University of Washington
Department of Economics

Spring 2015
Eric Zivot

Econ 424
Midterm 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 andwrite all answers directly on the exam in the space provided. If you need more space,you may use extra sheets of paper. Time limit is 1 hour and 50 minutes. Total points =116.

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 Apple stock (AAPL). Suppose you buy one share of the S&P 500 fund andone share of Apple stock at the end of January, 2010 forP vfinx ,t 60  89.91, Paapl ,t 60  25.88 , and then sell these shares at the end of January, 2015

for P vfinx ,t  184.2, Paapl ,t  116.7 . (Note: these are actual adjusted closing prices takenfrom Yahoo!). In this question, you will see how much money you could have made if
you invested in these assets right after the financial crisis.
a. What are the simple 60-month (5-year) returns for the two investments?
> r.vfinx = (p.vfinx.2 - p.vfinx.1)/p.vfinx.1> r.AAPL = (p.AAPL.2 - p.AAPL.1)/p.AAPL.1> r.vfinx[1] 1.05> r.AAPL[1] 3.51
b. What are the continuously compounded (cc) 60-month (5-year) returns for the twoinvestments?
> log(1 + r.vfinx)[1] 0.717> log(1 + r.AAPL)[1] 1.51
c. Suppose you invested $1,000 in each asset at the end of January, 2010. How muchwould each investment be worth at the end of January, 2015?
> w0 = 1000
> w1.vfinx = w0*(1 + r.vfinx)> w1.AAPL = w0*(1 + r.AAPL)> w1.vfinx[1] 2049> w1.AAPL[1] 4509

d. What is the compound annual return on the two 5 year investments?
> r.vfinx.a = (1 + r.vfinx)^(1/5) - 1> r.AAPL.a = (1 + r.AAPL)^(1/5) - 1> r.vfinx.a[1] 0.154> r.AAPL.a[1] 0.352

e. At the end of January, 2010, suppose you have $1,000 to invest in VFINX and AAPL
over the next 60 months (5 years). Suppose you purchase $400 worth of VFINX and theremainder in AAPL. 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.
> w0 = 1000
> x.vfinx = 400/w0> x.AAPL = 1 - x.vfinx> x.vfinx[1] 0.4> x.AAPL[1] 0.6> r.p = x.vfinx*r.vfinx + x.AAPL*r.AAPL> r.p[1] 2.53> log(1 + r.p)[1] 1.26

II. Probability Theory (20 points, 4 points each)
Let Rvfinx and RAAPL denote the monthly simple returns on VFINX and AAPL andsuppose that Rvfinx ~ iid N (0.013, (0.037) 2 ), Raapl ~ iid N (0.028, (0.073) 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?

AAPL 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 AAPL.

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

AAPL 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% monthlyZ 2.326 ).
Value-at-Risk values for each asset. (Hint: q0.01> w0 = 1000
> q.vfinx.01 = mu.vfinx + sigma.vfinx*(-2.326)> q.AAPL.01 = mu.AAPL + sigma.AAPL*(-2.326)> VaR.01.vfinx = w0*q.vfinx.01> VaR.01.AAPL = w0*q.AAPL.01> VaR.01.vfinx[1] -73.1> VaR.01.AAPL[1] -142
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 $73.1 or more.
With 1% probability (or one month in every 100 months), a one month $1000 investment
in AAPL will lose $142 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 two
problems, 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 )   2
cov(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).

c. The CER model for cc returns
rt     t ,  t ~ iid N (0,  2 ) ,implies that the log price follows a random walk with drift
tln Pt  ln Pt 1  rt  ln P0   t    s .
s 1

Show that E[ln Pt ] and var(ln Pt ) depend on t so that ln Pt is non-stationary.
tE  ln Pt   E[ln P0 ]  E[  t ]   E  s   ln P0   t which depends on t.
s 1t
 t
 tvar(ln Pt )  var  ln P0   t    s   var    s    var( s )  t 2 which depends on t.
s 1
 s 1  s 1d. Suppose the time series { X t } is independent white noise. That is,
X t ~ iid (0, 2 )Define two new time series {Yt } and {Z t } where Yt  X t2 and Z t | Yt | . Are {Yt } and{Z t } also independent white noise processes? Why or why not?
{Yt } and {Z t } are independent processes because any function of independent randomvariables are also independent. The process are covariance stationary because anyfunction of covariance stationary processes are covariance stationary (provided themean and variance and autocovariances are finite). Hence, the processes will haveconstant variance. The processes will not have mean zero. So they will behave like awhite noise process with a non-zero mean.

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) =  i and2cov(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[ Rp , y ]   p , y  yμb. 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) andApple stock (AAPL) over the 5-year period January 2010, through January 2015. For
this period there are T=60 monthly returns.

