University of Washington
Department of Economics

Fall 2010
Eric Zivot
Economics 424
Midterm Exam Solutions

This is a closed book and closed note exam. However, you are allowed one page of notes(double-sided). Answer all questions and write all answers in a blue book or on separate
sheets of paper. Time limit is 1 hours and 50 minutes. Total points = 108.

I. Return Calculations (20 pts, 4 points each)
Consider a 1-month investment in two assets: the Vanguard S&P 500 index (VFINX) andthe Vanguard Emerging Markets Stock Index (VEIEX). Suppose you buy one share ofthe S&P 500 fund and one share of the emerging markets fund at the end of September,2010 for P vfinx ,t −1 = 105.06, Pveiex ,t −1 = 28.64 , and then sell these shares at the end ofOctober, 2010 for P vfinx ,t = 109.04, Pveiex ,t = 29.49 . (Note: these are actual adjusted closing
prices taken from Yahoo!)a. What are the simple 1-month returns for the two investments?
> p.vfinx.1 = 105.06> p.vfinx.2 = 109.04> p.veiex.1 = 28.64> p.veiex.2 = 29.49> r.vfinx = (p.vfinx.2 - p.vfinx.1)/p.vfinx.1> r.veiex = (p.veiex.2 - p.veiex.1)/p.veiex.1> r.vfinx[1] 0.03788> r.veiex[1] 0.02968

b. What are the continuously compounded (cc) 1-month returns for the two investments?
> log(1 + r.vfinx)[1] 0.03718> log(1 + r.veiex)[1] 0.02925

c. Assume you get the same monthly returns from part a. and b. every month for the next
year. What are the annualized simple and cc returns?
> r.vfinx.a = (1 + r.vfinx)^12 - 1> r.veiex.a = (1 + r.veiex)^12 - 1> r.vfinx.a[1] 0.5624> r.veiex.a[1] 0.4204

d. Continuing with c., how much will $10,000 invested in each fund be worth after 1year?
> w0 = 10000> w1.vfinx = w0*(1 + r.vfinx.a)> w1.veiex = w0*(1 + r.veiex.a)> w1.vfinx[1] 15624> w1.veiex[1] 14204

e. At the end of September, 2010, suppose you have $10,000 to invest in VFINX andVEIEX over the next month. Suppose you purchase $2,000 worth of VFINX and theremainder in VEIEX. What are the portfolio weights in the two assets? Using the resultsfrom parts a. and b. compute the 1-month simple and cc portfolio returns.
> w0 = 10000> x.vfinx = 2000/w0> x.veiex = 1 - x.vfinx> x.vfinx[1] 0.2> x.veiex[1] 0.8> r.p = x.vfinx*r.vfinx + x.veiex*r.veiex> r.p[1] 0.03132> log(1 + r.p)[1] 0.03084

II. Probability Theory (20 points, 4 points each)
Let rvfinx and rveiex denote the monthly continuously compounded returns on VFINX andVEIEX and suppose that rvfinx ~ iid N (0.001, (0.05) 2 ), rveiex ~ iid N (0.01, (0.09) 2 ).

a. Sketch the normal distributions for the two assets on the same graph. Show the meanvalues and the ranges mean ± 2 sd. Which asset appears to be the most risky?
VEIEX has the larger sd and appears to be riskier.

b. Plot the risk-return tradeoff for the two assets. That is, plot the mean values of eachasset on the y-axis and plot the sd values on the x-axis. What relationship do you see?

Here we see a positive relationship between risk and return.

c. Let W0 = $100,000 be the initial wealth invested in each asset. Compute the 5%Zmonthly Value-at-Risk for each asset. (Hint: q0.05= −1.645 )> mu.vfinx = 0.001> mu.veiex = 0.01
> sigma.vfinx = 0.05
> sigma.veiex = 0.09
> w0 = 100000> q.vfinx = mu.vfinx + sigma.vfinx*(-1.645)> q.veiex = mu.veiex + sigma.veiex*(-1.645)> VaR.05.vfinx = w0*(exp(q.vfinx) - 1)
> VaR.05.veiex = w0*(exp(q.veiex) - 1)
> VaR.05.vfinx[1] -7804> VaR.05.veiex[1] -12894

d. Give an expression for the 12-month (annual) continuously compounded return, rt(12),in terms of the monthly continuously compounded returns. Using this expression
compute E[ rt (12)], and SD(rt (12)) for each asset.

rt ~ iid N ( μ , σ 2 )rt (12) = rt + rt −1 ++ rt −11

E [ rt (12) ] = E [ rt + rt −1 +var(rt (12)) = var(rt + rt −1 ++ rt −11 ] = E[rt ] + E[rt −1 ] ++ E[rt −11 ] = 12μ+ rt −11 ) = var(rt ) + var(rt −1 ) +SD (rt (12) = 12σUsing the above formulas, we have
> mu.vfinx.a = 12*mu.vfinx> sigma.vfinx.a = sqrt(12)*sigma.vfinx> mu.vfinx.a[1] 0.012> sigma.vfinx.a[1] 0.1732> mu.veiex.a = 12*mu.veiex> sigma.veiex.a = sqrt(12)*sigma.veiex> mu.veiex.a[1] 0.12> sigma.veiex.a[1] 0.3118+ var(rt −11 ) = 12σ 2

e. Continuing with d., let W0 = $100,000 be the initial wealth invested in each asset.
Compute the 5% annual Value-at-Risk for each asset.
> w0 = 100000> q.vfinx.a = mu.vfinx.a + sigma.vfinx.a*(-1.645)> q.veiex.a = mu.veiex.a + sigma.veiex.a*(-1.645)> VaR.05.vfinx.a = w0*(exp(q.vfinx.a) - 1)
> VaR.05.veiex.a = w0*(exp(q.veiex.a) - 1)
> VaR.05.vfinx.a[1] -23885> VaR.05.veiex.a[1] -32488

III. Matrix Algebra (16 points, 4 points each)
Let Ri denote the simple return on asset i (i = 1,2,…,n) with E[Ri] = μi, var(Ri) = σ i2 andcov(Ri, Rj) = σij.
a. Define the n × 1 vectors containing the returns and expected returns (call these R andμ), and define the n x n covariance matrix (call this Σ).
⎛ σ 12⎛ R1 ⎞⎛ μ1 ⎞⎜⎜ ⎟⎜ ⎟R = ⎜ ⎟, μ = ⎜ ⎟, Σ = ⎜⎜R ⎟⎜μ ⎟⎜σ⎝ n⎠⎝ n⎠⎝ 1,nσ 1,n ⎞⎟⎟σ n2 ⎟⎠

b. Let xi and yi (i = 1,2,…,n) denote the shares of wealth invested in asset i in twoportfolios. Define the n x 1 vectors of portfolio weights for these two portfolios. Denotethese vectors x and y.

⎛ x1 ⎞⎛ y1 ⎞⎜ ⎟⎜ ⎟x = ⎜ ⎟, y = ⎜ ⎟⎜x ⎟⎜y ⎟⎝ n⎠⎝ n⎠c. Using matrix algebra, give expressions for the returns, expected returns and variances
for the two portfolios.

R p , x = x′R, R p , y = y ′R,

μ p , x = x′μ , μ p , y = y′μ ,σ p2 , x = x′Σx, σ p2 , y = y′Σyd. Using matrix algebra, give expressions for the restriction that the weights in eachportfolio must sum to one, and give an expression for the covariance between the returns
on portfolio x and the returns on portfolio y.
x′1 = 1, y′1 = 1, 1′ = (1,… ,1)
σ x , y = x′Σy

IV. Time Series Concepts (8 points, 4 points each)a. Let {Yt } represent a stochastic process. Under what conditions is {Yt } covariancestationary?
E[Yt ] = μvar(Yt ) = σ 2cov(Yt , Yt − j ) = γ j (depends on j and not t )b. Realizations from four stochastic processes are given in Figure 1 below. Whichprocesses appear to be covariance stationary and which processes appear to be nonstationary? Briefly justify your answers.

