January 2, 2020

Chapter 13

Performance Evaluation and

Risk Management

Slides

13-1. Chapter 13

13-2. Performance Evaluation and Risk Management

13-3. Learnig Objectives

13-4. Performance Evaluation and Risk Management

13-5. Performance Evaluation Measures, I.

13-6. Performance Evaluation Measures, II.

13-7. Performance Evaluation Measures, III.

13-8. Performance Evaluation Measures, IV.

13-9. Jensen’s Alpha

13-10. Another Method to Calculate Alpha, I.

13-11. Another Method to Calculate Alpha, II.

13-12. Estimating Alpha Using Regression, I.

13-13. Estimating Alpha Using Regression, II.

13-14. The Information Ratio

13-15. R-Squared, I.

13-16. R-Squared, II.

13-17. Investment Performance Measurement on the Web

13-18. Comparing Three Well-Known Performance Measures: Be Careful

13-19. Comparing Performance Measures, I.

13-20. Comparing Performance Measures, II.

13-21. Global Investment Performance Standards

13-22. Sharpe-Optimal Portfolios, I.

13-23. Sharpe-Optimal Portfolios, II.

13-24. Sharpe-Optimal Portfolios, III.

13-25. Example: Solving for a Sharpe-Optimal Portfolio

13-26. Example: Using Excel to Solve for the Sharpe-Optimal Portfolio

13-27. Example: Using Excel to Solve for the Sharpe-Optimal Portfolio, Cont.

13-28. Investment Risk Management

13-29. Value-at-Risk (VaR)

13-30. Example: VaR Calculation, I.

13-31. Example: VaR Calculation, II.

13-32. Example: VaR Calculation, III.

13-33. Example: A Multiple Year VaR, I.

13-34. Example: A Multiple Year VaR, II.

13-35. Example: A Multiple Year VaR, III.

13-36. Computing Other VaRs

13-37. Useful Websites

13-38. Chapter Review

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Education.

Performance Evaluation and Risk Management 13-2

Chapter Organization

13.1 Performance Evaluation

A. Performance Evaluation Measures

B. The Sharpe Ratio

C. The Treynor Ratio

D. Jensen’s Alpha

E. Another Method to Calculate Alpha

F. Information Ratio

G. R-Squared

13.2 Comparing Performance Measures

A. Global Investment Performance Standards

B. Sharpe-Optimal Portfolios

13.3 Investment Risk Management

A. Value-at-Risk

13.4 More on Computing Value-at-Risk

13.5 Summary and Conclusions

Selected Web Sites

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

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

www.gipsstandards.org (learn more about performance reporting)

www.gloria-mundi.org (learn all about Value-at-Risk)

www.garp.org (The Global Association of Risk Professionals)

www.andreassteiner.net (Performance Analysis)

www.finplan.com (FinPlan – financial planning web site)

www.alternativesoft.com (learn about modified VaR)

Annotated Chapter Outline

13.1 Performance Evaluation

Performance Evaluation: Concerns the assessment of how well a money

manager achieves a balance between high returns and acceptable risks.

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Education.

Performance Evaluation and Risk Management 13-3

In general terms, performance evaluation focuses on assessing how well a

money manager (either a professional or an individual investor) achieves high

returns balanced with acceptable risks.

Performance evaluation is particularly significant when we consider efficient

markets. In a previous chapter, we raised the question of risk-adjusted

performance and whether anyone can consistently earn an "abnormal" return,

thereby "beating" the market. Our goal here, however, is to introduce the primary

assessment tools, rather than discuss whether we should entrust our investment

funds with fund managers.

A. Performance Evaluation Measures

Raw Return: States the total percentage return on an investment without

any adjustment for risk or comparison to any benchmark.

The raw return on a portfolio is a naive measure of performance evaluation. The

fact that a raw portfolio return does not reflect any consideration of risk suggests

that its usefulness is limited when making investment decisions.

So, we shall examine some of the best-known and most popular measures that

include an adjustment for risk.

