University of Washington
Department of Economics

Summer 2014
Eric Zivot

Economics 424/CFRM 462
Final Exam Solutions

This is a closed book and closed note exam. However, you are allowed one page of notes (8.5” by
11” or A4 double-sided) and the use of a calculator. Answer all questions and write all answers inthe space provided. If you need additional space, you may attach extra sheets of paper with yourcalculations. Time limit is 2 hours and 10 minutes. Total points = 100.

I. Constant Expected Return Model (32 points, 4 points each)
Consider the constant expected return (CER) modelrit  i   it ,  it ~ iid N (0,  i2 )cov(rit , rjt )   ij , cor (rit , rjt )  ij
for the monthly simple returns on the Vanguard S&P500 index (VFINX) and Amazon stock (AMZN) . The
following R output gives the estimates of  i ,  i ,  ij and  ij for the S&P 500 index (VFINX) and Amazon
stock (AMZN) from the T=60 months of data:> muhat.vals
vfinx amzn
0.016 0.036> sigmahat.vals
vfinx amzn
0.044 0.085> covhat.vals
vfinx,amzn
0.001> rhohat.vals
vfinx,amzn
0.343
a. For each asset, compute estimated standard errors for ˆ , ˆ and ˆ . Create a table showing theestimates in one row and the standard errors in another row. Are the estimates precise? Briefly justify youranswers.
The standard estimated standard error formulas are:1  ˆ 2ˆˆSEˆ ( ˆ ) , SEˆ (ˆ ) , SEˆ ( ˆ ) T2TT> nobs = nrow(ret)> se.muhat = sigmahat.vals/sqrt(nobs)> se.sigma = sigmahat.vals/sqrt(2*nobs)> se.rho = (1 - rhohat.vals^2)/sqrt(nobs)> rbind(muhat.vals, se.muhat)
vfinx amzn
muhat.vals 0.0157 0.036
se.muhat
0.0057 0.011

> rbind(sigmahat.vals, se.sigma)
vfinx
amzn
sigmahat.vals 0.04415 0.08489
se.sigma
0.00403 0.00775> rbind(rhohat.vals, se.rho)
vfinx,amzn
rhohat.vals
0.343
se.rho0.114
Standard errors for means are about 1/3 the size of the estimates which indicates that the means are
estimated with moderate imprecision. The standard errors for the volatilities are much smaller than theestimates which indicates precise estimation. The correlation standard error is about 1/3 the size of theestimate and indicates moderate imprecistion.
b. For the S&P 500 index (VFINX) only, compute 95% confidence intervals for  ,  . Briefly
comment on the precision of the estimates. In particular, note if both positive and negative values are in therespective confidence intervals.
The rule‐of‐thumb formula for a 95% confidence interval for a parameter  is

 (ˆ)
ˆ  2  SE
For VFINX, the estimated standard errors for  and  are> # 95% ci for mu> upper = muhat.vals + 2*se.muhat> lower = muhat.vals - 2*se.muhat> cbind(lower[1],upper[1])
[,1]
[,2]
vfinx 0.00431 0.0271> # 95% ci for sigma> upper = sigmahat.vals + 2*se.sigma> lower = sigmahat.vals - 2*se.sigma> cbind(lower[1],upper[1])
[,1]
[,2]
vfinx 0.0361 0.0522
The 95% CI for is fairly wide from an economic point of view (0.5% to 3%), whereas the 95% CI for is fairly tight from an economic point of view (3.6% to 5.2%). Only positive values are in the confidence
intervals.
c. Compute a 95% confidence interval for  vfinx , amzn . Use this interval to test the hypotheses

H 0 : vfinx ,amzn  0.5 vs. H1 : vfinx ,amzn  0.5 using a 5% significance level.

