CHAPTER 22



CHAPTER 26

EVALUATION OF PORTFOLIO PERFORMANCE

Answers to Questions

1. The two major factors would be: (1) attempt to derive risk-adjusted returns that exceed a naive buy-and-hold policy and (2) completely diversify - i.e., eliminated all unsystematic risk from the portfolio. A portfolio manager can do one or both of two things to derive superior risk-adjusted returns. The first is to have superior timing regarding market cycles and adjust your portfolio accordingly. Alternatively, one can consistently select undervalued stocks. As long as you do not make major mistakes with the rest of the portfolio, these actions should result in superior risk-adjusted returns.

2. Treynor (1965) divided a fund’s excess return (return less risk-free rate) by its beta. For a fund not completely diversified, Treynor’s “T” value will understate risk and overstate performance. Sharpe (1966) divided a fund’s excess return by its standard deviation. Sharpe’s “S” value will produce evaluations very similar to Treynor’s for funds that are well diversified. Jensen (1968) measures performance as the difference between a fund’s actual and required returns. Since the latter return is based on the CAPM and a fund’s beta, Jensen makes the same implicit assumptions as Treynor - namely, that funds are completely diversified. The information ratio (IR) measures a portfolio’s average return in excess of that of a benchmark, divided by the standard deviation of this excess return.

3. For portfolios with R2 values noticeably less than 1.0, it would make sense to compute both measures. Differences in the rankings generated by the two measures would suggest less-than-complete diversification by some funds - specifically, those that were ranked higher by Treynor than by Sharpe.

4. Jensen’s alpha (α) is found from the equation Rjt – RFRt == αj + βj[Rmt – RFRt] +ejt. The aj indicates whether a manager has superior (αj > 0) or inferior (αj < 0) ability in market timing or stock selection, or both. As suggested above, Jensen defines superior (inferior) performance as a positive (negative) difference between a manager’s actual return and his CAPM-based required return. For poorly diversified funds, Jensen’s rankings would more closely resemble Treynor’s. For well-diversified funds, Jensen’s rankings would follow those of both Treynor and Sharpe. By replacing the CAPM with the APT, differences between funds’ actual and required returns (or “alphas”) could provide fresh evaluations of funds.

5. The Information Ratio (IR) is calculated by dividing the average return on the portfolio less a benchmark return by the standard deviation of the excess return. The IR can be viewed as a benefit-cost ratio in that the standard deviation of return can be viewed as a cost associated in the sense that it measures the unsystematic risk taken on by active management. Thus IR is a cost-benefit ratio that assesses the quality of the investor’s information deflated by unsystematic risk generated by the investment process.

6. Since the return for selectivity is the difference between overall performance and the required return for risk, if the overall performance exceeds the required return for risk, the portfolio experiences a positive return for selectivity. In the example, the required return would have to be less than -0.50 in order to experience a positive selectivity value. Common sense tells us that a negative required return for assuming risk is not realistic.

6. A high R2 value of .95 implies that the portfolio is highly diversified and, thus, the diversification term will be minimal. By definition, if we have a selectivity value of a positive 2.5 percent and a minimal diversification term, net selectivity will be a positive value.

CHAPTER 26

Answers to Problems

2(a).

[pic]

2(b).

[pic]

| |Sharpe |Treynor |

|P |2 |3 |

|Q |4 |2 |

|R |5 |5 |

|S |1 |1 |

|Market |3 |4 |

2(c). It is apparent from the rankings above that Portfolio Q was poorly diversified since Treynor ranked it #2 and Sharpe ranked it #4. Otherwise, the rankings are similar.

3. CFA Examination I (1994)

3(a). The Treynor measure (T) relates the rate of return earned above the risk-free rate to the portfolio beta during the period under consideration. Therefore, the Treynor measure shows the risk premium (excess return) earned per unit of systematic risk:

Ri- Rf

Ti =

(i

where: Ri = average rate of return for portfolio i during the specified period

Rf = average rate of return- on a risk-free investment during the specified period

(i = beta of portfolio i during the specified period.

Treynor Measure Performance Relative to the Market (S&P 500)

10% - 6%

T = = 6.7% Outperformed

0.60

Market (S&P 500)

12% - 6%

TM = = 6.0%

1.00

The Treynor measure examines portfolio performance in relation to the security market line (SML). Because the portfolio would plot above the SML, it outperformed the S&P 500 Index. Because T was greater than TM, 6.7 percent versus 6.0 percent, respectively, the portfolio clearly outperformed the market index.

