LO 70.3: Describe the uses for the Modigliani-squared and Treynor s measure in

LO 70.3: Describe the uses for the Modigliani-squared and Treynor s measure in comparing two portfolios, and the graphical representation of these measures.
Universe Comparisons
Portfolio rankings based merely on returns ignore differences in risk across portfolios. A popular alternative is to use a comparison universe. This approach classifies portfolios according to investment style (e.g., small cap growth, small cap value, large cap growth, large cap value) and, then, ranks portfolios based on rate of return within the appropriate style universe. The rankings are now more meaningful because they have been standardized on the investment style of the funds. This method will fail, however, if risk differences remain across the funds within a given style.
The Sharpe Ratio
The Sharpe ratio uses standard deviation (total risk) as the relevant measure of risk. It shows the amount of excess return (over the risk-free rate) earned per unit of total risk. Hence, the Sharpe ratio evaluates the performance of the portfolio in terms of both overall return and diversification.
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The Sharpe ratio is defined as:
c _ R a – R f —————
where: RA = average account return R f = average risk-free return ctA = standard deviation of account returns
Professors Note: Again, the risk measure, standard deviation, should ideally he the actual standard deviation during the measurement period.
The Treynor Measure
The Treynor measure is very similar to the Sharpe ratio except that it uses beta (systematic risk) as the measure of risk. It shows the excess return (over the risk-free rate) earned per unit of systematic risk.
The Treynor measure is defined as: – p _ R a – R f < - where: R a = average account return R f = average risk-free return PA = average beta Professors Note: Ideally, the Treynor measure should be calculated using the actual beta for the portfolio over the measurement period. Since beta is subject to change due to varying covariance with the market, using the premeasurement period beta may not yield reliable results. The beta for the measurement period is estimated by regressing the portfolios returns against the market returns. For a well-diversified portfolio, the difference in risk measurement between the Sharpe ratio and the Treynor measure becomes irrelevant as the total risk and systematic risk will be very close. For a less than well-diversified portfolio, however, the difference in rankings based on the two measures is likely due to the amount of diversification in the portfolio. Used along with the Treynor measure, the Sharpe ratio provides additional information about the degree of diversification in a portfolio. Sharpe vs. Treynor. If a portfolio was not well-diversified over the measurement period, it may be ranked relatively higher using Treynor than using Sharpe because Treynor considers only the beta (i.e., systematic risk) of the portfolio over the period. When the Sharpe ratio is calculated for the portfolio, the excess total risk (standard deviation) due to diversifiable risk will cause rankings to be lower. Although we do not get an absolute measure of the lack of 2018 Kaplan, Inc. Page 121 Topic 70 Cross Reference to GARP Assigned Reading - Bodie, Kane, and Marcus, Chapter 24 diversification, the change in the rankings shows the presence of unsystematic risk, and the greater the difference in rankings, the less diversified the portfolio. Jensens Alpha Jensens alpha, also known as Jensens measure, is the difference between the actual return and the return required to compensate for systematic risk. To calculate the measure, we subtract the return calculated by the capital asset pricing model (CAPM) from the account return. Jensens alpha is a direct measure of performance (i.e., it yields the performance measure without being compared to other portfolios). A = RA - E (Ra ) where: a A Ra E(RA) = RF +pA[E(RM) - RF] = alpha = alpha = the return on the account A superior manager would have a statistically significant and positive alpha. Jensens alpha uses the portfolio return, market return, and risk-free rate for each time period separately. The Sharpe and Treynor measures use only the average of portfolio return and risk-free rate. Furthermore, like the Treynor measure, Jensens alpha only takes into account the systematic risk of the portfolio and, hence, gives no indication of the diversification in the portfolio. Information Ratio The Sharpe ratio can be changed to incorporate an appropriate benchmark instead of the risk-free rate. This form is known as the information ratio or appraisal ratio: i r a = where: R a = R b = cta - b = R a - R b cta - b average account returnstandard deviation of excess returns measured as the difference : average account return = average benchmark return = standard deviation of excess returns measured as the difference between account and benchmark returns The information ratio is the ratio of the surplus return (in a particular period) to its standard deviation. It indicates the amount of risk undertaken (denominator) to achieve a certain level of return above the benchmark (numerator). An active manager makes specific cognitive bets to achieve a positive surplus return. The variability in the surplus return is a measure of the risk taken to achieve the surplus. The ratio computes the surplus return relative to the risk taken. A higher information ratio indicates better performance. Page 122 2018 Kaplan, Inc. Topic 70 Cross Reference to GARP Assigned Reading - Bodie, Kane, and Marcus, Chapter 24 Professor's Note: The version o f the information ratio presented here is the most common. However, you should he aware that an alternative calculation o f this ratio exists that uses alpha