Figure 2: 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 (both series fluctuate up and down about the meanvalue). The volatilities of VFINX and AAPL look fairly constant over time. Overall, bothseries look like they could be realizations from a covariance stationary 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.
AAPL looks to have a slightly larger mean than VFINX and clearly a larger volatility.
AAPL also has many more large returns (positive and negative) than VFINX. The tworeturns do not look like they move very closely together (e.g. when VFINX goes up(down) it is not always the case that AAPL goes up (down)). Hence, it looks like thecorrelation between VFINX and AAPL 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 AAPL clearly did better than a $1 investment in VFINX. For AAPL $1grew to about $5, and for VFINX $1 grew to only about $2.

Figure 3 Cumulative simple returns on two assets.

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

The following table summarizes some sample descriptive statistics of the T=60 monthlyreturns.
Table 1 Descriptive Statistics and CER model estimates

Statistic
Mean (  )Standard Deviation ( ˆ )Skewness
Excess Kurtosis
1% empirical quantile ( q̂0.01 )VFINX
0.01270.0374-0.2634
0.286-0.0744

AAPL
0.02800.0725-0.0738-0.165-0.1289Lag 1 autocorrelation ( ̂1 )-0.1100.136

Lag 2 autocorrelation ( ̂ 2 )-0.031-0.015

Lag 3 autocorrelation ( ̂3 )Correlation between VFINXand AAPL ( ˆVFINX , AMZN )

0.011

0.070

0.435

d. Do the returns on VFINX and AAPL look normally distributed? Briefly justify youranswer.
Both the returns on VFINX and AAPL look pretty normally distributed. The histograms
are roughly bell shaped (VFINX has a slight negative skewness but not too large). Also,
the normal qq-plots for both assets are mostly linear and the boxplots only show one
moderate outlier for VFINX. Finally, the excess kurtosis values for both series are closeto zero (about 0.286 for VFINX and -0.165 for AAPL) which is consistent with the normal
distribution.
e. Which asset appears to be riskier? Briefly justify your answer.
AAPL has a larger SD than VFINX (0.0725 vs. 0.0374) so there is more uncertainty inthe return for AAPL. Also, the 1% quantile for for AAPL (-0.1289) is more negative than
the 1% quantile for VFINX (-0.0744) This means there is more tail risk with AAPL.
f. Does there appear to be any contemporaneous linear dependence between the returns
on VFINX and AAPL? Briefly justify your answer.
The scatterplot of the AMZN and VFINX returns looks pretty much like a shotgun blast
with a moderate rightward tilt indicating weak to moderate positive linear dependence.
This is confirmed by the sample correlation value of 0.435.
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. (Recall, the data are simple returns here)> W0 = 1000
> q.01 = apply(ret, 2, quantile, probs=0.01)> VaR.vfinx.01 = W0*q.01["VFINX"]> VaR.AAPL.01 = W0*q.01["AAPL"]> VaR.vfinx.01
VFINX-74.4> VaR.AAPL.01
AAPL-129

VI. Constant Expected Return Model (16 points)
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 Apple stock(AAPL) presented in part V above. Below are simulated returns and some graphicaldescriptive statistics for VFINX and AAPL from the CER model calibrated using thesample estimates of the CER model parameters for the two assets (these are the sample
statistics from the table of the previous question).

a. Briefly describe how to create the simulated returns on VFINX and AAPL from theCER model. That is, tell me the algorithm for creating the simulated data. You don’t haveto show me R code to answer this question. Just tell me the steps required (e.g. pseudocode) to simulate the data.
Here we want to do a multivariate simulation from the CER model in matrix form
R t  μ  ε t , εt ~ N (0, Σ)
To do this we need values for the 2 x 1 mean vector μ and 2 x 2 covariance matrix Σ .
We can get these values from the sample statistics in Table 12  vfinx
 vfinx   0.0127 ˆμˆ  
Σand   aapl   0.0280  vfinx , aapl

 vfinx ,aapl   0.0014 0.00118 2  0.00118 0.00526  aapl
22
 (0.0374) 2 , 0.00526   aapl
 (0.0725) 2 andwhere 0.0014   vfinx

0.00118   vfinx , aapl   vfinx ,aapl vfinx aapl . Then we can simulate T observation on

ε t ~ N (0, Σ) (using the R function mvtnorm, for example) giving ε t and create thesimulated returns using rt  μˆ  ε t .

b. 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 AAPL 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 (AAPL has
higher mean and volatility than VFINX; returns do not move very closelytogether) Histograms are roughly bell shaped and qq-plots are mostly linear. The negativeskewness of actual VFINX returns are not shown in the simulated returns. Thishistogram of the simulated VFINX returns shows positive skewness.
 Boxplots of simulated returns match closely the boxplots of the actual returns.
 Scatterplot of simulated returns looks like scatterplot of actual returns
 SACF of simulated returns look like SACF of actual returns.
c. Does the CER model appear to be a good model for VFINX and AAPL 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.