University of Washington
Department of Economics

Fall 2009
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 = 84.

I. Return Calculations (16 pts, 4 points each)
Consider a one year investment in two assets: the Vanguard S&P 500 index (VFINX) andthe Vanguard Short Term Bond mutual fund (VBISX). Suppose you buy one share of theS&P 500 fund and one share of the bond fund at the end of September, 2008 for
P sp 500,t −1 = 104.61, Pbond ,t −1 = 9.70 , and then sell these shares at the end of September,2009 for P sp 500,t = 97.45, Pbond ,t = 10.46 . (Note: these are actual closing prices taken fromYahoo!)a. What are the simple annual returns for the two investments?
> p.sp500.1 = 104.61> p.sp500.2 = 97.45> p.bond.1 = 9.70
> p.bond.2 = 10.46# a) simple returns on sp500 and bond> r.sp500 = (p.sp500.2 - p.sp500.1)/p.sp500.1> r.bond = (p.bond.2 - p.bond.1)/p.bond.1> r.sp500[1] -0.06844> r.bond[1] 0.07835
b. What are the continuously compounded annual returns for the two investments?
# b) cc returns on Amazon and sp500> log(1 + r.sp500)[1] -0.0709> log(1 + r.bond)[1] 0.07543

c. Assume you get the same annual returns from part a. every year for the next 10 years.
How much will $10,000 invested in each fund be worth after 10 years?
> w0 = 10000> w1.sp500 = w0*(1 + r.sp500)^10> w1.bond = w0*(1 + r.bond)^10> w1.sp500[1] 4921> w1.bond[1] 21262

d. The annual inflation rate between September 2008 and September 2009 was about-1% (yes, we actually had deflation!). Using this information, determine the simple andcontinuously compounded real annual returns on S&P 500 and the bond fund.
> inflat = -0.01# simple real returns> r.sp500.real = (1+r.sp500)/(1+inflat) - 1> r.bond.real = (1+r.bond)/(1+inflat) - 1> r.sp500.real[1] -0.05904> r.bond.real[1] 0.08924# cc real returns> log(1+r.sp500.real)[1] -0.06085> log(1+r.bond.real)[1] 0.08548

II. Probability Theory and Matrix Algebra (16 points, 4 points each)
Let rt denote the monthly continuously compounded return on an asset and suppose thatrt ~ iid N (0.01, (0.05) 2 ) .
a. What is the relationship between the continuously compounded return rt and the simple
return Rt? Given this relationship, what is the probability distribution of 1 + Rt?
rt = ln(1 + Rt ) ⇒ Rt = e rt − 1 ⇒ 1 + Rt = e rt

Because rt ~ iid N (0.01, (0.05) 2 ) , it follows that 1 + Rt ~ lognormal(0.01, (0.05) 2 ) .

b. Give an expression for the 6-month continuously compounded return, rt(6), in terms ofthe monthly continuously compounded returns. Using this expression computeE[rt (6)], var(rt (6)), and SD(rt (6)). What is the probability distribution of rt(6)?
5rt (6) = rt + rt −1 + " + rt −5 = ∑ rt − jj =055j =0j =0E[rt (6)] = ∑ E[rt − j ] = ∑ 0.01 = 6 × (0.01) = 0.06
⎛ 5
⎞ 5var(rt (6)) = var ⎜ ∑ rt − j ⎟ = ∑ var(rt − j ) = 6 × (0.05) 2 = 0.015⎝ j =0⎠ j =0SD (rt (6)) = 6(0.05) = 0.015 = 0.1224
Therefore, rt (6) ~ N (0.06, (0.1224) 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 the threeassets).

c. Using matrix algebra, give expressions for the returns, expected returns and variancesfor the 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 ′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.

3

3

i =1

i =1

∑ xi = x′1 = 1, ∑ yi = y′1 = 1
cov( R p , x , R p , y ) = cov(x′R, y ′R ) = x′ΣyIII. Time Series Concepts (16 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 20-2510y20-1204060

80

1000204060

TimeTimeProcess 3

Process 480

10080

100y4-4-2-2-10y3214200y1115220

Process 102040

60

8010002040

TimeFigure 1: Realizations from four stochastic processes.

