University of Washington
Department of Economics

Summer 2011
Eric Zivot

Economics 424/Applied Mathematics 540
Final Exam Solutions

I. Constant Expected Return Model and Hypothesis Testing (35 points, 5 points each)
Figure 1 shows monthly continuously compounded returns on the Vanguard S&P 500 index (VFINX) andthe Vanguard Emerging Markets index (VEIEX) over the 5‐year period September 2005, throughSeptember 2010. For this period there are T=60 monthly observations.

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-0.3

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ret.z

0.00.1

S&P 500Emerging Markets20062007200820092010Index

Figure 1 Monthly Continuously Compounded Returns on Two Vanguard Funds

Consider the constant expected return modelrit  i   it ,  it ~ iid N (0,  i2 )cov(rit , rjt )   ij , cor (rit , rjt )  ij
for the monthly continuously compounded returns on the Vanguard S&P500 index and the Vanguard
emerging markets index. The following R output gives the estimates of  i ,  i ,  ij and  ij for the S&P
500 index (VFINX) and the emerging markets index (VEIEX) from the T=60 months of data:> muhat.valsvfinx
veiex
0.0004694 0.0094886> sigmahat.vals
vfinx
veiex
0.05195 0.08549> covhat.vals
vfinx,veiex
0.003809> rhohat.vals
vfinx,veiex
0.8576a. For the S&P 500 index (vfinx), compute 95% confidence intervals for  ,  . Briefly comment on theprecision of the estimates. In particular, note if both positive and negative values are in the respective
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 ( ˆ )  ˆ / T  0.05195 / 60  0.0067
SE (ˆ )  ˆ / 2T  0.05195 / 120  0.0047
SEThe 95% confidence intervals for  and  arevfinx : 0.0004698  2  0.0067  [0.01294, 0.01388]
 vfinx : 0.05195  2  0.0047  [0.04247, 0.06144]
The mean is not estimated very precisely. Both negative and positive values are in the confidence
interval and the width of the interval is fairly big. The sd is estimated much more precisely.

b. For the S&P 500 index, test the hypotheses H 0 :   0 vs. H1 :   0 using a 5% significance level.
We can do this test in two equivalent ways. For both tests, we use the asymptotic normal distributionfor ̂ based on the Central Limit Theorem. First, we can see if   0 lies in the 95% confidence interval
we computed in part a. Since   0  [ 0.01294, 0.01388] we do not reject the null at the 5% level.
Alternatively, we can compute the t‐ratiot 0 ˆSE ( ˆ )

0.004694
 0.0706
0.0067And see if it’s absolute value is bigger than 2. Since it is not, we do not reject the null at the 5% level.
c. Compute a 95% confidence interval for  vfinx ,veiex . Use this interval to test
the hypotheses H 0 : vfinx ,veiex  0.5 vs. H1 :  vfinx ,veiex  0.5 using a 5% significance level.
For VFINX, the estimated standard error for  vfinx ,veiex is ( ˆ )  1  ˆ  1  (0.8575)  0.03417
SET6022

The 95% confidence intervals for  vfinx ,veiex is ( ˆ )  0.8575  0.03417  [0.7891, 0.9258]
 vfinx ,veiex : ˆ  2  SE
This interval is quite tight around 0.8575 and indicates a precise estimate. To test
H 0 : vfinx ,veiex  0.5 vs. H1 : vfinx ,veiex  0.5 we check to see if 0.5 lies in the 95% confidence
interval. Since 0.5 is not in the confidence interval we reject at the 5% significance level.

d. For vfinx and vieix Consider testing the hypothesesH 0 : rt ~ normal vs. H1 : rt is not normally distributed
using a 5% significance level. The JB statistics are JBvfinx  15.11 and JBvfinx  28.82 . 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 JB > 6 for both assets wereject the normality assumption at the 5% significance level for both assets.

e. Suppose you currently hold $2M (million) in the S&P 500 Index. That is, your initial wealth at thebeginning of the month is W0  $2 M . Using the estimates from the CER model compute the 5%value‐at‐risk (VaR) associated with a one‐month investment in the S&P 500 index.

 0.05  $2M   exp(qˆ R )  1  $2M   exp(0.08498)  1  0.1629  $162,900
VaR0.05
Rqˆ0.05 ˆ  ˆ  (1.645)  0.0004698  0.05195  (1.645)  0.08498

f. Describe briefly how you could use the bootstrap to compute an estimated standard error and 95%
confidence interval for the 5% VaR found in part f 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 deviation of these B 5% VaR values.

g. The 24‐month rolling estimates of m and s are illustrated in and below. The CER model assumes
that m and s are constant over time. Is this a reasonable assumption? Why or why not.

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returns

0.00

0.05

0.10

24 month rolling means and sds for VFINX

Rolling mean
Rolling sd
Monthly returns
2006

2007

2008

2009

Index

Figure 2 24‐month rolling estimates for VFINX

2010

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returns

0.0

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24 month rolling means and sds for VEIEX

Rolling mean
Rolling sd
Monthly returns
2006

2007

2008

2009

2010

Index

The rolling estimates of  and  for both assets are very similar. The rolling estimates of  start out
slightly positive and then dip negative during the financial crisis. There is clearly a structural break in themean from positive values to negative values around the end of 2008. The rolling estimates of  start toincrease at the end of 2008 and then level off at in the middle of 2009. The values at the end of thesample are almost twice as big as the values at the beginning of the sample. The rolling estimates showa non‐constant value for throughout the sample.