over the expected level o f unsystematic risk over the time period, a A ct(a ) M-Squared (M2) Measure A relatively new measure of portfolio performance developed by Leah Modigliani and her grandfather, 1985 Nobel Prize recipient Franco Modigliani, has become quite popular. The M 2 measure compares the return earned on the managed portfolio against the market return, after adjusting for differences in standard deviations between the two portfolios. Professor's Note: There are no squared terms in the M -squared calculation. The term M -squared" merely refers to the last names o f its originators (Leah and Franco Modigliani). The M 2 measure can be illustrated with a graph comparing the capital market line for the market index and the capital allocation line for managed Portfolio P. In Figure 2, notice that Portfolio P has a higher standard deviation than the market index. But, we can easily create a Portfolio P that has standard deviation equal to the market standard deviation by investing appropriate percentages in both the risk-free asset and Portfolio P. The difference in return between Portfolio P and the market portfolio, equals the M 2 measure for Portfolio P. Figure 2: The M2 Measure of Portfolio Performance Return 2018 Kaplan, Inc. Page 123 Topic 70 Cross Reference to GARP Assigned Reading - Bodie, Kane, and Marcus, Chapter 24 Example: Calculating the M2 performance measure Calculate the M2 measure for Portfolio P: Portfolio P mean return Portfolio P standard deviation Market portfolio mean return Market portfolio standard deviation Risk-free rate 10% 40% 12% 20% 4% Answer: To answer the question, first note that a portfolio, P, can be created that allocates 50/50 to the risk-free asset and to Portfolio P such that the standard deviation of Portfolio P equals the standard deviation of the market portfolio: cjp, = WpCTp = 0.50(0.40) = 0.20 Therefore, a 50/50 allocation between Portfolio P and the risk-free asset provides risk identical to the market portfolio. What is the difference in return between Portfolio P and the market portfolio? To answer this question, first we must derive the mean return on Portfolio P: Rp, = WpRp + WpRp = 0.50(0.04) + 0.50(0.10) = 0.07 Alternatively, the mean return for Portfolio P can be derived by using the equation of the capital allocation line for Portfolio P: Rp> Rp + R d R e
CTp R p +
R d Re V dp
Therefore, we now have created a portfolio, P, that matches the risk of the market portfolio (standard deviation equals 20%). All that remains is to calculate the difference in returns between Portfolio P and the market portfolio:
0.20 = 0.04 + (0.15)0.20 = 0.07
0.04 +
M2 = Rp, – RM = 0.07 – 0.12 = -0.05
Clearly, Portfolio P is a poorly performing portfolio. After controlling for risk, Portfolio P provides a return that is 5 percentage points below the market portfolio.
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Professors Note: Unfortunately, a consistent definition ofM 2 does not exist. Sometimes M 2 is defined as equal to the return on the risk-adjusted Portfolio P rather than equal to the difference in returns between P and M. However, portfolio rankings based on the return on P or on the difference in returns between P and M will be identical. Therefore, both definitions provide identical portfolio performance rankings.
M2 will produce the same conclusions as the Sharpe ratio. As stated earlier, Jensens alpha will produce the same conclusions as the Treynor measure. However, M2 and Sharpe may not give the same conclusion as Jensens alpha and Treynor. A discrepancy could occur if the manager takes on a large proportion of unsystematic risk relative to systematic risk. This would lower the Sharpe ratio but leave the Treynor measure unaffected.
Example: Risk-adjusted performance appraisal measures
The data in Figure 3 has been collected to appraise the performance of four asset management firms:
Figure 3: Performance Appraisal Data
Return Beta
Standard deviation Standard deviation of excess returns
Fund 1 6.45% 0.88 2.74%
Fund 2 8.96% 1.02 4.54%
Fund 3 9.44% 1.36 3.72%
Fund 4 5.82% 0.80 2.64%
Market Index
6% 1.00 2.80%
The market index return and risk-free rate of return for the relevant period were 6% and 3%, respectively. Calculate and rank the funds using Jensens alpha, the Treynor measure, the Sharpe ratio, the information ratio, and M2.
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Topic 70 Cross Reference to GARP Assigned Reading – Bodie, Kane, and Marcus, Chapter 24
Evaluation Tool
Fund 1
Jensens Alpha
6.45 – 5.64 =
Fund 2
8.96-6.06 =
Fund 3
9.44 – 7.08 =
Fund 4
5.82-5.40 =
6-45 3 – 3 S P
8-9 6 -3 _
9-44 – 3 _ i 7/
5’82 3 – 3 53
6-45-3 2.74 O J L O
8-96-3 _
9-44 – 3
3.72 / X /
5-82 2.64
3 _
i .u/
6-45 -6
5.6 u.uo _ n n s
8 -9 6 -6
9.44 6
_ PI ^8 U jLi O
5 .8 2 -6
PI 03
U V /
4 6.53% – 6% 3 + (1.26) x (2.8) = 6.53% – 6%
3 + (1.31) x (2.8) = 6.67% 6%
=0.67% 7.84% – 6% 3 + (1.73) x (2.8) = 7.84% – 6%
3 + (1.07) x (2.8) = 6% 6% =0
Note that Jensens alpha and the Treynor measures give the same rankings, and the Sharpe and M2 measures give the same rankings. However, when comparing the alpha/Treynor rankings to the Sharpe/M2 measures, Funds 2 and 3 trade places.
Fund 2 has a much higher total risk (standard deviation) than Fund 3 but has a much lower beta. Relatively speaking, a smaller proportion of Fund 2s total risk relates to systematic risk, which is reflected in the low beta. Compared to Fund 3, it must have a bigger proportion of risk relating to non-systematic risk factors.
Hence, Fund 2 does better in the alpha/Treynor measures, as those measures only look at systematic risk (beta). It fares less well when it comes to the Sharpe/M2 measures that look at total risk.
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St a t is t ic a l Sig n if ic a n c e o f Al p h a Re t u r n s