UCLA Anderson – MGMT237B: Fundamentals in Finance (Fall 2015)
Professor Eduardo Schwartz
Week #4
November 30, 2015
Handout written by Shenje Hshieh†
1
Mini-Case 1, Question 8
Suppose that you cannot borrow at the risk-free rate. You wish to construct a portfolio with an
expected return of 50%. What are the portfolio weights and the resulting standard deviation? What
reduction in standard deviation could you attain if you were allowed to borrow at the risk-free rate?
Assuming no riskless asset,
E[r] = 0.5 = wMSFT E[rMSFT ] + (1 − wMSFT )E[rGM ]
0.5 − E[rGM ]
≈ 1.484743
wMSFT =
E[rMSFT ] − E[rGM ]
wGM ≡ 1 − wMSFT ≈ −0.4847433
2
2
V ar(r) = wMSFT
V ar(rMSFT ) + wGM
V ar(rGM ) + 2wMSFT wGM Cov(rMSFT , rGM ) ≈ 0.3428892
(1)
Assuming we can now invest in the riskless asset,
E[r] = 0.5 = wT E[rT ] + (1 − wT )rf
0.5 − rf
wT =
≈ 1.553406
E[rT ] − rf
wMSFT = wT wT,MSFT ≈ 1.336773
wGM = wT wT,GM ≈ 0.2166328
wf ≡ 1 − wT ≈ −0.5534057
V ar(r) = wT2 V ar(rT ) ≈ 0.291194
(2)
where wT,MSFT and wT,GM are portfolio weights in the tangency portfolio, which are computed in
question 4. The reduction in standard deviation is
√
2
0.3428892 −
√
0.291194 ≈ 0.04594344
(3)
Global Minimum Variance Portfolio and the Mutual Fund
Theorem
All investors who choose portfolios by examining only mean and variance can be satisfied by holding
different combinations of only a few, in this case two, mutual funds regardless of their preferences.
All of the original assets, therefore, can be purchased by just two mutual funds, and the investors
can just buy these. Here, we want to show that any minimum variance portfolio can be expressed
with two funds.
†
Please email me at [email protected] if there are any errors.
1
2.1
Mean-Variance Analysis without a Riskless Asset Revisited
Recall the setup of the minimization problem:
w∗ = arg min w0 Σw
w
subject to
w0 R = µ
w0 1 = 1
(4)
We can solve this via Lagrange multiplier method. First, we set up the Lagrangian and take first
order conditions with respect to w:
L = w0 Σw − λ1 (w0 R − µ) − λ2 (w0 1 − 1)
∂L
= 2Σw − λ1 R − λ2 1 = 0
∂w
λ1
1 −1 ∗
R 1
w = Σ
λ2
2
(5)
Next, we solve for λ1 and λ2 :
0 ∗ 0
1 R0
R
µ
Rw
∗
−1
R
=
0 w =
0 ∗ =
0 Σ
1
1w
1
2 1
1 R0 Σ−1 R R0 Σ−1 1 λ1
=
=
2 10 Σ−1 R 10 Σ−1 1 λ2
λ1
1
λ2
1
λ
A 1
λ2
2
(6)
Solving for λ1 and λ2 , we get:
λ1
−1 µ
= 2A
λ2
1
(7)
Substituting λ1 and λ2 into w∗ we finally solve for the optimal weights for our portfolio:
∗
w =Σ
2.2
−1
−1 µ
R 1 A
1
(8)
Global Minimum Variance Portfolio
Let us define A = 10 Σ−1 1, B = 10 Σ−1 R, and C = R0 Σ−1 R, which are all scalars. Then, A−1 can
be rewritten as
−1
A
C B
=
B A
−1
1
A −B
=
AC − B 2 −B C
2
(9)
The optimal portfolio weights can be written as
w∗ = λ1 Σ−1 R + λ2 Σ−1 1
(10)
Aµ − B
AC − B 2
C − Bµ
λ2 =
AC − B 2
(11)
where
λ1 =
The variance of the mean-variance efficient optimal portfolio is therefore
0
σ 2 ≡ w∗ Σw∗ =
Aµ2 − 2Bµ + C
AC − B 2
(12)
Taking first order conditions writh respect to µ, we arrive at
∂σ 2
2Aµ − 2B
=
=0
∂µ
AC − B 2
B
µg =
A
Substituting µg into λ1 and λ2 above, we get λ1 = 0 and λ2 =
portfolio is
wg =
2.3
Σ−1 1
Σ−1 1
=
10 Σ−1 1
A
(13)
1
.
A
The global minimum variance
(14)
Two Fund Separation Theorem
From Equation 10, we can see that all minimum-variance portfolios are portfolio combinations of
only two distinct portfolios. Since the global minimum-variance portfolio corresponds to the second
term in Equation 10, we choose for simplicity, the portfolio corresponding to the first term as the
diversified portfolio. Assuming B 6= 0, we define
wd =
Σ−1 R
Σ−1 R
=
10 Σ−1 R
B
(15)
The Equation 10 can be written as
w∗ = (λ1 B)wd + (λ2 A)wg
(16)
Since λ1 B + λ2 A = 1, the two fund separation holds. Note that wd and wg are not unique. We
can take any two portfolios to generate all other portfolios on the frontier.
3
3
3.1
Practice Problems
Chapter 8, Questions 8 (Bodie, Kane, and Marcus 2014)
Consider the two (excess return) index model regression results for A and B:
RA = 1% + 1.2RM
R2 = .576
Residual standard deviation = 10.3%
RB = −2% + .8RM
R2 = .436
Residual standard deviation = 9.1%
3.1.1
(17)
Which stock has more firm-specific risk?
Firm-specific risk is measured by the residual standard deviation. Thus, stock A has more firmspecific risk: 10.3% > 9.1%.
3.1.2
Which has greater market risk?
Market risk is measured by beta, the slope coefficient of the regression. A has a larger beta
coefficient: 1.2 > 0.8.
3.1.3
For which stock does market movement explain a greater fraction of return
variability?
R2 measure the fraction of total variance of return explained by the market return. A’s R2 is larger
than B’s: 0.576 > 0.436.
3.1.4
If rf were constant at 6% and the regression had been run using total rather
than excess returns, what would have been the regression intercept for stock
A?
rA − rf = α + β(rM − rf )
rA = α + rf (1 − β) + βrM
(18)
α + rf (1 − β) = 1% + 6%(1 − 1.2) = −0.2%
(19)
The intercept is now equal to:
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3.2
Chapter 8, Questions 16 (Bodie, Kane, and Marcus 2014)
Based on current dividend yields and expected growth rates, the expected rates of return on stocks
A and B are 11% and 14%, respectively. The beta of stock A is .8, while that stock B is 1.5. The
T-bill rate is currently 6%, while the expected rate of return on the S&P 500 index is 12%. The
standard deviation of stock A is 10% annually, while that of stock B is 11%. If you currently hold
a passive index portfolio, would you choose to add either of these stocks to your holdings?
For stock A:
αA = E[rA ] − (rf + βA (E[rM ] − rf )) = 0.11 − (0.06 + 0.8(0.12 − 0.06)) = 0.002
(20)
For stock B:
αB = E[rB ] − (rf + βB (E[rM ] − rf )) = 0.14 − (0.06 + 1.5(0.12 − 0.06)) = −0.01
(21)
Stock A would be a good addition to a well-diversified portfolio. A short position in stock B may
be desirable.
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