skewness ... Expected Annual Return Standard Deviation of Return

Copyright © 2006 by Krzysztof Ostaszewski

Equilibrium Pricing Utility function

Risk = future outcome is uncertain, i.e., more than one outcome is possible.

Simple prospect (Bernoulli trial in probability, basically a bet): initial wealthis placed at risk, and there are only two possible outcomes.

Is this a good bet?

The expected wealth at the end of the year is:

The variance of this final wealth is:

and the standard deviation is:

Is this a good bet?

Need a benchmark …

W1,1 = $150,000 with probability p = 0.60

W0 = $100, 000

W1,2 = $80,000 with probability 1 − p = 0.40

E(W) = pW1,1 + 1− p( )W1,2 = 0.60 ⋅150,000 + 0.40 ⋅80,000 = $122,000

2 = p W1,1 − E(W )( )2 + 1− p( ) W1,2 − E(W )( )2 =

0.60 ⋅ 150,000 −122,000( )2 + 0.40 ⋅ 80,000−122,000( )2 =1,176,000,000

= 34,292.86


Suppose that the risk-free interest rate is 5%, then you have this decisiontree:

Initial wealth = $100,000

Possible actions:

- Invest in risky prospectProfit = $50,000 with probability p = 0.60Profit = -$20,000 with probability 1 – p = 0.40

- Invest in risk-free Treasury BillProfit = $5,000 with certainty

Expected profit in the risky prospect equals $22,000, and this represents$17,000 risk premium over the risk-free investment.

Risk premium = compensation for assuming the risk of an investment, itequals the difference between the investment’s expected return and the risk-free rate. It is a prospective concept. Retroactively it is called excess return.

An investor is called risk averse if he/she is only willing to assume risk,when compensated for it (i.e., when he/she can reasonably expect a riskpremium).

Speculation = assumption of considerable business risk with reasonableprobability of obtaining commensurate gain. Risk aversion and speculationare not inconsistent.

Gamble is the assumption of risk for no purpose but enjoyment of risk itself.


Risk neutral investorValues risky and risk-free prospects with the same expected outcomeequally.

Risk averse investorAlways prefers risk-free prospect to a risky prospect with the sameexpected outcome (i.e., requires risk premium to enter into aspeculative investment).

Risk loverAlways prefers risky prospect to a risk-free prospect with the sameexpected outcome.

Why would people have such differing attitudes towards risk? Because theydo not necessarily value final wealth, or profits, the same way. They do notattach the same value to money. We all know that your second milliondollars is much easier to make than the first one. You will see that once youpass all actuarial exams.

Utility function is the “scoring” system for valuing wealth, or investmentreturns. One example of a utility function is:

If A = 0, this investor is risk neutral. If A is negative, the investor is riskloving. If A is positive, this investor is risk averse, and larger A indicatelarger degree of risk aversion.

Certainty equivalent: risk-free rate considered equally attractive by aninvestor as the risky portfolio. For example, if A = 3, and

then the utility value is:

22% − 0.005 ⋅3⋅ 34%( )2 = 4.66%,

which is the certainty equivalent.

U r,( ) = E(r) − 0.005A 2

E(r) = 0.22 = 22%, = 0.34 = 34%,


The utility function presented above has the property that all of investor’spreferences about an available investment are summarized in two parametersof the uncertain investment return: mean and variance. In general, we saythat one uses the mean-variance criterion for comparing availableinvestments if one says that A is better than B when E r

A( ) ≥ E r

B( ) and

A≤

B. An investor who has the utility function given above uses the

mean-variance criterion.

Note that measuring risk is a challenge, because individual investors mayknow how much risk they can afford (this can be calculated), but they reallydo not know prospectively how much risk they can tolerate emotionally andpsychologically.

All portfolios of investments with the same certainty equivalent, if plotted inthe r -coordinate system, form a curve, called indifference curve, becausethe investor is indifferent between portfolios along the curve. All thoseportfolios have the same utility value.

Return, E(r)

Risk, σ

IndifferenceCurve


Portfolio Risk

Note that the expected return of a portfolio of investments is always equal tothe weighted expected rate of return of its components, with weights beingexactly the same as component weights.

Consider the following stock of Best Candy:

Normal year for sugar Abnormal year

Bullish stockmarket

Bearish stockmarket Sugar Crisis

Probability 0.50 0.30 0.20

Return 25% 10% -25%

Then

and

If this stock is combined with Treasury Bills yielding 5%, in an equallyweighed portfolio of each, then

But the variance of the portfolio is not as simple as this. In general, thevariance of the portfolio whose portion w1 is invested in asset 1, and portionw2 is invested in Asset 2, equals:

E(rBC ) =10.50%

2BC = 357.25, = 357.25 =18.9%

E(rPortfolio ) = 0.50 ⋅10.50% + 0.50 ⋅5% = 7.575%


where the last tern is the covariance of the two investments. Here is the wayto calculate covariance (defined here for discrete distributions):

Important: The risk-free investment (Treasury Bill) has zero variance andzero covariance with any risky investment. Therefore for the portfolio wejust analyzed,

Now consider the following stock of Sugar Kane company:

Normal year for sugar Abnormal year

Bullish stockmarket

Bearish stockmarket Sugar Crisis

Probability 0.50 0.30 0.20

Return 1% -5%% 35%

Then

But if we consider the portfolio of 50% Best Candy and 50% Sugar Kane:

1,2 = p(s) E(r1 ) − r1(s)( )s∈S∑ E(r2 ) − r2 (s)( )

2Portfolio = w1

2 21 + w2

2 22 + 2w1w2 1 , 2

Portfolio = 0.50 ⋅18.90% = 9.45%

E(rSK ) = 6%, SK =14.73%

E(rPortfolio ) = 8.25%, Portfolio = 4.53%


Here is the summary of these portfolios:

Portfolio Expected Return Standard Deviation

100% Best Candy 10.50% 18.90%

50% Best Candy50% T-Bill 7.575% 9.45%

50% Best Candy50% Sugar Kane 8.25% 4.83%

This is happening because:

Negative covariance means that these assets’ returns tend to move inopposite directions. Best mathematical measure of such a relationship is thecorrelation coefficient, defined as:

In this case, the correlation coefficient of Best Candy and Sugar Kane equals(-240.50)/(18.9 X 14.73)= - 0.83. Correlation coefficient varies between –1and 1, with –1 indicating “perfect” negative correlation, +1 indicating“perfect” positive correlation, and 0 indicating no correlation. Note that therisk-free investments has zero correlation with all other assets.

Cov rBC,rSK( ) = 0.50 25% −10.50%( ) 1% − 6%( ) +0.30 10% − 10.50%( ) −5% − 6%( ) + 0.20 −25% −10.50%( ) 35% − 6%( )= −240.50

1,2 = 1,2

1 ⋅ 2


One more important measure of probability distributions, in addition to theexpected value (mean) and variance, is its skewness (defined here fordiscrete distributions):

Positive skewness (graph of a PDF in the case of a continuous distribution):

Normal distribution is fully defined by its expected value (mean) andvariance. If we assume that returns are normal, then we can describe themwith mean return and variance of returns. This is also true if we assume thatthe continuously compounded annual rate of return is normal (then theeffective annual rate of return has lognormal distribution).

M 3 = p(s) r(s) − E(r)( )s∈S∑ 3

Probability

Returns


Also note the shape of the utility function of wealth:

Wealth

Utility of wealthRisk Neutral

Risk Averse

Risk Lover


Appendix A: A Defence of Mean-Variance Analysis

The structure of an investment’s uncertain returns is described by theprobability distribution of the rate of return random variable. A measure ofthe rate of return is its mean, or median, or mode, but the mean is used mostoften. There may be many ways to characterize risk:- Mean Absolute Deviation (MAD):

MAD = Pr s( )s=1

n

∑ r s( ) − E r( ) (for a discrete probability distribution)

- Variance- Standard deviation- Skewness (has some good sides: preservation of signs distinguishes good

from bad deviations, the third power gives greater weight to largerdeviations, and risk-averse investors do exhibit preference for positiveskewness).

In general:- First moment is a measure of the payoff,- Higher moments measures the uncertainty of the payoff. Even moments

measure the probability of extreme outcomes, and odd moments measureasymmetry of the payoff distribution.

Utility can be approximated by the expansion:U = E r( ) − b

0

2 + b1M

3− b

2M

4+ b

3M

5− ...

Higher moments terms tend to be of lesser significance. In practice, mostanalysis ignores moments other than the mean and the variance. But, asSamuelson pointed out, variance is as important as the mean.

Continuity of stock prices produces a continuous distribution of returns.Risk actually approaches zero as the holding period approaches zero (if youcan sell instantly at the current price, you do not have any instantaneousrisk). Investors can control risk by rebalancing their portfolios (this islimited by transaction costs). Rebalancing makes higher momentsunimportant. This is the argument presented by Paul Samuelson (1970). Aportfolio is considered compact if its risk approaches zero as length ofholding period approaches zero. Samuelson argued that rebalancing impliesthat portfolios can be made compact, and this implies:- All higher moments beyond variance have lesser significance- Variance is as important as returns


Appendix B: Risk aversion, expected utility, and the St. Petersburg Paradox

Consider a game where a coin is tossed repeatedly until the first headsappears and the player is paid R n( ) = 2n upon such appearance, with ndenoting the toss number when the heads appears. Then the expected payoffis infinite. Yet participants willing to pay a finite entry fee to pay it.Bernoulli resolved this St. Petersburg Paradox by pointing out what we nowcall diminishing marginal utility of extra payoff dollars. Thus, a concavefunction should be used to describe utility function.

In 1946 Von Neumann and Morgenstern used these ideas in game theory. Ifthe utility function is concave, in a fair game, loss in utility from losing Xexceeds the gain utility from winning the same amount X. Expected utilityfrom a fair game is less than the utility of not playing the game (for a risk-averse investor, of course, but this is what a concave utility function means= risk aversion). History of financial markets shows risk-aversion common.

In general, penalty for risk can be evaluated as:Penalty = Expected Profit – Certainty Equivalent


Exercise 1.A Treasury Bill pays 6 percent rate of return. Would a risk-averse investorinvest in a risky portfolio that pays 12 percent with a probability of 40percent or 2 percent with a probability of 60 percent?

Solution.The expected return on the risky asset is 0.40(12%) + 0.60(2%) = 6.00%.This is the same return as that of the T-Bill. But the investor is risk-averse,and he/she will prefer the T-Bill.


Exercise 2.You are given the following information:

Effective Annual ReturnProbability Stock ABC Stock DEF T-Bills

Scenario 1 0.60 25% 5% 4%Scenario 2 0.40 5% 15% 4%

Mix of InvestmentsStock ABC Stock DEF T-Bills

Portfolio I 100% 0% 0%Portfolio II 75% 0% 25%Portfolio III 75% 25% 0%

Calculate the expected rate of return and the standard deviation of eachportfolio.

Solution.

