Risk Return & The Capital Asset Pricing Model (CAPM)
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Transcript of Risk Return & The Capital Asset Pricing Model (CAPM)
Jacoby, Stangeland and Wajeeh, 2000
1
Risk Return & The Capital Asset Pricing Model (CAPM)
To make “good” (i.e., value-maximizing) financial decisions, one must understands the relationship between risk and return
We accept the notion that investors like returns and dislike risk
Consider the following proxies for return and risk:
Expected return - weighted average of the distribution of possible returns in the future.
Variance of returns - a measure of the dispersion of the distribution of possible returns in the future.
Chapter 10
2
Expected (Ex Ante) Return
An ExampleConsider the following return figures for the following year on stock XYZ under three alternative states of the economy Pk Rk
Probability Return inState of Economy of state k state k
+1% change in GNP 0.25 -5%
+2% change in GNP 0.50 15%
+3% change in GNP 0.25 35%
1.00
SS
S
kkk PRPRPRPRRE
22111
][
where, Rk = the return in state k (there are S states)
Pk = the probability of return k (state k)
3
Q. Calculate the expected return on stock XYZ for the next
year
A.
Expected Returns - An Example
Or, use the formula:
Use the following table Pi Ri Pi Ri
Probability Return inState of Economy of state i state i
State 1: +1% change in GNP 0.25 -5%
State 2: +2% change in GNP 0.50 15%
State 3: +3% change in GNP 0.25 35%
Expected Return =
332211
3
1
][ PRPRPRPRREi
ii
4
Variance and Standard Deviation of Returns
An Example
Recall the return figures for the following year on stock XYZ under three alternative states of the economy Pk Rk
Probability Return inState of Economy of state k state k
State 1: +1% change in GNP 0.25 -5%
State 2: +2% change in GNP 0.50 15%
State 3: +3% change in GNP 0.25 35%
Expected Return = 15.00%
where, Rk = the return in state k (there are S states)
Pk = the probability of return k (state k) and
= the standard deviation of the return:
2222
211
1
22
][][][
][)(V
RERPRERPRERP
RERPRar
SS
S
kkk
2
5
Q. Calculate the variance of and standard deviation of
returns on stock XYZ
A.
Variance & Standard Deviation - An Example
Or, use the formula:
=
Standard deviation:
Use the following table
State of Economy Pk X
(Rk - E[R])2 = Pk(Rk - E[R])2
+1% change in GNP 0.25 0.04
+2% change in GNP 0.50 0.00
+3% change in GNP 0.25 0.04
Variance of Return =0.02
3
1
22 ][ k
kk RERP
%14.141414.002.02
Jacoby, Stangeland and Wajeeh, 2000
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Q. Calculate the expected return on assets A and B for the next
year, given the following distribution of returns:
A. Expected returns
E(RA) = (0.400.30) + (0.60(-0.10)) = 0.06 = 6%
E(RB) = (0.40(-0.05)) + (0.600.25) = 0.13 = 13%
State of the Probability Return on Return oneconomy of state asset A asset B
Boom 0.40 30% -5%Bust 0.60 -10% 25%
Portfolio Return and Risk
Jacoby, Stangeland and Wajeeh, 2000
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Q. Calculate the variance of the above assets A and B
A. Variances
Var(RA) = 0.40(0.30 - 0.06)2 + 0.60(-0.10 - 0.06)2 = 0.0384
Var(RB) = 0.40(-0.05 - 0.13)2 + 0.60(0.25 - 0.13)2 = 0.0216
Q. Calculate the standard deviations of the above assets A and B
A. Standard Deviations
A = (0.0384)1/2 = 0.196 = 19.6%
B = (0.0216)1/2 = 0.147 = 14.7%
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Expected Return on a Portfolio
The Expected Return on Portfolio p with N securities
where,
E[Ri]= expected return of security i
Xi = proportion of portfolio's initial value invested in security i
Example -
Consider a portfolio p with 2 assets: 50% invested in asset A and 50%
invested in asset B. The Portfolio expected return is given by:
E(RP) = XAE(RA) + XBE(RB)
=
][...][][][][ 22111
NN
N
iiip REXREXREXREXRE
Returns and Risk for Portfolios - 2 Assets
Jacoby, Stangeland and Wajeeh, 2000
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Variance of a Portfolio
The variance of portfolio p with two assets (A and B)
where,
Standard Deviation of a Portfolio
The standard deviation of portfolio p with two assets (A and B)
ABBABBAA
pp
XXXX
RVar
2
)( 2222
2
][][),( 1
,,
S
iBiBAiAiBAAB RERRERPRRCOV
5.02222 2
)(
ABBABBAA
pp
XXXX
RVar
10
Q. Calculate the variance of portfolio p (50% in A and 50% in B)
A. Recall: Var(RA) = 0.0384, and Var(RB) = 0.0216
First, we need to calculate the covariance b/w A and B:
The variance of portfolio p
Q. Calculate the standard deviations of portfolio pA. Standard Deviations
p = (0.0006)1/2 = 0.0245 = 2.45%
][][),( 2
1,,
i
BiBAiAiBAAB RERRERPRRCOV
ABBABBAA
pp
XXXX
RVar
2
)( 2222
2
11
Note: E(RP) = XAE(RA) + XBE(RB) = 9.5%, but
Var(Rp) =0.0006 < XAVar(RA) + XBVar(RB)
= (0.500.0384) + (0.500.0216) = 0.03 This means that by combining assets A and B into portfolio p, we eliminate
some risk (mainly due to the covariance term)
Diversification - Strategy designed to reduce risk by spreading the portfolio across many investments
Two types of Risk:
Unsystematic/unique/asset-specific risks - can be diversified away
Systematic or “market” risks - can’t be diversified away
In general, a well diversified portfolio can be created by randomly combining 25 risky securities into a portfolio (with little (no) cost).
