Chapter 10. 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 - PowerPoint PPT Presentation
Jacoby, Stangeland and Wa jeeh, 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
Transcript
Jacoby, Stangeland and Wajeeh, 2000
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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
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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)
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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
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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
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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
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Q. Calculate the variance of the above assets A and B
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
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•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
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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
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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
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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:
