LAB 3 Report

Done by:

Lang, Qin 1988-12-05

Low Lihui Valerie1989-09-24

Q1

Before we evaluate the actual investment performance of the five constructed portfolios for period 1992.02-2008.07, we firstly calculate the mean, variance and standard deviation of each of the portfolio using Excel. The results are generated as below: Portfolios| Z1| Z2| Z3| Z4| Z5| Zm|

Mean| 0.008490| 0.003843| 0.009980| 0.000141| 0.004840| 0.0066| Variance| 0.005335| 0.007238| 0.004951| 0.008652| 0.00491| 0.003932| Standard Deviation| 0.073039| 0.085075| 0.070361| 0.093016| 0.070071| 0.062707|

Sharpe Ratio

According to Keith Cuthbertson and Dirk Nitzsche (2004), Sharp ratio measures the slope of the transformation line that, through constructing the tangency portfolio we can find the optimum weight to invest in risky assets. It is defined for any portfolio i as: sri = ERi -Rfσi

where E[Ri] is the expected return on portfolio i, σi denotes the variance of portfolio i, and Rf is the risk-free rate. To estimate the Sharpe ratio based on one period returns, we have sri= Mean (Zit)Std (Zit)

For aggregation over time, the Sharpe ratio for q period returns is q times the Sharpe ratio computed above, if serially uncorrelated returns exist. Therefore, we have sri (q) = q sri

The corresponding results generated by Excel are presented as follows, Portfolios| Z1| Z2| Z3| Z4| Z5|

Sharp ratio| 0.116237| 0.045171| 0.14184| 0.001515| 0.069077| Sharp ratio (yearly)| 0.402655| 0.156477| 0.49135| 0.005247| 0.239288| Next, we perform a significance test for the estimated Sharpe ratios. Assume that returns are IID. The hypotheses for the test statistic are: H0: sri = 0

H1: sri ≠ 0 (sri < 0 or sri > 0)

The test statistic is then defined as

sri ~ N(0, 1+12sri2T)

Assume 5% significance level, we calculate the P-value in Excel for each portfolio using P-value = 2* (1-Normsdist (abs (T1+ 12sri2sri)

Thus, we get the following results

Portfolios| Z1| Z2| Z3| Z4| Z5|

Sharpe test| 1.630096| 0.635287| 1.985908| 0.021314| 0.970837| P-value| 0.103081| 0.525241| 0.047044| 0.982995| 0.33163| As can be seen, under the assumption, only Z3 is significantly different from 0 (P-value < 0.05). Thus, for Z3, H0 is rejected. As for the others, we do not reject H0. In other words, Z3 is the optimum portfolio regarding the Sharpe ratio as there is significant excess return for Z3. Treynor Ratio

Based on Keith Cuthbertson and Dirk Nitzsche (2004), Treynor’s index is a measure of excess return per unit of risk. Different from Sharpe ratio, the denominator is changed to beta as the measure of risk, while the numerator is unchanged. It is shown as follows, Tri = ERi- Rfβi

We compute βi as

βi= cov ( Zit, Zmt)var(Zmt)

Based on the mean value calculated in the first place, we generate Treynor ratio by tri = Mean (Zit)βi

Thus, the results are presented as below,

Portfolios| Z1| Z2| Z3| Z4| Z5|

Treynor ratio| 0.007759| 0.003265| 0.013343| 0.000121| 0.004826| As Keith Cuthbertson and Dirk Nitzsche (2004) suggest, since Treynor ratio is used to compare the historic performance of alternative portfolios, the best portfolio is the one with the highest Treynor ratio. Therefore, we choose Z3 accordingly in our case. Jensen’s alpha

B.P.S. Murthi et al. (1997) define Jensen’s alpha as the difference between the actual portfolio return and the estimated benchmark return. It is given by Zit = αi + βiZmt + εit

In Excel, OLS is applied to generate the estimated value of αi using αi= Intercept (Zit, Zmt)

Thus, we get the following results

Portfolios| Z1| Z2| Z3| Z4| Z5|

Jensen’s Alpha| 0.001282| -0.00391| 0.005053| -0.007552| -0.001767| David A. Sauer (1997) suggests that a statistically significant positive alpha would imply superior investment performance. As can be seen, among the five portfolios...