University of Washington
Department of Economics

Spring 2006
Eric Zivot

Economics 424
Final Exam Solutions

This is a closed book and closed note exam. However, you are allowed one page of handwritten
notes. Answer all questions and write all answers in a blue book. Total points = 100.

I. Matrix Algebra and Portfolio Math (25 points)
Let Ri denote the continuously compounded return on asset i (i = 1, …, N) with E[Ri] = μi,var(Ri) = σ 2 and cov(Ri, Rj) = σij. Define the (N × 1) vectors R = ( R ,… , R )′ , μ = ( μ ,… , μ )′ ,i1N1Nm = ( m1 ,… , mN )′ , x = ( x1 ,… , xN )′ , y = ( y1 ,… , y N )′ , t = ( t1 ,… , t N )′ , 1 = (1,… ,1)′ and the (N ×N) covariance matrix⎛ σ 12⎜σΣ = ⎜ 12⎜⎜⎜⎝ σ 1Nσ 12σ 22σ 2Nσ 1N ⎞⎟σ 2N ⎟⎟⎟σ N2 ⎟⎠.

The vectors m, x, and y contain portfolio weights that sum to one. Using simple matrix algebra,answer the following questions.
a. For the portfolio defined by the vector x, give the expression for the portfolio return (Rp,x), theportfolio expected return (μp,x), and the portfolio variance ( σ p2 , x ).
R p , x = R′x, μ p , x = x′μ, σ p2 , x = x′Σxb. Write down the optimization problem used to determine the global minimum variance
portfolio assuming short sales are allowed. Let m denote the vector of portfolio weights in theglobal minimum variance portfolio.
min m′Σm s.t. m′1 = 1mc. Write down the optimization problem used to determine an efficient portfolio with target
return equal to μ 0 assuming short sales are allowed. Let x denote the vector of portfolio weights
in the efficient portfolio.
min x′Σx s.t. x′1 = 1 and x′μ = μ0x

d. Using the results from parts b. and c., briefly describe how you can compute the efficient
frontier containing only risky assets (Markowitz bullet) using just two efficient portfolios. Givethe expressions for the expected return and variance for portfolios on the efficient frontier.
Any portfolio on the Markowitz bullet can be computed as a convex combination of any othertwo portfolios on the bullet. For example, one portfolio could be the global minimum variance
portfolio m and the other could be the portfolio x that has target return μ 0 . Then, any efficient
portfolio has the form

z = α m + (1 − α ) x
μ p,z = αμ p,m + (1 − α ) μ p,xσ p2 , z = α 2σ p2 ,m + (1 − α ) 2 σ p2 , x + 2α (1 − α )σ mx
σ mx = m′Σxe. Write down the optimization problem used to determine the tangency portfolio, assuming
short sales are allowed and the risk free rate is give by rf . Let t denote the vector of portfolio
weights in the tangency portfolio.

max
tt ′μ - rf( t ′Σt )1/ 2

s.t. t ′1 = 1

II. Efficient Portfolios (40 points)
The graph below shows the efficient frontier computed from three Vanguard mutual funds: S&P
500 index (vfinx), European stock index (veurx), and long-term bond index (vbltx).

Portfolio Frontier
3.50%

3.00%

2.50%
vfinx
veurx
2.00%

vbltxERtangency port

global min
frontier

veurx

1.50%

T-bills
tangencyvfinx

1.00%

global min port
0.50%

vbltx

T-Bills
0.00%
0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

SD

Figure 1 Markowitz Bullet

8.00%

9.00%

Expected return and standard deviation estimates for specific assets are summarized in the tablebelow. These estimates are based on monthly continuously compounded return data over thethree year period January 2003 – January 2006.
Table 1 Portfolio Statistics

AssetMean (E[R])Standarddeviation (SD(R))

VFINXVEURX
VBLTX
T-Bills
Global Min
Portfolio
Tangency Portfolio

1.11%
1.68%
0.51%
0.42%
0.69%

2.60%
3.70%
2.85%
0%
1.83%

1.88%

4.23%

Weight in
Global Min
Portfolio
83%
-27%
44%

Weight in
Tangency
portfolio
-44%
139%
5%

The pair-wise scatter plots of the returns and sample correlation matrix are given below:
0.0

0.05

0.10

0.02

0.06

-0.05

0.05

0.10

-0.02

vfinx

-0.02

0.04

-0.05

0.0veurx

-0.08

vbltx

-0.02

0.02

0.06

-0.08

Figure 2 Return Scatter Plots> print(cor.mat, digits=4)
vfinx
veurx
vbltx
vfinx 1.00000 0.85441 -0.01635-0.04

