University of Washington
Department of Economics

Summer 2013
Eric Zivot

Economics 424/Applied Mathematics 462
Final Exam Solutions

I. 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
 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. 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
 ep  xtan tan ,  tan
 t tII. Efficient Portfolios (32 points points, 4 points each)
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
ER

veiex
1.50%

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%

SD

Figure 1 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

Asset

VPACX
VBLTX
VEIEX
T-Bills
Global Min
Portfolio

Mean
E[R])

0.43%
0.49%
1.28%
0.08%
0.40%

Standard
deviation(SD(R))
5.59%
2.90%
8.45%
0%
2.84%

Weightin Global
Min
Portfolio
23%
87%
-10%

Weight in
WeightEfficientin Tangency
Portfolio
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 continuously
compounded)
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 is
SRA  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
SRA,vbltx  0.4856
0.10050.1536  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.

h. Suppose you want to find the efficient portfolio of risky assets only that has an expected
monthly return equal to 2% but that you are prevented from short selling. Is it possible to find suchan 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 to1.28% has a short position in vpacx and the tangency portfolio (with mean 1.78%) has a shortposition in vpacx as well. Hence, for an efficient portfolio with mean 2% we would need a shortposition in vpacx.

III. Empirical Analysis of the single index model and the CAPM (36 points, 4 points
each)

The single index model for asset returns has the form
Rit   i   i RMt   iti  1, , n assetst  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 i ?
i measures the contribution of asset i to the risk of the market index measured by thestandard deviation of the market index. Explicitly,

i 

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 2004 – September 2009. In the regressions, the market index is
the Vanguard S&P 500 index (vfinx).
> summary(vpacx.fit)
Coefficients:
Estimate Std. Error t value Pr(>|t|)(Intercept) 0.00356
0.00385
0.93
0.36
vfinx
1.001390.0819412.22<2e-16 ***--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.0298 on 58 degrees of freedom
Multiple R-squared: 0.72,
Adjusted R-squared: 0.715
F-statistic: 149 on 1 and 58 DF, p-value: <2e-16> summary(vbltx.fit)
Coefficients:
Estimate Std. Error t value Pr(>|t|)(Intercept) 0.00479
0.00364
1.32
0.193
vfinx
0.166390.077442.15
0.036 *--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 0.0282 on 58 degrees of freedom
Multiple R-squared: 0.0737,
Adjusted R-squared: 0.0578
F-statistic: 4.62 on 1 and 58 DF, p-value: 0.0358> summary(veiex.fit)
Coefficients:
Estimate Std. Error t value Pr(>|t|)(Intercept)
0.01170.00572.04
0.046 *
vfinx
1.52620.121412.57<2e-16 ***--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.0442 on 58 degrees of freedom
Multiple R-squared: 0.732,
Adjusted R-squared: 0.727
F-statistic: 158 on 1 and 58 DF, p-value: <2e-16

b. Make a table (see example below) showing the estimated values of β, its estimated standard
error, the estimate of   , and the R2 values from the three regression equations. How accurate are
the estimates of β?
Asset
vpacx
vbltx
veiex


1.001
0.16639
1.5262SE()0.08194
0.07744
0.1214

R2

0.02980.02820.04420.72
0.0737
0.732

SE() is largest for veiex and smallest for vbltx. However, the values of SE() are fairly small
for vpacx and veiex relative to the value of , so I would say that  is estimated reasonably
precisely for these assets. However, the value of SE() for vbltx is about half the value of  and so is not estimated as precisely.
c. What can you say about the risk characteristics of the three assets relative to the S&P 500
index (vfinx)?
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, give the percentage of total risk due to the market (non-diversifiable risk) andthe percentage not due to the market (diversifiable risk). Which asset is most beneficial to hold interms of diversification?
The market risk is measured by R2 and non-market risk is measured by 1 – R2. The following
table summarizes the results for the three assets

Asset
vpacx
vbltx
veiex

R2 (market risk)
0.72
0.07370.732

1-R2 (non-market risk)
0.28
0.92630.268

e. For each asset, test the null hypothesis that  = 1 against the alternative that   1 using a
5% significance level. What do you conclude?

vpacx : t 1 

ˆ  1 1.001309  1
 0.0169.08194
SE ( ˆ )vbltx : t 1 

ˆ  1 0.16639  1
 10.76.07744
SE ( ˆ )veiex : t 1 

ˆ  1 1.5262  1
 4.33.1214
SE ( ˆ )We reject the null hypothesis that  = 1 against the alternative that   1 using a 5%
significance level whenever | t 1 | 2 . Therefore, we reject the null for vbltx and veiex but not for
vpacx.
If an asset has  = 1 then it has the same risk as the market index: adding the asset to themarket index does not change the volatility of the index.
f. Using the information listed in Table 1 (from Portfolio theory section), what is the  of the
global minimum variance portfolio?

 p  mvpacx  vpacx  mvbltx  vbltx  mveiex  veiex (.23)(1.001)  (.87)(0.1664)  (.10)(1.5262)  0.22
g. Using the information in the table in question b. above and the standard deviation of vfinx of
0.04737, show how you could estimate the 3 x 3 covariance matrix of the return vector consistingof vpacx, vbltx and veiex using the single index model where the market return is vfinx. You only
have to write out the matrix algebra formula with the appropriate information from the table in thevectors and matrices. You do not have to numerically compute the covariances.
Using matrix algebra, the single index covariance has the formΣSI =  M2 ββ + D  2,vpacx00   vpacx 




2
0 
  ,vbltx
β    vbltx  , D   0


 0
0
 2,veiex  veiex Using the information from the table in question b we have  0.0298 2
 1.001  M2  (0.04737) 2 , β   0.16639  , D  0 1.5262 



0


0

 0.0282 
0




0

2
 0.0442  
02

h. In the Capital Asset Pricing Model (CAPM), what is the relationship between the expected
return i on any asset and its risk as measured by i ?
E[ Ri ]  rf   i ( E[ RM ]  rf )i. Using the estimated expected returns ˆ i from Table 1 above and the estimated ˆi values
from the R output, plot ˆ vs. ˆ . Is this plot consistent with the CAPM? Why or why not?
ii

The CAPM predicts an exact upward sloping linear relationship between average return andbeta (with slope equal to the market risk premium). Here we don’t see an exact linear relationshipbut the asset with the highest beta (veiex) does have the highest average return. However, the asset
with the lowest beta (vbltx) has an average return slightly higher than vpacx whose beta is 1. Dueto estimation errors in average returns and betas it is not possible to rule out the CAPM
relationship from the above plot.