I. Matrix Algebra and Portfolio Math (30 points, 5 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

N1

Nm   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
 22


 2N

  1N   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. 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  1mL(m,  )  m Σm   (m 1  1)

c. 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 )d. 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 one efficient portfolio to be the global
minimum variance portfolio and choose the other efficient portfolio to be the efficient portfoliowith target return equal to the highest expected return of the available assets.
e. 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/ 2s.t. t 1  1 and t i  0, i  1,, N

f. 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

II. Efficient Portfolios (28 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
3.00%

2.50%

2.00%

vpacx
vbltx

tangency port
ERveiex1.50%

global min
frontierveiex

T-bills
1.00%

tangency

0.50%

global min port

vbltx

vpacx

T-Bills
0.00%
0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

SDFigure 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 period September 2004 – September 2009.
Table 1 Portfolio Statistics

AssetVPACX
VBLTX
VEIEX
T-Bills
Global Min
Portfolio

MeanE[R])

0.43%
0.49%
1.28%
0.08%
0.40%

Standarddeviation(SD(R))
5.59%
2.90%
8.45%
0%
2.84%

Weightin GlobalMin
Portfolio
23%
87%
-10%

WeightWeight inin Tangency
Efficientportfolio
Portfolio
Mean = 1.28%
-120%
-197%
129%
151%
91%
145%

Efficient
Portfolio with
Mean=1.28%
Tangency
Portfolio

1.28%

4.77%

1.76%

6.53%

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 vpacx, vbltx, and veiex (Remember, the returns are continuouslycompounded)
The annualized cc mean return is  A  12   m . For the three portfolios we have

 A,vpacx  12  (.0043)  .0516
 A,vbltx  12  (.0049)  .0588
 A,veiex  12  (.0128)  .1536
The annualized cc standard deviation is  A  12 m

 A,vpacx  12  (.0559)  .1936
 A,vbltx  12  (.0290)  .1005
 A,veiex  12  (.0845)  .2927
The annualized T-Bill rate is rA, f  12  0.0008  0.01
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 isSRA  A  rf , A, rf , A  12  rf ,mAUsing the results from part a., we get

0.0516  0.01
 0.2149
0.1936
0.0588  0.01
 0.4856
SRA,vbltx 0.1005
0.1536  0.01
SRA,veiex  0.4906
0.2927
SRA,vpacx The asset with the highest annual Sharpe ratio is veiex.

c. Find the efficient portfolio of risky assets only (e.g. a portfolio on the Markowitz bullet) thathas an expected monthly return equal to 1%. 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 p , z   p ,m  (1   )  p ,tSetting  p , z  0.01 and using  p ,m  0.0040,  p ,t  0.0176 we can solve for : p , z   p ,t0.01  0.0176
 0.56
 p ,m   p ,t 0.0040  0.0176

1    0.44
The weights on vfinx, veurx and vbltx in this portfolio arez   m  (1   )t

 .23  1.97   .74  
 (0.56)  .87   (0.44)  1.51    1.16  .10  1.45   .59  
We can get the same answer if we use the global minimum variance portfolio and the efficient
portfolio with expected return equal to 1.28%. The calculations are essentially the same: p , z   p ,m  (1   )  p ,eSetting  p , z  0.01 and using  p ,m  0.0040,  p ,e  0.0128 we can solve for : p , z   p ,e0.01  0.0128
 0.32 p ,m   p ,e 0.0004  0.0128

1    0.68
The weights on vfinx, veurx and vbltx in this portfolio arez   m  (1   ) x

 .23  1.20   .74  
 (0.32)  .87   (0.68)  1.29    1.16  .10 
 .91   0.59 

 


d. How much should be invested in T-bills and the tangency portfolio to create an efficient
portfolio with expected return equal to 1%? What is the standard deviation of this efficient
portfolio? Transfer the graph of the Markowitz bullet to your blue book and indicate the location ofthis efficient portfolio on the graph.
xtan .01  rf

 tan  rf

.01  0.0008
 0.55.0176  0.0008

x f  1  xtan  1  0.55  0.45

 p ,e  xtan tan  (0.55)(0.0653)  0.0358

e. In the efficient portfolio you found in part d, what are the shares of wealth invested in
T-Bills, vpacx, vbltx, and veiex?

vpacx
0.55*(-1.97)=-1.08

vbltx
0.55*(1.51)=0.83

veiex
0.55*(1.45)=0.80

T-Bills
0.45

f. 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 5% quantiles for the global minimum variance portfolio
port : q.05  0.0004  (0.0284)(1.645)  0.0427

Then we compute the 5% VaR
port : VaR.05  $100, 000   e 0.0427  1  $4,182.95

g. The efficient frontier of risky assets shown in Figure 1 allows for short sales (see the weights
in the portfolios listed in Table 1). Transfer this graph to your blue book. On this graph, indicate
roughly the location of the efficient frontier of risky assets that does not allow short sales.