0.00.04

veurx 0.85441 1.00000
vbltx -0.01635 0.04798

0.04798
1.00000

Using the above information, please answer the following questions.
a. Based on the pair-wise scatterplots and sample correlations, do you think there will bediversification benefits from forming portfolios of the three mutual funds? If yes, which asset
should provide the largest diversification benefit? Briefly justify your answer.
Diversification works best when assets are not perfectly positively correlated. Here the 3 assets
are imperfectly correlated so some diversification should be possible. Since vbltx is essentially
uncorrelated with vfinx and veurx, the biggest diversification benefit should come from vbltx.
b. Find the efficient portfolio of risky assets only (e.g. a portfolio on the Markowitz bullet) thathas an expected monthly return equal to 1.5%. In this portfolio, how much is invested in vfinx,veurx, and vbltx? (Hint: recall, any portfolio on the Markowitz bullet can be expressed as aconvex combination of any other two portfolios on the Markowitz bullet)
Here, we make use of the fact that the global minimum variance portfolio and the tangency
portfolio are on the Markowitz bullet. Therefore, the expected return for any portfolio on theMarkowitz bullet can be expressed as

μ p , z = αμ p ,m + (1 − α ) μ p ,tSetting μ p , z = 0.015 and using μ p ,m = 0.0069, μ p ,t = 0.0188 we can solve for α:

α=μ p , z − μ p ,t0.015 − 0.018== 0.32
μ p ,m − μ p ,t 0.0069 − 0.018

1 − α = 0.68

The weights on vfinx, veurx and vbltx in this portfolio arez = α m + (1 − α ) x⎛ .83 ⎞
⎛ −.44 ⎞ ⎛ 0.03 ⎞⎜⎟= (0.32) ⎜ −.27 ⎟ + (0.68) ⎜⎜ 1.39 ⎟⎟ = ⎜⎜ 0.86 ⎟⎟⎜ .44 ⎟
⎜ .05 ⎟ ⎜ 0.17 ⎟
⎝
⎠⎝⎠ ⎝
⎠
c. Transfer the graph of the Markowitz bullet to your blue book. On this graph indicate theefficient combinations of T-bills and risky assets. Also, indicate a portfolio that would bepreferred by a very risk averse investor and a portfolio that would be preferred by a risk tolerant
investor.

d. How much should be invested in T-bills and the tangency portfolio to create an efficient
portfolio with the same expected return as vfinx? What is the standard deviation of this efficient
portfolio? Transfer the graph of the Markowitz bullet to your blue book and indicate the locationof this efficient portfolio on the graph.
xtan =

μvfinx − rf .0111 − 0.0042== 0.47μtan − rf .0188 − 0.0042x f = 1 − xtan = 1 − 0.47 = 0.53

σ p ,e = xtanσ tan = (0.47)(0.0423) = 0.02
e. In the efficient portfolio you found in part d, what are the shares of wealth invested in vfinx,veurx, and vbltx?
vfinx
0.47*(-0.44)=-0.21

veurx
0.47*1.39=0.66

vbltx
0.47*(0.05)=0.02

f. Assume that the continuously compounded returns for all assets are normally distributed. For
an initial investment of $100,000, compute the 1% value-at-risk (VaR) on vfinx and the efficientportfolio that has the same expected return as vfinx. (Hint: the 1% quantile of a standard normal
random variable is -2.33). Which portfolio has the smallest 1% VaR?
First we find the 1% quantiles for the 2 portfolios
vfinx : q.01 = 0.011 + (0.026)( −2.33) = −0.0495
port : q.01 = 0.011 + (0.020)( −2.33) = −0.0356
Then we compute the 1% VaR

vfinx : VaR.01 = $100,000 × ( e0.0495 − 1) = −$4,833
port : VaR.01 = $100,000 × ( e0.0356 − 1) = −$3, 499
g. How much should be invested in T-bills and the tangency portfolio to create an efficient
portfolio with the same standard deviation as vfinx? What is the expected return on thisportfolio? Transfer the above graph to your blue book and indicate the location of this efficient
portfolio on the graph.
xtan =

σ vfinx 0.0260== 0.62σ tan 0.0423

x f = 1 − xtan = 1 − 0.62 = 0.38

μ p ,e = rf + xtan ( μtan − rf ) = 0.0042 + 0.62(0.0423 − 0.0042) = 0.0132
h. The global minimum variance portfolio has a negative weight on veurx. What is theinterpretation of a negative weight? Is the negative weight problematic? Why or why not?
A negative weight represents a short-sale of the asset. That is, you borrow the asset; sell it; use
the proceeds to purchase more of the other assets. A short sale is sometime problematic.
Sometimes you are prevented from short selling certain assets. For this example, short selling isproblematic because you cannot short-sell a mutual fund.
III. Empirical Analysis of the single index model (35 points)
The single index model for asset returns has the form
Rit = α i + β i RMt + ε iti = 1,… , n assets
t = 1,… , T time periods

a. What are the assumptions of the SI model?

ε it ~ iid N(0, σ ε2 )i

RMt ~ iid N( μ M , σ M2 )cov( RMt , ε it ) = 0
cov(ε it , ε jt ) = 0 for all i ≠ jcov(ε it , ε js ) = 0 for all i ≠ j; t ≠ sb. What is the interpretation of ε it in the SI model?

