Performance Evaluation Measures, I.
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Transcript of Performance Evaluation Measures, I.
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?
13-10
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.
13-12
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.
13-13
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?
13-14
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.
13-15
Comparing Three Well-Known Performance Measures: Be Careful
13-16
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.
13-17
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).
13-18
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.
13-19
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.
13-21
Sharpe-Optimal Portfolios, III.
13-22
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.
13-27
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)?
13-28
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
13-29
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
13-30
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.
13-31
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%.
13-32
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%.
13-33
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: