University of Washington
Department of Economics

Summer 2014
Eric Zivot
Econ 424/CFRM 462
Midterm Exam Solutions

I. Return Calculations (20 pts, 4 points each)
Consider a 60-month (5 year) investment in two assets: the Vanguard S&P 500 index(VFINX) and Amazon stock (AMZN). Suppose you buy one share of the S&P 500 fundand one share of Amazon stock at the end of January, 2009 forP vfinx ,t 60  68.18, Pamzn ,t 60  58.82 , and then sell these shares at the end of January, 2014
for P vfinx ,t  164.1, Pamzn ,t  403 . (Note: these are actual adjusted closing prices taken fromYahoo!). In this question, you will see how much money you could have made if youinvested in these assets right after the financial crisis.
a. What are the simple 60-month (5-year) returns for the two investments?
> p.amzn.1 = 58.82> p.amzn.2 = 403> p.vfinx.1 = 68.18> p.vfinx.2 = 164.1> r.vfinx = (p.vfinx.2 - p.vfinx.1)/p.vfinx.1> r.amzn = (p.amzn.2 - p.amzn.1)/p.amzn.1> r.vfinx[1] 1.407 = 140.7%> r.amzn[1] 5.851 = 585.1%
b. What are the continuously compounded (cc) 60-month (5-year) returns for the twoinvestments?
> log(1 + r.vfinx)[1] 0.8783 = 87.83%> log(1 + r.amzn)[1] 1.924 = 192.4%
c. Suppose you invested $1,000 in each asset at the end of January, 2009. How much
would each investment be worth at the end of January, 2014?
> w0 = 1000
> w1.vfinx = w0*(1 + r.vfinx)> w1.amzn = w0*(1 + r.amzn)> w1.vfinx[1] 2407> w1.amzn[1] 6851

d. What is the simple compound annual return on the two 5 year investments?
> r.vfinx.a = (1 + r.vfinx)^(1/5) - 1> r.amzn.a = (1 + r.amzn)^(1/5) - 1> r.vfinx.a[1] 0.192 = 19.2% per year
> r.amzn.a[1] 0.4695 = 46.95% per year (wow!!)e. At the end of January, 2009, suppose you have $1,000 to invest in VFINX and AMZN
over the next 60 months (5 years). Suppose you purchase $400 worth of VFINX and theremainder in AMZN. What are the portfolio weights in the two assets? Using the resultsfrom parts a. and b. compute the 5-year simple and cc portfolio returns.
# portfolio weights> w0 = 1000
> x.vfinx = 400/w0> x.amzn = 1 - x.vfinx> x.vfinx[1] 0.4> x.amzn[1] 0.6# portfolio returns> r.p = x.vfinx*r.vfinx + x.amzn*r.amzn> r.p[1] 4.074 = 407.4%> log(1 + r.p)[1] 1.624 = 162.4%

II. Probability Theory (20 points, 4 points each)
Let Rvfinx and Ramzn denote the monthly simple returns on VFINX and AMZN andsuppose that Rvfinx ~ iid N (0.016, (0.044) 2 ), Ramzn ~ iid N (0.036, (0.084) 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?

AMZN has a higher SD which means its normal pdf is more spread out than the normal
pdf of VFINX. There is more uncertainity, hence more risk, in the return for AMZN.
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?

AMZN has both a higher expected return and standard deviation (risk) than VFINX. Thisis the stylized risk-return tradeoff.
c. Let W0 = $1,000 be the initial wealth invested in each asset. Compute the 1% monthly
Z
Value-at-Risk values for each asset. (Hint: q0.01
 2.326 ).
> q.vfinx.01 = mu.vfinx + sigma.vfinx*(-2.326)> q.amzn.01 = mu.amzn + sigma.amzn*(-2.326)> VaR.01.vfinx = w0*q.vfinx.01> VaR.01.amzn = w0*q.amzn.01> VaR.01.vfinx[1] -86.34> VaR.01.amzn[1] -159.4
d. Continuing with c., state in words what the 1% Value-at-Risk numbers represent (i.e.,explain what 1% Value-at-Risk for a one month $1,000 investment means)With 1% probability (or one month in every 100 months), a one month $1000 investmentin VFINX will lose $86.34 or more.
With 1% probability (or one month in every 100 months), a one month $1000 investment
in AMZN will lose $159.4 or more.
e. The normal distribution can be used to characterize the probability distribution ofmonthly simple returns or monthly continuously compounded returns. What are two
problems with using the normal distribution for simple returns? Given these twoproblems, why might it be better to use the normal distribution for continuously
compounded returns?
Problem 1: Simple returns are bounded from below by -1. The normal distribution isdefined over -∞ to +∞ and so it is possible returns to be less than -1 with positive
probability if they are normally distributed
Problem 2: Multi-period simple returns are multiplicative, not additive. So if simple
returns are normally distributed then multi-period returns are not normally distributed(because the product of two normal random variables is not normally distributed).
The normal distribution is more appropriate for continuously compounded returns
because continuously compounded returns are defined over -∞ to +∞ and multi-period
continuously compounded returns are additive.

