University of Washington
Department of Economics

Fall 2008
Eric Zivot
Economics 424
Midterm ExamThis 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 = 100.
I. Return Calculations (20 pts, 5 points each)
1. Consider a one year investment in two assets: Amazon stock and the S&P 500 index.
Suppose you buy Amazon and S&P 500 at the end of September 2007 at
P A,t −1 = 93.15, PS ,t −1 = 1526.75 and then sell at the end of September 2008 for
P A,t = 72.76, PS ,t = 1164.74 . (Note: these are actual closing prices taken from Yahoo!)a. What are the simple annual returns for the two stocks?
> pa.1 = 93.15> pa.2 = 72.76> ps.1 = 1526.75> ps.2 = 1164.74# a) simple returns on Amazon and sp500> ra = (pa.2 - pa.1)/pa.1> rs = (ps.2 - ps.1)/ps.1> ra[1] -0.2188943> rs[1] -0.2371115
b. What are the continuously compounded annual returns for the two stocks?
> log(1 + ra)[1] -0.2470447> log(1 + rs)[1] -0.2706434
c. The annual inflation rate between September 2007 and September 2008 was about 5%.
Using this information, determine the simple and continuously compounded real annual
returns on Amazon and S&P 500.
> inflat = 0.05
> # simple real returns

> ra.real = (1+ra)/(1+inflat) - 1> rs.real = (1+rs)/(1+inflat) - 1> ra.real[1] -0.2560898> rs.real[1] -0.2734395> # cc real returns> log(1+ra.real)[1] -0.2958349> log(1+rs.real)[1] -0.3194336
d. At the end of September, 2006, suppose you have $100,000 to invest in Amazon andS&P 500 over the next year. Suppose you sell short $60,000 in S&P 500 and use theproceeds to buy $160,000 in Amazon. Using the results from part a, compute the annual
simple and continuously compounded return on the portfolio.
> xs = -60000/100000> xa = 160000/100000> xa[1] 1.6> xs[1] -0.6> rp = xa*ra + xs*rs> rp[1] -0.2079639> # cc portfolio return> log(1 + rp)[1] -0.2331483
II. Probability Theory and Matrix Algebra (20 points, 5 points each)
1. Suppose you currently hold $2M (million) in Starbucks stock. That is, your initial
wealth at the beginning of the month is W0 = $2 M . Let Rsbux denote the monthly simple
return on Starbucks stock, and assume that Rsbux ~ N (0.03,(0.20) 2 ) . Let
W1 = W0 (1 + RSBUX ) be a random variable representing your wealth at the end of themonth.
a) Compute E [W1 ] , var(W1 ) and SD (W1 )> w0 = 2> e.rsbux = 0.03
> sd.rsbux = 0.20
> e.w = w0 + w0*e.rsbux> e.w[1] 2.06

> var.w = w0*(w0*sd.rsbux*sd.rsbux)> var.w[1] 0.16> sd.w = w0*sd.rsbux> sd.w[1] 0.4b) What is the probability distribution of W1 ? Sketch the distribution, indicating thelocation of E [W1 ] and E[W1 ] ± 2 ⋅ SD (W1 ) .
Since R is normally distributed and W1 is a linear
function of R, W1 is also normally distributed withmean $2.06M and SD $0.4M.
> e.w + 2*sd.w[1] 2.86> e.w - 2*sd.w[1] 1.26

0.0

0.20.4

pdf

0.6

0.8

1.0

Normal distribution for Wealth

1.0

1.5

2.0w.vals

2.5

3.0c) Briefly explain why the normal distribution may not be appropriate for describing
the distribution of simple returns.
The normal distribution is defined from −∞ to ∞ . Simple
returns cannot be smaller than -1. Also, multi-period
simple returns are multiplicative (geometric average). Thatis, the 2 period return is a geometric average(multiplicative) of two 1 period returns. If the 1 period
returns are normally distributed then the 2 period return
will not be normal.
2. Let Ri denote the continuously compounded return on asset i (i = 1,2,3) with E[Ri] = μi,var(Ri) = σ i2 and cov(Ri, Rj) = σij. Define the 3 × 1 vectors⎛1⎞⎛ y1 ⎞⎛ x1 ⎞⎛ μ1 ⎞⎛ R1 ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟R = ⎜ R2 ⎟, μ = ⎜ μ 2 ⎟, x = ⎜ x 2 ⎟, y = ⎜ y 2 ⎟, 1 = ⎜1⎟
⎜1⎟
⎜y ⎟
⎜x ⎟
⎜μ ⎟
⎜R ⎟⎝ ⎠
⎝ 3⎠
⎝ 3⎠
⎝ 3⎠
⎝ 3⎠
and the 3 × 3 covariance matrix⎛ σ 12 σ 12 σ 13 ⎞⎜⎟
Σ = ⎜ σ 12 σ 22 σ 23 ⎟⎜σ2 ⎟⎝ 13 σ 23 σ 3 ⎠The vectors x and y represent portfolio weights (i.e., shares of wealth invested in thethree assets).