The 95% confidence intervals for  vfinx , amzn is> upper = rhohat.vals + 2*se.rho> lower = rhohat.vals - 2*se.rho> cbind(lower,upper)
lower uppervfinx,amzn 0.116 0.571
Since zero is not in the 95% CI for , we can reject the null hypothesis that  = 0 at the 5% level.
d. Let W0 = $100,000 be the initial wealth invested in each asset. Estimate the monthly 5%
Value-at-Risk for each asset assuming each asset is normally distributed.
Because we have simple returns, the 5% VaR is computed using
Z
VaRˆ.05   ˆ  ˆ  q.05
  W0> VaR.05 = (muhat.vals + sigmahat.vals*qnorm(0.05))*100000> VaR.05
vfinx
amzn-5692 -10366

e. Briefly explain how you can use the bootstrap to compute a standard error for the monthly 5% normal
VaR for VFINX you computed in part d. above.
Bootstrapping involves sampling with replacement from the original data to create B different samples.
On each sample the 5% VaR is computed, and the estimated bootstrap SE is the sample standard deviationof these B 5% VaR values.

f. For VFINX and AMZN Consider testing the hypotheses

H 0 : rt ~ normal vs. H1 : rt is not normally distributed
using a 5% significance level. The JB statistics are JBvfinx  1.847 and JBamzn  1.512 . What do you
conclude?
Under the null of normality, the JB statistic has a chi-square distribution with 2 degrees of freedom.
The 5% critical value is the 95% quantile of the chi-square distribution with 2 degrees of freedom. Thisquantile is 6. Hence, we reject the null at the 5% significance level if JB > 6. Since both JB values are lessthan 6, we cannot reject the hypothesis that the returns on both assets are normally distributed.
g. The 24-month rolling estimates of  and  are illustrated in Figure 1 and Figure 2 below (on nextpage). The CER model assumes that  and  are constant over time. Is this a reasonable assumption? Whyor why not.
For both VFINX and AMZN, the rolling mean and sd values behave similarly. The means do not move
too much (they both dip a bit in late 2011 and then recover) . The sd values, however, show a slow decline

over the sample reflecting a drop in market volatility over 2012-2014 compared with 2009-2011. The meanand sd values are not constant over time.

Figure 1 Rolling estimates for VFINX

Figure 2 Rolling estimates for AMZN

h. The CER model can be used for continuously compounded returns or for simple returns. What are
the advantages and disadvantages of using the CER model for continuously compounded returns? What are
the advantages and disadvantages of using the CER model for simple returns?

The advantages of the CER model for cc returns: (1) cc returns are more likely normally distributed
than simple returns; (2) cc returns are additive over time. The main disadvantage of the CER model for cc
returns involves the application to portfolios. The cc return on a portfolio is not a weighted average of the
cc returns on the individual assets so the CER model does not hold exactly for portfolios. For simple
returns, the main disadvantage is that multiperiod returns are not additive. The CER model does not holdfor multiperiod returns. The main advantage, is that the portfolio return is a weighted average of asset
returns. Here the CER model holds for portfolios.

II. Matrix Algebra and Portfolio Math (36 points, 4 points each)
Let Ri denote the continuously compounded return on asset i (i = 1, , N) with E[Ri] = i,var(Ri) =  2 and cov(Ri, Rj) = ij. Define the (N  1) vectors R   R , , R  ,   ( ,,  ) ,
i

1

N

1

N

m   m1 , , mN  , x   x1 , , xN  , y   y1 , , y N  , t   t1 , , t N  , 1  1, ,1 and the (N  N)

covariance matrix  12   12 
  1N 12   1N 
 22   2 N 


 2N

  

  N2 .

The vectors m, x, y and t contain portfolio weights that sum to one. Using simple matrix
algebra, answer the following questions.
a. For the portfolios defined by the vectors x and y give the expression for the portfolio returns,(Rp,x and Rp,y), the portfolio expected returns (p,x and p,y), the portfolio variances (  p2 , x and  p2 , y ),and the covariance between Rp,x and Rp,y (  xy ).
R p , x  Rx,  p , x  x μ,  p2 , x  x ΣxR p , y  Ry,  p , y  y μ,  p2 , y  y Σy xy  xyb. For portfolio x, derive the n x 1 vector of marginal contributions to portfolio volatility
defined by
MCR 
By the chain rule we have p (x)x.

 p (x)
x


1
1/2
1/2
 xΣx    xΣx  2Σx
x
2
Σx
Σx


1/2
 xΣx   p (x)c. Write down the optimization problem and give the Lagrangian 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.
min m Σm s.t. m 1  1
m

