University of Washington
Department of Economics

Fall 2008
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 i2 and cov(Ri, Rj) = σij. Define the (N × 1) vectors R = ( R1 ,… , RN )′ , μ = ( μ1 ,… , μ N )′ ,m = ( 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 1N

σ 12
σ 22
σ 2N

σ 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 not allowed. Let m denote the vector of portfolio weights in
the global minimum variance portfolio.

min m′Σm s.t. m′1 = 1 and mi ≥ 0, i = 1,…, Nmc. Write down the optimization problem used to determine an efficient portfolio with target
return equal to μ 0 assuming short sales are not allowed. Let x denote the vector of portfolio
weights in the efficient portfolio.

min x′Σx s.t. x′1 = 1 and x′μ = μ0 , xi ≥ 0, i = 1,…, Nx

d. Briefly describe how you would compute the efficient frontier containing only risky assets(Markowitz bullet) when short sales are not allowed.
Because no short sales are allowed, the Markowitz bullet cannot be constructed using a convex
combination of any two efficient (frontier) portfolios. Instead, a brute force method is required
where minimum variance portfolios with no short sales are computed (as in part c. above) for an
increasing grid of target returns starting at the expected return of the global minimum variance
portfolio with no short sales. The picture below illustrates the method.

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.

maxtt ′μ - rf( t ′Σt )1/ 2s.t. t ′1 = 1 and t i ≥ 0, i = 1,…, N

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
2.00%

frontierveiex

T-bills

1.50%

tangency

vpacx
1.00%

global min port
0.50%

vbltx
0.00%
0.00%

T-Bills
2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

SD

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

Asset

Mean (E[R])Standard
deviation(SD(R))

Weight in
Global Min
Portfolio

VPACX
VBLTX
VEIEX
T-Bills
Global Min
Portfolio
Tangency
Portfolio
Efficient
Portfolio withMean=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) = .1308
rA,vbltx = 12 ⋅ (.0054) = .0648
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 × .1308) = $3.70
FVvbltx = exp(10 × .0648) = $1.91
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 arez = 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 efficientportfolio 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. How much should be invested in T-bills and the tangency portfolio to create an efficient
portfolio with standard deviation equal to 4%? What is the expected return 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.
x tan =σ ep
0.04== 1.34, x f = 1 − x tan = −0.34,
σ tan 0.0298

μ ep = r f + xtan ( μ tan − r f ) = 0.0025 + 1.34(0.0135 − 0.0025) = 0.0172

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 − 0.0025= 2.51.0135 − 0.0025

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. Assuming an initial $100,000 investment for one month, compute the 5% value-at-risk on theglobal 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 = αi + 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 indexportfolio at time t.
a. In the SI model, what is the interpretation of βi ?

βi 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 )The following represents R linear regression output from estimating the single index model for
the Vanguard Pacific Stock Index (vpacx), the Vanguard long-term bond index (vbltx) and theVanguard Emerging Markets Fund (veiex) using monthly continuously compounded return dataover the 5 year period September 2001 – September 2006. In the regressions, the market index isthe Vanguard S&P 500 index (vfinx).
> summary(vpacx.fit)
Call: lm(formula = vpacx ~ vfinx, data = final.z)
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)
Call: lm(formula = vbltx ~ vfinx, data = final.z)
Residuals:
Min
1Q Median
3Q
Max-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)
Call: lm(formula = veiex ~ vfinx, data = final.z)
Residuals:
Min1Q
Median3Q
Max-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

b. 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?

βˆvpacx
vbltx
veiex

.493
-.183
1.117

SE( βˆ )
.131
.092
.120

σˆε

R2.0374
.0262
.0343

.196
.0642
.596

c. What can you say about the risk characteristics of the three assets relative to the S&P 500
index (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.
d. 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.
e. 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
f. 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 + βi ( 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