Process 20.8
y20.0-2-1

0.4

020

4060

80

100

04060
Time

Process 3

Process 480

10080

100

1201

y40-4 -3 -2 -1

y320Time200204060
Time80100-4 -3 -2 -1

y1121.2

Process 10204060
Time

Figure 1: Realizations from four stochastic processes.

••Processes 1 and 3 appear to be covariance stationary. The means and volatilities
appear constant over time.
Processes 2 and 4 appear to be non-stationary. Process 2 has a clear
deterministic trend, so the mean is not constant over time. Process 4 appears tohave two distinct means. The first mean is roughly zero for the first half of thesample, and the second mean is roughly -2 for the second half of the sample.

V. Descriptive Statistics (24 points, 4 points each)
Figure 3 shows monthly continuously compounded returns on the Vanguard S&P 500
index (VFINX) and the Vanguard Emerging Markets index (VEIEX) over the 5-year
period September 2005, through September 2010. For this period there are T=60
monthly observations.

-0.1
-0.3

-0.2

ret.z

0.0

0.1

S&P 500
Emerging Markets2006

2007

2008

2009

2010

Index
Figure 2: Monthly continuously compounded returns on two Vanguard Funds.

a. Do the monthly continuously compounded returns from the two funds look likerealizations from a covariance stationary stochastic process? Why or why not?
Recall, covariance stationarity implies that the mean, variance and autocovariances areconstant over time. Both processes appear to have changing means and variances. Thecorrelation between the processes may also be changing over time. Clearly thevolatilities are higher and the means are lower during the crisis period. The series seemto move together more closely during the crisis period as well. Given these observations,
it does not appear that these two series are covariance stationary.
b. Comment on any common features, if any, of the two return series. Recall, thefinancial crisis reached its peak toward the end of 2008.

Both series tend to move together, indicating a positive correlation. The positive
correlation appears to be greater during and after the crisis period. Also, the volatilities
of both series seem to increase around the same time.
The figures below gives some graphical diagnostics of the return distributions for theS&P 500 index and the emerging markets fund.

0-0.15-0.05

cc return642

cc return

80.05

10

S&P 500 monthly cc returns

-0.20

-0.10

0.00

0.10

vfinx.mat

Normal Q-Q Plot
0.05
-0.05-0.15

Sample Quantiles

86420density estimate

10

Smoothed density

-0.2

-0.1

0.0

cc return0.1-2-1

01

Theoretical Quantiles

2

-0.4

-0.3

-0.2

-0.1

0.0

0.1

-0.3 -0.2 -0.1 0.0

cc return201

cc return

30.1

4

Emerging markets fund monthly cc return

0.2

veiex.mat

Normal Q-Q Plot

-0.2

0.0

0.1

0.1

-2

-1

00.0
-0.1
-0.2

S&P 500

1

Theoretical Quantiles

0.1

cc return0.2-0.3 -0.2 -0.1 0.0

Sample Quantiles

43210-0.4-0.3

density estimate

5Smoothed density

Emerging Markets

20.1
0.0
-0.1
-0.3

-0.2

Emerging markets returns

-0.15

-0.10

-0.05

0.00

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0.10

S&P 500 returns-0.2 0.2 0.6 1.0

ACF

Vanguard S&P 500 Index

051015Lag-0.2 0.2 0.6 1.0

ACF

Vanguard Emerging M arkets Index

051015Lag

c. Do the returns on the S&P500 index and the emerging markets fund look normally
distributed? Briefly justify your answer.
The returns on both the S&P 500 index and the emerging markets index exhibit nonnormal behavior. The histograms show a long left tail (negative skewness) and theboxplot indicates a some large negative outliers (fat tails). The normal QQ-plot is not
linear in the left tail. All of this is strong evidence against the normal distribution.

d. Which asset appears to be riskier? Briefly justify your answer.
The volatility of the emerging markets index is much larger than the volatility of the S&P
500 and appears to be riskier.
e. Based on the scatterplot of returns, does there appear to be any linear dependence
between the returns on the S&P500 index and the emerging markets fund? Briefly justify
your answer.
There is a strong positive linear association in the scatterplot. When the S&P 500 returns
increase, the emerging markets returns also increase and the scatter lies quite close to astraight line.
f. Based on the sample autocorrelation functions (SACFs), do the returns show any strong
temporal dependence?
Both sample autocorrelation plots show a positive first lag autocorrelation that is
marginally statistically different from zero (the estimate of ρ1 is just above the dottedline). The remaining sample autocorrelations are not very large. Hence there is weakevidence of temporal dependence that lasts at most one period.

VI. Constant Expected Return Model (20 points, 4 points each)
Consider the constant expected return model

rit = μi + ε it , ε it ~ iid N (0, σ i2 )cov(rit , rjt ) = σ ij , cor (rit , rjt ) = ρij
for the monthly continuously compounded returns on the Vanguard S&P500 index andthe Vanguard emerging markets index presented in part V. The following R output gives
the estimates of μ i , σ i , σ ij and ρ ij for the S&P 500 index (VFINX) and the emerging
markets index (VEIEX) from the T=60 months of data:> muhat.vals
vfinx
veiex
0.0004698 0.0094863> sigmahat.vals
vfinx
veiex
0.05195 0.08549> covhat.vals
vfinx,veiex
0.003809

> rhohat.vals
vfinx,veiex
0.8575a. For the S&P 500 index, compute 95% confidence intervals for μ , σ . Briefly
comment on the precision of the estimates. In particular, note if both positive andnegative values are in the respective confidence intervals.
The rule-of-thumb formula for a 95% confidence interval for a parameter θ isθˆ ± 2 × SE (θˆ)
For VFINX, the estimated standard errors for μ and σ are
SE ( μˆ ) = σˆ / T = 0.05195 / 60 = 0.0067
SE (σˆ ) = σˆ / 2T = 0.05195 / 120 = 0.0047

The 95% confidence intervals for μ and σ are

μvfinx : 0.0004698 ± 2 × 0.0067 = [−0.01294, 0.01388]
σ vfinx : 0.05195 ± 2 × 0.0047 = [0.04247, 0.06144]
The mean is not estimated very precisely. Both negative and positive values are in theconfidence interval and the width of the interval is fairly big. The sd is estimated much
more precisely.
b. Compute a 95% confidence interval for ρ vfinx ,veiex . Use this interval to comment on theprecision of the correlation estimate.
For VFINX, the estimated standard error for ρ vfinx ,veiex isSE ( ρˆ ) =1 − ρˆ 2 1 − (0.8575) 2== 0.03417
T60

The 95% confidence intervals for ρ vfinx ,veiex isρ vfinx ,veiex : ρˆ ± 2 × SE ( ρˆ ) = 0.8575 ± 0.03417 = [0.7891, 0.9258]
This interval is quite tight around 0.8575 and indicates a precise estimate.

c. Suppose you currently hold $2M (million) in the S&P 500 Index. That is, your initial
wealth at the beginning 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 inthe S&P 500 index. Hint: 5% quantile of the standard normal distribution is -1.645,respectively.
R) − 1) = $2M × ( exp(−0.08498) − 1) = −0.1629 = $162,900
VaR 0.05 = $2M × ( exp(qˆ0.05R
qˆ0.05= μˆ + σˆ × (−1.645) = 0.0004698 + 0.05195 × (−1.645) = −0.08498

d. 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 c above.
Bootstrapping involves sampling with replacement from the original data to create Bdifferent 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.

Below are simulated returns for the S&P 500 index and the Emerging Markets index
from the CER model calibrated using the sample estimates of the CER model parameters
for the two assets.

S&P 500
Emerging Markets

0.0
-0.1

cer.sim.z

0.1

0.2

Simulated Returns

2006

2007

2008
Index

2009

2010

0.1
0.0-0.1

Emerging markets returns

0.2

Simulated Returns

-0.10

-0.05

0.00

0.05

0.10

S&P 500 returns

e. Which features of the actual returns are captured by the simulated returns and whichfeatures are not?
The features of the actual returns captured by the simulated returns are:•••

The mean of VEIEX is slightly higher than VFINXVEIEX has higher volatility than VFINX
Strong positive correlation between VEIEX and VFINX

The features not captured by the simulated returns:•The changing volatility and correlation during and after the crisis period.