FIN 422: Student Managed Investment Fund

FIN 422: Student Managed Investment Fund
Topic 10: Evaluation of Portfolio Performance Larry Schrenk, Instructor

Overview 18.1 The Two Questions of Performance Measurement
18.2 Simple Performance Measurement Techniques 18.3 Risk-Adjusted Portfolio Performance Measures 18.4 Application of Portfolio Performance Measures 18.5 Holdings-Based Portfolio Performance Measurement 18.6 The Decomposition of Portfolio Returns 18.7 Factors That Affect Use of Performance Measures 18.8 Reporting Investment Performance

Learning Objectives @

Readings Reilley, et al., Investment Analysis and Portfolio Management, Chap. 18

18.1 The Two Questions of Performance Measurement

The actual return a manager produces over an investment horizon can be split into: The return that should have been earned given the capital commitment and the amount of risk in the portfolio Any incremental return due to superior investment skills (alpha) There are three ways that investors can estimate expected returns: The average contemporaneous return to a peer group of comparably managed portfolios The contemporaneous return to an index (or index fund) serving as a benchmark for the managed portfolio The return estimated by a risk factor model, such as the CAPM or multifactor model

There are two main questions that an investor attempts to answer when assessing the performance of an investment manager: How did the portfolio manager actually perform? Why did the portfolio manager perform as he or she did?

18.2 Simple Performance Measurement Techniques

18.2 Simple Performance Measurement Techniques
Developments in portfolio theory in the early 1960s showed investors how to quantify risk in terms of the variability of returns No single measure combined both return and risk and the two factors had to be considered separately

18.2.1 Peer Group Comparisons
Collects the returns produced by a representative set of investors over a specific period of time and displays them in a simple boxplot format Potential problems No explicit adjustment for risk level of the portfolios Difficult to form a comparable peer group that is large enough to make the percentile rankings meaningful By just focusing on relative returns, the comparison loses sight of whether the investor in question has accomplished his individual objectives and satisfied his investment expectations

18.2.1 Peer Group Comparisons

Portfolio Drawdown Portfolio drawdown measures how well he has protected the investor against losses over time Maximum drawdown calculates the largest percentage decline in value (from peak to trough) wherever during the horizon that occurs

Portfolio Drawdown

18.3 Risk-Adjusted Portfolio Performance Measures

18.3 Risk-Adjusted Portfolio Performance Measures
There are five major portfolio performance measures that combine risk and return performance into a single statistic: Sharpe Portfolio Performance Measure Treynor Portfolio Performance Measure Jensen Portfolio Performance Measure The Information Ratio Performance Measure Sortino Performance Measure

18.3.1 Sharpe Portfolio Performance Measure
This performance measure seeks to measure the total risk of the portfolio by using the standard deviation of returns Where:

Demonstration of Comparative Sharpe Measures Suppose that during the most recent 10-year period, the average annual total rate of return (including dividends) on an aggregate market portfolio, such as the S&P 500, was 14 percent and the average nominal rate of return on government T-bills was 8 percent The standard deviation of the annual rate of return for the market portfolio over the past 10 years was 20 percent

Examine the risk-adjusted performance of the following portfolios: The Sharpe measures for each of these funds are as follows:

18.3.2 Treynor Portfolio Performance Measure
Postulated two components of risk: Risk produced by general market fluctuations Risk resulting from unique fluctuations in the portfolio securities Introduced the characteristic line Building on capital market theory, he introduced a risk free asset that could be combined with different portfolios to form a portfolio possibility line He showed that rational investors would always prefer the portfolio line with the largest slope

The slope of this portfolio possibility line (designated T) is: Where: βM = slope of the fund’s characteristic line during that time period

Comparing a portfolio’s T value to a similar measure for the market portfolio indicates whether the portfolio would plot above the security market line (SML) Calculate the T value for the aggregate market as follows: Where: βM = 1.0 (the market’s beta) TM = slope of the SML

18.3.2 Treynor Portfolio Performance Measure (slide 4 of 6)
Demonstration of Comparative Treynor Measures Assume again that RM = 0.14 and RFR = 0.08 You are deciding between three different portfolio managers, based on their past performance:

Compute T values for the market portfolio and for each of the individual portfolio managers as follows:

18.3.3 Jensen Portfolio Performance Measure
The Jensen measure (Jensen, 1968) was originally based on the capital asset pricing model (CAPM), which calculates the expected one-period return on any security or portfolio by the following expression: αj indicates whether the portfolio manager is superior or inferior in her investment ability A superior manager has a significant positive α (alpha) value, while an inferior manager’s returns consistently fall short of expectations based on the CAPM, leading to a significant negative α