Portfolio IExpected return = 0.60(25%) + 0.40(5%) = 17%.Second moment = 0.60(0.252) + 0.40(0.052) = 0.0385.Variance = 0.096. Standard deviation = 0.097976 = 9.80%

Portfolio IIReturn in scenario 1: 19.75%, return in scenario 2: 4.75%Expected return = 0.60(19.75%) + 0.40(4.75%) = 13.75%Second moment = 0.60(0.19752) + 0.40(0.04752) = 0.02430625Variance = 0.054. Standard deviation = 0.07348469 = 7.35%

Portfolio IIIReturn in scenario 1: 20%, return in scenario 2: 7.50%Expected return = 0.60(20%) + 0.40(7.50%) = 15.00%Second moment = 0.60(0.202) + 0.40(0.0752) = 0.02625Variance = 0.00375. Standard deviation = 0.06123724 = 6.12%


Exercise 3.A portfolio has an expected rate of return of 15% and a standard deviation of15% The risk-free rate is 6 percent. An investor has the following utilityfunction: U = E(r) – 0.0001(A/2)σ2. Which value of A makes this investor

indifferent between the risky portfolio and the risk-free asset? For thisproblem, whole numbers for percentages are used, i.e., expected return of9% is written as 9, and standard deviation of 10% is written as 10.

Solution.The portfolio’s certainty (i.e., risk-free) equivalent is 15% -0.0001(A/2)225% = 15% - 1.125A and it should be 6%, so 9% = 1.125A,thus A = 8.


Society of Actuaries May 2003 Course 6 Examination, Problem No. A-2(5 points)You are given the following information for Stock X:

Probability

0.60

0.20

0.20

One -period rate of return

10%

5%

-10%

For Stock Y we have:

Probability

0.75

0.25

One -period rate of return

20%

-20%

The risk-free rate is 4%, the investor has a one-year horizon and the investoris indifferent between investing in Stock Y and earning the risk-free rate.Determine whether or not the investor would purchase Stock X.

Solution.The expected rate of return over one period for Stock X is:

0.60 ⋅10% +0.20⋅5 % +0.20 ⋅(-10%) = 6 % + 1 %- 2 %= 5%.The expected rate of return over one period for Stock Y is:

0.75 ⋅20%+0.25 ⋅(-20%) =15%- 5 %=10%.Thus the investor requires 6% risk premium for investing in Stock Y versusthe risk-free rate. If the required risk premium for Stock X is the same as forStock Y, the investor will clearly not invest in Stock X. But it might makesense to compare the risk of Stock X and the risk of Stock Y. The varianceof the one period rate of return of Stock X is:

0.60 ⋅ 10% - 5 %( )2 + 0.20 ⋅ 5% - 5 %( )2 + 0.20 ⋅(-10%-5%)2 =

= 1 5%( )2

+ 0 %( )2

+ 45 %( )2

= 60 %( )2

.Its standard deviation is approximately 7.75%. On the other hand, thevariance of the one period rate of return of Stock Y is:

0.75 ⋅ 20%-10%( )2 + 0.25 ⋅(-20%-10%)2 =

= 75 %( )2

+ 225 %( )2

= 300 %( )2

.


The standard deviation of the rate of return is 17.32%. Thus Stock X offers1% risk premium for 7.75% standard deviation, and Stock Y offers 6% riskpremium for 17.32% standard deviation. This again makes it unlikely thatthe investor will invest in Stock X. We can also address this issue by a moredetailed analysis. If we assume that the utility function of the investor is ofthe form:

U r,( ) = E(r) − 0.005A 2

where for Stock Y we have E r( ) = 10%, 2 = 300 %( )2, and we also know

that its utility is the same as that of a risk-free investment, then we have:

4% = 10% − 0.005 ⋅ A ⋅300 %( )2 ⇒ A =6%

0.005 ⋅300 %( )2 = 4 %( )−1.

Given that, the utility of Stock X is:5% − 0.005 ⋅4 %( )−1 ⋅60 %( )2 = 5% −1.2% = 3.8%.

Clearly this is less than the utility of the risk-free investment, and theinvestor will not invest in Stock X.


May 2003 SOA Course 6 Examination, Problem No. B-2You are given the following with respect to a portfolio consisting of twomutual funds:

Mutual Fund Weight Variance

Stock 25% 100

Bond 75% 36

The correlation coefficient between the fund returns is –0.5. Calculate thevariance of the portfolio.

A. 3.91 B. 15.25 C. 26.50 D. 37.25 E. 40.75

Solution.Let R

S be the random rate of return of the stock fund, R

B be random rate of

return the bond fund, and RP be the portfolio rate of return. Then

RP

= 0.25RS+ 0.75R

B,

andVar R

P( ) = 0.252 Var RS( ) + 0.752 Var R

B( ) + + 2⋅0.25 ⋅0.75 ⋅Cov R

S,R

B( ) = = 0.0625 ⋅100 +0.5625⋅36 +

+2*0.25*0.75*(-0.5)*10∗6=15.25.

Answer B.


Capital allocation between the risky asset and the risk-free asset

Risky asset can be any risky asset, i.e., asset offering a range of futurepossible returns.

Risk-free asset is represented by Treasury Bills.

We will study the relationship of the expected rate of return of a portfoliocomposed of the two choices to the standard deviation of such a portfolio.We will graph and study their relationship in a two-dimensional coordinatesystem, with the x-axis given to standard deviation and the y-axis given toexpected return. What does this mean?- Investor chooses a level of risk she is comfortable with;- The graph gives the expected rate of return available for this level of risk.

We will assume that investor’s preferences concerning investments are fullydescribed by expected return and standard deviation ( E(r)and ), and thatwe can describe investor’s utility with the formula:

U r,( ) = E(r) − 0.005A 2

We first derive the set of risk/return graphical representations for a portfolioof a generic risky asset and a risk-free asset. The opportunity set so createdis called the Capital Allocation Line.

If we replace the generic risky asset by a unit holding in the overall market,the resulting opportunity set is called Capital Market Line. This is illustratedin the later part.

Note the following terminology:Capital allocation decision: allocation of invested funds between low-risk(or risk-free), lower yield, assets and high-risk assets, with higher expectedreturns.Asset allocation decision: decision about allocation of risky investmentsacross asset classes.Security selection decision: decision as to which individual securities shouldbe purchased in each asset class.


Portfolios of one risky asset and one risk-free asset

Using the following notation:r

C- return of the complete portfolio,

rF - risk-free rate

y - fraction of the portfolio invested in the risky asset Pand with standard notation for the expected return and the standarddeviation, we arrive at the following parameters of the portfolio:Expected return: E r

C( ) = r

F+ y E r

P( ) − r

F( )Standard deviation:

C= y

P

Capital Allocation Line:

Equation of the line Ca E r

C( ) is:

E rC

( ) = rF

+ C

P

E rP

( ) − rF( )

P

E rC

( )

rF


Reward-to-variability ratio = slope of the CAL

S =E r

P( ) − r

F

P

Portfolios on the CAL to the left of P are weighted averages of P and thersik-free asset, while the portfolios to the right of P are created by borrowingat the risk-free rate and leveraging the position in P.

Risk tolerance and asset-allocationInvestor wants to maximize utility while taking a portfolio lying on theCAL. Utility maximization problem is:Maximize U = E r

C( ) − 0.005A

C

2 = rF

+ y E rP

( ) − rF( ) − 0.005Ay 2

P

2

The solution gives the optimal allocation to the risky asset:

y* =E r

P( ) − r

F

0.01AP

2 .

Graphically, one can present the solution by comparing the CAL line withindifference curves, the point of tangency of the two is optimal:

E r( ) Moredesirableportfolios


Note that higher indifference curves represent higher utility, and the investoris trying to reach the point on the highest possible indifference curve, whilestaying on the CAL line. The shape of indifference curves depends on risk-preferences of the investor, more risk-averse investors have steeperindifference curves, which result in optimal asset allocation with greaterrisk-free position.

Passive strategies: The Capital Market Line

Types of investment strategies:- Active management: actively vary allocations in order to “beat the

market”, i.e., achieve portfolio returns more than commensurate withrisk.

- Passive management: buying a well-diversified portfolio matching theoverall market in performance, or matching a prescribed market index.This approach does not try to time the market, or search mis-pricedsecurities.

Reasons why one might want to pursue a passive strategy:- This is a low-cost alternative;- Get a free ride: Benefit from market efficiency, i.e., from research and

analysis done by others;

Capital Market Line: capital allocation line created with the use of aTreasury Bill and a broad index of common stocks.

Criticisms of index funds:- Most commonly used index: S&P 500 is too heavily concentrated in

large companies and technology;- Too heavily concentrated in big-company stocks, of course one can use

another index (e.g., Wilshire 5000) to include small-company stock;- As “hot” stocks climb in value, index funds automatically buy them, and

as stocks drop, index funds sell them, thus index funds are too inclined tofollow trends;

- Many claim index can be beaten, but history refutes them (to my bestknowledge, there is only one fund in the U.S. that has beaten S&P 500every year over the last ten years).


Note: In Bodie/Kane/Marcus book "Investments"mean and volatility are writtenas whole numbers not fractions, when they express percentages, e.g., 20% iswritten as 20 in the utility formula, instead of 0.20. In the above formula, we writeit as 0.20. It is simply a unit adjustment. Use fractions for this problem.

Exercise 1.It is common to use the following utility function for modeling utility of aninvestment:

U ,( ) = −12

A 2 ,

where is the expected rate of return of an investment, and is thestandard deviation of that rate of return. A is an index or investor’s riskaversion. You are given that a portfolio has an expected rate of return of20%, and a standard deviation of that rate of return of 20%. The risk-freerate is 7%. Which investment alternative (risky or risk-free) will be chosenby an investor whose A = 4?

Solution.Note that since the utility of a risk-free investment is exactly its rate ofreturn, this utility function gives exactly the certainty-equivalent of anyinvestment, and compares various investment alternatives in terms of theircertainty equivalents.Here we have, for the risky investment:

U ,( ) = 0.20 − 0.5 ⋅4 ⋅0.202 = 0.12.

Since this is more that the risk-free rate of 7%, the risky investment ispreferable.


May 2003 SOA Course 6 Examination, Problem No. B-1You are given the following with respect to an optimal risky portfolio:expected return is 7%, risk premium is 3%, and the variance is 100.Calculate the slope of the capital allocation line (CAL) for this portfolio.

A. 0.03 B. 0.04 C. 0.30 D. 0.40 E. 3.00

Solution.The slope of the CAL line is the ratio of the risk premium to the standarddeviation. The risk premium is 3%, and the standard deviation is 10 (also %,even though the problem does not say that), so that the slope is 0.3.

Answer C.


Optimal Risky Portfolios

Can you diversify all risk by using the insurance principle, i.e., the law oflarge numbers?

You can diversify away the nonsystematic risk, which is firm specific, butnot systematic risk, which is by its very nature common to all marketparticipants.

How is the risk reduction in a portfolio achieved? This is the consequence ofthe formula for the variance of returns of a portfolio of two assets.

