University of Washington
Department of Economics

Spring 2006
Eric Zivot

Economics 424
Final ExamThis 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 ).
b. 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 the
global minimum variance portfolio.
c. 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.
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.
e. 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.
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%

vbltx

ER

tangency port

global min
frontier

veurx

1.50%

T-bills
tangency

vfinx

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%

8.00%

9.00%

SDFigure 1 Markowitz Bullet

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

Asset

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

VFINX
VEURX
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 inTangency
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

0.0

veurx

-0.08

vbltx

-0.02

0.02

0.06

-0.08

-0.04

0.0

0.04

Figure 2 Return Scatter Plots> print(cor.mat, digits=4)
vfinx
veurx
vbltx
vfinx 1.00000 0.85441 -0.01635
veurx 0.85441 1.00000 0.04798
vbltx -0.01635 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.
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)
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.
e. In the efficient portfolio you found in part d, what are the shares of wealth invested in vfinx,veurx, and vbltx?
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?
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.
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?

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?
b. What is the interpretation of ε it in the SI model?
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
Median3QMax-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

0.0

0.02
S&P 500 returns

0.04

0.06

0.08

> summary(veurx.fit,cor=F)
Call: lm(formula = veurx ~ vfinx, data = final.ts)
Residuals:
Min1QMedian3QMax-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?
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?
e. For each regression, what is the proportion of market or systematic risk and what is theproportion of firm specific or unsystematic risk?

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

Aug

Sep

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

Jan

Feb

Mar

Apr

May

Jun

Jul
2005

Aug

Sep