Using matrix algebra, give expressions for the returns, expected returns and variances forthe two portfolios.
R p , x = x′R, R p , y = y ′R,

μ p , x = x′μ, μ p , x = y ′μ,
σ p2 , x = x′Σx, σ p2 , x = y ′ΣyIII. Time Series Concepts (15 points)
1. Let {Yt } represent a stochastic process. Under what conditions is {Yt } covariancestationary? (5 points)
E[Yt ] = μvar(Yt ) = σ 2cov(Yt , Yt − j ) = γ j (depends on j and not t )2. Consider the random walk modelt

Yt = Yt −1 + ε t = Y0 + ∑ ε j , Y0 = constant.
j =1ε t ~ iid N (0, σ 2 )Is {Yt } a covariance stationary stochastic process? Why or why not? (5 points)
No. The random walk process is not stationary. The varianceof the random walk process depends on time:tt

j =1j =1var(Yt ) = ∑ var(ε j ) = ∑ σ 2 = tσ 2

3. The figure below shows annual observations on the dividend yield of the S&P 500
index over the period 1871 through 2000 along with the sample ACF. (5 points)

0.02

0.06

0.10

Annual dividend price ratio on S&P 500

1870

1880

1890

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

0.4
-0.2

0.2

ACF

0.6

0.8

1.0

Series : dpratio

05101520Lag

Assume the dividend yield is covariance stationary. Based on the shape of the sample
autocorrelation function, would an MA(1) process or an AR(1) process best describe thedata? Briefly justify your answer.
The SACF decays toward zero and does not cut off at lag 1.
Therefore, it looks more like an AR(1) process than anMA(1) process.

IV. Constant Expected Return Model (45 points, 5 points each)
Consider the constant expected return model

rit = μi + ε it , ε it ~ iid N (0, σ i2 )cov( rit , rjt ) = σ ij , corr ( rit , rjt ) = ρ ij
for the monthly continuously compounded returns on the Dow Jones Industrial Average(dji) and the Vanguard long-term bond index (vbltx) over the period September 2003
through September 2008. For this period there are T=60 monthly observations. The data
are shown in the graphs below.

0.00
-0.05
-0.10

Monthlyccreturn

0.05

Dow Jones 30

2004

2005

2006
Index

2007

2008

0.00
-0.02
-0.04

Monthlyccreturn

0.02

0.04

Vanguard Long T erm Bond2004

2005

2006

2007

2008

Indexa) Do the monthly continuously compounded return data look like they come from theCER model? Why or why not?
The CER model postulates that cc returns are (covariancestationary) iid normal random variables with constantmeans, variances and covariances (correlations). The above
two return series look a bit like computer simulations fromthe CER model. The returns appear to fluctuate randomly
about a constant mean. The mean and the variance for theVanguard bond index appears to be constant over time.
However, for the Dow, the mean appear appears to beslightly lower and the volatility appears to be slightly
higher after 2007 suggesting that the mean and variance ofthe returns are not constant through time.

b) The figures below gives some graphical diagnostics of the return distributions for dji
and vbltx.

-0.100.00 0.05

cc return840

cc return12

dowjones monthly cc returns

-0.10

-0.05

0.00

0.05

dowjones.mat

-0.15

-0.05

0.00 0.05-0.10

Sample Quantiles

12

Normal Q-Q Plot

0 2 4 6 8

density estimate

Smoothed density

0.05

-2

cc return-1

012

Theoretical Quantiles

-0.04

0.00

cc return

151005cc return

20250.04

vbltx monthly cc returns

-0.06

-0.04

-0.02

0.00

0.02

0.04

vbltx.mat

Normal Q-Q Plot

0.00
-0.04

Sample Quantiles

15
10
50density estimate

200.04

Smoothed density

-0.08

-0.04

0.00
cc return

0.04

-2

-1

01

Theoretical Quantiles

2

0.05
0.00
-0.05
-0.10

DJIVBLTX

Sample descriptive statistics are (note: the reported kurtosis is excess kurtosis)> apply(ret.z, 2, mean)
dowjones
vbltx
0.002614947 0.003035592> apply(ret.z, 2, sd)
dowjones
vbltx
0.03014207 0.01990333> apply(ret.z, 2, skewness)dowjones
vbltx-0.6686971 -0.7185054> apply(ret.z, 2, kurtosis)
dowjones
vbltx
1.76144376 0.07748463

Based on this information, do you think the monthly cc returns on dji and vbltx arenormally distributed? Briefly justify your answer.
Recall, the normal distribution is symmetric (zeroskewness) and has a kurtosis equal to three.
Dow Jones: The histogram and qq-plot show a negative
skweness. The boxplot indicates one large negative outlier.
The sample skewness is moderate and negative and the excess

kurtosis is large. However, these values appear to be
driven by the one large negative return. The negativeskewness and kurtosis greater than three cast doubt on theappropriateness of the normal distribution.
Vanguard long term bond: The histogram and boxplot showsthat the returns are highly negatively skewed, and the qqplot has a curved shape that reflects the negativeskewness. The sample skewness is negative while the excess
kurtosis is close to zero. The large negative skewness
casts doubt on the appropriateness of the normal
distribution.