L(m,  )  m Σm   (m 1  1)
d. Continuing with part b., derive the first order conditions for determining the minimum
variance portfolio m. Write these first order conditions as a system of linear equations in the form
Az = b and show how the portfolio m can be determined from this system.
The first order conditions areL 2 Σm    1  0
m
L m1  1  0
These first order conditions can be written in matrix form as 2 Σ 1  m   0 
 1 0      1  or Az  b   This linear system has solutionz  A -1b

The vector m is the first n elements of the n+1 x 1 vector z.

e. Write down the optimization problem and give the Lagrangian used to determine an efficient
portfolio with target return equal to  0 assuming short sales are allowed. Let x denote the vectorof portfolio weights in the efficient portfolio.

min xΣx s.t. x1  1 and xμ  0x

L(x, 1 , 2 )  xΣx  1 (x1  1)  2 (xμ  0 )f. Briefly describe how you would compute the efficient frontier containing only risky assets(Markowitz bullet) when short sales are allowed.
Because short sales are allowed, the Markowitz bullet can be constructed using a convex
combination of any two efficient (frontier) portfolio: z    m  (1   )  x , where m and x are any
two frontier portfolios. To get a nice picture, choose portfolio m to be the global minimum
variance portfolio and choose the other efficient portfolio x to be the efficient portfolio with target
return equal to the highest expected return of the available assets. Then, for example, vary  from
1/2
1 to -1 in increments of 0.1 and compute z. For each z, compute  p , z  zμ and  p , z   z Σz  andthen plot these pairs of points.
g. Write down the optimization problem used to determine the tangency portfolio, assuming
short sales are not allowed and the risk free rate is given by rf . Let t denote the vector of portfolio
weights in the tangency portfolio.
maxtt  - rf

 t Σt 

1/ 2

s.t. t 1  1 and t i  0, i  1,, N

h. Continuing with part g., is there an analytical solution (i.e., matrix algebra mathematicalformula) for the tangency portfolio when short sales are not allowed?
There is no analytical solution for the tangency portfolio when short sales are not allowed. The
inequality constraints do no allow the method of Lagrange multipliers to be used to determine ananalytic solution using matrix algebra.
i. Write down the equations for the expected return (  ep ) and standard deviation (  ep ) ofefficient portfolios consisting of the tangency portfolio and T-bills, where the T-bill rate (risk-freerate) is given by rf and t denotes the vector of portfolio weights in the tangency portfolio.

 ep  rf  xtan (  tan  rf ),  tan  t 2
 t t ep  xtan tan ,  tan

III. Efficient Portfolios (32 points, 4 points each)
The graph below shows the efficient frontier computed from three Vanguard mutual funds:
S&P 500 Index (vfinx), US Long Term Bond Index (vbltx), and European Stock Index (veurx).

Figure 3 Markowitz Bullet

Expected return and standard deviation estimates for specific assets are summarized in thetable below. These estimates are based on monthly continuously compounded return data over thefive year (60 month) period January 2009 – January 2014 (same data as in the class project).

Table 1 Portfolio Statistics

Asset

VFINX
VEURX
VBLTX
T-Bills
Global Min
Portfolio
Efficient
Portfolio with
Mean=0.0148
Tangency
Portfolio

Mean(E[R])

Standard
deviation(SD(R))

0.01480.01160.00830.00100.01140.03870.05740.02550.00000.01590.01480.02320.01240.0166Weight in
Global Min
Portfolio
0.58
-0.19
0.61

Weight in
Weight in
Efficient
Tangency
Portfolio
portfolio
Mean = 1.48%
1.35
0.80
-0.69
-0.33
0.34
0.53

Using the above information, please answer the following questions.
a. Compute annualized means and standard deviations from the monthly statistics in Table 1
for the three portfolios vfinx, vbltx, and veurx using the square-root-of-time-rule (this is only anapproximation because we have simple returns). Also, compute the annualized T-Bill rate.
The annualized cc mean return is  A  12   m . For the three portfolios we have

 A,vfinx  12  (.0148)  .177
 A,veurx  12  (.0116)  .139
 A,vbltx  12  (.0083)  .099
The annualized cc standard deviation is  A  12 m