2Portfolio = w

1

2 21 + w

2

2 22 + 2w

1w

2 1,2

Number ofSecurities

St. Deviation

Market Risk

Unique Risk


In general

Var wiR

ii=1

n

∑( ) = wi

2Vari =1

n

∑ Ri

( ) + 2wi

i < j∑ w

jCov R

i,R

j( )This means that if you combine two risky assets the variance of returns oftheir combination can vary depending on the weights in the two assets anddepending of the relationship of returns of the assets. If you have two riskyassets, this is illustrated below:

Thick Line = Portfolio Opportunity Set. Minimum variance portfolio alwaysto the left of both assets as long as correlation is less that the ratio of twovariances.• Relationship depends on correlation coefficient -1.0 < ρ < +1.0

• The smaller the correlation, the greater the risk reduction potential• If r +1.0, no risk reduction is possible• If there is perfect negative correlation, assets can be combined to create aportfolio with zero variance, by having the following weights:

w1= 2

1+

2

,w2

= 1

1+

2

= 1− w1.

Each asset then serves as a hedge for the other one. Hedge is an asset thatcombined with a given one eliminates or at least lowers the overall portfoliorisk.

ρ = 1

13%

8%

E(r)

St. Dev12% 20%

ρ =

0.3

ρ = -1

ρ = -1


Minimum variance portfolio: the problem is to minimize

Min w1

2 21 + w

2

2 22 + 2w

1w

2 1 2,

subject to the sum of weights being one, and given assets standardsdeviations and correlation coefficient. There is always a solution, given bythe formula:

w1=

22 −

1 2

21 + 2

2 − 21 2

To determine portfolio with highest utility for a given level of risk, investorssolve the following optimization problem:

Max E rP

( ) − 0.005AP

2 .and this results in allocations:

w1= E r

1( ) − E r

2( ) + 0.01A

2

2 −1 2

( )0.01A

1

2 +2

2 − 21 2( ) ,

w2

= E r2

( ) − E r1

( ) + 0.01A1

2 −1 2

( )0.01A

1

2 +2

2 − 21 2( ) = 1− w

1.

Optimal risky portfolio with two risky assets and a risk-free asset

Reward to variability ratio: SP

=E r

P( ) − r

F

P

, it is the slope of the Capital

Allocation Line. Start the process of optimization with the CAL connectingthe risk-free asset and the minimum variance portfolio. Then look for aportfolio with the highest possible value of S

P. If there are only two risky

assets, the optimal weights are:

w1=

E r1( ) − r

F( ) 2

2 − E r2( ) − r

F( ) 1 2( )E r

1( ) − rF( ) 2

2 + E r2( ) − r

F( ) 1

2 − E r1( ) − r

F+ E r

2( ) − rF( ) 1 2( ) ,

w2

=1− w1.

Optimal risky portfolio P – best combination of risky assets to be latercombined with the risk-free asset to form a complete portfolio. Usinginvestor’s risk preferences one can arrive at the familiar formula for the

allocation to the risky portion: y∗ =E r

P( ) − r

F

0.01AP

2 (rest should be invested in

the risk-free asset).


Extending concepts to all securitiesPortfolio opportunity set: all combinations of expected return and standarddeviation that can be constructed from a given set of assets.

- Optimal combinations result in highest expected return for a givenlevel of risk

- The optimal set of portfolios is called the efficient frontier- The portfolios on the efficient frontier are dominant- The entire market portfolio lies on the efficient frontier, and

security with the highest available return lies on the efficientfrontier

- Two funds theorem: There exist two assets which generate theefficient frontier

Extending to include riskless asset:- The optimal combination becomes linear- A single combination of riskless and risky asset will dominate

E(r)

Efficientfrontier

Globalminimum

varianceportfolio

Minimumvariancefrontier

Individualassets

St. Dev.


CAL(P) produces an optimal solution among various possible CapitalAllocation Lines, as it has the highest slope.

But this does not take into consideration investors’ risk preferences. In orderto incorporate those, we would need to maximize utility given the efficientfrontier. Next graphs shows the result.

M

E(r)

CAL (Globalminimum variance)

CAL (A)CAL (P)

P

A

F

P P&F A&FM

A

G

P

M


If lending and borrowing happen at different interest rates, the efficientfrontier is changed, as in the next figure:

E(r)

Efficientfrontier ofrisky assets

Morerisk-averseinvestor

U’ U U

Q

PS

St. Dev

Lessrisk-averseinvestor

E(r)

Frf

A

P

Q

B

CAL

St. Dev


Steps in construction of the efficient frontier of risky assets:Input list:

- Expected return for each security;- Variance for each security;- Covariance for each pair of securities

Number of inputs needed for n securities: n + n + 12

n n −1( ). Variances and

covariances usually are estimated using historical data, but one can also usescenario analysis. Using that data, one can determine the expected return andvariance for any risky portfolio with weights w

i in security i, i = 1,...,n.

Then using optimization one can identify the efficient frontier. Note thatExcel’s Solver will handle this easily.

Additional constraints may be incorporated into the model, such as:- No short positions;- Ensure the minimum level of dividend yield;- Socially responsible investing.

Note that we are able to separate portfolio selection into two independenttasks: determination of the optimal risky portfolio, and determination of thecomplete portfolio. Determination of the optimal risky portfolio (efficientfrontier) is purely technical, the choice of the complete portfolio depends onthe level of risk aversion. This is called the Separation Property.

Why would we distinguish between asset allocation and security selection?- Increased demand for sophisticated investment management- More complex spectrum of financial instruments- Economies of scale in investment analysis


Appendix A: The power of diversification

If one considers an equally-weighted portfolio with weight wi=

1n

in each

security then the portfolio variance is: P

2 =1n

2 +n −1

nCov (the bar, as

always in statistics, refers to the average sample value). The first term of thisexpression represents the firm-specific risk and it approaches zero as n tendsto infinity. The second term represents systematic risk, and it cannot beeliminated through diversification.

If the securities in the portfolio were uncorrelated, portfolio variancecould be made as small as we wish through diversification. If the securitieswere perfectly correlated, diversification would not offer any benefits. Thehigher the correlation coefficient, the more limited the power ofdiversification becomes. For an individual security its contribution toportfolio risk depends mainly on its covariance with other securities in theportfolio, NOT on its individual variance (unless the number of securities inthe portfolio is very small or the portfolio is concentrated in a small numberof issues).


Apendix B: The insurance principle: Risk-sharing versus risk-pooling

In insurance one assumes that a large number of independent policies isneeded, and it is indeed sufficient, for insurers to eliminate risk. Thereasoning is based on viewing the business as a series of independent bets ofroughly same size, with same mean and variance. Standard deviation of thesample mean declines with the square root of n, the number or bets.

But consider a general situation when you accumulate a series ofindependent risky prospects. This is called in general risk pooling. Focus ondollar profit, not the rate of return or average return. Expected total profit isE r( ) = nP , where P is the expected profit from a single bet. The standard

deviation of the dollar profit is R

n( ) =R

n , where R is the standard

deviation of the dollar profit of a single bet.. Of course, this standarddeviation increases with the square root of n, the number or bets.

We only should use the average rate of return when evaluating mutuallyexclusive portfolios of equal size. This is analogous to the use of NPVinstead of IRR when choosing between different size projects. Correctanalysis of risk involves comparing portfolios of the same size notincreasing the size of the portfolio. For example, it would be useful toreplace one bet by a small share of a diversified portfolio of many large bets.

Risk sharing, not risk pooling, is what makes the insurance industry work.What happens is that individual investors in insurance companies can holdsmall stakes, and as long as the expected return is acceptable to them (mustbe greater that r

F), investors will bear the risk.


Appendix C: The fallacy of time diversification

It is a common misconception that time diversification reduces risk. Thereasoning assumes that:

- Rates of return over successive periods are independent;- Good and bad periods cancel each other out.

But overall uncertainty increases as the holding period length increases.Total return becomes more uncertain, even though the annual average rate ofreturn has smaller volatility estimate. Note that in the market the price of aput option with an exercise price equal to the underlying’s current pricecompounded by the annual return desired increases with investment horizon.

This is not in the book, but may help in understanding the issue:A very relevant article on the subject of time diversification is: “WhatPractitioners Need to Know About Time Diversification,” Mark Kritzman,Financial Analysts Journal, AIMR, January/February 1994. Here is theoutline of its ideas:

It is commonly assumed that investors with longer horizons should allocate alarger fraction of their savings to risky assets than investors with shorterhorizons. The rationale is that if our assets are invested in the risky assetsuch as stocks, over a short horizon we could lose a substantial part of oursavings. Over a long horizon, however, favorable short-term stock returnsare likely to offset poor short-term returns, so in that case it is more likelythat stocks will realize an actual return close to their expected return.

• The notion that above-average returns tend to offset below-average returnsover long horizons is called time diversification. Indeed, if returns inconsecutive periods follow the same probability distribution and areindependent of each other, Central Limit Theorem implies that cumulativereturn over longer period of time has standard deviation which, in relation toits mean, declines by the same order as the square root of time.

• But ... the same standard deviation, while it declines with respect to themean, in absolute terms it increases by the same order as the square root oftime. Therefore, while you may be less likely to lose money over a long timehorizon, the magnitude of your potential losses increases with time.

• But ... the expected value of your cumulative return also increases withtime, in fact directly with time, not with the square root of time. What


happens, however, is that at the same time lower boundary of the 99%confidence (or most other confidence intervals for the mean) interval looksless and less appealing.

Assumptions required to demonstrate mathematically that risk tolerance isindependent of investment horizon

• Your risk aversion is invariant to changes in your wealth;• You believe that risky returns are random;• Your future wealth depends only on investment returns.

But• People often tend to be risk-averse: satisfaction they derive from increasesof wealth is not linearly related to increases in your wealth (your first millionis much more fun than the second one! – since you will be actuaries, youwill experience that). Note that even risk-averse investors, e.g., with logutility function, may, mathematically, have risk tolerance independent oftime horizon.• You may not believe that risky returns are random, you may think they aremean-reverting (people who grew up under Capitalism tend to believe inmean-reversion, people who grew up under socialism are not always so sure,they always worry there might be a commie with a Kalashnikov around thecorner). If your believe in mean reversion, you should take more risk overlong term.• Over long time horizon, you also have a chance to adjust your work andconsumption patterns. Thus, your terminal wealth may depend on yourhuman capital, and your consumption.• You may also be extremely vulnerable to your wealth dropping below acertain level (poverty), and unwilling to accept any chance of this happening(and this has higher probability over a longer time period if you assumerisky investments).• Your utility function may be discontinuous (e.g., you become a millionaire,you start viewing the world differently).

Society of Actuaries Course 6 examination, May 2001, Problem No. 27You are given the following data:

Expected Annual Return Standard Deviation of Return

Stock I 10% 25%Stock II 35% 60%

The correlation coefficient between Stock I and Stock II is –0.2. Annualrisk-free rate is 5%. Calculate the difference between the weights of StockI and Stock II in the optimal portfolio consisting of only these two stocks,but allowing for inclusion of the risk-free security in the optimalitycalculation. Assume that you can borrow and lend at the risk-free rate, andthe optimal portfolio can theoretically include the risk-free security, but youchoose that one among optimal portfolios, which does not contain anyallocation to the risk-free security, neither short nor long.

