# Document

```Chapter Three
Capital Asset Pricing Model
Perfect Market Conditions
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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=1N 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)
r  ri  (1   )rM .
Then
  i  (1   )M ,
 2   2 i2  (1  )2 M2  2 (1  ) iM .
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
d 2
d
d 
 i   M .
d
 [2 i2  2(1   ) M2  2(1  2 ) iM ].
Since  = (2)½, we have
d 
d
1
1 2  2 d 2
 (  )
2
d
 [ i2  (1   ) M2  (1  2 ) iM ]
Therefore,
Now
d 
d
 0
 iM   M2

.
M
d  d 


d   d 
 d  

.
 d 
1

.
Thus,
d
d 
But
d
d 
 0
 d 
 d  




d

d


  0

  0
   M   .
 i
M
 iM   M2
 Slope of theCML
 0
 M  rf
.
M

 i   M  M

,
 iM   2
 iM   2
  M  rf 
2

Therefore,
 M  rf
M
i   M
M
M
M
Thus,
 i  rf
 iM 
  M  rf  2  1.
  M

 iM 
  M  rf  2 
  M 
  iM  M  rf .
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 = iM2 M2 + i2
= 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 M2 .
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 (rq) 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(0.080.03)/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 =
• 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?
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
• 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
1
2
3
4
5
6
7
8
9
10
ABC
14
10
19
-8
23
28
20
14
-9
19
S&P500
12
7
20
-2
12
23
17
20
-5
16
T-Bill
7
7.5
7.7
7.5
8.5
8
7.3
7
7.5
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 riskfree 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
Compound Interest and
Dividend Yield
•At present, t=0. Invest \$1 at an annual risk-free rate of
r. You receive \$(1+r)T after T years. If the interest is
compounded m periods per year, then you get
\$(1+r/m)mT in T years.
•Suppose the interest is compounded continuously over
time, that is, let m tend to , then recall the fact that
(1+x/m)mT tends to exT as m tends to  from elementary
calculus, we get the compound interest formula that at
the final period T years after, your investment becomes
F = erT.
• Furthermore, if your initial investment is \$P, then your
final value will become F = P erT. We usually call P
the present value and F the final value.
• This idea can also be represented by the net zero cash
flow, or present value analysis. In other words, the
present value must be related to the final value by a
discount factor via P = F erT. The factor erT is known
as the discount factor.
 Consider the dividend yield of a stock (or stock
index). At time t=0, suppose the stock is traded at
\$S per share, suppose the dividend yield is q% for
the period T, and the risk-free rate is r% for the
same period. At t=0, buy eqT share of the stock
(assuming we can buy a fraction of a share,
otherwise, scale it up and the argument remains the
same). At t=T, this investment grows to eqT eqT =1
share of the stock.
• At t=0, short 1 future for 1 share at price F.
How does the cash flow change?
• At t=0, you have an outflow of S eqT .
• At t=T, you receive F from the proceeds of the
future. But this has to be discounted at time 0.
In other words, you should have F erT at time 0.
• For an arbitrage free market, the net cash flow at time
0 must be zero. Therefore, the relationship
SeqT = FerT must hold. That is, F = Se(r  q)T.
• The dividend yield can also be thought of the cost of
carry for the storage of an asset. Since you are
shorting the future, the factor –qT represents the cash
from the dividend you haven’t received and hence has
to be discounted. But in physical assets, this amount
is really the storage cost. This is known as the cost of
carry.
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