The Effect of Diversification on Portfolio Risk
Jacoby, Stangeland and Wajeeh, 2000
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Portfolio Diversification
Average annualstandard deviation (%)
Number of stocksin portfolio
Diversifiable (nonsystematic) risk
Nondiversifiable(systematic) risk
49.2
23.9
19.2
1 10 20 30 40 1000
Jacoby, Stangeland and Wajeeh, 2000
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Beta and Unique Risk Total risk = diversifiable risk + market risk
We assume that diversification is costless, thus diversifiable (nonsystematic) risk is irrelevant
Investors should only care about nondiversifiable (systematic) market risk
Market risk is measured by beta - the sensitivity to market changes
Example: Return (%)
State of the economy TSE300 BCE
Good 18 26
Poor 6 -4
14
Interpretation: Following a change of +1% (-1%) in the market return,
the return on BCE will change by +2.5% (-2.5%)
NOTE: If the security has a -ve cov w/ TSE 300 =>
Beta and Market Risk
300TSEr
BCEr
• (6%, -4%)
• (18%, 26%)
The Characteristic Line
badgood
badgood
TSETSE
BCEBCEBCE rr
rr
,,
,,
0BCE
Jacoby, Stangeland and Wajeeh, 2000
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Beta and Unique Risk Market Portfolio - Portfolio of all assets in the economy. In practice a
broad stock market index, such as the TSE300, is used to represent the market
Beta ()- Sensitivity of a stock’s return to the return on the market portfolio
2m
imi
Covariance of security i’s return with the market return
Variance of market return
Jacoby, Stangeland and Wajeeh, 2000
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Markowitz Portfolio Theory We saw that combining stocks into portfolios can reduce standard
deviation
Covariance, or the correlation coefficient, make this possible:
The standard deviation of portfolio p (with XA in A and XB in B):
Note: , or
Thus,
212 2222ABBABBAAp XXXX
BA
ABAB
BAABAB
21)(2 2222BAABBABBAAp XXXX
Jacoby, Stangeland and Wajeeh, 2000
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Markowitz Portfolio Theory - An Example Consider assets Y and Z, with
Consider portfolio p consisting of both Y and Z. Then, we have:
Expected Return of p
Standard Deviation of p
21)(2 2222ZYYZZYZZYYp XXXX
10247.00105.0 , %20][ Y YRE
012.0000144.0 , %4.14][ Z ZRE
][][][ ZZYYp REXREXRE
%4.14%20 ZY XX
21012.010247.02000144.00105.0 22YZZYZY XXXX
18
Look at the next 3 cases (for the correlation coefficient):
Where
Expected Return of Portfolio
Standard deviation of a portfolio
Portfolio YZ = -1
YZ = +1 YZ = 0
1 18.6% 7.38% 7.98% 7.69% 2 17.2 4.52 5.72 5.16 3 15.8 1.66 3.46 2.72
Portfolio
1 2 3
YX 0.75 0.50 0.25 ZX 0.25 0.50 0.75
11generalIn ij
Jacoby, Stangeland and Wajeeh, 2000
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.
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
22%
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%
][ pRE
p
The Shape of the Markowitz Frontier - An Example
Z.