B. The Sharpe Ratio

Sharpe Ratio: Measures investment performance as the ratio of portfolio

risk premium over portfolio return standard deviation. This ratio was

originally proposed by Nobel Laureate William F. Sharpe.

p

fp

σ

RR

Ratio Sharpe

The portfolio risk premium, Rp – Rf, is the basic reward for bearing risk, while the

return standard deviation is a measure of the total risk for a security or a portfolio.

We referred to the risk premium as “excess return” in a previous chapter.

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

systematic risk principle states that the reward for bearing risk depends only on

the systematic risk of an investment. So no matter how much total risk an asset

has, only the systematic portion is relevant in determining the expected return

(and the risk premium) on that asset. Hence, the Sharpe ratio is probably most

appropriate for evaluating relatively diversified portfolios, since it penalizes non-

diversified portfolios by also taking into account unsystematic risk.

The Sharpe ratio uses total standard deviation, but investors are likely only

concerned about downside volatility (i.e., returns below expectation). The Sortino

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Education.

Performance Evaluation and Risk Management 13-4

ratio is calculated just like the Sharpe ratio, but the deviation is calculated using

only returns that are below the mean. Thus, the Sortino ratio does not penalize

for upside volatility.

C. The Treynor Ratio

Treynor Ratio: Measures investment performance as the ratio of portfolio

risk premium over portfolio beta. Jack L. Treynor originally proposed this

measure.

p

fp

β

RR

Ratio Treynor

As with the Sharpe ratio, the Treynor ratio is a reward-to-risk ratio. The key

difference is that the Treynor ratio looks at systematic risk only, not total risk.

D. Jensen’s Alpha

Jensen's Alpha: Measures investment performance as the raw portfolio

return less the return predicted by the Capital Asset Pricing Model

(CAPM). This measure was originally proposed by Michael C. Jensen.

fMpfp RREβRRE:Return Expected Portfolio the CAPM, the From

fMpfpppp

RREβRRRERα Alpha,sJensen'

Jensen's alpha is simply the abnormal return above or below the security market

line. In this sense, it can be interpreted as a measure of how much the portfolio

"beat the market." A positive alpha is a good thing because the portfolio has a

relatively high return given its level of systematic risk.

E. Another Method to Calculate Alpha

Following the same approach as we used to find beta, alpha can be found using

a regression of the security returns (above the risk free rate) on the market or

index (excess return). The resulting intercept is an estimate of the alpha. One

advantage of this approach is that the regression output provides a measure of

the significance of this estimate.

F. Information Ratio

Information Ratio: A security’s alpha divided by its tracking error.

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Performance Evaluation and Risk Management 13-5

Tracking error is the deviation of the annual difference between the security’s

excess return and the market’s excess return. Dividing alpha by tracking error

essentially penalizes the fund manager for excessive volatility of returns.

G. R-squared

Squaring the correlation of the fund’s returns to the market’s returns gives R-

squared. This is the percentage of the fund’s returns that are driven by the

market. A higher R-squared gives confidence to the alpha estimate.

13.2 Comparing Performance Measures

As illustrated by the example in the text, the three basic performance measures

can yield substantially different performance rankings. So, which measure should

be used to evaluate portfolio performance? The simple answer is, "It depends."

To evaluate an entire portfolio held by an investor, the Sharpe ratio is

appropriate. To evaluate securities or portfolios for possible inclusion in a broader

or "master" portfolio, either the Treynor ratio or Jensen's alpha is appropriate.

The Treynor ratio and Jensen's alpha are really very similar; the only difference

being that the Treynor ratio standardizes everything, including any excess return,

relative to beta. To summarize:

The Sharpe ratio is appropriate for the evaluation of an entire

portfolio. The Sharpe ratio does not require a beta estimate. The

Sharpe ratio penalizes a portfolio for being undiversified, because

standard deviation measures total risk. Total risk approximates

systematic risk only for relatively well-diversified portfolios.

