Chapter Three

Capital Asset Pricing Model

Perfect Market Conditions• Look for higher return, nonsatiation.

• Look for lower risk, risk aversion.

• Assets are infinitely divisible.

• Taxes and transaction costs are negligible

• All investors have the same one-period.

• Mean variance analysis of a single period.

• Risk-free rate is the same for all.

• Information is freely and instantly available.

• Investors all have the same perceptions about expected returns, standard deviations and covariance of securities.

These collectively represent the behavior of all investors in the market place, i.e., they represent certain kind of equilibrium conditions.

• From the one fund theorem, every investor will purchase a single fund of risky assets. Since every investor uses the same mean, variance, and covariance, everyone will choose the same combination of risky assets. Each investor will spread her investment among risky assets in the same relative proportions, adding risk-free lending and borrowing to achieve a personally preferred combination of risk and return according to her indifference curves.

Theorem. (Separation Theorem) The optimal combination of risky assets for an investor can be determined without any knowledge of the investor’s preference towards risk and return. This is the tangency portfolio.

One Fund = Market Portfolio• What should that one fund be?• Every security must have a non-zero proportion in

the tangency portfolio. In equilibrium, the proportions of the tangency portfolio must correspond to the proportions of the market portfolio. Therefore the one fund must be the market portfolio.

Theorem. The market portfolio is a portfolio consisting of all securities, where the proportion invested in each security corresponds to its relative market value. The relative market value (capitalization weight) of a security is simply equal to the aggregate market value of the security divided by the sum of the aggregate values of all securities.

In theory, the optimal (tangency) portfolio is found by solving a system of linear equations based on the given means and variance covariance matrix.

Claim: We don’t need to solve this system to identify the optimal portfolio. In practice, everyone is solving it using the same set of parameters. Everyone is trading according to the (same) solution (tangency portfolio) that he/she obtained. If prices do not match the solution, prices will be changed, which in turn affect the estimates of the asset returns. Hence, investor will recalculate their optimal portfolios. This process continues until demand exactly matches supply; in equilibrium.

In that case, everyone buys the same tangency portfolio, and that must be the market portfolio.

Capital Market Line

• In the world of CAPM, the efficient set can be determined easily. It consists of an investment in the market portfolio, coupled with a desired amount of risk-free lending and borrowing. Since (M M and rf both lie on this line, any efficient portfolio P P must satisfy the linear equation, known as the Capital Market Line (CML):

P rf = (P/M)(M rf ), or P = rf + P (M rf )/ M = rf + K P.

• All portfolios other than those using the market portfolio and the risk-free asset would lie below the CML. The slope of this line, K= (M rf )/ M , is known as the price of risk.

The Sharpe ratio is equal to ( rf )/ , which is the expected excess return per unit risk.

According to the capital market line, the Sharpe ratio of any efficient portfolio is the same as that of the market portfolio, that is, K (the price of risk).

The expected excess return (i rf ) of asset i is also known as the risk premium. The CAPM relates the risk premium to the market via the market beta through the so-called security market line, which is defined later.

Market Portfolio

• Let the return of the market portfolio be written as

rM = i=1N XiM ri. Then it has risk

2M i j XiM XjM ij. On the other hand,

iM = cov(ri , rM) = cov(ri , j=1N XjM rj)

= j=1N XjM ij.

Therefore,

2M X1M j=1

N XjM 1j + … + XNM j=1N XjM Nj

= X1M 1M + … + XNM NM .

Security Market Line

• In CAPM, everyone is concerned with M since its value affects the slope K, the price of risk. The preceding equation relates the contribution of each security to the risk of the market, M, through the size of its covariance with the market, iM . Intuitively, securities with larger iM are considered as contributing more to the risk of the market, and hence should provide higher return, i.e., i iM . This leads to the security market line (SML) in CAPM:

i rf = (iM/M2)(M rf )

= (iM )(M rf ).

• This quantity iM is the same beta of security i given in the market model.

• By the same token, we define the beta of the portfolio as the weighted average of the betas of its component securities as:

PM = i=1N XiM iM .

• This is the celebrated Capital Asset Pricing Model:Theorem.(CAPM) If the market portfolio is efficient,

the expected return of any asset must satisfy the SML

i rf = (iM )(M rf ).

Proof. For any , consider the portfolio ( can be negative).)1( Mi rrr

Then,)1( Mi

.)1(2)1( 22222iMMi

As varies, ( ,) traces out a curve in the - diagram as shown in Figure 3.1.

In particular, for = 0, we have r=rM and this corresponds to the market portfolio M. In general, this curve cannot cross the CML. Otherwise, we would have a portfolio above the CML, contradicting to the notion that the CML constitutes an efficient set. As passes through zero, this curve must be tangent to the CML. We shall exploit this tangency to establish the theorem.

The tangency condition translated into the condition that the slope of the curve is equal to the slope of the CML at the point M. Consider

.Mid

d

].)21(2)1(22[ 222

iMMid

d

Since = (2)½, we have

.1

])21()1([

)(2

1

22

22

12

iMMi

d

d

d

d

Therefore,

.2

0 M

MiM

d

d

Now.

d

d

d

d

d

d

.

