Chapter

Performance Evaluation and

Risk Management

McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

13

13-2

Performance Evaluation Measures, I.

• The raw return on a portfolio, RP, is simply the total percentage return on a portfolio.

• The raw return is a naive performance evaluation measure because:

– The raw return has no adjustment for risk.– The raw return is not compared to any

benchmark, or standard.

• Therefore, the usefulness of the raw return on a portfolio is limited.

13-3

Performance Evaluation Measures, II.The Sharpe Ratio

• The Sharpe ratio is a reward-to-risk ratio that focuses on total risk.

• It is computed as a portfolio’s risk premium divided by the standard deviation of the portfolio’s return.

p

fp

σ

RRratio Sharpe

13-4

Performance Evaluation Measures, III.The Treynor Ratio

• The Treynor ratio is a reward-to-risk ratio that looks at systematic risk only.

• It is computed as a portfolio’s risk premium divided by the portfolio’s beta coefficient.

p

fp

β

RRratio Treynor

13-5

Performance Evaluation Measures, IV.Jensen’s Alpha

• Jensen’s alpha is the excess return above or below the security market line. It can be interpreted as a measure of by how much the portfolio “beat the market.”

• It is computed as the raw portfolio return less the expected portfolio return as predicted by the CAPM.

Actual return

CAPM Risk-Adjusted ‘Predicted’ Return“Extra” Return

RREβ R Rα fMpfpp

13-6

Jensen’s Alpha

13-7

Another Method to Calculate Alpha, I.

• Recall that the characteristic line graphs the relationship between the return of an investment (on the y-axis) and the return of the market or benchmark (on the x- axis).

• The slope of this line represents the investment’s beta.

• With only a slight modification, we can extend this approach to calculate an investment’s alpha.

• Consider the following equation,

PM,RPP,RP

PfMfP

βRERE

:premiums riskof terms in or,

β RRE R RE

13-8

Another Method to Calculate Alpha, II.• Suppose an actively managed fund took on the same amount of risk as the

market (i.e., a beta of 1), but the fund earned exactly 2% more than the market every period?

• If we graph this hypothetical situation, the slope (i.e., beta) is still 1, but the x intercept is now 2. This intercept value is the fund’s alpha.

13-9

Estimating Alpha Using Regression, I.• Suppose we want to estimate a fund’s alpha.

• Unlike the previous situation, we do know whether its returns are consistently higher (or lower) than the market by a fixed amount.

• In this case, we apply a simple linear regression.

– X-variable: the excess return of the market

– Y-Variable: the excess return of the investment

• The intercept of this estimated equation is the fund’s alpha.

• Consider the following Spreadsheet Analysis slide.

– In the spreadsheet example, the intercept estimate is -.0167, or -1.67 percent per year.

– The beta of this security is .96, which is the coefficient estimate on the x variable.

How would you characterize the performance of this fund?

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Estimating Alpha Using Regression, II.

13-11

The Information Ratio

• Suppose a mutual fund reports a positive alpha.

• How do we know whether this alpha is statistically significantly different from zero or simply represents a result of random chance?

– One way: Evaluate the significance level of the alpha estimate using a regression. – Another way: Calculate the fund’s information ratio.

• The information ratio is a fund’s alpha divided by its tracking error.

• Tracking error measures the volatility of the fund’s returns relative to its benchmark.

• Consider the fund we evaluated in the Spreadsheet Analysis in the previous slide.– Over the five years, the excess return differences are 2%, 4%, -20%, 14%, and -10%. – The tracking error is the standard deviation of these return differences, which is

13.2%.– The information ratio for this fund is -1.67% divided by 13.2%, which is -0.13.

• The information ratio allows us to compare investments that have the same alpha: A higher information ratio means a lower tracking error risk.

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R-Squared, I.

• Recall that correlation measures how returns for a particular security move relative to returns for another security. Correlation also plays a key role in performance measurement.

• Suppose a particular fund has had a large alpha over the past three years.

– All else equal, we might say that this fund is a good choice. – Suppose this fund is a sector-based fund that invests only in gold. – It is possible that the large alpha is simply due to a run-up in gold

prices over the period and is not reflective of good management or future potential.

• To evaluate this type of risk we can calculate R-squared, which is simply the squared correlation of the fund to the market.

• For example, if this fund’s correlation with the market was .60, then the R-squared value is .36.

