Exercise 1: Effectiveness of Treynor-Black Strategy
At Jan. 1st, 2011, the optimal Treynor-Black portfolio is as below.

After holding the four portfolios for 36 months, the performances of the four portfolios are shown in the following tables. Wi of each portfolio represents the characteristic of the certain portfolio and M2 measures how much the portfolio outperform the market.

The results show that Portfolio 1 which has the aforementioned Treynor-Black portfolio weight distribution has the best performance. The M2 is 0.0347. Portfolio 2 has equal weights on each of the six stocks, which is unreasonable. As a result, its performance is not good, with a 0.0019 M2. Portfolio 3 has equal weights on the positive alpha stocks, which makes more sense than portfolio 2. The portfolio is better accordingly. Portfolio is 100% invested in S&P 500 Index, which is a passive portfolio. So we only use S(m)2 as a performance measurement.

Exercise 2: Analysis of Risk-Free Rate Total period | | | | Price-based | Yield-based | Mean | 0.002% | 0.130% | S.D. | 0.060% | 0.155% | Median | -0.002% | 0.019% | | | | A HPR | 0.220% | 15.604% | A HPR(annualized) | 0.022% | 1.560% | | | | AMRy(annualized) | 0.022% | 1.560% | AMRy=(∑Ri/Q)*P | | | ASDy(annualized) | 0.209% | 0.537% | ASDy=STDEV.S*(P^(1/2)) | | |

Pre-crisis years | | | | Price-based | Yield-based | Mean | -0.028% | 0.265% | S.D. | 0.044% | 0.124% | Median | -0.031% | 0.269% | | | | A HPR | -0.994% | 9.540% | A HPR(annualized) | -0.099% | 0.954% | | | | AMRy(annualized) | -0.099% | 0.954% | AMRy=(∑Ri/Q)*P | | | ASDy(annualized) | 0.151% | 0.428% | ASDy=STDEV.S*(P^(1/2)) | | |

Crisis years | | | | Price-based | Yield-based | Mean | 0.034% | 0.160% | S.D. | 0.092% | 0.160% | Median | 0.004% | 0.127% | | | | A HPR | 1.217% | 15.286% | A HPR(annualized) | 0.122% | 1.529% | | | | AMRy(annualized) | 0.122% | 1.529% | AMRy=(∑Ri/Q)*P | | | ASDy(annualized) | 0.317% | 0.526% | ASDy=STDEV.S*(P^(1/2)) | | |

Post-crisis years | | | | Price-based | Yield-based | Mean | 0.000% | 0.007% | S.D. | 0.008% | 0.004% | Median | -0.001% | 0.006% | | | | A HPR | -0.003% | 0.318% | A HPR(annualized) | 0.000% | 0.032% | | | | AMRy(annualized) | 0.000% | 0.032% | AMRy=(∑Ri/Q)*P | | | ASDy(annualized) | 0.027% | 0.014% | ASDy=STDEV.S*(P^(1/2)) | | |

We can see that the mean and median of risk free rates calculated by price-based method would be negative occasionally whether in total period or in sub-periods. And the numbers from yield-based method are almost greater than those from price-based method by 100 basis points. These attribute to the larger number of holding period return calculated by yield-based, which is 15.604% in total period and in crisis years. I think the positive of risk free rates would be more accepted by people, so yield-based method would be more popular among the investors.

Exercise 3: Performance & Efficiency Analysis of ETFs (a) Mean and standard deviation

In general, all ETFs have higher mean returns than indexes, except Vanguard Total International Stock. This shows that ETFS outperform the underlying indices overall. Besides, the standard deviation of each ETF and its underlying index is very close, which indicates similar volatility between the ETFs and the underlying indices.

(b) Tracking error | TE | Russell 3000 | 0.134% | QQQ | 0.170% | Dow Jones | 0.195% | S&P 500 | 0.196% | S&P/TSX 60 | 0.268% | MSCI | 0.546% |
The ranking of tracking errors of the six ETFs is listed above. Tracking error is a divergence between the price behavior of a position or a portfolio and the price behavior of a benchmark. This is often in the context of a hedge or mutual fund that did not work as effectively as intended. So less TE is preferred. It means the ETF imitates the benchmark better. In the above table, iShare Russell 3000 Index has the least TE and Vanguard Total International Stock has the largest TE.

(c) Regression Regressions of each pair of ETF and its underlying index are done and the result is ranked as shown above. Alpha shows how much the ETF outperforms the benchmark. From this prospective, Dow Jones Industrial Average ETF has done the best job and Vanguard Total International Stock underperformed the index. Beta shows how much the ETF is related to the underlying benchmark. The table shows that Vanguard Total International Stock is most volatile and has the biggest beta among the six. All other ETFs has beta that is around one, which indicates that the ETFs imitate their underlying indices quite well.