Department of Economics
University of California, Berkeley
November 9, 2003
Financial Economics
Economics 136
Fall 2006
Economics 136: Financial Economics
Section Notes for Week 11
1
Capital Allocation Between a Risky Portfolio and a Risk-Free
Asset
This material is in BKM chapter 7.
Assume for the moment that an investor must decide how to invest all of her wealth and has
only two options: a risk-free asset such as Treasury Bills (T-Bills) and a risky portfolio of
stocks (such as a mutual fund). Since all of her wealth must be invested, the decision that
she makes can be summarized by one parameter, the fraction of her wealth that she invests
in the risky portfolio, w. Since she must allocate all of her wealth to either the mutual fund
or T-Bill, the fraction of her wealth invested in T-Bill must be, 1 − w.
If we assume a functional form for the investors objective (or utility) function, then we can
determine the optimal fraction of wealth for the investor to put into the risky portfolio, w∗ .
Let RP denote the returns of the combined T-Bill and mutual fund portfolio and RM F denote
the returns of the mutual fund. Assuming that people like high mean return and dislike high
return variance then they would like to solve the following optimization,
1
1 2
max E[RP ] − A · V ar(RP ) = max Rf + w(E[RM F ] − Rf ) − w · A · V ar(RM F )
w
w
2
2
Then the first order condition (foc) is
0 = E[RM F ] − Rf − w · A · V ar(RM F )
E[RM F ] − Rf
w∗ =
A · V ar(RM F )
Note: w∗ gives the optimal fraction of wealth invested in the risky portfolio. The total
portfolio which invests w∗ in the risky portfolio and 1 − w∗ in the risk-free asset will result
in the optimal portfolio which has an expected return and standard deviation such that the
investor’s indifference curve is tangent to the CAL at this point. This is why it is optimal.
It gives the investor the highest possible utiity subject to the mean and standard deviation
of investment possibilities (the CAL is the frontier of these possibilities).
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2
Finding the Optimal Risky Prtfolio
This material is in BKM chapter 8.
In the last section, we took the expected return and standard deviation of the risky portfolio
as given. Now we consider how to construct the best risky portfolio from a number of risky
assets (2 in this case, but N in general). The notion of a “best” risky portfolio can have
several interpretations. First, I consider finding the risky portfolio that has the smallest
possible variance. Then, I look at the risky portfolio that results in the maximum invetor
utility when it is subsequently combined with a risk-free asset. This will be the risky portfolio which maximizes the slope of the CAL, or equivalently, has the maximum Sharpe ratio.
Assume there are only two risky assets which have returns RE and RD (think of a risky
equity fund, and a risky bond (debt) fund). The expected portfolio return and standard
deviation of the risky portfolio RRP are
E[RRP ] = wE E[RE ] + (1 − wE )E[RD ]
p
V ar(RRP )
σ(RRP ) =
2
1
= wE V ar(RE ) + (1 − wE )2 V ar(RD ) + 2wE (1 − wE )Cov(RE , RD ) 2
To find the minimum variance portfolio we minimize the variance of the risky portfolio,
min V ar(RRP ) = min wE2 V ar(RE ) + (1 − wE )2 V ar(RD ) + 2wE (1 − wE )Cov(RE , RD )
wE
wE
the first order condtion (foc) sets
dV ar(RRP )
wE
= 0.
2wE V ar(RE ) + 2(wE − 1)V ar(RD ) + 2(1 − 2wE )Cov(RE , RD ) = 0
wE V ar(RE ) + wE V ar(RD ) − 2wE Cov(RE , RD ) = V ar(RD ) − Cov(RE , RD )
∗
wE,min−var
=
V ar(RD ) − Cov(RE , RD )
V ar(RE ) + V ar(RD ) + 2Cov(RE , RD )
To find the fraction of wealth to invest in equity that will result in the risky portfolio with
the maximum Sharpe ratio
max SP =
wE
=
E[RP ] − Rf
σ(RP )
wE E[RE ] + (1 − wE )E[RD ] − Rf
1
[wE2 V ar(RE ) + (1 − wE )2 V ar(RD ) + 2wE (1 − wE )Cov(RE , RD )] 2
the first order condtion (foc) sets
dSP (wE )
wE
=0
dSP (wE )
E[RE ] − E[RD ]
=
1
wE
[wE2 V ar(RE ) + (1 − wE )2 V ar(RD ) + 2wE (1 − wE )Cov(RE , RD )] 2
1
wE E[RE ] + (1 − wE )E[RD ] − Rf
− ·
2 [w2 V ar(RE ) + (1 − wE )2 V ar(RD ) + 2wE (1 − wE )Cov(RE , RD )] 23
E
·(2wE V ar(RE ) − 2(1 − wE )V ar(RD ) + 2(1 − wE )Cov(RE , RD )) = 0
2
(E[RE ] − E[RD ])
2
2
· wE V ar(RE ) + (1 − wE ) V ar(RD ) + 2wE (1 − wE )Cov(RE , RD ) =
1
· (wE E[RE ] + (1 − wE )E[RD ] − Rf )
2
·(2wE V ar(RE ) − 2(1 − wE )V ar(RD ) + 2(1 − wE )Cov(RE , RD ))
Fill in the steps here. Let me know if I’ve made any mistakes. The next line is correct.
wE∗ =
(E[RD ] − Rf )V ar(RE ) − (E[RE ] − Rf )Cov(RE , RD )
(E[RD ] − Rf )V ar(RE ) + (E[RE ] − Rf )V ar(RD ) − (E[RD ] + E[RE ] − 2Rf )Cov(RE , RD )
While this first order condition is rather messy, the concept of maximizing the Sharpe ratio
by finding the risky portfolio that results in the Capital Allocation Line (CAL) with the
highest slope is easy to see graphically.