III. Time Series Concepts (16 points, 4 points each)a. Let {Yt } represent a stochastic process. Under what conditions is {Yt } covariancestationary?
E[Yt ]  var(Yt )   2cov(Yt , Yt  j )   j (depends on j and not t )b. Realizations from four stochastic processes are given in Figure 1 below.

Figure 1: Realizations from four stochastic processes.

Which processes appear to be covariance stationary and which processes appear to benon-stationary? Briefly justify your answers.
Processes 1 and 3 appear to be covariance stationary. The means and volatilities
appear constant over time and the series exhibit mean reversion (when the series
gets above or below the mean it reverts back to the mean)
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 volatilities (low in the first half and high in the second half).

Figure 2 below shows a realization of a stochastic process representing a monthly timeseries of overlapping 2-month continuously compounded returns rt(2), where the 1-monthcontinuously compounded returns follow a Gaussian White noise process.

1 2 3-3-1rt(2)Overlapping 2-month cc return020406080

100Time0.60.2-0.2

ACF

1.0Overlapping 2-month cc return

051015LagFigure 2 Overlapping 2-month returns

c. Based on the sample autocorrelations, which time series process is most appropriate fordescribing the series: MA(1) or AR(1)? Justify your answer.
Based on the sample autocovariance function (SACF), an MA(1) model looks moreand  j  0 for j  1appropriate than an AR(1) model. In an MA(1) model, 1 1 2

and in an AR(1) model  j   j . The SACF shows ˆ1  0.42 and ˆ j  0 for j  1 which isconsistent with an MA(1) model.
d. If you think the process is an AR(1) process, what do you think is the value of theautoregressive parameter  ? If you think the process is a MA(1) process, what do you
think is the value of the moving average parameter  ?
If the processes was an AR(1), then   ˆ1  0.42 . If the process was an MA(1) then 1  0.42    0.42  (1   2 )  0.42  0.42   2  0.42   2    0.42  021
This is a quadratic equation in  . Using the quadratic formula, there are two solutions1  1  4(0.42) 2 1.836 and 0.554 .
2(0.42)

IV. Matrix Algebra (16 points, 4 points each)
Let Ri denote the simple return on asset i (i = 1, , N) with E[Ri] = i, var(Ri) =  i2 andcov(Ri, Rj) = ij. Define the (N  1) vectors R   R1 , , RN  ,   (1 ,,  N ) ,x   x1 , , xN  , y   y1 , , y N  , and 1  1, ,1 and the (N  N) covariance matrix  12   12   1N 12   1N 
 22   2 N 


 2N

 

  N2 .

The vectors x and y contain portfolio weights (investment shares) 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 matrix algebra expression
for the portfolio returns, (Rp,x and Rp,y) and the portfolio expected returns (p,x and p,y).

R p , x  xR, R p , y  y R
E[ R p , x ]   p , x  xμ, E[ R p , y ]   p , y  yμb. For the portfolios defined by the vectors x and y give the matrix algebra expression
for the constraint that the portfolio weights sum to one.

x1  1 and y1  1

c. For the portfolios defined by the vectors x and y give the matrix algebra expression
for the portfolio variances (  p2 , x and  p2 , y ), and the covariance between Rp,x and Rp,y ( xy ).
 p2 , x  var( R p , x )  xΣx,  p2 , y  var( R p , y )  yΣy
 xy  cov( R p , x , R p , y )  xΣyd. In the expression for the portfolio variance  p2 , x , how many variance terms are there?
How many covariance terms are there?
There are N variance terms and N(N-1) total covariance terms (N(N-1)/2 uniquecovariance terms).

V. Descriptive Statistics (32 points, 4 points each)
Figure 3 shows monthly simple returns on the Vanguard S&P 500 index (VFINX) andAmazon stock (AMZN) over the 5-year period January 2009, through January 2014. For
this period there are T=60 monthly returns.

Figure 3: Monthly simple returns on two assets.

a. Do the monthly returns from the two assets look like realizations from a covariancestationary stochastic process? Why or why not?
Recall, covariance stationarity implies that the mean, variance and autocovariances areconstant over time. Visually it looks like both mean values are constant over time andboth series exhibit mean reversion. The volatility of VFINX looks fairly constant overtime whereas the volatility of AMZN look a bit higher in the first half of the sample but it
is not obvious. Overall, both series look like they could be realizations from a covariancestationary process.

b. Compare and contrast the return characteristics of the two assets. In addition,comment on any common features, if any, of the two return series.
AMZN looks to have a slightly larger mean than VFINX and clearly a larger volatility.
AMZN also has many more large returns than VFINX. The two returns do not look likethey move very closely together (e.g. when VFINX goes up (down) it is not always thecase that AMZN goes up (down)). Hence, it looks like the correlation between VFINX andAMZN is a small positive number.
c. Figure 4 below gives the cumulative simple returns (equity curve) of each fund which
represents the growth of $1 invested in each fund over the sample period. Which fundperformed better over the sample?
A $1 investment in AMZN clearly did better than a $1 investment in VFINX. For AMZN
$1 grew to about $7, and for VFINX $1 grew to only about $2.

Figure 4 Cumulative simple returns on two assets.

The figures below give some graphical diagnostics of the return distributions for VFINXand AMZN.

The following table summarizes some sample descriptive statistics of the monthly returns
Statistic
Mean (  )Standard Deviation ( ˆ )Skewness
Excess Kurtosis
1% empirical quantile ( q̂0.01 )VFINX
0.015710.04415-0.42240.1587-0.09083

AMZN0.03597
0.08489
0.38110.1550-0.11182

Lag 1 autocorrelation ( ̂1 )-0.0730.122

Lag 2 autocorrelation ( ̂ 2 )-0.096-0.099Lag 3 autocorrelation ( ̂3 )0.010-0.185The sample correlation between VFINX and AMZN returns is 0.3435.

d. Do the returns on VFINX and AMZN look normally distributed? Briefly justify youranswer.
Both the returns on VFINX and AMZN look pretty normally distributed. The histograms
are roughly bell shaped (VFINX has a slight negative skewness and AMZN has a slight
positive skewness). Also, the normal qq-plots for both assets are mostly linear. The qqplot for AMZN shows a slightly fatter right tail than implied by the normal distribution
and there are two large outliers in the boxplot. Finally, the excess kurtosis values forboth series are close to zero (about 0.15) which is consistent with the normal distribution.
e. Which asset appears to be riskier? Briefly justify your answer.
AMZN has a larger SD than VFINX (0.08 vs. 0.04) so there is more uncertainty in thereturn for AMZN. This means there is more risk with AMZN. However, VFINX has anegative skewness whereas AMZN has a positive skewness so it appears that VFINX hasa larger tail risk than AMZN.

f. Does there appear to be any contemporaneous linear dependence between the returns
on VFINX and AMZN? Briefly justify your answer.
The scatterplot of the AMZN and VFINX returns looks pretty much like a shotgun blast
with a slight rightward tilt indicating very weak positive linear dependence. This isconfirmed by the sample correlation value of 0.34.

g. Do the monthly returns show any evidence of (linear) time dependence? Briefly justify
your answer.
For both series the SACF plots show no significant autocorrelations for lags less than 15(no bars are above the dotted lines). All of the sample autocorrelations are small (closeto zero) and show no distinct pattern over time. Hence, there does not appear to be any
linear time dependence in the returns.
h. Let W0 = $1,000 be the initial wealth invested in each asset. Compute the 1% monthly
historical Value-at-Risk for each asset.
> W0 = 1000
> q.01 = apply(ret, 2, quantile, probs=0.01)> VaR.vfinx.01 = W0*q.01["VFINX"]> VaR.amzn.01 = W0*q.01["AMZN"]> VaR.vfinx.01
VFINX-90.83> VaR.amzn.01
AMZN-111.8

VI. Constant Expected Return Model (8 points)
Consider the constant expected return (CER) modelrit  i   it ,  it ~ iid N (0,  i2 )cov(rit , rjt )   ij , cor (rit , rjt )  ijfor the monthly simple returns on the Vanguard S&P500 index (VFINX) and Amazon
stock (AMZN) presented in part V above. Below are simulated returns and somegraphical descriptive statistics for VFINX and AMZN from the CER model calibratedusing the sample estimates of the CER model parameters for the two assets (these are thesample statistics from the table of the previous question).

a. Which features of the actual returns shown in part V are captured by the simulated
CER model returns and which features are not?
Almost all features of the VFINX and AMZN returns are captured by the simulated CERmodel returns. There are only a few notable differences Time plot of simulated returns looks very close to actual returns (AMZN has
higher mean and volatility than VFINX; returns do not move very closely
together) Histograms are roughly bell shaped and qq-plots are mostly linear. The negativeskewness of actual VFINX returns and the positive skewness of the AMZN returns
are not shown in the simulated returns
 Boxplots of simulated returns match closely the boxplots of the actual returns.
However, the boxplot of the simulated returns do not show the positive outliers forAMZN
 Scatterplot of simulated returns looks like scatterplot of actual returns
 SACF of simulated returns look like SACF of actual returns.

b. Does the CER model appear to be a good model for VFINX and AMZN returns? Whyor why not?
Yes, the CER model appears to be a good model for VFINX and AMZN returns. Most of
the important features of the actual data are captured by simulated data from the CERmodel.