The Treynor ratio is appropriate for the evaluation of securities or

portfolios for possible inclusion in a broader or master portfolio.

The Treynor ratio requires a beta estimate, and betas from different

sources can be quite different.

Jensen's alpha is appropriate for the evaluation of securities or

portfolios for possible inclusion in a broader or master portfolio.

Jensen’s alpha is easy to interpret. However, Jensen’s alpha

requires a beta estimate, and betas from different sources can be

quite different.

A. Global Investment Performance Standards

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Performance Evaluation and Risk Management 13-6

The CFA Institute developed the Global Investment Performance Standards

(GIPS) to standardize the way portfolio managers report their performance. This

allows for better comparability across investment managers.

B. Sharpe-Optimal Portfolios

A fund’s allocation with the highest possible Sharpe ratio is said to be "Sharpe-

optimal." The procedure for finding the Sharpe-optimal portfolio is similar to the

procedure for identifying the Markowitz efficient frontier.

Consider a plot of expected return versus standard deviation for the investment

opportunity set of risk-return possibilities for a portfolio.

The slope of a straight line drawn from the risk-free rate to a portfolio, say

portfolio A, is the Sharpe ratio for that portfolio.

A

fA

σ

RRE

Slope

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Education.

Expecte

d

Return

Standard

deviation

×

×

×

×

×

×

×

×

×

×

×

××

×

×

×

Expected

Return

Standard

deviation

A

E(RA

)

A

Rf

×

Performance Evaluation and Risk Management 13-7

Hence, the portfolio on the line with the steepest slope is the Sharpe-optimal

portfolio. This steepest-slope line is always going to be the one that is tangent to

the Markowitz efficient frontier. Hence, one of the "efficient" portfolios on the

Markowitz efficient frontier is "the very best," at least in the sense of being

Sharpe-optimal.

13.3 Investment Risk Management

Investment Risk Management: Concerns a money manager's control

over investment risks, usually with respect to potential short-run losses.

The book focuses on the Value-at-Risk approach.

A. Value-at-Risk (VaR)

Value-at-Risk (VaR): Assesses risk by stating the probability of a loss a

portfolio may experience within a fixed time horizon.

Normal Distribution: Represents a statistical model for assessing

probabilities related to many phenomena, including security returns.

In essence, the VaR method involves evaluating the probability of a significant

loss. We will assume that security returns follow a normal distribution. Since a

normal distribution can be completely specified by its mean and standard

deviation, these are all that we need to state the VaR "statistic" for a stock

portfolio. For example, a VaR risk assessment may be "a return of –.07 or worse

with a probability of 17%," i.e., Prob(Rp –.07) = 17%.

13.4 More on Computing Value-at-Risk

The main task here involves evaluating horizons that are shorter or longer than a

year. This translates to calculating the expected return and standard deviation

corresponding to the period being studied.

In general, if T is the number of years (or fraction of a year),

TRERE

pTp,

and

Tσσ

pTp,

Example: VaR for a Sharpe-Optimal Portfolio: The key here is that one must first

calculate the expected return and variance (or standard deviation) of the

portfolio. In a Sharpe-Optimal portfolio using two assets, one uses the expected

return and standard deviation formulas for a two asset portfolio:

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Performance Evaluation and Risk Management 13-8

In the case of just two assets, the formulas for the portfolio weights for the

optimal Sharpe portfolio are known to be:

Once the weights are calculated, there is enough information to calculate the

expected return and standard deviation for the Sharpe-Optimal portfolio.

13.5 Summary and Conclusions

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E(Rp)=xAE(RA)+xBE(RB)

σP

2=xA

2σA

2+xB

2σB

2+2x AxBσAσBCorr (RARB)

xA=σB

2×E(RA-RF)− Corr(RARB)×σA×σB×E(RB-R F)

σA

2×E(RB-RF) + σB

2×E(RA-R F) −

[

E(RA)+E(RB)-2×RF

]

×Corr (RARB)×σA×σB

xB=1−xA