3
Example: BKM chapter 8, problems 1-8
A pension fund manager is considering three mutual funds. The first is a stock fund, the
second is a long-term government and corporate bond fund, and the third is a T-Bill money
market fund. The correlation betwenn the stock fund and the bond fund is ρ(RS , RB ) = 0.1.
Stock Fund (RE )
Bond Fund (RD )
T-Bill Fund (Rf )
Expected Return Standard Deviation
20%
30%
12%
15%
8%
0%
1. What are the investment proportions in the minimum-variance portfolio of the two risky
funds, and what is the expected value and standard deviation of its rate of return?
ans: First let’s find the covariance of the debt and equity fund returns
Cov(RE , RD ) = σ(RE )σ(RD )ρ(RE , RD )
= 0.30 ∗ 0.15 ∗ 0.10 = 0.0045
V ar(RD ) − Cov(RE , RD )
V ar(RE ) + V ar(RD ) + 2Cov(RE , RD )
0.152 − 0.0045
=
= 0.1739
0.302 + 0.152 + 2 ∗ 0.0045
∗
wE,min−var
=
E[RRP ] = 0.1739 ∗ 0.20 + (1 − 0.1739) ∗ 0.12 = 13.39%
p
σ(RRP ) =
V ar(RRP )
1
= 0.17392 ∗ 0.302 + (1 − 0.1739)2 ∗ 0.152 + 2 ∗ 0.1739 ∗ (1 − 0.1739) ∗ 0.0045 2
= 13.92%
2. Tabulate and draw the investment opportunity set of the two risky funds. Use investment
proportions for the stock funds of zero to 100% in increments of 20%.
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ans:
w Expected Return Standard Deviation
0
12%
15%
0.2
13.6%
13.9%
0.4
15.2%
15.7%
0.6
16.8%
19.5%
0.8
18.4%
24.5%
20%
30%
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3. Draw a tangent from the risk-free rate to the opportunity set. What does your graph
show for the expected return and standard deviation of the optimal risky portfolio.
ans:
4. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky porfolio.
ans: The fraction of the portfolio invested in equity is
(E[RD ] − Rf )V ar(RE ) − (E[RE ] − Rf )Cov(RE , RD )
(E[RD ] − Rf )V ar(RE ) + (E[RE ] − Rf )V ar(RD ) − (E[RD ] + E[RE ] − 2Rf )Cov(RE , RD )
(0.12 − 0.08)0.302 − (0.20 − 0.08)0.0045
=
(0.12 − 0.08)0.302 + (0.20 − 0.08)0.152 − (0.12 + 0.20 − 2 ∗ 0.08)0.0045
0.0036 − 0.00054
=
= 0.5484
0.0036 + 0.0027 − 0.00072
wE∗ =
E[RRP ] = wE∗ E[RE ] + (1 − wE∗ )E[RD ] = 16.4%
1
σ(RRP ) = (wE∗ )2 V ar(RE ) + (1 − wE∗ )2 V ar(RD ) + 2wE∗ (1 − wE∗ )Cov(RE , RD ) 2
= 18.4%
5. What is the Sharpe ratio of the best feasible CAL?
ans:
SP∗ =
0.164 − 0.08
E[RP ] − Rf
=
= 0.4565
σ(RP )
0.184
6. You require that your total portfolio (TP) yield an expected return of 14%, and that it
be efficient on the best feasible CAL.
a) What is the standard deviation of your total portfolio?
ans:
4
E[RT P ] = 0.14 = Rf + w(E[RP ] − Rf ) = 0.08 + w(0.164 − 0.08)
0.04
= 0.4762
w =
0.084
σ(RT P ) = w · σ(RP ) = 0.4762 ∗ 0.184 = 8.76%
b) What is the proportion invested in the T-Bill fund and each of the two risky funds?
ans:
Proportion
T-Bill
1 − 0.4762 = 0.5238
Stock Fund
0.4762 ∗ 0.5484 = 0.2611
Bond Fund 0.4762 ∗ (1 − 0.5484) = 0.2151
7. If you were to use only the risky funds, and still require an expected return of 14%, what
must be the investment proportions of your portfolio? Compare its standard deviation to
that of the optimized portfolio you found in 6. What do you conclude?
ans:
14% = wE E[RE ] + (1 − wE )E[RD ]
= wE 20% + (1 − wE )12%
wE = 0.25
1
σ(RP ) = (wE∗ )2 V ar(RE ) + (1 − wE∗ )2 V ar(RD ) + 2wE∗ (1 − wE∗ )Cov(RE , RD ) 2
= 14.1%
The standard deviation in 6 is 8.76% which is lower than 14.1%. By combining the optimal
risky portfolio with the risk-free asset it is possible to acheive the same expected return with
a lower standard deviation, this can be seen graphically.
4
Example: BKM chapter 7, problem 13
13. Consider the following information about a risky portfolio that you manage and a riskfree asset: E[RP ] = 11%, σ(RP ) = 15%, Rf = 5%.
(a) Your client wants to invest a portion of her total investment budget in your risky fund to
provide an expected rate of return on her overall or complete portfolio equal to 8%. What
proportion should she invest in the risky portfolio, P, and what proportion in the risk-free
asset?
ans:
8% = Rf + w(E[RP ] − Rf )
= 5% + w(11% − 5%)
w = 0.5
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(b) What will be the standard deviation of the rate of return on her portfolio?
ans: σ(RT P ) = w · σ(RP ) = 0.5 ∗ 15% = 7.5%
(c) Another client wants the highest return possible subject to the constraint that you limit
his standard deviation to be no more than 12%. Which client is more risk averse?
ans: The first client is more risk averse.
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