ε it represents random news that is unrelated to the market.
The following represents S-PLUS linear regression output from estimating the single index
model for the Vanguard long-term bond index (vbltx) and the Vanguard European Equity index(veurx) using monthly continuously compounded return data over the 3 year period January
2003 –January 2006. In the regressions, the market index is the Vanguard S&P 500 index(vfinx).

> summary(vbltx.fit,cor=F)
Call: lm(formula = vbltx ~ vfinx, data = final.ts)
Residuals:
Min1Q
Median3Q
Max-0.09582 -0.0164 0.008568 0.01776 0.04875
Coefficients:
Value Std. Error t value Pr(>|t|)(Intercept) 0.0053 0.0053
1.0080 0.3206
vfinx -0.0179 0.1880-0.0953 0.9246
Residual standard error: 0.02894 on 34 degrees of freedom
Multiple R-Squared: 0.0002673
F-statistic: 0.009091 on 1 and 34 degrees of freedom, the p-value
is 0.9246

0.02
0.0
-0.04
-0.08

long-term bond index returns

0.04

SI regression for long-term bonds-0.02

0.00.02
S&P 500 returns

0.04

0.06

0.08

> summary(veurx.fit,cor=F)
Call: lm(formula = veurx ~ vfinx, data = final.ts)
Residuals:
Min1Q
Median3Q
Max-0.04203 -0.01184 -0.001937 0.0134 0.03397
Coefficients:
Value Std. Error t value Pr(>|t|)(Intercept) 0.0033 0.0035
0.9411 0.3533
vfinx 1.2155 0.1268
9.5881 0.0000
Residual standard error: 0.01952 on 34 degrees of freedom
Multiple R-Squared: 0.73
F-statistic: 91.93 on 1 and 34 degrees of freedom, the p-value is3.387e-011

0.05
0.0-0.05

European stock index returns

0.10

SI regression for European stock index

-0.02

0.0

0.02

0.04

0.06

0.08

S&P 500 returns

c. For the long-term bond index and the European equity index, what are the estimated values of
α and β and what are the estimated standard errors for these estimates? Is β for the Europeanequity index estimated more precisely than β for the long-term bond index? Why or why not?

Long-term bondEuropean equity

α̂SE (αˆ )βˆ0.00530.00330.00530.0035-0.0179
1.2155SE ( βˆ )0.18800.1268The value of SE ( βˆ ) is smaller for the European stock index, which indicates that βveurx is
estimated more precisely. This is to be expected since the European stock index is a welldiversified portfolio whereas the long-term bond index is not very diversified. Diversification
concentrates the risk of the European stock index on market risk and forces the returns to moreclosely follow the market index return. This leads to a smaller error variance and hence a more
precise estimate of β.
d. Based on the estimated values of β, what can you say about the risk characteristics of the longterm bond index and the European equity index relative to the market index?

The estimated beta for the long-term bond is negative and close to zero. This means that adding
the short-term bond index to the market portfolio will lower the variability of the market index.
Thus the long-term bond is much less risky than the extended market index. Since the Europeanequity beta is about 1.2, adding it to the market index will raise the variability. Thus theEuropean equity index is more risky than the market index.

e. For each regression, what is the proportion of market or systematic risk and what is theproportion of firm specific or unsystematic risk?

Long-term bondEuropean Equity

R-square (% market risk)
0.00
0.73

1-(R-square) (% non-market risk)
1.00
0.27

f. Why should the European equity index have a greater proportion of market risk than the longterm bond index?

The European equity index is a large diversified portfolio of stocks, whereas the long-term bondindex is a small, not diversified, index of long-term bonds. The diversification effect concentrates
the market risk of the European equity index and reduces the amount of non-market risk. Thistends to increase the R-square (% of market risk). Long-term bonds generally do not have muchmarket risk since long-term interest rates are fairly stable. Hence the R-square of the bond index
is expected to be low (close to zero). This is what we see in the data.
g. The following graphs show the 24-month rolling estimates of β for the SI models for the longterm bond index and the European equity index. Using these graphs, what can you say about thestability of β over time?

-0.20

-0.15

-0.10

-0.05

0.00

24 month rolling betas for long-term bond index

Dec
2004

JanFebMar

AprMayJunJul
2005

AugSep

Oct

Nov

Dec

Oct

Nov

Dec

1.05

1.10

1.15

1.20

1.25

1.30

1.35

24 month rolling betas for European equity index

Dec
2004

JanFebMar

AprMayJun

Jul2005

AugSep

The rolling estimates reveal instability in the estimated b’s. The rolling b’s on the long-termbond index start out close to zero, dip to negative values in the middle of the sample, and thenreturn to zero at the end of the sample. The rolling b’s for the European stock index steadily
decline over the sample from about 1.3 down to 1.