finance homework

Comparison The Fama-French market factor (Mkt) is better diversified than the SP 500, containing about 10 times as many stocks. However, the additional stocks in Mkt are relatively small, and hence the performance of the value weighted portfolios is expected to be quite similar, and the correlation of the excess returns very high. Notice also that a sample of 81 annual returns with as large a SD as we have here is still quite small.l comparing the continuously compounded excess returns we see that the difference between the two portfolios is indeed quite small, and the correlation coefficient between their returns is 0.99, almost perfect. The SD of the two portfolios is indistinguishable. Both deviate from the normal distribution as seen from the negative skew and positive kurtosis (that is, they have fat tails -- although not large.) Accordingly, the VaR (5% percentile) of the two is smaller than what is expected from a normal with same mean and SD. This is also indicated by the lower minimum excess return for the period. The serial correlation is also small and indistinguishable across the portfolios. As a result of all this, we expect the risk premium of the two portfolios to be similar, as we find from the sample. Notice that the SD of the 81-year average for the SP500 is somewhat higher (because of the small positive serial correlation) than 20.46/sqrt(81) = 2.27, and almost identical for the Mkt. This SD is far larger than the difference in the average of the two portfolios (0.02%). Notice also that the excess return of both portfolios have a small negative correlation with the risk-free rate. Since we expect the risk-free rate to be highly correlated with the rate of inflation, this suggests that equities are not a perfect hedge against inflation. More rigorous analysis of this point is important, but beyond the scope of this question.

Analysis The bond portfolio is less risky, its SD is less than half that of the stock portfolio. Yet, as the portfolio table shows, mixing 0.87% of bonds with 13% (riskier) stocks would have produced a portfolio less risky than bonds. In the particular sample of these 20 years, the average return on the less risky portfolio (bonds) was higher than that of the riskier portfolio (stocks). This is exactly what is meant by 'risk,' expectation are not going to always be realized.

Analysis (1) Five-year betas. The beta of Ford has been quite aggressive, representing the sensitivity of sales to economic conditions. The beta of GM is much lower than that of Ford, representing the large part of the finance arm of the company. The Beta of Toyota is defensive for U.S, investors, reflecting the large part of Toyota's business abroad. (2) Five-year alpha estimates. The large negative alpha values of Ford and GM reflect the greater than expected deterioration of these companies. Toyota's positive alpha is due to the pre-crisis years, when its performance was stellar, better than expected. (3) Beta over two-year (first and last) subperiods. As a first pass we note that large standard deviation of the beta estimates. Non of the subperiod estimates deviates from the overall period estimate by more than two standard deviation. That is, the t-statistic of deviation from the overall period is not significant for any of the companies subperiod beta estimates. Looking beyond the aforementioned observation, the differences can be attributed to different alpha values during the subperiods. The case of Toyota is most revealing: The alpha estimate for the first two years is positive and for the last two years negative (both large). Following a good performance in the "normal" years prior to the crisis, Toyota surprised investors with a negative perofrmance, beyond what could be expected from the index. This suggests that a beta of around 0.5 is more reliable. The shift of the intercepts from positive to negative when the index moved to largely negative returns, explains why the line is steeper when estimated for the overall period. Draw a line in the positive qudrant for the index with a slope of 0.5 and positive intercept. Then draw a line with similar slope in the negative quadrant of the index with a negative intercept. You can see that a line that reconciles the observations for both quadrants will be steeper. The same logic explains part of the behavior of subperiod betas for Ford and GM.

Index level

26-week MA

Week of Year

Index and index MA

SP 500 index and 26-week moving average

1

1

FSRBX/SPY

26-week MA

Year.week

FSRBX/SP 500, MA

Relative strength FSRBX vs. SP 500

1

1