Performance Evaluation

Introduction• Complicated subject• Theoretically correct measures are difficult to

construct• Different statistics or measures are

appropriate for different types of investment decisions or portfolios

• Many industry and academic measures are different

• The nature of active managements leads to measurement problems

Abnormal PerformanceWhat is abnormal?Abnormal performance is measured:• Benchmark portfolio• Market adjusted• Market model / index model adjusted• Reward to risk measures such as the Sharpe

Measure:E (rp-rf) / p

Factors That Lead to Abnormal Performance

• Market timing• Superior selection

– Sectors or industries– Individual companies

Risk Adjusted Performance: Sharpe

1) Sharpe Index

rrpp - r - rff

rrpp = Average return on the portfolio= Average return on the portfolio

rrff = Average risk free rate= Average risk free rate

pp= Standard deviation of portfolio = Standard deviation of portfolio

returnreturn

pp

Risk Adjusted Performance: Treynor

2) Treynor Measure rrpp - r- rff

ßßpp

rrpp = Average return on the portfolio = Average return on the portfolio

rrff = Average risk free rate = Average risk free rate

ßßp p = Weighted average = Weighted average for portfoliofor portfolio

= r= rpp - [ r - [ rff + ß + ßpp ( r ( rmm - r - rff) ]) ]

3) 3) Jensen’s MeasureJensen’s Measure

pp

p

rrpp = Average return on the portfolio = Average return on the portfolio

ßßpp = Weighted average Beta = Weighted average Beta

rrff = Average risk free rate = Average risk free rate

rrmm = Avg. return on market index port. = Avg. return on market index port.

Risk Adjusted Performance: Jensen

= = Alpha for the portfolioAlpha for the portfolio

M2 Measure

• Developed by Modigliani and Modigliani• Equates the volatility of the managed portfolio

with the market by creating a hypothetical portfolio made up of T-bills and the managed portfolio

• If the risk is lower than the market, leverage is used and the hypothetical portfolio is compared to the market

M2 Measure: Example

Managed Portfolio Market T-bill

Return 35% 28% 6%

Stan. Dev 42% 30% 0%

Hypothetical Portfolio: Same Risk as Market

30/42 = .714 in P (1-.714) or .286 in T-bills

(.714) (.35) + (.286) (.06) = 26.7%

Since this return is less than the market, the managed portfolio underperformed

T2 (Treynor Square) Measure

• Used to convert the Treynor Measure into percentage return basis

• Makes it easier to interpret and compare• Equates the beta of the managed portfolio with the

market’s beta of 1 by creating a hypothetical portfolio made up of T-bills and the managed portfolio

• If the beta is lower than one, leverage is used and the hypothetical portfolio is compared to the market

T2 ExamplePort. P. Market

Risk Prem. (r-rf) 13% 10%

Beta 0.80 1.0

Alpha 5% 0%

Treynor Measure 16.25 10

Weight to match Market w = M/P = 1.0 / 0.8

Adjusted Return RP* = w (RP) = 16.25%

T2P = RP

* - RM = 16.25% - 10% = 6.25%

T2P = RP/Bp – RM = 13/.8 – 10 = 6.25%

Which Measure is Appropriate?It depends on investment assumptions1) If the portfolio represents the entire

investment for an individual, Sharpe Index compared to the Sharpe Index for the market.

2) If many alternatives are possible, use the Jensen or the Treynor measureThe Treynor measure is more complete because it adjusts for risk

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• Other Measures of Risk – A very popular measure of risk is called Value at

Risk or VAR. It is generally the amount you can lose at a particular confidence interval. It is only concerned with loss.

– At the 95% level, E(R) – 1.65(standard dev)– At the 99% level, E(R) – 2.33(standard dev)

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– VAR Example

– Invest $100 at 10% with S.D. of 15%. What is the 95% Var over the year?

– 10% - 1.65(15%) = -14.75%– Var = 100 * -.1475 = $14.75

Semi-variance or semi-standard deviation

This statistic only measures variations below the mean or some threshold.

Semi-variance = 1/n * (summation of the average – rt)^2 where Where: n = the total number of observations below the meanrt = the observed valueaverage = the mean or target value of the data set. Standard deviation is just the square root of the above.

Semi-variance or semi-standard deviation

Example:You note the returns 3, 8, 3, 5, 11. Mean is 6. Thus

3 returns are less than the mean. The semi variance is [(6-3)^2 + (6-3)^2 + (6-5)^2]/3

= 6.33 and the semi-standard deviation is 6.33^.5 = 2.51. Compare this to the population standard deviation which is [(6-3)^2 + (6-8)^2 + (6-3)^2 + (6-5)^2+(6-11)^2]/5 = 9.6. Square root of 19.6 is 3.1%. Note that deviations above the mean are not considered, thus the large deviation(11 compared to 6) is ignored and the semi-standard deviation is actually smaller in this case.

Monte Carlo Methods• Monte Carlo methods can be used to ask what if

questions based on thousands of simulations. The range of values found during the simulations can be used to estimate possible outcomes and thus the risk associated with any position.

• For example, using the mean and standard deviation of the stock market, one could simulate possible return patterns over many years to see how a portfolio may be affected.

Limitations

• Assumptions underlying measures limit their usefulness

• When the portfolio is being actively managed, basic stability requirements are not met

• Practitioners often use benchmark portfolio comparisons to measure performance

Performance Attribution

• Decomposing overall performance into components

• Components are related to specific elements of performance

• Example components– Broad Allocation– Industry– Security Choice– Up and Down Markets

Process of Attributing Performance to Components

Set up a ‘Benchmark’ or ‘Bogey’ portfolio• Use indexes for each component• Use target weight structure

Appropriate Benchmark• Unambiguous – anyone can reproduce it• Investable• Measurable• Specified in Advance

Process of Attributing Performance to Components

• Calculate the return on the ‘Bogey’ and on the managed portfolio

• Explain the difference in return based on component weights or selection

• Summarize the performance differences into appropriate categories

Example, Performance Attributuion

Actual Ret Act. Wt Bmk Wt Index Ret

Semi-C 2% 0.7 0.6 2.5

Energy 1% 0.2 0.3 1.2

Bio-Tech 0.50% 0.1 0.1 0.5

Actual: (0.70 ´ 2.0%) + (0.20 ´ 1.0%) + (0.10 ´ 0.5%) = 1.65%

Benchmark: (0.60 ´ 2.5%) + (0.30 ´ 1.2%) + (0.10 ´ 0.5%) = 1.91%

Underperformance = 1.91% - 1.65% = 0.26%

Security Selection

Portfolio Sector Excess Man. Port

Sector Performance

Performance

Performance

Weight Contribution

Semi-C 2.00% 2.50% -0.50% 0.7 -0.35%

Energy1.00% 1.20% -0.20% 0.2 -0.04%

Bio-Tech

0.50% 0.50% 0.00% 0.1 0.00%

Contribution of security selection:

-0.39%

Sector Allocation

Actual Benchmark

Excess

Sector Weight Weight Weight Sector Ret. Contribution

Semi-C 0.7 0.6 0.1 2.50% 0.25%

Energy 0.2 0.3 -0.1 1.20% -0.12%

Biotech 0.1 0.1 0 0.50% 0.00%

Contribution of sector allocation:

0.13%

Lure of Active ManagementAre markets totally efficient?• Some managers outperform the market for extended

periods• While the abnormal performance may not be too

large, it is too large to be attributed solely to noise• Evidence of anomalies such as the turn of the year

existThe evidence suggests that there is some role

for active management