Solution.Consider a portfolio P with portion w in Stock I and 1 – w in Stock II. LetRI be the random rate of return on Stock I, RII be the random rate of returnon Stock II, and

RP = wRI + 1 w( )RIIbe the rate of return of the portfolio. Then the portfolio expected return is

E RP( ) = 0.10w + 0.35 1 w( ) = 0.35 0.25w,

and its variance is:

Var RP( ) = w2 0.252 + 1 w( )20.62 + 2 w 1 w( ) 0.2( ) 0.25 0.6 =

= 0.0625w2+ 0.36 0.72w + 0.36w2 0.06w + 0.06w2

=

= 0.4825w2 0.78w + 0.36.Portfolios made out of the risk-free security and any of the P portfolios lie inthe -µ coordinate system on a straight line connecting the µ -intercept atthe risk-free rate rF = 5% and a point on the curve created by all Pportfolios. The specific portfolio we are looking for, denoted by P*,corresponds to the point of tangency of the line and the curve. This isillustrated in the graph below. The portfolio we are looking for is the lineconnecting the risk-free rate on the producing the largest possible slope. Wewill use that maximality of the slope to find the portfolio P *. Let us denotethe weight w invested in Stock I in portfolio P * by w *.

The slope of any line connecting the risk free asset and a portfolio Pcomposed of a portion w invested in Stock I and portion 1 w invested inStock II is:

E RP( ) rF

P

=0.35 0.25w 0.05

0.4825w2 0.78w + 0.36=

=0.30 0.25w

0.4825w2 0.78w + 0.36.

We want this quantity maximized. We take its derivative with respect to wand obtain

d

dw

0.30 0.25w

0.4825w2 0.78w + 0.36=

=0.25 0.4825w2 0.78w + 0.36

0.4825w2 0.78w + 0.36

0.30 0.25w( ) 0.4825w 0.39( )

0.4825w2 0.78w + 0.36( )3

=

=0.027 0.04725w

0.4825w2 0.78w + 0.36( )3

.

This is equal to zero when0.027 0.04725w,

or

I

E RI( )

rF

II

E RII( )

I

P *II

µ

w = w* =0.027

0.047250.57142857.

We know that this corresponds to a maximum from the graph illustrating theproblem. This means that the portfolio P * should be invested approximately57.14% in Stock I and the rest, approximately 42.86% in Stock II. Thedifference of the two is

57.14% 42.86% = 14.28%.This problem can also be solved by the formula below (we use the following

notation:Var RI( ) = I2 ,Var RII( ) = II

2 , =Cov RI ,RII( )

Var RI( ) Var RII( ) in it)

w* =

=E RI( ) rF( ) II

2 E RII( ) rF( ) I II

E RI( ) rF( ) II2+ E RII( ) rF( ) I

2 E RI( ) rF + E RII( ) rF( ) I II

.

In this case we have

w* =0.05 0.36 0.30 0.2( ) 0.25 0.6

0.05 0.36 + 0.30 0.0625 0.35 0.2( ) 0.25 0.6

=0.018 + 0.009

0.018 + 0.01875 + 0.0105=0.027

0.047250.5714286.

This is, of course, exactly the same answer.

NEAS Spring 2003 Notes for CAS Exam 8, Copyright © 2003 by Krzysztof Ostaszewski - 135 -

dP

dP

= dP

/ dwd

P/ dw

= −0.25P

0.4825w− 0.39= P

− rF

P

= 0.30 − 0.25w

P

.

Consequently,−0.25 0.4825w 2 − 0.78w +0.36( ) = 0.30 − 0.25w( ) 0.4825w − 0.39( ),and this quadratic equation solves to w = 0.57142857. The question asks forw – (1 – w) = 0.57142857 – 0.42857143 = 0.14285714.

Exercise 2.Consider two perfectly negatively correlated risky securities A and B. A hasan expected rate of return of 10% and a standard deviation of 16%. B has anexpected rate of return of 8% and a standard deviation of 12%. What will bethe rate of return of the risk-free portfolio that can be formed with the twosecurities?

Solution.Minimum variance portfolio created out of these two securities will havezero variance. Its weight in the first asset will be:

w1=

22 −

1 2

21 + 2

2 − 21 2

.

This gives

w1=

0.122 + 0.16 ⋅0.120.162 + 0.122 + 2 ⋅0.16 ⋅0.12

= 0.4286,w2

= 0.5714.

The return of the minimum variance portfolio will be:0.4286 ⋅10% + 0.5714 ⋅8% = 8.86%.

The result is, of course, the risk-free rate of return.


The Capital Asset Pricing Model

Most important equilibrium model of capital markets. William Sharpe, oneof its creators, received Nobel Prize in Economics for it in 1990. You canlook up what William Sharpe does now at http://www.financialengines.com.

Demand for stocks and equilibrium price- Mutual funds’ demand for shares

• Determining how much to buy of each stock: Plot data in/ E r( ) plane to generate efficient frontier, draw the Capital

Asset Line and find the point of tangency, it determines theoptimal portfolio. Then determine buy orders for each stockbased on its amount in the optimal portfolio.

• Demand schedule for a stock is the number of shares a fundwould want to hold at each price. Its derivation assumes thatprice and returns of other stocks remain the same. Like mostdemand functions, it slopes downward because of income effect(at lower price, fund can buy more shares with same budget)and substitution effect (lower price raises expected return,making this stock more attractive relative to other ones). By theway, for stocks, theoretically, negative demand is possible (viaa short position).

- Index funds’ demand for stocks• Demand curve is inelastic (steep), as index funds must buy

regardless of price.- Equilibrium price is determined by supply (vertical, in the short run)

and demand (horizontal aggregation of individual demand curves). Ina competitive market, each stock’s demand curve will reflect actualprices of all other stocks, and demand by one entity for a stock willnot have much effect in the stock’s price (everyone is a price-taker).


Assumptions of CAPM:- Individual investors are price-takers- Single-period investment horizon, same for everyone- Investments are limited to traded financial assets (may also borrow or

lend at the risk-free rate)- No taxes, and transaction costs- Information is costless and available to all investors- Investors are rational mean-variance optimizers- Homogeneous expectations

Results:- All investors will hold the same portfolio for risky assets – marketportfolio. They may also hold a riskless asset.- Market portfolio contains all securities and the proportion of each securityis its market value as a percentage of total market value.- Risk premium on the market depends on the average risk aversion of allmarket participants.- Risk premium on an individual security is a function of its covariance withthe market.

Resulting Efficient Frontier and Capital Market Line are given in the figure:

E(r)

E(rM)

rf

MCML

σm

σ


Slope of the CAPM line is important:E r

M( ) − rf

M

The risk premium on the market portfolio will be proportional to its risk andto the degree of risk aversion across investors. Specifically:

E rM

( ) − rf

= 0.01A M

2

The risk premium on individual securities will be proportional to the riskpremium on the market portfolio, and the beta coefficient of the securityrelative to the market portfolio. Beta is defined as:

i=

Cov ri,r

M( )

M

2

The risk premium on an individual security is a function of that security’scontribution to the risk of the market portfolio: its covariance of returns withthe assets that make up the market portfolio. Key Formula (expected rate ofreturn of an individual security):

E ri

( ) = rf+

iE r

m( ) − r

f( )Security Market Line:

E(r)

E(rM)

r

SML

M

ßß = 1.0


Examples:

E(rm) - rf = 0.08 rf = 0.03

βx = 1.25

E(rx) = 0.03 + 1.25(0.08) = 0.13 or 13%

βy = 0.6

E(ry) = 0.03 + 0.6(0.08) = 0.078 or 7.8%

Graphically:

E(r)

Rx=13%

SML

m

ß

ß

1.0

Rm=11%

Ry=7.8%

3%

xß

1.25

yß

.6

.08


Disequilibrium – mispriced security (its price will increase until thisrelationship disappears):

Why do all investors hold the market portfolio?- They all have same information, same time-horizon, etc., thus arrive at

the same risky portfolio;- Aggregate market portfolio includes the entire wealth of the economy;- If a stock were excluded, its prices would drop until it is included.

Important implication: Passive investing is efficient. Mutual fund theorem:“investors will choose to invest their entire risky portfolio in a marketindex.”

An individual stock’s contribution to portfolio risk is: wiCov r

i,r

M( ) .

The expression E r

M( ) − r

F

M

2 is called the market price of risk. We also define

the marginal price of risk: incremental risk premium required for eachadditional unit of risk. For the market portfolio, the marginal price of risk is

E(r)

15%

SML

ß

1.0

R =11

rf=3%

1.25


∆E r( )∆

M

2=

E rM

( ) − rF

2M

2. For an individual security, its marginal price of risk is

∆E ri

( )∆

i

2=

E ri

( ) − rF

2Cov ri,r

M( ) . This security’s price continues to adjust until the

marginal price of risk for it is the same as for the market:

E ri

( ) − rF

=Cov r

i,r

M( )

M

2E r

M( ) − r

F( ) . This basic reasoning shows how the

key formula is derived.

There have been many criticisms of CAPM formula, as it does not appear toreally work in the market. However, one must note that the assumptions donot work in reality, and so the formula is a theoretical one. To adjust it toreality, must consider generalizations, which adjust the unrealisticassumptions. For example: Brennan examined the impact of personal taxrate on the theory, while Mayers examined the impact of non-traded assets(Social Security benefits, human capital). One important observation aboutthe CAPM formula: it holds for individual assets and for portfolios.

Difference between CML and SML: the x-axis for SML is in terms of beta,not the standard deviation as in the case of AML.

Alpha: expected return in excess of that required by the CAPM:

= E ri

( ) − rf+

iE r

m( ) − r

f( )( )or, the portion of return unexplained by the CAPM. This concept, alsoknown as Jensen’s alpha, is commonly used in the analysis of performanceof investment managers.

Uses for CAPM:- Investment strategy (buy market, add long underpriced, short

overpriced);- Capital budgeting (required rate of return, hurdle rate, for a project)

The project-specific required return is determined by the project betacoupled with the market risk premium and the risk-free rate. The CAPMtells us that an acceptable expected rate of return for the project is:

rF

+ E rM

( ) − rF( )

and this becomes the project hurdle rate.- Utility rate-making.


Extensions of the CAPM

Black’s Zero Beta Model:Market portfolio is no longer optimal for everyone if borrowing is restrictedand/or borrowing rate exceeds lending rate. Black incorporates thefollowing:- Combinations of portfolios on the efficient frontier are efficient.- All frontier portfolios have companion portfolios that are uncorrelated.

The zero-beta portfolio is the minimum variance portfolio uncorrelatedwith a chosen efficient portfolio. To find the standard deviation of thezero-beta portfolio, draw a horizontal line from the y-intercept to theminimum-variance frontier. To find its expected return, draw a tangentfrom the efficient portfolio to the y-axis.