2

000

MMiM

Mi

d

d

d

d

d

d

Thus,

But

.

CML theof Slope0

M

fM rd

d

Therefore,

.1

,

2

2

2

2

M

M

M

M

iMfM

iMfMMi

iM

MMi

M

fM

r

r

r

Thus, .

2

fMiM

iMfMfi

r

rrM

This completes the proof of the result.

Remarks

1. The SML implies that the mean return of any efficient asset is linearly proportional to iM = cov(ri , rM) and thus iM, i.e.,

i iM and i iM= iM /M2 .

2. Suppose that we rewrite the market model as ri - rf = iM(rM – rf ) i, Then taking expectation on both sides, we see E(i)=0. By taking the covariance with rM on both sides of this equation, we get cov(i , rM) = 0 and therefore

i2 = iM

2 M2 + i

2

= Systematic risk + Idiosyncratic Risk.

3. Consider an asset lying on the CML with a given beta value, . It has mean

i - rf = (M – rf ) and risk i2 = 2 M

2 . Consider a group of assets with the same . According to CAPM, they all have mean i = rf + (M – rf ). However, if they carry idiosyncratic risks, they cannot lie on the CML. They will lie to the right to the asset lying on the CML that only has systematic risk.

Hedging with CAPM

• Suppose that an asset is priced at S0 per share today and suppose that we invest it for T years, say. Let the annual risk-free rate be r. What should be the value of a future contract that allows you to buy or sell this share in T years? Let this value be denoted by F0.

• The answer is given by

F0 = S0 e r T .• This formula can be established by using a no

arbitrage argument as follows.

• To be specific, let a stock be trading at $30 per share today and let r=5% per annum. Suppose you enter into a forward contract that allows you to buy or sell the share at $35 two years from today. What would you do as an investor?

• Borrow $3000 at a rate of 5% per year.• Buy 100 shares of the stock today financed by the

loan.• Enter into a forward contract to sell 100 shares at

$35 in two years.• The proceeds in two years equals to 3500 – 3000 e0.05x2 = 3500 3316 = $181.

• On the other hand, suppose that you enter into a forward contract which allows you to buy or sell at $31 per share in two years. What would you do again?

• Short sell 100 shares today.• Invest the proceeds into the risk-free market for two

years.• Enter into a forward contract to buy 100 shares at

$31 per share in two years.• The proceeds in two years equals to 3000 e0.05x2 3100 = 3316 –3100 = $216.• In either case, you lock in an arbitrage risk-less

profit. Therefore, the only reasonable value of the contract must be $33.16 per share, i.e.,

33.16 = 30 e0.05x2 or in general F0 = S0 e r T .

• When the stock has an annual dividend yield of q%, the above formula can be modified to

F0 = S0 e (rq) T . (Exercise)• In practice, investment managers often use stock

index futures to hedge against the risk of her underlying portfolio. For example, the S&P500 is worth US$250 times the level of the index while the Hang Seng index is worth HK$50 times the level of the index.

• Consider a 3-month future contract of S&P500. Suppose that the stocks in the index provide a dividend yield of 3% per year, the risk-free rate is 8% per year, and the current index level is 900. Then in 3 months, the future contract should be priced at the level 900e(0.080.03)/4 = 911.32. Therefore, each future contract would be worth US$ 250x911.32 = US$227,830.

• Consider a portfolio with =1. Since such a portfolio mirrors the market, the position of the future contract should be chosen so that the value of the stocks underlying the future contract should equal to the total value of the portfolio being hedged.

• For a portfolio with =2, the portfolio is twice volatile than the market. The position of the future contract should be chosen so that the value of the stocks underlying the future contract should equal to twice the total value of the portfolio being hedged.

• For a portfolio with =1/2, the portfolio is half volatile than the market. The position of the future contract should be chosen so that the value of the stocks underlying the future contract should equal to half the total value of the portfolio being hedged.

• In general, let A denote the value of the assets underlying the future contract, P denote the total value of the portfolio, and the beta of the portfolio with the market from CAPM. The correct number of contracts to short, N, is given by

NA = P or N = P/A.• Suppose that you want to hedge the risk of a $2

million portfolio in 3 months. Using an S&P500 future contract maturing in 4 months, we can hedge against the risk of this portfolio as follows. Let the current index level be 1,000, = 1.5, q=2% and rf = 4% per annum. Then the number of contracts required is given by N = 1.5x2,000,000/ (1000x250) = 12.

• Suppose S&P drops to a level of 900 in 3 months, bad news for your portfolio. That’s why you need to hedge against the risk.

• The value of the future contract in 4 months is (using arbitrage free argument) is

1000 e(0.04-0.02)/3 = 1006.7.• After three months, the price of the contract for the

remaining one month is

900 e(0.04-0.02)/12 =901.5.• Therefore, the gain on the future position is

(1006.7– 901.5) x 250 x 12 = $315,600.• Index lost 10% in three months (from 1000 to 900),

and there is a dividend yield for the index of 2% per year (which means 0.5% per three months). Therefore, the actual three months return of the index is –9.5%.