 A,vfinx  12  (.0387)  .134
 A,veurx  12  (.0574)  .199
 A,vbltx  12  (.0255)  .088
The annualized T-Bill rate is rA, f  12  0.001  0.012

b. Using the annualized information from part a., compute the annualized Sharpe ratios/slopes
for each of the three portfolios. Which portfolio is ranked best using the Sharpe ratio?
The annualized Sharpe ratio is
SRA  A  rf , A, rf , A  12  rf ,mAUsing the results from part a., we get

0.177  0.012
 1.233
0.134
0.139  0.012
SRA,veurx  0.6400.199
0.099  0.012
SRA,vbltx  0.986
0.088
SRA,vfinx The asset with the highest annual Sharpe ratio is vfinx.
c. Find the efficient portfolio of risky assets only (i.e., a portfolio on the Markowitz bullet) thathas an expected monthly return equal to 0.015 (1.5%). In this portfolio, how much is invested in
vfinx, vbltx, and veurx? Show this portfolio on Figure 3.
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 p ,z   p ,m  (1   )  p ,tSetting  p , z  0.015 and using  p ,m  0.0114,  p ,t  0.0124 we can solve for : p , z   p ,t
0.015  0.0124
 2.71 p ,m   p ,t 0.0114  0.0124

1    3.71
The weights on vfinx, veurx and vbltx in this portfolio arez   m  (1   )t
 .58  .80   1.40 



 

 (0  2.71)  .19   (3.71)  .33    .72  .61  .53   .32 



 

This portfolio is plotted as the point E.015 in Figure 3.

We can get the same answer if we use the global minimum variance portfolio and the efficient
portfolio with expected return equal to 1.48%. The calculations are essentially the same: p ,z   p,m  (1   )  p ,eSetting  p , z  0.015 and using  p ,m  0.0114,  p ,e  0.0148 we can solve for : p , z   p ,e 0.015  0.0148
 0.06
 p ,m   p ,e 0.0114  0.0148

1    1.06
The weights on vfinx, veurx and vbltx in this portfolio arez   m  (1   ) x
 .58  1.35   1.40 



 

 (0.06)  .19   (1.06)  .69    .72  .61  .34   0.32 



 

d. How much should be invested in T-bills and the tangency portfolio to create an efficient
portfolio with expected return equal to 0.015 (1.5%) ? What is the standard deviation of thisefficient portfolio? Indicate the location of this efficient portfolio on Figure 3.

xtan 

.015  rf

 tan  rf

.01  0.001
 1.23.0124  0.001

x f  1  xtan  1  1.23  0.23

 p ,e  xtan tan  (1.23)(0.0166)  0.024
This portfolio is plotted as the point E2.015 in Figure 3.
e. In the efficient portfolio you found in part d, what are the shares of wealth invested in
T-Bills, vfinx, vbltx, and veurx?
vfinx
1.23*(0.80)=0.99

veurx
1.23*(-0.33)=-0.41

vbltx
1.23*(0.53)=0.65

T-Bills-0.23

f. Assuming an initial $100,000 investment for one month, compute the 5% value-at-risk basedon the normal distribution for the global minimum variance portfolio.
First we find the 5% quantiles for the global minimum variance portfolio
port : q.05  0.0114  (0.0159)(1.645)  0.0147
Then we compute the 5% VaR

port : VaR.05  $100, 000  (0.0147)  $1, 473> w0 = 100000> q.05 = gmin.port$er + gmin.port$sd*qnorm(0.05)> q.05[1] -0.0147> VaR.05 = q.05*w0> VaR.05[1] -1473

g. The efficient frontier of risky assets shown in Figure 3 allows for short sales (see the weights
in the portfolios listed in Table 1). Using the space below, copy the efficient frontier from Figure 3
and indicate roughly the location of the efficient frontier of risky assets that does not allow shortsales.

h. Suppose you want to find the efficient portfolio of risky assets only that has an expected
monthly return equal to 0.02 (2%) but that you are prevented from short selling. Is it possible tofind such an efficient portfolio? Briefly explain why or why not.
It is not possible to find an efficient portfolio of risky assets only that has an expected return of
2%. To see this, from Table 1 above notice that the efficient portfolio with expected return equal to
1.48% has a short position in veurx and the tangency portfolio (with mean 1.24%) has a shortposition in veurx as well. Hence, for an efficient portfolio with mean 2% we would need a shortposition in veurx.