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R-Squared, II.• R-squared represents the percentage of the fund’s movement that can be

explained by movements in the market.

• Because correlation ranges only from -1 to +1, R-squared values will range from 0 to 100 percent. – An R-squared of 100 indicates that all movements in the security are

driven by the market, indicating a correlation of -1 or +1. – A high R-squared value (say greater than .80) might suggest that the

performance measures (such as alpha) are more representative of potential longer term performance.

• Excel reports correlation as “Multiple R” and R-squared as “R Square.”

• Verify that the squared value for “Multiple R” equals “R Square” in the previous Spreadsheet Analysis slide.

• How do you interpret an R-squared of 0.549?

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Investment Performance Measurement on the Web

• Consider the information below for the Fidelity Low-Priced Stock Fund, a small-cap value fund.

• We obtained the numbers from www.morningstar.com by entering the fund’s ticker symbol (FLPSX) and following the “Ratings and Risk” link.

• How do you judge the manager’s of this fund? For this fund:― Beta is 1.10, so the degree of market risk is above average. ― Alpha is 3.51 percent, which could indicate superior past performance. ― Standard deviation is 22.90 percent. ― Mean of -1.08 percent is the geometric return of the fund over the past three years. ― Sharpe ratio is -.14. ― R-squared is 91, which means that 91% of its returns are driven by the market’s return.

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Comparing Three Well-Known Performance Measures: Be Careful

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Comparing Performance Measures, I.

• Because the performance rankings can be substantially different, which performance measure should we use?

Sharpe ratio:

• Appropriate for the evaluation of an entire portfolio.

• Penalizes a portfolio for being undiversified, because, in general, total risk systematic risk only for relatively well-diversified portfolios.

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Comparing Performance Measures, II.Treynor ratio and Jensen’s alpha:

• Appropriate for the evaluation of securities or portfolios for possible inclusion into an existing portfolio.

• Both are similar, the only difference is that the Treynor ratio standardizes returns, including excess returns, relative to beta.

• Both require a beta estimate (and betas from different sources can differ a lot).

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Global Investment Performance Standards

• Comparing investments can be difficult: various performance measures can provide different rankings.

• Even if the various performance measures provide a similar ranking, we could still make an incorrect choice in selecting an investment manager. How is this possible?

• Performance measures often rely on self-reported returns (many funds do not have public portfolios).

• The CFA Institute developed the Global Investment Performance Standards (GIPS).

• Goal: Provide a consistent method to report portfolio performance to prospective (and current) clients

• By standardizing the way portfolio performance is reported, GIPS provide investors with the ability to make comparisons across managers.

– Firms are not required by law to comply with GIPS, compliance is voluntary.

– Firms that do comply, however, are recognized by the CFA Institute.

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Sharpe-Optimal Portfolios, I.

• Allocating funds to achieve the highest possible Sharpe ratio is said to be Sharpe-optimal.

• To find the Sharpe-optimal portfolio, first look at the plot of the possible risk-return possibilities, i.e., the investment opportunity set.

ExpectedReturn

Standard deviation

××

××

×

×

××

×

×

×× ×

×

×

×

13-20

ExpectedReturn

Standard deviation

× A

Rf

A

fA

σ

RREslope

Sharpe-Optimal Portfolios, II.• The slope of a straight line drawn from the risk-free rate to where the

portfolio plots gives the Sharpe ratio for that portfolio.

• The portfolio with the steepest slope is the Sharpe-optimal portfolio.

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Sharpe-Optimal Portfolios, III.

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Example: Solving for a Sharpe-Optimal Portfolio

• From a previous chapter, we know that for a 2-asset portfolio:

So, now our job is to choose the weight in asset S that maximizes the Sharpe Ratio.

We could use calculus to do this, or we could use Excel.

)R,CORR(Rσσx2xσxσx

r -)E(Rx)E(Rx

σ

r-)E(RRatio Sharpe

)R,CORR(Rσσx2xσxσxσ : VariancePortfolio

)E(Rx)E(Rx)E(R :Return Portfolio

BSBSBS2B

2B

2S

2S

fBBSS

P

fp

BSBSBS2B

2B

2S

2S

2P

BBssp

13-23

Example: Using Excel to Solve for the Sharpe-Optimal Portfolio

DataInputs:

ER(S): 0.12 X_S: 0.250STD(S): 0.15

ER(B): 0.06 ER(P): 0.075STD(B): 0.10 STD(P): 0.087

CORR(S,B): 0.10R_f: 0.04 Sharpe

Ratio: 0.402

Suppose we enter the data (highlighted in yellow) into a spreadsheet.