• Recall risk free rate is 1% in three months. According to CAPM, the expected return of your portfolio satisfies the CML,

p rf = p(m rf) so that

p = 1 + 1.5(– 9.5 – 1) = –14.75%.

• Thus, the expected value of your portfolio after three months (inclusive of dividends) is

• 2,000,000 x (1 – 0.1475) = $1.705 millions.• The balance of your position after three months

becomes $1.705 + $0.3156 = $2.0206 millions.

• The return of your portfolio including hedging is 0.02 million, about 1%, which matches the risk free rate for three months.

• The loss in the portfolio is 0.1475x2 million = $295,000, resulting in a net gain of

315,600 – 295,000 = $20,600, about 1% of the value of the portfolio, i.e., the risk-free rate in 3 months.

• If the market goes up in 3 months instead, we will gain in the portfolio but lose in the future contracts. However, it can be shown that the same return, which equals to the risk-free rate, is realized regardless of the market performance. This is known as the perfect hedge case.

Summary

• P=2.0 million, rf = 4% annual, or 1% in 3 months. Dividend yield q = 2% annual or 0.5% in 3 months.

• Future contract matures in 4 months, you only want to hedge the risk of your portfolio for 3 months. There is a one month time lag.

• S&P500 current level is 1,000, N =12 contracts and each point in S&P500 equals to US$250.

• Holding on to a 2 million portfolio and getting worried.

• Market goes down 9.5% in 3 months accounting for dividends, which means 10.5% below risk-free rate.

• Portfolio with a beta =1.5, shrinking by 1.5x10.5= 15.75% below risk-free rate or 14.75% by itself.

• Portfolio is expected to shrink by 2x0.1475 = $295,000 in 3 months.

• A 2 million portfolio is expected to shrink to 2(1 – 0.1475) = $1,705,000 in 3 months.

• You are really worried and you want to seek help. Is there something you can do?

• Ask your friend from RMS for help. He said Hedge! But how?!!

• You worry about the market going down, so short sell it to make a profit.

• Recall the market goes down 10.5% below risk-free rate after accounting for a dividend yield of 0.5% in 3 months.

• The index drops from 1000 to 900 in 3 months, resulting in a return of –10%. When a dividend yield of 0.5% in is taken into account, the final return of the index is –9.5%.

• But according to no arbitrage argument, the future contract in 4 months should be equal to 1000e(0.04-0.02)/3 = 1006.7.

• If you enter into a future contract which sells the level at 1006.7, and if the market falls below it, then you can profit from the spread. This is known as a short selling technique.

• However, there is still one month time spread between your holding period and the maturity of the contract.

• We need to adjust for it by 900e(0.04-0.02)/12 = 901.5.

• The actual spread becomes 1006.7 – 901.5 = 105.2. This is the profit that will be made from the short selling of the future contract.

• Since each point is worth $250 and there are 12 contracts, you will realize a profit of 250x12x105.2 = $315,600.

• The actual gain is this amount minus the loss from the portfolio and is equal to315,600– 295,000 =$20,600.

• This is almost equal to $2 M x1% = 20,000.

Hedging with Future Contract

Today, managing a 2.0mil portfolio, and S&P at 900. The current future price of a 4 month contract is 1006.7.

3 months after, market goes down 10% and S&P drops to 900. Portfolio is expected to shrink to 1.705M. The future price of the same 4-month contract, which matures in one month from now, becomes 901.5.

Close out your position and taking a profit of (1006.7-901.5) = 105.2.

Performance Evaluation

We can use the CAPM to evaluate the performance of a fund. Suppose that there is an ABC fund with the10-year record of rates of return as shown in Table 3.1.

Year ABC S&P500 T-Bill

1 14 12 7

2 10 7 7.5

3 19 20 7.7

4 -8 -2 7.5

5 23 12 8.5

6 28 23 8

7 20 17 7.3

8 14 20 7

9 -9 -5 7.5

10 19 16 8

• We can calculate the mean, variance, and covariance, and beta according the sample version formulae.

• According to the market model, ABC – rf = J + (M – rf). If CAPM was true, then J=0. Note that this is very

similar to the SML, except we replace i by ABC , but that’s the best we can do in this situation. The quantity J is known as the Jensen index. It represents the deviation from the CAPM for the fund. J>0 implies that the fund is performing better than the market. For the ABC fund, J = 0.00104 > 0, see Figure 3.2. Can we conclude that this is a better fund than the market and the one fund should just be ABC, not the market?

Figure 3.2

• Since n=10, it is not clear how reliable are our estimates. More importantly, is ABC efficient? To answer this question, consider the straight line

ABC – rf = S.• S is the slope of the line drawn between the risk-

free point and the ABC fund on the (-) diagram. S is also known as the Sharpe ratio or Sharpe index. If ABC were efficient, S=K= price of risk, which equals to the slope of the CML. If ABC were efficient, then this slope would be maximum.

• However, for ABC, S=0.43566 while for S&P, S=0.46669. ABC is not as efficient as the market S&P, see Figure 3.3.

• In conclusion, while ABC is worth holding, it is not quite efficient. It would be necessary to supplement it with other assets to attain efficiency.

Figure 3.3