c) The following R output gives the estimates of μ i , σ i , σ ij and ρ ij for dji and vbltx fromthe 5 years of monthly data.
> muhat.vals
dowjones
vbltx
0.002614947 0.003035592> sigmahat.vals
dowjones
vbltx
0.03014207 0.01990333> covhat.vals
dji,vbltx-3.683218e-05> rhohat.vals
dji,vbltx-0.06139438

Which asset appears to be riskier? Do you think there is any benefit of holding a portfolio
consisting of dji and vbltx? Justify your answer.
Risk is generally measured by the return standard
deviation. Here, the sd for dji is 0.030 and the sd forvbltx is 0.020. Therefore dji is the riskier asset. In a
two asset portfolio, the portfolio variance isx12σ 12 + x 22σ 22 + 2 x1 x 2σ 12 . If the two assets are negatively
correlated then this reduces the portfolio variance(diversification effect). Here, dji and vbltx are slightly
negatively correlated so there is a clear risk reduction
benefit of holding these two assets in a portfolio.

d) Consider a 12-month (one year) investment. Let rt (12) denote the 12-month (annual)
continuously compounded return. Using the monthly CER estimates for dji and vbltx,
give the estimates for the annual mean and standard deviation.
> muhat.vals*12
dowjones
vbltx
0.03137937 0.03642711> sigmahat.vals*sqrt(12)dowjones
vbltx
0.10441518 0.06894717

e) Using the above output, compute for both assets estimated standard errors for
μ , σ and ρ . Briefly comment on the precision of the estimates.
> nobs = length(ret.z[,1])> se.muhat = sigmahat.vals/sqrt(nobs)> se.sigma = sigmahat.vals/sqrt(2*nobs)> se.rho = (1 - rhohat.vals^2)/sqrt(nobs)> se.muhat
dowjones
vbltx
0.003891324 0.002569509> se.sigma
dowjones
vbltx
0.002751582 0.001816917> se.rho
dji,vbltx
0.1286128
The standard errors for the mean are about the same
magnitude as the estimates for the mean indicating that themeans are not estimated well. The standard errors for thestandard deviations are much smaller than the estimatedvalues and indicate that the standard deviations are
estimated more precisely than the means. The standard error
for the estimated correlation is quite large relative tothe estimate and shows that the correlation is not
estimated precisely.

f) For dji only compute 95% confidence intervals for μ and σ . Briefly comment on theprecision of the estimates. In particular, note if both positive and negative values are in
the respective confidence intervals.
> # 95% ci for mu

> upper = muhat.vals + 2*se.muhat> lower = muhat.vals - 2*se.muhat> cbind(lower[1],upper[1])
[,1]
[,2]
dowjones -0.005167701 0.01039760> # 95% ci for sigma> upper = sigmahat.vals + 2*se.sigma> lower = sigmahat.vals - 2*se.sigma> cbind(lower[1],upper[1])
[,1]
[,2]
dowjones 0.02463890 0.03564523
The wide confidence interval for the mean containing
positive and negative values indicates imprecise
estimation. In contrast, the confidence interval for the sd
is fairly narrow indicating a precise estimation.

g) Below are the sample autocorrelation functions for dji and vbltx. Using theinformation in these graphs, would you say that the CER model assumption that returns
are uncorrelated over time is appropriate? Briefly justify your answer.

-0.2 0.0 0.2

ACF

Dow Jones Returns

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.2

1.4

Lag

-0.2 0.0 0.2

ACFLong T erm Bond Returns

0.2

0.4

0.6

0.8

1.0Lag

The estimates autocorrelations are small for both dji andvbltx. Note of the estimates are bigger than 0.2. Thereappears to be no compelling evidence for exploitable
autocorrelation in these returns.

h) Suppose you currently hold $2M (million) in the Dow Jones 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 1% and 5% value-at-risk (VaR) associated with a one-monthinvestment in the Dow. Hint: the 1% and 5% quantiles of the standard normal distribution
are -2.326 and -1.645.
> W0 = 2> q.05 = muhat.vals + sigmahat.vals*qnorm(0.05)> q.01 = muhat.vals + sigmahat.vals*qnorm(0.01)> VaR.05 = (exp(q.05[1]) - 1)*W0> VaR.01 = (exp(q.01[1]) - 1)*W0> VaR.05
dowjones-0.09175716> VaR.01
dowjones-0.1305558i) Describe briefly how you could compute an estimated standard error for the estimateof 5% VaR found in part (h) above.
You could use bootstrapping to obtain an estimated SE forthe 5% VaR. This involves sampling with replacement fromthe original data to create B different samples. On eachsample the 5% VaR is computed, and the estimated bootstrap
SE is the sample standard deviation of these B 5% VaRvalues.