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Chapter 13Performance Evaluation and Risk Management

• Active and Passive Portfolio Management

• Performance evaluation• Comparing performance

measures• Investment risk management• More on computing value-at-risk

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It is Not the Return On My Investment...“It is not the return on my investment that I am concerned about. It is the return of my investment!”

– Will Rogers

However,

“We’ll GUARANTEE you a 25% return of your investment!”

– Tom and Ray Magliozzi

Investors can either follow an active strategy or a passive strategy after they form their portfolio.

Investors who choose passive management do not change the composition of their portfolio. A Buy and Hold strategy involves buying financial instruments, forming the portfolio, and then holding the chosen securities in the original percentage of the portfolio and not changing the composition of the portfolio.

Another technique of passive portfolio management is buying shares of a market index fund that mimics a market index (e.g., S&P/TSX Composite Index) and holding it.

Investors do not pay transaction fees for frequent buying and selling. They also do not spend a lot of time gathering information to find underpriced securities.

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Passive Portfolio Management

In contrast to passive management, active management involves changing the composition of portfolios.

Some active managers follow a market timing strategy. There are mutual fund managers who specialize in market timing. This strategy involves using various techniques and predicting future movements in financial instruments, especially stocks, and buying and selling them at the right time. Also, market timers may advocate moving into a stock market or an industry at a particular time.

Other managers attempt to find undervalued and overvalued securities, buy the undervalued ones and short sell the overvalued ones to try and obtain abnormal profits.

Active management is costly, time consuming, and risky. Investors need to evaluate managers of actively managed funds to find out whether or not their strategies are successful. The Sharpe ratio, Treynor ratio, and all the other risk measures are important tools for these investors.

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Active Portfolio Management

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Performance Evaluation Measures Performance evaluation is a term for assessing how

well a money manager achieves a balance between high returns and acceptable risks.

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.

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Performance Evaluation Measures

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 for the portfolio’s return.

p

fp

σ

RRratio Sharpe

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Performance Evaluation MeasuresThe 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

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Performance Evaluation Measures

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 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

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Jensen’s Alpha

Another Method to Calculate Alpha 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

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Another Method to Calculate Alpha 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.

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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?12

Estimating Alpha Using Regression

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Estimating Alpha Using Regression

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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The Information Ratio

R-Squared 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 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? 16

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Performance Evaluation Measures M2 Measure It is the excess return of a hypothetical portfolio over

the market portfolio. For a portfolio p, we can create a hypothetical portfolio

by combining a percentage of portfolio p with risk free asset (T-bill) such that the hypothetical portfolio will have the same standard deviation as the market portfolio.

We can then compare two same risk portfolios using their returns.

M2 = Rhp- Rm

where hp represents the hypothetical portfolio and m represents the market index.

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Portfolio Performance MeasurementTable 13.1 and Table 13.2

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 Performance Measures 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.

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).

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 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

××

××

×

×

××

×

×

×× ×

×

×

×

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ExpectedReturn

Standard deviation

× A

Rf

A

fA

σ

RREslope

Sharpe-Optimal Portfolios 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

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Solving for a Sharpe-Optimal Portfolio

Two asset portfolio return is given by:

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

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

)R,CORR(Rσ)σω(12ωσ)ω(1σω

r -))E(Rω(1)E(Rω

σ

r-)E(RRatio Sharpe

)R,CORR(Rσ)σω(12ωσ)ω(1σωσ : VariancePortfolio

))E(Rω(1)E(Rω)E(R :Return Portfolio

BABAAA2B

2A

2A

2A

fBAAA

P

fp

BABAAA2B

2A

2A

2A

2P

BAAAp

weight in asset B

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Using Excel to Solve the Sharpe-Optimal Portfolio

DataInputs:

ER(A): 0.12 W_A: 0.250STD(A): 0.15

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

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

Ratio: 0.402

Suppose we enter the following into a spreadsheet, guessing that W_A = 0.25 is a “good” portfolio.

Using formulas for portfolio return and standard deviation, we can compute a Sharpe Ratio:

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Using Excel to Solve the Sharpe-Optimal Portfolio

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

We use the Solver, found under Tools.

Data ChangerInputs: Cell:

ER(A): 0.12 W_A: 0.700STD(A): 0.15

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….

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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.

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 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)? 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 deviations

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.

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Example:VaR Calculation

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

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

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• 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• 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

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 one are:

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Useful Websites

www.stanford.edu/~wfsharpe (visit Professor Sharpe’s homepage)

www.morningstar.com (Comprehensive source of investment information)

www.gloriamundi.org (learn all about Value-at-Risk) www.garp.org (The Global Association of Risk

Professionals) www.riskmetrics.com (Check out risk grades)