We “guess” that Xs = 0.25 is a “good” portfolio.

Using formulas for portfolio return and standard deviation, we compute Expected Return, Standard Deviation, and a Sharpe Ratio:

13-24

Example: Using Excel to Solve for the Sharpe-Optimal Portfolio, Cont.

• Now, we let Excel solve for the weight in portfolio S that maximizes the Sharpe Ratio.

• We use the Solver, found under Tools.

Data ChangerInputs: Weight In: Cell:

ER(S): 0.12 Stocks 0.700STD(S): 0.15 Bonds 0.300

ER(B): 0.06 ER(P): 0.102STD(B): 0.10 STD(P): 0.112

CORR(A,B): 0.10R_f: 0.04 Sharpe

Ratio: 0.553

Solving for the Optimal Sharpe Ratio

Given the data inputs below, we can use theSOLVER function to find the Maximum Sharpe Ratio:

Target Cell

Well, the “guess” of 0.25 was a tad low….

13-25

Investment Risk Management

• Investment risk management concerns a money manager’s control over investment risks, usually with respect to potential short-run losses.

• We will focus on what is known as the Value-at-Risk (VaR) approach.

13-26

Value-at-Risk (VaR)

• Value-at-Risk (VaR) is a technique of assessing risk by stating the probability of a loss that a portfolio may experience within a fixed time horizon.

• If the returns on an investment follow a normal distribution, we can state the probability that a portfolio’s return will be within a certain range, if we have the mean and standard deviation of the portfolio’s return.

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Example: VaR Calculation, I.

• Suppose you own an S&P 500 index fund.

• What is the probability of a return of -7% or worse in a particular year?

• That is, one year from now, what is the probability that your portfolio value is down by 7 percent (or more)?

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Example: VaR Calculation, II.

• First, the historic average return on the S&P index is about 13%, with a standard deviation of about 20%.– A return of -7 percent is exactly one standard deviation

below the average, or mean (i.e., 13 – 20 = -7).– We know the odds of being within one standard

deviation of the mean are about 2/3, or 0.67.

• In this example, being within one standard deviation of the mean is another way of saying that:

Prob(13 – 20 RS&P500 13 + 20) 0.67

or Prob (–7 RS&P500 33) 0.67

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Example: VaR Calculation, III.

• That is, the probability of having an S&P 500 return between -7% and 33% is 0.67.

• So, the return will be outside this range one-third of the time.

• When the return is outside this range, half the time it will be above the range, and half the time below the range.

• Therefore, we can say: Prob (RS&P500 –7) 1/6 or 0.17

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Example: A Multiple Year VaR, I.

• Once again, you own an S&P 500 index fund.

• Now, you want to know the probability of a loss of 30% or more over the next two years.

• As you know, when calculating VaR, you use the mean and the standard deviation.

• To make life easy on ourselves, let’s use the one year mean (13%) and standard deviation (20%) from the previous example.

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Example: A Multiple Year VaR, II.

• Calculating the two-year average return is easy, because means are additive. That is, the two-year average return is:

13 + 13 = 26%

• Standard deviations, however, are not additive.

• Fortunately, variances are additive, and we know that the variance is the squared standard deviation.

• The one-year variance is 20 x 20 = 400. The two-year variance is:

400 + 400 = 800.

• Therefore, the 2-year standard deviation is the square root of 800, or about 28.28%.

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Example: A Multiple Year VaR, III.

• The probability of being within two standard deviations is about 0.95.

• Armed with our two-year mean and two-year standard deviation, we can make the probability statement:

Prob(26 – 228 RS&P500 26 + 228) .95

or

Prob (–30 RS&P500 82) .95

• The return will be outside this range 5 percent of the time. When the return is outside this range, half the time it will be above the range, and half the time below the range.

• So, Prob (RS&P500 –30) 2.5%.

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Computing Other VaRs.

• In general, for a portfolio, if T is the number of years,

5%Tσ1.645TRERProb

2.5%Tσ1.96TRERProb

1%Tσ2.326TRERProb

ppTp,

ppTp,

ppTp,

TRERE pTp, Tσσ pTp,

Using the procedure from before, we make make probability statements. Three very useful ones are: