University of Washington
Department of Economics

Fall 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 ).
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.

maxtt ′μ - rf( t ′Σt )1/ 2

s.t. t ′1 = 1

II. Efficient Portfolios (35 points)
The graph below shows the efficient frontier computed from three Vanguard mutual funds:
Pacific Stock Index (vpacx), US Long Term Bond Index (vbltx), and Emerging Markets Fund(veiex).

Portfolio Frontier
4.50%
4.00%
3.50%
vpacx

3.00%

vbltx
veiex

ER

2.50%global mintangency port

frontier

2.00%

veiex

T-bills
tangency

1.50%

vpacx
1.00%

global min port
0.50%

vbltx
T-Bills
0.00%
0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

SD

Figure 1 Markowitz Bullet

14.00%

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 the five
year period September 2001 – September 2006.
Table 1 Portfolio Statistics

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

Weight in
Global MinPortfolio

VPACX
VBLTX
VEIEX
T-Bills
Global Min
Portfolio
Tangency
Portfolio
Efficient
Portfolio with
Mean=2.03%

1.09%
0.54%
2.03%
0.25%
0.83%

4.14%
2.68%
5.36%
0%
2.16%

19%
69%
12%

1.35%

2.98%

2.03%

5.21%

Weight in
Efficient
Portfolio
Mean =
2.03%
-35%
22%
113%

Weight in
Tangency
portfolio

-4%
49%
56%

Using the above information, please answer the following questions.
a. Compute annualized means from the monthly means for the three portfolios vpacx, vbltx, andveiex. For each asset, assuming you get the same annual return for 10 years, how much will $1
grow to after 10 years?
The annualized cc return is rA = 12 ⋅ rm . For the three portfolios we haverA,vpacx = 12 ⋅ (.0109) = .1313
rA,vbltx = 12 ⋅ (.0054) = .0653
rA,veiex = 12 ⋅ (.0203) = .2436
The future value after 10 year with continuous compounding is FV = e10⋅rA . For the threeportfolios we have
FVvpacx = exp(10 × .1313) = $3.72
FVvbltx = exp(10 × .0653) = $1.92
FVveiex = exp(10 × .2436) = $11.43
b. Find the efficient portfolio of risky assets only (e.g. a portfolio on the Markowitz bullet) thathas an expected monthly return equal to 3%. In this portfolio, how much is invested in vpacx,vbltx, and veiex?
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 = 0.03 and using μ p ,m = 0.0083, μ p ,t = 0.0135 we can solve for α:

α=μ p , z − μ p ,t0.03 − 0.0135== −3.18
μ p ,m − μ p ,t 0.0083 − 0.0135

1 − α = 4.18
The weights on vfinx, veurx and vbltx in this portfolio arez = α m + (1 − α ) x⎛ .19 ⎞
⎛ −.04 ⎞ ⎛ −.77 ⎞⎜⎟= ( −3.18) ⎜ .69 ⎟ + (4.18) ⎜⎜ .49 ⎟⎟ = ⎜⎜ −.16 ⎟⎟⎜ .12 ⎟⎜ .56 ⎟ ⎜ 1.94 ⎟⎝ ⎠⎝⎠ ⎝⎠We can get the same answer if we use the global minimum variance portfolio and the efficient
portfolio with expected return equal to 2.03%. The calculations are essentially the same:

μ p , z = αμ p ,m + (1 − α ) μ p ,eSetting μ p , z = 0.03 and using μ p ,m = 0.0083, μ p ,e = 0.0203 we can solve for α:

α=μ p , z − μ p ,e0.03 − 0.0203== −0.81
μ p ,m − μ p ,e 0.0083 − 0.0203

1 − α = 1.81
The weights on vfinx, veurx and vbltx in this portfolio arez = α m + (1 − α ) x⎛ .19 ⎞
⎛ −.35 ⎞ ⎛ −.77 ⎞⎜⎟= ( −0.81) ⎜ .69 ⎟ + (1.81) ⎜⎜ .22 ⎟⎟ = ⎜⎜ −.16 ⎟⎟⎜ .12 ⎟
⎜ .113 ⎟ ⎜ 1.94 ⎟
⎝ ⎠⎝
⎠ ⎝
⎠
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 tolerantinvestor.

d. How much should be invested in T-bills and the tangency portfolio to create an efficient
portfolio with expected return equal to 3%? 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 =.03 − rf

μtan − rf=.03 − 0.0042= 2.51.0188 − 0.0042

x f = 1 − xtan = 1 − 2.51 = −1.51

σ p ,e = xtanσ tan = (2.51)(0.0298) = 0.0746

e. In the efficient portfolio you found in part d, what are the shares of wealth invested in vpacx,vbltx, and veiex?
vpacx
2.51*(-0.04)=-0.11

vbltx
2.51*0.49=1.22

veiex
2.51*(0.56)=1.39

f. Suppose you are not allowed to sell any of the assets short? Can you find an efficient portfolioof the three assets that has an expected return of 3%?
No. One way to see this is to note that the tangency portfolio, which has an expected return of
1.35%, already has a negative weight in VPACX. Since an efficient portfolio is a convex
combination of T-bills and the tangency portfolio and the weight on the tangency portfolio needs
to be greater than 1 to achieve an expected return of 3% we cannot find a combination of thethree assets to achieve a target return of 3% not allowing short sales.

g. Assuming an initial $100,000 investment for one month, compute the 5% value-at-risk on the
global minimum variance portfolio.
First we find the 1% quantiles for the global minimum variance portfolio
port : q.05 = 0.0083 + (0.0216)( −1.645) = −0.0272
Then we compute the 1% VaR

port : VaR.05 = $100,000 × ( e −0.0272 − 1) = −$2,687.61
III. Empirical Analysis of the single index model and the CAPM (40 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

where Rit denotes the cc return on asset i at time t, and RMt denotes the return on a market index
portfolio at time t.
a. In the SI model, what is the interpretation of β i ?