- Returns on individual assets can be expressed as linear combinations ofany two efficient portfolios:

E ri

( ) = E rQ( ) +

E rP( ) − E r

Q( )( ) Cov ri,r

P( ) − Cov rQ,r

P( )( )P

2 − Cov rQ,r

P( )Versions of the Black model:- No risk-free asset available- Risk-free lending but no risk-free borrowing possible

E ri

( ) = E rZ M( )( ) +

E rM( ) − E r

Z M( )( )( )Cov ri,r

M( )M

2

- Borrowing at a rate higher than the risk-free rate.


Efficient Portfolios and Zero Companions:

Lifetime consumption model:This version of CAPM recognizes that investors have different holdingperiods. Fama showed that in this case single-period CAPM may still apply.His key assumptions are: investors preferences do not change over time, andthe risk-free rate, as well as the probability distribution of the risky returnsdo not change in an unpredictable fashion over time.

Note that the section on CAPM and liquidity is not on the syllabus of CASExam 8. Here is its summary:Empirical research supports a premium for illiquidity of investments(Amihud and Mendelson), and we arrive at CAPM with Illiqudity Premium:

E ri

( ) − rF

=i

E rM

( ) − rF( ) + f c

i( ) ,

where: f ci

( )= liquidity premium for security i, f ci

( ) increases at adecreasing rate (as a function of that security’s illiquidity).

Q

P

Z(Q)

Z(P)

E[rz (Q)]

E[rz (P)]

E(r)

σ


Exercise 1.Security X has a beta of 1.5. The risk-free rate is 3.00% and the marketexpected rate of return is 9.00%. According to the Capital Asset PricingModel, what is this security’s expected rate of return?

Solution.E r( )= r

f+ E r

M( )− rf( ) =

= 3.00%+1.5 9.00%− 3.00%( ) == 12.00%.

Exercise 2.What is the beta of a portfolio with expected return of 18% if the risk-freerate is 6%, and the market expected return is 14%?

Solution.We are given that 18% = 6% + 14% − 6%( ) and we need to find . Itfollows that 12% = ⋅8% and = 1.5.


Exercise 3.The market price of a security is $50. Assume that the Capital Asset PricingModel holds. The expected rate of return of this security is 14%. The risk-free rate is 6%, and the market risk premium (over the risk-free rate) is8.50%. What will be the market price of the security if the covariance of itsreturns with the market doubles, while no other parameters are changed?Assume that the security is a perpetuity of a constant dividend.

Solution.Basic CAPM formula is

E r( )− rf

= E rM

( ) − rf( )

where

=Cov r,r

M( )Var r

M( ) .

When the covariance doubles, beta doubles. We need to find the originalbeta first. It must satisfy14%− 6% = 8.50%( )so that

= 8085

= 1617

.

After beta doubles, = 3217

.

Therefore, new rate of return is

rnew

= 6%+ 3217

8.50%( ) = 22%.

What is the dividend D of this security? At 14% expected rate, perpetuity of

D is worth $50, and this must equal D

0.14, so that D = $7. At 22% expected

return, perpetuity of $7 is worth $31.82.


Exercise 4.If I am buying a firm with an expected perpetual cash flow of $1,000 but Iam unsure of its risk. If I think the beta of the firm is 0.5, when in fact thebeta is really 1, how much more will I offer for the firm than it is trulyworth. Assume that the risk-free rate is 6%, and the market expected rate orreturn is 16%.

Solution.If beta is 0.5, the discount rate is 6% + 0.5 (16% - 6%) = 11%. If beta is 1,the discount rate is 16%. The value of the perpetuity of $1,000 is $1000/0.11= 9,090.91 at 11%, and $1000/0.16 = $6,250 at 16%. The difference,$2,840.91, is the amount of overpayment.

Exercise 5.A share of stock sells for $50 today. The risk-free rate of interest is 3%, andthe expected rate of return of the market is 9%. The share of stock underconsideration will pay a dividend of $3 per share at the end of the year, andits beta is 1.2. What is the expected price of the stock at the end of the year(ex-dividend, i.e., after the payment of the dividend) as given by the CapitalAsset Pricing Model?

Solution.Using the basic CAPM formula, we have the following:

E Ri

( ) = rF

+ E RM

( ) − rF( )

with rF = 0.03, E R

M( ) = 0.09, = 1.2, so that

E Ri

( ) = 0.03 +1.2 0.09 − 0.03( ) = 0.102 .

This means that $50 is expected to grow to $55.10 (total rate of return of10.20% on $50). But of this $55.10, $3 will be paid in a dividend, and theexpected price of the stock at the end of the year is $52.10.


Exercise 6.The risk-free rate is 7%. The expected market rate of return is 15%. If youexpect stock X with a beta of 1.3 to offer a rate of return of 12%, is thisstock under-priced or over-priced based on your expectation?

Solution.Using the basic CAPM formula, we get: 7% + 1.3(15% - 7%) = 17.40% >12%, therefore this security is overpriced. It should be shorted against a longmarket position.


Single-Index and Multifactor Models

There are problems with implementation of the classical Markowitz model:- It requires a large number of inputs;- Statistical estimates of the inputs are often unreliable, e.g., correlation

coefficients that are inconsistent;- Correlations tend to be positive for all stocks, as all stocks are affected

by same macroeconomic factors.

Common macroeconomic factors affect performance of firms:- business cycles- interest rates- technological changes- cost of labor and materials

Single factor model:r

i= E r

i( ) +

iF + e

i

where

iis the responsiveness of this security to macro events,

F is the unanticipated components of the macro factor,e

i is the impact of unanticipated firm-specific events.

Asingle-index model is a variation of a single factor approach: we use acertain broad market index (e.g., S&P 500) as a proxy for the systematicmacroeconomic factor. The rate of return on a stock is the sum of threecomponents:

i= the stock’s expected return if the market is neutral, i.e., if

market excess return is zero,

ir

M− r

F( ) = expected excess return attributable to the movements of

the overall market,e

i = unexpected component of the security return due to firm specific

events.

We get this formula:r

i− r

F=

i+

ir

M− r

F( ) + e

i


We’ll write RM

= rM

− rF and R

i= r

i− r

F for the excess returns, then the

key formula becomes:

Ri=

i+

iR

M+ e

i

It follows from this formula that:Var R

i( ) =

i

2Var RM

( ) + Var ei

( )or equivalently:

i

2 =i

2

M

2 + 2 ei

( ) .

This decomposes the security risk into the market risk and firm-specific risk,with the two assumed uncorrelated in the model. We actually not onlyassume that firm specific events are independent of the market, we alsoassume that they are uncorrelated among themselves. Thus we also have:

Cov Ri,R

j( ) = CoviR

M,

jR

M( ) =i j M

2.

This allows us to greatly reduce the number of calculations to arrive atefficient portfolios: instead of calculating correlations of all securities amongthemselves, we can calculate individual securities’ correlations with themarket.

Advantages of the single-index model:- We only need 3n + 1 estimates for the inputs (for each stock, we

must have the estimate of its expected excess return, its beta withrespect to the index, and its firm-specific variance, and we alsoneed an estimate of the overall market variance) to create anefficient frontier,

- No need to calculate covariances between securities, just betweeneach security and the market.

But …- Is it really possible that returns are driven by one factor? There

may be industry-specific events affecting only one industry.Should we use a multifactor model instead?

- There may be correlation between individual ei’s.

- What is the index representing that single factor? Do we reallyknow it is S&P 500?


Estimating the index model:The process of estimation can be done by performing a linear regression ofsecurity’s excess returns on the market’s excess returns. The slope is

i, and

the intercept is i (average firm-specific return when the market excess

return is zero). We denote the residuals by ei, ideally they should be

uncorrelated with each other and with RM

. The estimate of the firm-specific

variance is: 2 ei

( ) =1

n − 2e

i

2

i =1

n

∑ .

An example of such an estimation in the text: regression of excess returns ofGM against excess returns of S&P 500

RGM ,t

=GM

+GM

RM ,t

+ eGM ,t

Regression results:- Alpha = -2.590- Beta = 1.1357- R-Squared = 0.575- Residuals represent firm-specific returns, their variance is 12.601,

standard deviation 3.550

The index model gives an insight into portfolio diversification. If we have aportfolio of n securities, each equally weighted, then for each security

Ri=

i+

iR

M+ e

i

and for the portfolioR

P=

P+

PR

M+ e

P

But RP

=1n

Ri

i =1

n

∑ , P

=1n i

i=1

n

∑ , P

=1n i

i =1

n

∑ so that eP

=1n

ei

i=1

n

∑ , and

P

2 =P

2

M

2 +1n2

2 ei

( )i =1

n

∑ =P

2

M

2 +1n

2 ei

( ).

Therefore we arrive at another mathematical justification for the distinctionbetween the diversifiable risk and the systematic risk. First componentrepresents systematic risk and is not diversifiable. Second componentreflects non-systematic risk, and it approaches zero as n tends to infinity.


In this context, diversifiable risk is quantified as: 1n

2 ei

( ), and the

systematic risk is quantified as: P

2

M

2.

The CAPM and the index model

Actual vs. expected returns- To test that the market portfolio is mean-variance efficient, we

would need to construct a market-value weighted portfolio of allrisky assets, and this is just not feasible. Then we would need toshow that such a portfolio’s reward to variability ratio is higherthan that of any other portfolio. But CAPM is not about realizedreturns, but about theoretical expected returns, so we would onlybe able to use estimates.

- To test the expected return-beta relationship predicted by CAPM,we still deal with data, which gives actual returns not expectedreturns. More assumptions are necessary to make CAPM testableand implementable.

Number ofSecurities

Portfolio Variance

Systematic Risk

Diversifiable Risk


CAPM says ex ante (before the fact, as it is about expected returns):

E ri

( ) − rF

=i

E rM

( ) − rF( ). To make the leap from expected to realized

returns, we study single-index model Ri=

i+

iR

M+ e

i. If we take the

covariance of the security with the market return on both sides of thisexpression, we get: Cov R

i,R

M( ) =

i M

2 , and this is consistent with the

previous definition: i=

Cov Ri,R

M( )

M

2.

Compare CAPM E ri

( ) − rF

=i

E rM

( ) − rF( ) with single-index model

ri− r

F=

i−

ir

M− r

F( ) + e

i and you see that CAPM basically says that the

expected alpha for every asset should be zero. In a sample period studied, itis possible that some stocks will exhibit positive or negative alphas.But:

- Realized values of alpha should average to zero for a sample ofobserved returns.

- Sample alphas should be unpredictable, independent from oneperiod to another.

One more model worth noting is the market model:

ri− E r

i( ) =

ir

M− E r

M( )( ) + e

i

This is a model in which the return surprise of any security is proportional tothe return surprise of the market plus a firm-specific surprise. If CAPM isvalid, this is equivalent to the single index model, and the two concepts canbe used interchangeably.


The industry version of the index model

Merrill Lynch, in its publication Security Risk Evaluation (known as: BetaBook) uses the following model: r = a + br

M+ e*

, and estimates the valueof the b parameter for various securities. We really have:

r = rf+ + r

M− r

f+ e = + r

f1−( ) + r

M+ e .

and if the risk-free rate is constant over the study period, there is nodifference between estimating beta and estimating b. The estimation is doneusing linear regression on a monthly basis. In that regression, the portion ofthe outcome variance attributable to the model, know as R 2 or coefficient ofdetermination, is equal to

R 2 =2

M

2

2= 1−

2 e( )2

=Systematic Variance

Total Variance.

Recall that 2 e( ) is the variance of the residuals. Generally, higher R-squared indicates better regression model, but … it is the F-statistic whichreally tells you if the model is statistically significant. Also note that thesquare root of R-squared is the correlation coefficient between the returns ofthe security and the market.

There is a problem with practical calculations of beta. Beta changes overtime. Most of such calculations make some form of adjustment to historicalbeta before using it as a predictor. Merrill Lynch uses “adjusted beta”defined as:

Adjusted Beta =23

⋅Sample Beta +13

⋅1This approach basically assumes that betas move towards one from theirhistorical values. Another approach would be to study how beta evolvedover time, and determine:

Current Beta = a + b (Past Beta)

Then we would apply the same adjustment for the prediction:

Forecast Beta = a + b (Current Beta)


An alternative approach would be to incorporate data on firm size and debtratio:

Current Beta = a + b1⋅Past Beta + b

2⋅Firm Size +b

1⋅Debt Ratio

Rosenberg and Guy used such an approach with the following variables:- Variance of earnings,- Variance of cash flows,- Growth of earnings per share,- Firm size, as represented by market capitalization,- Dividend yield,- Debt-to-asset ratio.

They found that industry group helps predict beta, and one should apply anindustry group adjustment factor when creating adjusted beta.

Multifactor modelsSingle-factor models explain only a fraction of stock returns, and it is morerealistic to consider a multi-factor model. For example, one could use a two-factor model with GDP growth and interest rates as the factors.Advantages of multifactor models:

- Capture the fact different firms and industries have differentsensitivities to various factors,

- Can include variables related to the business cycle, as betas seemto vary over the business cycle,

- Useful in identifying the risk sources.One of the most successful multifactor models is that of Chen, Roll and Ross(1986):R

i,t=

i+

i,IPIP

t+

i,EIEI

t+

i,UIUI

t+

i,CGCG

t+

i ,GBGB

t+ e

i,t,

where:IP = % change in industrial productionEI = % change in expected inflationUI = % change in unanticipated inflationCG = excess return of long-corporate bonds over long-termgovernment bondsCG = excess return of long-term government bonds over T-Bills


Two principles for selecting macro factors:- include only the ones with predictive ability,- choose those that relate to risks for which investors would demand

a risk premium.Fama and French used the following factors:

- Market index,- Small minus big: return on a portfolio of small stocks minus the

return on a portfolio of large stocks,- High minus low: return on a portfolio of stocks with high book-to-

market ratios minus the return on a portfolio of stocks with lowratios.

Why would a multifactor model be more appropriate?- Because risk exposures could be multivariate.- Because various sources of risk may be priced into securities, due

to sources of uncertainty affecting consumers’ incomes (e.g.,uncertainties in labor income, prices of important consumptiongoods, changes in future investment opportunities – Merton’ sIntertemporal CAPM, or ICAPM, see below).

Theoretical foundations of multifactor models:- Should risk premiums be limited solely to the contribution to the

overall portfolio variance?- Young investors may value more those stocks that have low

correlation with future labor income, to hedge. CAPM holds ifsuch hedging demand affects all securities equally (unlikely).

- Merton’ s Intertemporal CAPM: incorporates demand forsecurities by investors concerned with lifetime consumption.Sources of uncertainty that may affect security returns: uncertainlabor income, prices of important consumption goods, changes infuture investment opportunities (e.g., riskiness of various assetclasses).

Empirical models and the ICAPM- Models using proxies for extra-market sources of risk are not doing

well, are of poor quality,- Why? Factors cannot be used to hedge, random outcomes may be

assigned too much meaning, past risk premiums related to size andbook-to-market ratios undercut the predictability of CAPM.


Exercise 1.Suppose that the correlation coefficient between returns of your mutual fundand the S&P 500 Index, representing the market, is 0.70. What can you sayabout the distribution of your mutual fund’s risk between systematic andsecurity-specific risk?

Solution.The square of the correlation coefficient is R-squared, so R-squared is 0.702

= 0.49. This means that 49% of this fund’s risk is attributable to the market,i.e., is systematic, and 51% is nonsystematic.

Exercise 2.The correlation coefficient between the Lapu-Lapu International Fund andthe EAFE Market Index is 1.0. The expected return of the EAFE Index is11%, and the expected return of the Lapu-Lapu International Fund is 15%.The risk-free rate of return in EAFE countries is 3%. Based on this analysis,what is the implied beta of the Lapu-Lapu International Fund?

Solution.Since the correlation coefficient is 1, so is R-squared in the market model,and therefore the residuals have zero variance, and zero mean, and they are

zero. So: ri− r

f=

ir

M− r

f( ) . Taking expected values we have:

15% − 3%( ) =i11% − 3%( ), so that

i= 1.5.

Exercise 3.The index model for stock A has been estimated with the following result:RA = 0.01 + 0.9RM + eA. If you are given that σM = 0.25, and R2 = 0.25 forthis model, then what is the standard deviation of return of stock A?

Solution.R2 is the ratio of systematic variance and total variance, so 0.25 == (0.92 0.252)/σ2, and therefore σ = 0.45.


Exercise 4.The index model has been estimated for stocks A and B with the followingresults:

RA = 0.01 + 0.5RM + eA

RB = 0.02 + 1.3RM + eB

σM = 0.25, σ(eA) = 0.20 σ(eB) = 0.10Based on this information, calculate the covariance between the returns onstocks A and B.

Solution.

Utilize the formula: Cov Ri,R

j( ) = CoviR

M,

jR

M( ) =i j M

2.

Substituting, we get:Covariance = (0.5)(1.3)(0.25)2 = 0.040625.

Exercise 5.Suppose that you hold a well-diversified portfolio with a very large numberof securities, and that the single factor model holds. If the standard deviationof your portfolio was 0.20, and the standard deviation of the market returnswas 0.16, what is the approximate value of the beta of your portfolio?

Solution.For a well-diversified portfolio, non-systematic risk can be assumed to be

zero so that: P

2

M

2= 2

. We have (0.2)2/(0.16)2 = 1.56, and beta equals 1.25.


Exercise 6.An index model regression applied to past monthly returns in Microsoftstock price produces the following estimates, which are believed to be stableover time:

rMSFT = 0.20% + 1.8 rM

If the market index subsequently rises by 8% in a month and Microsoft stockprice rises by 12% in the same month, what is the abnormal return ofMicrosoft stock?

Solution.Forecast rate of return on MSFT is

0.20% + 1.8(8%) = 14.60%MSFT’s actual return was 12%, meaning that the abnormal return wasnegative 2.60% (i.e., -2.60%), the difference of the two quantities.


Arbitrage Pricing Theory

APT is a theory created in 1976 by Stephen Ross.

Arbitrage arises if an investor can construct a zero investment portfolio witha sure profit. Since no investment is required, an investor can create largepositions to secure large levels of profit, theoretically infinite. Arbitrageopportunities occur when the law of one price is violated, i.e., when an assettrades at two different prices.

How market prices eliminate arbitrage opportunities:- If an arbitrage opportunity exists, investors aware of it will want to

take an infinitely large position (or at least as large of a position aspossible);

- But such activities will cause market prices to adjust until thearbitrage opportunity disappears.

Arbitrage is different than risk-return dominance. In the case of risk-returndominance, many individual investors make limited portfolio changes,depending on their level of risk preferences. In the case of arbitrage,everyone aware of it will want to take an infinite position. In efficientmarkets, profitable arbitrage opportunities will quickly disappear. Note thatCAPM is about risk-return dominance: all investors hold (or seek) mean-variance efficient positions, and equilibrium is restored by large number ofinvestors adjusting their portfolios.Risk arbitrage is not arbitrage, it is speculation on perceived mispricedsecurities, usually in connection with merger and acquisition targets.


Arbitrage pricing theory: single-factor version of APTAPT is derived from a factor model, based on diversification and arbitragearguments. Assumptions for single-factor APT:

- No opportunities for a risk-free arbitrage,- Two sources of uncertainty in asset returns: macroeconomic factors F

(not necessarily the market portfolio return, and has a zero expectedvalue, because its affect on prices comes only from unexpectedsurprises) and firm-specific factors e

i, which are uncorrelated among

themselves and uncorrelated with F.Under these assumptions we arrive at the following formula:

ri= E r

i( ) +

iF + e

i.

The coefficient i represents the sensitivity of the firm under consideration

to the factor F.

For well-diversified portfolios, the individual securities’ eI’s arenonsystematic returns, uncorrelated among themselves and uncorrelatedwith the factor F. If we construct an n-stock portfolio with weight w

i in

security i, then the rate of return on this portfolio isr

P= E r

P( ) +

PF + e

P

where P

= wi

i=1

n

∑ i. The portfolio variance is then:

P

2 =P

2

F

2 + 2 eP

( ),

and 2 eP

( ) = wi

2

i=1

n

∑ 2 ei

( ) . For sufficiently well-diversified portfolios,2 e

P( ) is negligible, so that

rP

= E rP

( ) +PF and

P=

P F.

Key insight: Because nonfactor risks can be diversified away, only factorrisks command a risk premium in market equilibrium.


Relationship for an individual security,

and for a diversified portfolio:

F

E(r)

F

E(r)


The Security Market LineAll diversified portfolios must plot on a straight line with y-intercept r

F,

slope of which is E rM

( ) − rF.

The Security Market Line is identical as in the CAPM. Comparison of APTand CAPM:

- APT applies to well-diversified portfolios and not necessarily toindividual stocks.

- With APT it is possible for some individual stocks to be mis-priced -not lie on the SML.

- APT is more general in that it gets to an expected return and betarelationship without the assumption of the market portfolio.

- APT can be extended to multifactor models.

APT is more practical than CAPM, as well-diversified portfolios can beconstructed, but CAPM market portfolio is really impossible to find. But …CAPM is more comprehensive than APT, because CAPM holdunequivocally, and APT may not hold for non-diversified portfolios.

Beta (Market Index)

rF

M

1.0

E rM

( ) − rF

Market RiskPremium


It is probably more useful to compare the index model and APT:

Index model:- Assumptions: same as CAPM, specified market index is virtually perfectlycorrelated with unobservable market portfolio, probability distribution ofreturns is stationary.- Under those assumptions, market index portfolio is efficient, and theCAPM formula holds for all assets.- The model is testable and based on mean-variance efficiency assumption.

APT- Assumes a single-factor securities market, based on a no-arbitragecondition, and cannot rule out a violation of the CAPM relationship forindividual securities.

Multifactor APT

In fact, APT makes most sense for multifactor models:

ri= E r

i( ) +

i ,1F

1+

i ,1F

2+ e

i

A factor portfolio is defined as a well-diversified portfolio constructed tohave a beta of 1 on one of the factors of the model, and 0 on all the otherfactors. The multifactor SML is:

E rP

( ) = rF

+P ,1

E r1

( ) − rF( ) +

P ,2E r

2( ) − r

F( ) + ...+P ,n

E rn

( ) − rF( )

where E rk

( ) − rF is the k-th factor portfolio’s risk premium.


Conclusions:• When securities are priced so that there are no risk-free arbitrageopportunities, we say that they satisfy the no-arbitrage condition. Pricerelationships that satisfy the no-arbitrage condition are important because weexpect them to hold in real-world markets.• In a single-factor security market, all well-diversified portfolios have tosatisfy the expected return-beta relationship of the security market line tosatisfy the no-arbitrage condition.• If all well-diversified portfolios satisfy the expected return-betarelationship, then all but a small number of securities must satisfy thisrelationship.• The assumption of a single-factor security market made in the simpleversion of the APT, together with the no-arbitrage condition, implies thesame expected return-beta relationship as does the CAPM, yet it does notrequire the restrictive assumptions of the CAPM and its (unobservable)market portfolio. The price of this generality is that the APT does notguarantee this relationship for all securities at all times.• A multifactor APT generalizes the single factor model to accommodateseveral sources of systematic risk.


Exercise 1.Suppose that the market can be described by the following three sources ofsystematic risk with associated risk premiums:

Factor Risk Premium

Industrial production (I) 6%Interest rates (R) 2%Consumer confidence (C) 4%

The return of a particular stock is generated according to the followingequation:

r = 15% +1.0I + 0.5R + 0.75C + e

where the T-Bill rate is 6%, and the variables represent surprises in thefactor returns (with expected value of a surprise equal to zero). Find theequilibrium rate of return on this stock using the APT. Is the stockoverpriced or underpriced?

Solution:The APT required rate of return based on the risk-free rate and the factorbetas is:

E r( ) = rF

+1F

1+

2F

2+

3F

3=

= 6% + 1.0 ⋅6% + 0.5 ⋅2% + 0.75 ⋅4% = 16.00%.On the other hand, according to the equation for the return on the stock, theactually expected return on the stock is 15% (because the expected surpriseson all factors by definition are zero – I know this is hard to read from theproblem, but the exercise is here to make this point for you). Because therequired rate of return based on risks exceeds the actually expected return,we conclude that the stock is overpriced.


Exercise 2.Assume that the security returns are generated by the single-index model,R

i=

i+

iR

M+ e

i, written in terms of excess returns. The risk-free rate

is 2%. Suppose also that there are three securities A, B, and C, characterizedby the following data:

Security βi E(ri) σ(ei)

A 0.8 10% 25%

B 1.0 12% 10%

C 1.2 14% 20%

If the market return has standard deviation of 20%, calculate the variance ofreturns of securities A, B, and C.

Now assume that there are infinitely many assets with return characteristicsidentical to those of A, B, and C, respectively. If one forms a well-diversified portfolio of type A securities, what will be the mean and varianceof the portfolio’s excess returns? What about portfolios composed only oftype B and C stocks?

Is there an arbitrage opportunity in the market? What is it?

Solution:Recall the formula: 2 = 2

M

2 + 2 e( ). Thus, using it for each securityseparately, we get:

A

2 = 0.82 ⋅202 + 252 = 881

B

2 = 1.02 ⋅202 +102 = 500

C

2 = 1.22 ⋅202 + 202 = 976

If there is an infinite number of assets with identical characteristics, a well-diversified portfolio of each type will have only systematic risk since thenon-systematic risk will approach zero with large n. The mean return willequal to that of the individual stocks.


Since E(αι) = 0, the expected risk premiums for the three securities are:

A: 10% - 2% = 8%B: 12% - 2% = 10%C: 14% - 2% = 12%

Taking the expectations of the model, we get:E R

i( ) =

iE R

M( )

BecauseE R

A( )

A

=8%0.8

= 10% = E RM

( )E R

B( )

B

=10%1.0

= 10% = E RM

( )E R

C( )

C

=12%1.2

= 10% = E RM

( )there are no arbitrage opportunities, as all securities lie on the same securitymarket line.


Exercise 3.Consider the multifactor APT. The risk premiums on the factor 1 and factor2 portfolios are 5% and 3%, respectively. The risk-free rate of return is 10%.Stock A has an expected return of 19% and a beta on factor 1 of 0.8. What isthe beta of A on factor 2?

Solution.For stock A, we have the APT formula:19% = 10% + 0.8(5%) + β2(3%), which solves for β2 = 1.67.

Exercise 4.Consider a single factor APT. Portfolio A has a beta of 1.0 and an expectedreturn of 16%. Portfolio B has a beta of 0.8 and an expected return of 12%.The risk-free rate of return is 6%. If you wanted to take advantage of anarbitrage opportunity, what should you do under these circumstances?

Solution.RA = 16% = 1.0 F + 6%, so F = 10.00%RB = 12% = 0.8 F + 6%, so F = 7.50%This is inconsistent, and we should be long A, short B to take advantage ofthis arbitrage opportunity.


Exercise 5.Consider a one factor APT. The variance of returns of the factor portfolio is6 %2( ). The beta of a well-diversified portfolio on the factor is 1.1. What isthe variance of returns on the well-diversified portfolio?

Solution.σ2

P = (1.1)2 6 %2( ).= 7.26 %2( ) (nonsystematic risk can be ignored forwell-diversified portfolios).

Exercise 6.Consider the multifactor model APT with two factors. Portfolio A has a betaof 0.75 on factor 1 and a beta of 1.25 on factor 2. The risk premiums on thefactor 1 and 2 portfolios are 1% and 7%, respectively. The risk-free rate ofreturn is 7%. What is the expected return on portfolio A if no arbitrageopportunities exist?

Solution.7% + 0.75(1%) + 1.25(7%) = 16.5%.


Exercise 7.There are three stocks: A, B, and C. You can either invest in these stocks, orshort them. There are three possible states of nature for economic growth inthe upcoming year: strong, moderate, or weak, all of them equally likely.The returns for the upcoming year on stocks A, B, and C for each of thesestates of nature are:

Stock StrongGrowth

ModerateGrowth

WeakGrowth

A 38% 18% -6%

B 20% 15% 0%

C 6% 14% 16%

Is there an arbitrage opportunity you can exploit? How would you constructan arbitrage portfolio?

Solution.If any of individual stocks performed better than some other stock undercircumstances, it would be possible to short the inferior one, and go long thesuperior one, resulting in a riskless arbitrage. This is not the case: A beatsothers under strong growth regime, B beats C under strong and moderategrowth, and C beats everyone under weak growth. However, if you combineA and C in equal amounts, the returns of this portfolio would be:- 22% under strong growth;- 16% under moderate growth;- 5% under weak growth.This turns out to be always better than B. Thus if you short B, and invest thesame amount as the amount shorted in an equally weighted portfolio of Aand C, you will earn profits on a net investment of zero, and those profitswill be earned on a net zero investment, i.e., pure arbitrage will be achieved.


Exercise 8.What is the difference between pure arbitrage and risk arbitrage?

Solution.Risk arbitrage involves searching for mispricings based on speculativeinformation that may or may not materialize. It involves risk. Pure arbitragemeans creating a zero net investment portfolio earning riskless profit.

Exercise 9.Discuss the advantages of arbitrage pricing theory (APT) over the CapitalAsset Pricing Model (CAPM).

Solution.The APT does not require that the benchmark portfolio in the SecurityMarket Line (SML) relationship be the true market portfolio. Any well-diversified portfolio lying on the SML may serve as a benchmark portfolio.Thus, the APT has more flexibility than the CAPM, as problems associatedwith an unobservable market portfolio are not a concern with APT. Inaddition, the APT provides further justification for the use of the indexmodel for practical implementations of the SML relationship. That is, if theindex portfolio is not a precise proxy for the true market portfolio, which is acause of considerable concern in the context of the CAPM, if an indexportfolio is sufficiently diversified, the SML relationship holds, according toAPT.


Market Efficiency

Random walks and the efficient market hypothesisRandom walk = theory that stock price changes are random andunpredictable. This idea gives credence to the notion that all availableinformation is already incorporated into the stock prices, and prices respondonly to new information, which is random and unpredictable.

Do security prices reflect information?One could argue that:

- All information needed to predict stock performance shouldalready be reflected in stock prices.

- Stock prices only respond to new information, which isunpredictable.

- Therefore movement of stock prices should be unpredictable.

Efficient Market Hypothesis: Stock prices already reflect all availableinformation.

Notable research by Grossman and Stiglitz: incentive to gather informationonly if it produces higher investment returns. Thus there may be differencesin efficiency between markets, as the bang for the research buck may vary.

Why look at market efficiency?- Implications for business and corporate finance.- Implications for investment.

Is information available to all investors? This is not necessary if themarginal investors, especially if they are large, benefit from uncovering theinformation, competition among them will force all relevant information tobe uncovered and represented in prices. But this also means that ifdiscovering the information is unprofitable, then inefficiencies will remains.

Random Walk - stock prices are random- Actually sub-martingale (there is a positive trend, due to time

value of money).- Expected price movement is positive over time.


Positive trend and random about the trend: Random Walk with positive trendis illustrated in the figure below. Note that the positive trend is linear, notexponential, the latter being more appropriate for security prices.

Versions of the Efficient Market Hypothesis:

- Weak form: Stock prices already reflect all information that can bederived be examining market trading data such as the history ofpart prices, trading volume, or short interest. This version of EMHimplies that trend analysis, or technical analysis in general, isfruitless

- Semistrong form: all publicly available information regarding theprospects of a firm must be reflected already in the stock price.This version of EMH implies that fundamental analysis is fruitless.

- Strong form: stock prices reflect all information relevant to thefirm, even including information available only to the companyinsiders.

Note: Rule 10b-5 of the SEC sets limits on trading by corporate officers,directors, and substantial owners, requiring them to report trades to the SEC.These insider, their relatives, and any associates who trade on informationsupplied by insiders are considered in violation of the law.

SecurityPrices

Time


Implications of Efficiency for Active and Passive Management

Active Management- Security analysis: technical analysis, fundamental analysis.- Market timing

Passive Management- Buy and Hold.- Index Funds.

Technical analysis:- Attempts to analyze supply and demand for stocks. Inherently

assumes that price of a given stock is determined by supply anddemand for that stock only.

- Seeks to uncovers patterns in stock prices, exploit a trend duringthe adjustment period.

- Dow Theory: primary trend, secondary trend, and tertiary or minortrends.

- Elliott Wave Theory/Kondratieff waves: broad waves lasting 48 to60 years in the economy and the market.

- Resistance and support levels. Moving averages.- Given market efficiency of any kind, technical analysis should be

futile

Fundamental analysis- Study of the firm’s financial statements, management, firm’s

position within the industry, and industry’s prospects.- If everyone else studies the same data just as hard, fundamental

analysis should be useless.- If you can indeed do a better job than others, you can earn extra

returns.

Note that extra returns have to be economically feasible, i.e., you shouldearn enough to justify all the effort. What’s a small investor to do?

- Buy mutual funds.- Buy and hold a diversified portfolio.- Buy an index fund.


Even if the market is efficient a role exists for portfolio management- Proper diversification.- Appropriate risk level, and appropriate for the risk profile and

combined portfolio of an investor.- Tax considerations.- Other considerations, e.g., asset-liability management, take into

account the time horizon and the age of the investor.

Empirical Tests of Market Efficiency- Event studies: examining price changes during the period in which

the event occurs (and stock price reacts to new information). Studyabnormal return: return on a stock beyond what would bepredicted by market movements alone.

- Assessing performance of professional managers.- Testing some trading rules.

Performing an event study:Examine prices and returns over time

Returns are adjusted to determine if they are abnormalMarket Model approach

a. Rt = at + btRmt + et

(Expected Return)b. Excess Return =

(Actual - Expected)et = Actual - (at + btRmt)

0 +-Announcement Date


Cumulate the excess returns over time (because information may have beenutilized in periods prior to announcement):

This pattern is present in studies of:- Takeover attempts,- Dividends increases,- Reversed for dividend decreases

- Lack of drift in Cumulative Abnormal Returns (CAR) following theannouncement date

- How does SEC feel about patterns exhibited: leakage of informationseems present, but it does not appear excessive? SEC seems to have someeffect, otherwise abnormal returns would appear much earlier and suddenjump on the event day would not occur.

Are markets efficient?Issues in examining the results of studies:

- Magnitude issue: only managers of large portfolios can earnenough profits to make the exploitation of minor mis-pricing worththe effort.

- Selection bias issue: schemes that really work will not bedisclosed, while schemes that fail will be disclosed by triumphantacademics.

- Lucky event issue: if many investors using a variety of schemesmake fair bets, some investors will be lucky and win a majority ofbets.

- Possible Model Misspecification: tests require a risk adjustment tocompare returns on risk-adjusted basis, and this risk adjustment isdone using beta, i.e., using CAPM. Thus the test ends us being ajoint test of efficiency, and the risk adjustment procedure.


Tests of predictability of stock market returns- Over short horizons, returns show slight positive autocorrelation. A

filter rule is a rule for buying or selling stocks based on past pricechanges. Trend following, derived from positive autocorrelation, isone. Jegadeesh and Titman’s study shows that while performanceof individual stocks is unpredictable, portfolios of best performingstocks have enough momentum to offer profit opportunities.

- Over long horizons: negative serial correlation. Appears that stockprices fluctuate around fair values. Does this mean markets areirrational and suffer from fads, bubbles, etc.? Or is this simply anindication of changing risk preferences. Do we really have enoughobservations to make conclusions about long-term patterns?

- Predictors of broad market returns: there is some evidence thateasily observable variables (dividend yield, earnings yield, yieldsspread between corporates and Treasuries) can predict market. Butthese variables may be just proxies for changing risk preferences.

Portfolio strategies and market anomalies- Fundamental analysis reveals anomalies that seem inconsistent

with market efficiency. But tests of risk-adjusted returns are jointtests of market efficiency and the risk-adjustment procedureapplied.

- P/E Ratio effect: Low P/E ratio stocks outperform (could it be thatthey have higher risk?).

- Small Firm Effect (January Effect): but small companies stocks aremore volatile, and thus affected more by tax-related selling beforethe end of the year.

- Neglected Firm Effect: this could be just a form of liquiditypremium.

- Market to Book Ratios Effect: firms with higher ratios of book tomarket values tend to earn higher returns. This is significant, aswhen we control for book-to-market, beta does not seem to matter.

- Reversals: Past winners tend to underperform, past losers tend tooutperform. This suggests that market overreacts to relevant news.But the effect may be overstated as it has not led to higher risk-adjusted returns.

- Post-Earnings Announcement Drift: market appears to adjust to theearnings information only gradually, and abnormal returnscontinue beyond the earnings announcement surprise.


- Market Crash of 1987: 23% one day decline, where was marketefficiency?

Risk premiums or anomalies?- Some of the suspected anomalies may be correlated: small firm,

low book-to-market, and reversal effects may be all distressedfirms.

- French and Fama study suggested that the effects aremanifestations of risk premiums. Proposed a three-factor model:market portfolio, portfolio reflecting relative returns of small vs.large firms, and a portfolio reflecting relative returns of firms withhigh vs. low book-to-market ratios. The model was effective inexplaining security returns, and suggests that the extra factors areproxies for underlying risk factors.

- Lakonishok, Shleifer and Vishney study suggested that anomaliesindicate inefficient markets. The problem is that analystsextrapolate past performance. When errors are recognized, pricesreverse (this is consistent with the reversal effect, and to somedegree with the size and book-to-market effects). Another study ofLa Porta seems to corroborate the findings: firms with lowexpected growth outperform those with high expected growth.

- Daniel and Titman study: concluded that the size and book-to-market ratios predict returns, and this is not the associated beta.The findings appear to contradict French and Fama.

- Abnormal returns could in all cases be explained by liquiditypremiums.

- Insider information: all insiders must register trades with SEC.There appear small abnormal returns from using that informationfor trading, but not enough to pay transaction costs.

- Post-earnings-announcement drift: slow reaction to earningsannouncements. An earnings surprise is the difference betweenactual earnings and those expected. Foster, Olsen and Shevlinfound strong correlation of positive abnormal returns and positiveearnings surprise. The reaction was, however, gradual, notimmediate as Efficient Market Hypothesis would suggest.

Are we maybe torturing the data into submission? Perhaps these anomaliesare only a result of too much analysis? There is a true puzzle here, not justdata mining. Two more comments:


- Value stocks (low P/E ration, high Book/Market, depressed prices)generally outperform glamorous stocks,

- Patterns show up even in un-tortured data in other countries.

Behavioral interprerationAnomalies are evidence of investors’ irrationality:

- Forecasting errors help explain the P/E ratio effect,- Overconfidence,- Regret avoidance (people blame themselves more when an

unconventional decision turns out to be wrong, not so forconventional ones),

- Framing and mental accounting (the way the decision is framedmay affect behavior, investment decisions are often segregated).

- We need to remember how much of stock gains/losses is merely amatter of luck, not skill.

Mutual funds performance- Do market professionals outperform market idenx?- Some evidence of persistent positive and negative performance,

but more consistency of negative performance, mostly due to highexpenses.

- Malkiel calculated alphas for a large sample of funds, and got anice bell-shaped distribution centered at zero.

- But the benchmark chosen matters. S&P 500 not appropriate forall. Still, Elton et al. study concludes in favor of passive investingusing a benchmark composed of S&P 500, equity index of non-S&P 500 small firms, and bond market index.

- Carhart study found little consistency in performance of managersexcept for top and bottom 10th deciles. Consistent bad performancewas mostly attributable to high expenses, high transaction costs,high portfolio turnover.

- There are not as many studies for bond fund performance, but mostindicate underperformance roughly equal to expenses.

What should an investor do when choosing a mutual fund?- Does it make sense for your portfolio?- How has the manager performed? What explains the manager’s

good performance? Has the manager performed consistently well?- Has the fund grown too large?


Exercise 1.If markets are efficient, what would be the correlation coefficient betweenstock returns for two non-overlapping time periods?

Solution.Zero. If not, returns from one period would provide information aboutreturns in the later periods, and abnormal profits could be made.

Exercise 2.“If all securities are fairly priced, all must offer equal expected rates ofreturn”. Is this statement true?

Solution.No. Returns will differ because of risk differentials.

Exercise 3.An index model regression applied to past monthly returns in GeneralMotors’ stock price produces the following estimates, which are believed tobe stable over time:

rGM

= 0.10% +1.1rM

.If the market index subsequently rises by 8% and General Motors’ stockprice rises by 7%, what is the abnormal change in General Motors’ stockprice?

Solution.Forecast rate of return on GM is

0.10% + 1.1⋅8% = 8.90%.GM’s actual return was 7%, meaning that the abnormal return was – 1.90%.


Exercise 4.An index model regression applied to past monthly returns in Microsoftstock price produces the following estimates, which are believed to be stableover time:

rMSFT

= 0.20% +1.8rM

.If the market index subsequently rises by 8% in a month and Microsoft stockprice rises by 12% in the same month, what is the abnormal return ofMicrosoft stock?

Solution.Forecast rate of return on MSFT is

0.20% + 1.8 ⋅8% = 14.60%.MSFT’s actual return was 12%, meaning that the abnormal return wasnegative 2.60% (i.e., -2.60%), the difference of the two quantities.

Exercise 5.What is an event study? It is a test of what form of market efficiency?Discuss the process of conducting an event study, including the bestvariable(s) to observe as tests of market efficiency.

Solution.A event study is an empirical test which allows the researcher to assess theimpact of a particular event on a firm's stock price. To do so, one often usesthe index model and estimates the residual term which measures the firm-specific component of the stock's return. This variable is the differencebetween the return the stock would ordinarily earn for a given level ofmarket performance and the actual rate of return on the stock. This measureis often referred to as the abnormal return of the stock. However, it is verydifficult to identify the exact point in time that an event becomes publicinformation; thus, the better measure is the cumulative abnormal return,which is the sum of abnormal returns over a period of time (a windowaround the event date).

This technique may be used to study the effect of any public event on afirm's stock price; thus, this technique is a test of the semi-strong form of theEMH.


Exercise 6.Describe all forms of the efficient market hypothesis, and the empiricalevidence regarding their validity.

Solution.Versions of the Efficient Market Hypothesis:- Weak form: Stock prices already reflect all information that can be

derived be examining market trading data such as the history of partprices, trading volume, or short interest. This version of EMH impliesthat trend analysis, or technical analysis in general, is fruitless. Empiricalstudies generally support this form of the EMH, when transaction costsare considered.

- Semistrong form: all publicly available information regarding theprospects of a firm must be reflected already in the stock price. Thisversion of EMH implies that fundamental analysis is fruitless. Empiricalstudies generally support this form of the EMH, when transaction costsare included.However, some anomalies exist which seem to contradictthe semi-strong form of the efficient market hypothesis.

- Strong form: stock prices reflect all information relevant to the firm, evenincluding information available only to the company insiders. Empiricalstudies generally do not support this form of the EMH, insiders dooutperform the market.


Exercise 7.Discuss the small firm effect, the neglected firm effect, and the Januaryeffect, the tax effect, and how the four effects may be related.

Solution.Studies have shown that small firms earn a risk-adjusted rate of returngreater than that of larger firms. Additional studies have shown that firmsthat are not followed by analysts (neglected firms) also have a risk-adjustedreturn greater than that of larger firms. However, the neglected firms tend tobe small firms; thus, the neglected firm effect may be a manifestation of thesmall firm effect. Finally, studies have shown that returns in January tend tobe higher than in other months of the year. This effect has been shown topersist consistently over the years. However, the January effect may be thetax effect, as investors may have sold stocks with losses in December for taxpurposes and reinvested in January. Small firms (and neglected firms) wouldtend to be more affected by this increased buying than larger firms, as theirprices tend to be more volatile.

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Equilibrium Pricing Utility function · Equilibrium Pricing ... expected value (mean) and variance, is its skewness ... Expected Annual Return Standard Deviation of Return