60Time

The processes that appear to be covariance stationary are Process 1 and Process 3. Bothprocesses are not trending, so the mean is not changing over time, and both processes
appear to have constant volatility over time. In fact, Process 1 is Gaussian White Noise,and Process 2 is an AR(1) process with φ = 0.9 .
Processes 2 and 4 appear to be non-stationary. Process 2 appears to wander away fromits initial starting point and exhibits an upward drift. This is non-stationary behavior. Infact, Process 2 is a random walk. Process 4 shows higher volatility in the first half of thesample than in the second half. Hence, the volatility of this process is not constant overtime which violates the stationarity conditions. In fact, Process 3 is a Gaussian White
Noise process with σ = 1 in the first half of the sample and σ = 0.25 in the second half ofthe sample.

Figure 2 below shows a realization of a stochastic process representing a monthly timeseries of overlapping 2-month continuously compounded returns, where the 1-monthcontinuously compounded returns follow a Gaussian White noise process.
c. Based on the sample autocorrelations, which time series process is most appropriate fordescribing the series: MA(1) or AR(1)?
Based on the sample autocovariance function (SACF), an MA(1) model looks moreθ
appropriate than an AR(1) model. In an MA(1) model, ρ1 =and ρ j = 0 for j > 11+θ 2
and in an AR(1) model ρ j = φ j . The SACF shows ρˆ1 ≈ 0.42 and ρˆ 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 φ ? If you think the process is a MA(1) process, what do you
think is the value of the moving average parameter θ ?
If the processes was an AR(1), then φ ≈ ρˆ1 ≈ 0.42 . If the process was an MA(1) thenθ= ρ1 ≈ 0.42 ⇒ θ = 0.42 × (1 + θ 2 ) = 0.42 + 0.42 × θ 2 ⇒ 0.42 × θ 2 − θ + 0.42 = 021+θ
This is a quadratic equation in θ . Using the quadratic formula, there are two solutions

θ=1 ± 1 − 4(0.42)2= 1.836 and 0.554 .
2(0.42)

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

020

40

60

80

100

Time

0.6
0.2-0.2

ACF

1.0Overlapping 2-month cc return

051015Lag
Figure 2: Stochastic process representing overlapping 2-month continuously compounded returns.

IV. Descriptive Statistics (16 points, 4 points each)
Figure 3 shows monthly continuously compounded returns on the Vanguard S&P 500index (VFINX) and the Vanguard Short Term Bond mutual fund (VBISX) over the 5year period October 2004, through October 2009. For this period there are T=60
monthly observations.

-0.050.010
0.000-0.010

Short term bond fund0.020 -0.15

S&P 500 index

0.05

Monthly cc returns

2005

2006

2007

2008

2009

Index

Figure 3: 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 stationary processes have time invariant means, variances andautocorrelations that only depend on the time between observations. Consider first theS&P 500 index. The returns are not obviously trending so it looks like the mean is notchanging over time. However, the volatility in the latter part of the sample appears to bemuch bigger than in the first part of the sample. This is evidence of non-stationarity. Next
consider the Bond fund. The returns are not obviously trending up or down, but it looks
like the mean is slightly higher in the latter part of the sample than in the first part whichis evidence of non-stationarity. Also, the volatility of the bond returns appears to beslightly higher in the second half of the sample but the difference is not as large as it wasfor the S&P 500 returns. Overall, there is some evidence that the bond returns may be
non-stationary.

The figures below gives some graphical diagnostics of the return distributions for theS&P 500 index and the bond fund.

-0.15-0.05

cc return6402

cc return

810

0.05

S&P 500 monthly cc returns

-0.20

-0.10

0.00

0.05

0.10

vfinx.mat

-0.05
-0.15

Sample Quantiles

86402

density estimate

0.05

Normal Q-Q Plot

10

Smoothed density

-0.2

-0.1

0.0

0.1

-2

cc return-101

2

Theoretical Quantiles

0.020
0.010
0.000
-0.010cc return10 20 30 40 50 60 70
0cc return

Bond fund monthly cc returns

-0.01

0.00

0.01

0.02

vbisx.mat

Normal Q-Q Plot

0.010
-0.0100.000

Sample Quantiles

5030
0 10density estimate

700.020

Smoothed density

-0.01

0.00

0.01

cc return

0.02

-2

-101

Theoretical Quantiles

2-0.010-0.005

0.000

0.005

Bond returns
0.010

0.015

0.020

S&P 500

-0.15

-0.10
Bond-0.05

S&P 500 returns

0.00

0.05

0.10

-0.15

-0.10

-0.05

0.00

0.05

0.10

b. Do the returns on the S&P500 index and the bond fund look normally distributed?
Briefly justify your answer.
Consider first the S&P 500 index. The returns exhibit non-normal behavior. The
histogram shows a long left tail (negative skewness) and the boxplot indicates a somelarge 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.
Next consider the bond fund. The histogram and smoothed histogram are somewhat
symmetric, but show an interesting bi-modality (e.g. two modes). The second mode isdistinctly positive around 2%. Recall, the time plot of the bond returns looks to have ahigher mean in the second part of the sample and this appears to be showing up in thehistogram as two modes. The boxplot shows some positive skewness and a few large
positive values. The QQ-plot is linear in the left tail but not linear in the right tail. All of
this is evidence that the bond returns are also not normally distributed.