β i measures the contribution of asset i to the risk of the market index measured by the standard
deviation of the market index. Explicitly,

βi =cov( Rit , RMt )var( RMt )b. How is the interpretation of ε it in the SI model different from its interpretation in the constant
expected return (CER) model?

ε it captures random news that is uncorrelated with the “market” as captured by RMt . In the CER
model, ε it captures all random news.
The following represents S-PLUS linear regression output from estimating the single index
model for the Vanguard Pacific Stock Index (vpacx), the Vanguard long-term bond index (vbltx)
and the Vanguard Emerging Markets Fund (veiex) using monthly continuously compounded
return data over the 5 year period September 2001 – September 2006. In the regressions, themarket index is the Vanguard S&P 500 index (vfinx).
> summary(vpacx.fit,cor=F)
Call: lm(formula = vpacx ~ vfinx, data = final.ts)
Residuals:
Min1Q
Median3Q Max-0.0822 -0.0258 -0.00312 0.0262 0.09
Coefficients:
Value Std. Error t value Pr(>|t|)(Intercept) 0.008 0.005
1.678
0.099
vfinx 0.493 0.131
3.758
0.000
Residual standard error: 0.0374 on 58 degrees of freedom
Multiple R-Squared: 0.196
F-statistic: 14.1 on 1 and 58 degrees of freedom, the p-value is 0.000399> summary(vbltx.fit,cor=F)
Call: lm(formula = vbltx ~ vfinx, data = final.ts)
Residuals:
Min
1Q Median
3QMax

-0.0936 -0.0171 0.00431 0.0151 0.0556
Coefficients:
Value Std. Error t value Pr(>|t|)(Intercept) 0.006 0.003
1.891
0.064
vfinx -0.183 0.092-1.994
0.051
Residual standard error: 0.0262 on 58 degrees of freedom
Multiple R-Squared: 0.0642
F-statistic: 3.98 on 1 and 58 degrees of freedom, the p-value is 0.0508> summary(veiex.fit,cor=F)
Call: lm(formula = veiex ~ vfinx, data = final.ts)
Residuals:
Min
1Q
Median3QMax-0.0949 -0.0178 0.000847 0.0274 0.0655
Coefficients:
Value Std. Error t value Pr(>|t|)(Intercept) 0.014 0.004
3.150
0.003
vfinx 1.117 0.120
9.287
0.000
Residual standard error: 0.0343 on 58 degrees of freedom
Multiple R-Squared: 0.598
F-statistic: 86.3 on 1 and 58 degrees of freedom, the p-value is 4.47e-013

c. Make a table showing the estimated values of β, its estimated standard error, the estimate ofσ ε , and the R2 values from the three regression equations. How accurate are the estimates of β?

βˆ
vpacx
vbltx
veiex

.493
-.183
1.117

SE( βˆ )
.131
.092
.120

σˆε.0374.0262.0343

R2.196.0642.596

d. What can you say about the risk characteristics of the three assets relative to the S&P 500index (vfinx)? Which asset is most beneficial to hold in terms of diversification?
Using β as a measure of risk, we see that vbltx has the lowest risk and veiex has the highest risk.
The bond index vbltx is the most beneficial asset to hold in terms of diversification since it has
the lowest β and the lowest R2.

e. For each asset, test the null hypothesis that β = 1 against the alternative that β ≠ 1 using a 5%
significance level. What do you conclude?
vpacx : tβ =1 =βˆ − 1 .493 − 1== −3.87.131
SE ( βˆ )vbltx : tβ =1 =βˆ − 1 −.183 − 1== −12.86.092
SE ( βˆ )veiex : tβ =1 =βˆ − 1 1.117 − 1== 0.98.120
SE ( βˆ )We reject the null hypothesis that β = 1 against the alternative that β ≠ 1 using a 5%
significance level whenever | tβ =1 |> 2 . Therefore, we reject the null for vpacx and vbltx but notfor veiex.
f. Using the information listed in Table 1 (from Portfolio theory section), what is the β of theglobal minimum variance portfolio?

β p = mvpacx β vpacx + mvbltx β vbltx + mveiex β veiex= (.19)(.493) + (.69)( −.183) + (.12)(1.117) = 0.10
g. If the monthly risk free rate is 0.25% (0.0025) and the monthly risk premium on the S&P 500
is index is .5% (0.005), what are the monthly expected returns predicted by the capital asset
pricing model (CAPM) for the three assets?
The CAPM pricing relationship isE[ Ri ] = rf + β i ( E[ RM ] − rf )Using rf = 0.0025 and E[ RM ] − rf = .005 we have

E[ Rvpacx ] = 0.0025 + (.493)(.005) = .00497
E[ Rvbltx ] = 0.0025 + ( −.183)(.005) = .00159
E[ Rveiex ] = 0.0025 + (1.117)(.005) = .00809
h. Using the information listed in Table 1 (from Portfolio theory section) and the regression
output, make a plot of the average return for against the estimate of β for the three assets. What
does the CAPM predict about the relationship between average return and β? Does thisrelationship appear to hold for the three assets?
ER vs Beta
2.50%

ER

2.00%
1.50%
1.00%
0.50%
0.00%
-0.5

00.5

11.5

Beta

There appears to be a strong upward sloping linear relationship between the average return andβ as predicted by the CAPM.