The Sharpe measure (S) relates the rate of return earned above the risk free rate to the total risk of a portfolio by including the standard deviation of returns. Therefore, the Sharpe measure indicates the risk premium (excess return) per unit of total risk:

Ri - Rf

S =

(i

where: Ri = average rate of return for portfolio i during the specified period

Rf = average rate of return on a risk-free investment during the specified period

(i = standard deviation of the rate of return for portfolio i during the specified

period.

Sharpe Measure Performance Relative to the Market (S&P 500)

10% - 6%

S = = 0.222% Underperformed

18%

Market (S&P 500)

12% - 6%

SM= = 0.462%

13%

The Sharpe measure uses total risk to compare portfolios with the capital market line (CML). The portfolio would plot below the CML, indicating that it underperformed the market. Because S was less than SM, 0.222 versus 0.462, respectively, the portfolio underperformed the market. I

3(b). The Treynor measure assumes that the appropriate risk measure for a portfolio is its systematic risk, or beta. Hence, the Treynor measure implicitly assumes that the portfolio being measured is fully diversified. The Sharpe measure is similar to the Treynor measure except that the excess return on a portfolio is divided by the standard deviation of the portfolio.

For perfectly diversified portfolios (that is, those without any unsystematic or specific risk), the Treynor and Sharpe measures would give consistent results relative to the market index because the total variance of the portfolio would be the same as its systematic variance (beta). A poorly diversified portfolio could show better performance relative to the market if the Treynor measure is used but lower performance relative to the market if the Sharpe measure is used. Any difference between the two measures relative to the markets would come directly from a difference in diversification.

In particular, Portfolio X outperformed the market if measured by the Treynor measure but did not perform as well as the market using the Sharpe measure. The reason is that Portfolio X has a large amount of unsystematic risk. Such risk is not a factor in determining the value of the Treynor measure for the portfolio, because the Treynor measure considers only systematic risk. The Sharpe measure, however, considers total risk (that is, both systematic and unsystematic risk). Portfolio X, which has a low amount of systematic risk, could have a high amount of total risk, because of its lack of diversification. Hence, Portfolio X would have a high Treynor measure (because of low systematic risk) and a low Sharpe measure (because of high total risk).

4(a). Portfolio MNO enjoyed the highest degree of diversification since it had the highest R2 (94.8%). The statistical logic behind this conclusion comes from the CAPM which says that all fully diversified portfolios should be priced along the security market line. R2 is a measure of how well assets conform to the security market line, so R2 is also a measure of diversification.

4(b).

Fund Treynor Sharpe Jensen

ABC 0.975(4) 0.857(4) 0.192(4)

DEF 0.715(5) 0.619(5) -0.053(5)

GHI 1.574(1) 1.179(1) 0.463(1)

JKL 1.262(2) 0.915(3) 0.355(2)

MNO 1.134(3) 1.000(2) 0.296(3)

4(c).

Fund t(alpha)

ABC 1.7455(3)

DEF -0.2789(5)

GHI 2.4368(1)

JKL 1.6136(4)

MNO 2.1143(2)

Only GHI and MNO have significantly positive alphas at a 95% level of confidence.

5(a). (Information ratio) IRj = (j/(u where (u = standard error of the regression

IRA = .058/.533 = 0.1088

IRB = .115/5.884 = 0.0195

IRC = .250/2.165 = 0.1155

5(b). Annualized IR = (T)1/2(IR)

Annualized IRA = (52)1/2(0.1088) = 0.7846

Annualized IRB = (26)1/2(0.0195) = 0.0994

Annualized IRC = (12)1/2(0.1155) = 0.4001

5(c). The higher the ratio, the better. Based upon the answers to part a, Manager C would be rated the highest followed by Managers A and B, respectively. However, once the values are annualized, the ranking change. Specifically, based upon the annualized IR, Manger A is rated the highest, followed by C and B. (In both cases, Manager C is rated last). Based upon the Grinold-Kahn standard for “good” performance (0.500 or greater), only Manager A meets that test.

Manager M:

Periods HPY

1. [(692,000 – (1 -.50)(-35,000)]/[700,000 + (.50)(-35,000)] – 1

= (692,000 + 17,500/(700,000 – 17,500) – 1 = 709,500/682,500 – 1 = .0396

2. (663,000 – (1 -.50)(-35,000)]/[692,000 + (.50)(-35,000)] – 1

= 680,500/674,500 – 1 = .0089

3. (621,000 – (1 -.50)(-35,000)]/[663,000 + (.50)(-35,000)] – 1

= 638,500/645,500 – 1 = -.0108

4. (612,000 – (1 -.50)(-35,000)]/[621,000 + (.50)(-35,000)] – 1

= 629,500/603,500 – 1 = .0431

5. (625,000 – (1 -.50)(-35,000)]/[612,000 + (.50)(-35,000)] – 1

= 642,500/594,500 – 1 = .0807

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