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Material Type: Exam; Class: ECONOMETRIC APP; Subject: Economics; University: University of Washington - Seattle; Term: Unknown 1989;
Typology: Exams
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University of Washington Fall 2003 Department of Economics Eric Zivot
Economics 483
Final Exam Suggested Solutions
This is a closed book and closed note exam. However, you are allowed one page of handwritten notes. Answer all questions and write all answers in a blue book. Total points = 100.
I. Matrix Algebra and Portfolio Math (20 points)
matrix
2 1 12 1 2 12 2 2
2 1 2
N N
N N N
σ σ σ σ σ σ
σ σ σ
Using simple matrix algebra, answer the following questions.
a. Write down the optimization problem used to determine the global minimum variance portfolio assuming short sales are allowed. Let m denote the vector of portfolio weights in the global minimum variance portfolio.
min s.t. 1 m m ′ Σm m 1 ′ =
b. Write down the optimization problem used to determine the global minimum variance portfolio assuming short sales are not allowed. Again, let m denote the vector of portfolio weights in the global minimum variance portfolio.
min s.t. 1, (^) i 0 1, , m m ′ Σm m 1 ′ = m ≥ i = … N
c. Write down the optimization problem used to determine an efficient portfolio with target
in the efficient portfolio.
d. Write down the optimization problem used to determine the tangency portfolio, assuming short sales are allowed and the risk free rate is give by rf. Let t denote the vector of portfolio
weights in the tangency portfolio.
max (^) 1/ 2 s.t. 1 f t
′ (^) r ′ (^) = ′
t - t 1 t Σt
24 month rolling means and stdevs for European equity index
Q1 Q2 1999 Q3 Q4 Q1 Q2 2000 Q3 Q4 Q1 Q2 2001 Q3 Q4 Q1 Q2 2002 Q3 Q4 Q1 Q2 2003 Q3 Q
-0.
-0.
returnsmeans stdevs
For the European equity returns, the rolling means start out slightly positive then dip to negative values and only become positive again at the end of the sample. The negative returns reflect the decline in equity markets that were generally experienced during the last two years as a result of the tech crash and the recession. The rolling standard deviations increases slightly over the sample (from about 4% per month to about 6% ). The switch in the means from positive to negative seems to represent a non-constant mean. The assumption of a constant mean and standard deviation does not seem to be reasonable for the European equity returns.
b. The following graphs show 24-month rolling correlations between the returns on the short- term bond index and the European equity index. Using these graphs, do you think that the
Returns on short-term bonds and European equity
2000 Q4^ Q1^ Q2^2001 Q3 Q4^ Q1^ Q2^2002 Q3^ Q4^ Q1^ Q2^2003 Q3^ Q
-0.
24 month rolling correlations b/w short-term bonds and European equity
2000 Q4^ Q1^ Q2^2001 Q3 Q4^ Q1^ Q2^2002 Q3^ Q4^ Q1^ Q2^2003 Q3^ Q
-0.
-0.
-0.
The rolling correlations start out slightly positive, around 0.1, and steadily decline to a negative value of about -0.3 at the end of the sample. This result is due to the fact that the mean returns on the European equity went from positive to negative during the sample period, whereas the
short-term bond returns had a stable positive mean. Thus, the assumption of a constant covariance between bond returns and equity returns does not seem to be supported by the data.
III. Empirical Analysis of the single index model (40 points)
The following represents S-PLUS linear regression output from estimating the single index model for the Vanguard short-term bond index (vbisx) and the Vanguard European Equity index (veurx) using monthly continuously compounded return data over the period November 1998 – October 2003. In the regressions, the market index is the Vanguard extended market index (vexmx).
vbisx.fit = lm(vbisx~vexmx) summary(vbisx.fit,cor=F)
Call: lm(formula = vbisx ~ vexmx) Residuals: Min 1Q Median 3Q Max -0.01654 -0.004081 -0.000257 0.004462 0.
Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 0.0042 0.0009 4.7468 0. vexmx -0.0282 0.0126 -2.2290 0.
Residual standard error: 0.00688 on 58 degrees of freedom Multiple R-Squared: 0. F-statistic: 4.969 on 1 and 58 degrees of freedom, the p-value is 0.
SI regression for short term bonds
extended market index
short term bond index
-0.1 0.0 0.
-0.
veurx.fit = lm(veurx~vexmx) summary(veurx.fit,cor=F)
Call: lm(formula = veurx ~ vexmx) Residuals: Min 1Q Median 3Q Max -0.1047 -0.02203 -0.0009752 0.02241 0.
Coefficients: Value Std. Error t value Pr(>|t|)
risk than the bond index and the power of diversification should concentrate the risk on market risk. This is verified by comparing the R-square values for the two regressions.
d. For each regression, what is the proportion of market or systematic risk and what is the proportion of firm specific or unsystematic risk? For each regression, what does the Residual Standard Error represent?
R-square (% market risk) 1-(R-square) (% non-market risk) Short-term bond 0.08 0. European Equity 0.47 0.
For each regression, the residual standard error is an estimate of the typical size of ε in the SI
e. Why should the European equity index have a greater proportion of systematic risk and a larger standard error than the short-term bond index?
The European equity index is a large diversified portfolio of stocks, whereas the short-term bond index is a small, not diversified, index of short-term bonds. The diversification effect concentrates the market risk of the European equity index and reduces the amount of non-market risk. This tends to increase the R-square (% of market risk). Short-term bonds generally do not have much market risk since short-term interest rates are fairly stable. Hence the R-square of the bond index is expected to be low (close to zero). This is what we see in the data.
f. For the short-term bond index and the European equity index, test the null hypothesis that β = 0 against the alternative hypothesis that β ≠ 0 using a 5% significance level. What do you conclude?
0
t SE β
=
Decision: reject if |t-stat| > 2
Short-term bonds 0.0282 0
European equity 0.5039 0
g. For Short term bond index and the European equity index, test the null hypothesis that β = 1 against the alternative hypothesis that β ≠ 1 using a 5% significance level. What do you conclude?
1
t SE β
=
Decision: reject if |t-stat| > 2
Short-term bonds 0.0282 1
European equity 0.5039 1
h. The following graphs show the 24-month rolling estimates of β for the SI models for the short-term bond index and the European equity index. Using these graphs, what can you say about the stability of β over time?
24 month rolling betas for short-term bond index
2000 Q4^ Q1^ Q2^2001 Q3^ Q4^ Q1^ Q2^2002 Q3 Q4^ Q1^ Q2^2003 Q3^ Q
-0.
-0.
-0.
-0.
The rolling betas for the short-term bond index start out slightly positive and decline steadily to about -0.08. There appears to be clear evidence that the beta has declined over the sample (is not constant).
24 month rolling betas for European equity index
2000 Q4^ Q1^ Q2^2001 Q3^ Q4^ Q1^ Q2^2002 Q3 Q4^ Q1^ Q2^2003 Q3^ Q