c. Which asset appears to be riskier? Briefly justify your answer.
If we measure risk by return volatility, then the S&P 500 index is clearly more risky than
the short term bond fund. The figure showing the side-by-side boxplots clearly shows thatthe bond fund returns are substantially less volatile than the S&P 500 index returns.
d. Based on the scatterplot of returns, does there appear to be any linear dependence
between the returns on the S&P500 index and the bond fund? Briefly justify youranswer.
The scatterplot of returns does not show any obvious linear pattern. The points are fairly
equally distributed in the four quandrants which would indicate that the covariance andcorrelation between the returns is close to zero (we will see this in the sample statistics inthe next problem). Hence, the S&P 500 and bond returns are not linearly related.

V. 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 ) = ρijfor the monthly continuously compounded returns on the Vanguard S&P500 index andthe Vanguard short term bond fund presented in part IV. 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 bond fund(VBISX) from the T=60 months of data:

> muhat.vals
sp500
bond0.0002158 0.0035687> sigmahat.vals
sp500
bond
0.047410 0.006237> covhat.vals
sp500,bond
3.236e-06> rhohat.vals
sp500,bond0.01094

a. Using the above output, compute for both assets estimated standard errors forμ and σ , and compute an estimated standard error for ρ . Briefly comment on theprecision of the estimates.
> nobs = nrow(ret.z)> se.muhat = sigmahat.vals/sqrt(nobs)> se.sigma = sigmahat.vals/sqrt(2*nobs)> se.rho = (1 - rhohat.vals^2)/sqrt(nobs)> se.muhat
sp500
bond0.0061206 0.0008052> se.sigma
sp500
bond0.0043279 0.0005694> se.rho
sp500,bond0.1291> nobs[1] 60The means are estimated less precisely than the standard deviation, and the mean andstandard deviation are estimated more precisely for the bond fund than for the S&P 500
index. The correlation is not estimated very precisely as the SE is much larger than theestimate.
b. 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.

> upper = muhat.vals + 2*se.muhat> lower = muhat.vals - 2*se.muhat> cbind(lower[1],upper[1])
[,1]
[,2]
sp500 -0.01203 0.01246# 95% ci for sigma> upper = sigmahat.vals + 2*se.sigma> lower = sigmahat.vals - 2*se.sigma> cbind(lower[1],upper[1])
[,1]
[,2]
sp500 0.03875 0.05607

The 95% confidence interval for μ contains both negative and positive values andindicates that the mean could be as low as -1.2% or as high as 1.2% per month. This is afairly wide range and indicates much uncertainty about the true value of the mean.
The 95% confidence interval for σ only contains positive values (which is good because
σ must be positive) and is fairly narrow: from 3.8% to 5.6%.

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 1% and 5% value-at-risk (VaR) associated with a one-monthinvestment in the S&P 500 index. Hint: the 1% and 5% quantiles of the standard normal
distribution are -2.326 and -1.645, respectively.
> 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
sp500-0.1496> VaR.01
sp500-0.2085

With 5% probability, you can lose $149,600 or more, and with 1% probability you canlose $208,500 or more.
d. Describe briefly how you could use the bootstrap to compute an estimated standarderror for the 5% VaR found in part c above.

Bootstrapping involves sampling with replacement from the original data to create B
different 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.
e) Below are the sample autocorrelation functions for the S&P 500 and the bond fund.
Using the information in these graphs, would you say that the CER model assumptionthat returns are uncorrelated over time is appropriate? Briefly justify your answer.

0.6
0.2
-0.2

ACF

1.0

Vanguard S&P 500 Index

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.2

1.4

Lag

0.6
0.2
-0.2

ACF

1.0

Vanguard Short Term Bond

0.0

0.2

0.4

0.6

0.8

1.0

Lag

The returns on the S&P 500 index display some large positive autocorrelations at lags 1and 4. This is evidence against the CER model assumption of uncorrelated returns. Thebond returns, in contrast, show no evidence of autocorrelation and exhibit behavior
consistent with the CER model assumption of no autocorrelation.