Y
= -1
= 0
= +1
Jacoby, Stangeland and Wajeeh, 2000
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Efficient Sets and Diversification
= -1
-1 <
= 1
][ pRE
p
21
The Efficient (Markowitz) FrontierThe 2-Asset Case
Stock Z
Stock Y
Standard Deviation
Expected Return (%)
75% in Z and 25% in Y
Expected Returns and Standard Deviations vary given different weighted combinations of the two stocks
The Feasible Set is on the curve Z-Y
The Efficient Set is on the MV-Y segment only
Minimum Variance Portfolio (MV)
MV
22Standard Deviation
Expected Return (%)
The Efficient (Markowitz) FrontierThe Multi-Asset Case
Each half egg shell represents the possible weighted combinations for two assets
The Feasible Set is on and inside the envelope curve
The composite of all asset sets (envelope), and in particular the segment MV-U constitutes the efficient frontier
Minimum Variance Portfolio (MV)
MV
U
Jacoby, Stangeland and Wajeeh, 2000
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Efficient Frontier
Goal is to move UP and LEFT.
WHY?
We assume that investors are rational (prefer more to less) and risk averse
Expected
Return (%)
Standard Deviation
(Risk)
Jacoby, Stangeland and Wajeeh, 2000
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Low Risk
High Return
High Risk
High Return
Low Risk
Low Return
High Risk
Low Return
Which Asset Dominates?
Expected
Return (%)
Standard Deviation
(Risk)
Jacoby, Stangeland and Wajeeh, 2000
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Short Selling Definition
The sale of a security that the investor does not own
How?
Borrow the security from your broker and sell it in the open market
Cash Flow
At the initiation of the short sell, your only cash flow, is the proceeds from selling the security
Closing the Short
Eventually you will have to buy the security back in order to return it to the broker
Cash Flow
At the elimination of the short sell, your only cash flow, is the price you have to pay for the security in the open market
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Short Selling A Treasury Bill - An Example The Security
A Treasury bill is a zero-coupon bond issued by the Government, with a face value of $100, and with a maturity no longer than one year
If the yield on a 1-year T-bill is 5%, then its current price is: 100/1.051 = $95.24
The Short sell
Borrow the 1-year T-bill from your broker and sell it in the open market $95.24
Cash Flow
The short sell proceeds: $95.24
Closing the Short
At the end of the year - buy the T-bill back (an instant before it matures) in order to return it to the broker
Cash Flow
The price you have to pay for the T-bill in the open market an instant before maturity (in 1 year): 100/1.050 = $100
Risk-Free Borrowing
This transaction is equivalent to borrowing $95.24 for one year, and paying back $ 100 in a year. The interest rate is: (100/95.24) -1 = 5% = the 1-year T-bill yield
27
•Lending or Borrowing at the risk free rate (Rf) allows us to exist outside the
Markowitz frontier.
•We can create portfolio A by investing in both Rf (lending money) and M
•We can create portfolio B by short selling Rf (borrowing money) and holding M
The Capital Market Line (CML)The Efficient Frontier With Risk-Free Borrowing and Lending
Expected returnof portfolio
Standarddeviation of
portfolio’s return.
Risk-freerate (Rf )
A
M.B
..
CML
CML is the new
efficient frontier
28
Note all securities are in M, and all investors have M in their portfolios since they are all on the new
efficient frontier - CML - investing in Rf and M.
Therefore
Investors are only concerned with and , and
with the contribution of each security i to M, in terms of contribution of systematic risk (measured by beta) contribution of expected return
According to the CAPM:
where,
m][ mRE
The Capital Asset Pricing Model (CAPM)
][][ fmifi RRERRE
1: 2
2
2
22
m
m
m
mmm
m
i
m
miim
m
im
NOTE
imi
29
The Security Market Line (SML)
The Capital Asset Pricing Model (SML):
Note:
(1) -> entire risk of i is diversified away in M
(2) -> security i contributes the average risk of M
fiii RRE ][0but 0 ][][1 mii RERE
][ iRE
i
. M
SML
1
][ mRE
fR
][][ fmifi RRERRE
30
The Security Market Line (SML)
The SML is always linear CML - just for efficient portfolios
SML - for any security and portfolio
(efficient or inefficient)
Example:
Consider stocks A and B, with: a = 0.8, b = 1.2,
let E[Rm] = 14% and Rf = 4%. By the SML:
E[Ra] =
E[Rb] =
Consider a portfolio p, with 60% invested in A and 40% invested in B, then:
E[Rp] = XaE[Ra] + XbE[Rb] = 0.612% + 0.416% = 13.6%,
and p = Xa a + Xb b = 0.60.8 + 0.41.2 = 0.96
By the CAPM: E[Rp] = 4% + 0.96[14% - 4%] = 13.6%
* If A and B are on the SML => P is also on SML