18.3.3 Jensen Portfolio Performance Measure
Applying the Jensen Measure The Jensen alpha measure of performance requires using a different RFR for each time interval during the sample period It does not directly consider the portfolio manager’s ability to diversify because it calculates risk premiums in terms of systematic risk The Jensen performance measure is flexible enough to allow for alternative models of risk and expected return than the CAPM Risk-adjusted performance (α) can be computed relative to any multifactor model:

18.3.4 Information Ratio Performance Measure
The information ratio measures a portfolio’s average return in excess of that for a benchmark portfolio divided by the standard deviation of this excess return: Where:

18.3.4 Information Ratio Performance Measure

18.3.5 Sortino Performance Measure
The Sortino measure is a risk-adjusted performance statistic that differs from the Sharpe ratio in two ways: It measures the portfolio’s average return in excess of a user-selected minimum acceptable return threshold, which is often the risk-free rate used in the S statistic although it need not be The Sharpe measure focuses on total risk—effectively penalizing the manager for returns that are both too low and too high—while the Sortino ratio captures just the downside risk (DR) in the portfolio

Sortino and Price (1994) calculate this measure as follows: Where: τ = minimum acceptable return threshold specified for the time period Dri = downside risk coefficient for Portfolio i during the specified time period

Downside risk: Is the volatility of the returns produced by a portfolio that fall below some hurdle rate that the investor chooses Attempts to measure the volatility associated with the shortfall that occurs if an investment produces a return that is lower than anticipated DR comes closer than measures of total risk (σ) to capturing what investors truly consider risky A popular measures is the semi-deviation, which uses the portfolio’s average (expected) return as the hurdle rate:

Comparing the Sharpe and Sortino Ratios Suppose that over the past 10 years, two portfolio managers have produced the following returns:

Using semi-deviation to compute DR for both portfolios leaves: When only the possibility of receiving a less-than-expected return is considered, Portfolio A now appears to be the riskier alternative due to the fact it has more extreme negative returns than Portfolio B

Assuming a minimum return threshold of 2 percent to match the Sharpe measure, the Sortino ratios for both portfolios indicate that, by limiting the extent of his downside risk, the manager for Portfolio B was actually the superior performer:

18.3.6 Summarizing the Risk-Adjusted Performance Measures
Each of the risk-adjusted performance statistics just described is widely used in practice and has strengths and weaknesses

18.3.6 Summarizing the Risk-Adjusted Performance Measures

18.4 Application of Portfolio Performance Measures

Total rate of return on a mutual fund where Rit = the total rate of return on Fund i during month t EPit = the ending price for Fund i during month t Divit = the dividend payments made by Fund i during month t Cap.Dist.it = the capital gain distributions made by Fund i during month t BPit = the beginning price for Fund i during month t

Total Sample Results Selected 30 open-end mutual funds from the nine investment style classes and used monthly data for the five-year period from July 2005 to June 2010 Active fund managers performed much better than earlier performance studies A primary factor for this outcome was the abnormally poor performance of the index during the middle of the sample period The various performance measures ranked the performance of individual funds consistently Exhibit 18.9, 18.10

Measuring Performance with Multiple Risk Factors Jensen measures calculated for the 30 mutual funds using two different versions of the Fama–French model to estimate expected returns:

Jensen’s alphas are computed relative to: A three-factor model including the market (Rm - RFR), firm size (SMB), and relative valuation (HML) variables A four-factor model that also includes the return momentum (MOM) variable The one-factor and multifactor Jensen measures produce similar but distinct performance rankings

Relationship among Performance Measures Implications of high positive correlations: Although the measures provide a generally consistent assessment of portfolio performance when taken as a whole, they remain distinct at an individual level Therefore it is best to consider these composites collectively The user must understand what each means

18.5 Holdings-Based Portfolio Performance Measurement

18.5 Holdings-Based Portfolio Performance Measurement
Two advantages to assessing performance with investment returns: Returns are usually easy for the investor to observe on a frequent basis They represent the investor actually benefits from the manager’s investing prowess It is also possible to view investment performance in terms of which securities the manager buys or sells from the portfolio Using a holdings-based measure can provide additional insights about the quality of the portfolio manager

18.5.1 Grinblatt-Titman Performance Measure
Assess the quality of the services provided by money managers by looking at adjustments they made to the contents of their portfolios For a particular reporting period t, their performance measure (GT) is: where: (wjt, wjt–1) = the portfolio weights for the jth security at the beginning of Period t and Period t - 1, respectively Rjt = the return to the jth security during Period t, which begins on Date t - 1 and ends on Date t

An advantage of the GT statistic is that it can be computed without reference to any specific benchmark, which was not true for returns-based measures such as the information ratio However, the GT measure fails to reward or penalize the manager for portfolio adjustments in which the share price change actually occurs in a later period

18.5.2 Characteristic Selectivity Performance Measure
The measure compares the returns of each stock held in an actively managed portfolio to the return of a benchmark portfolio that has the same aggregate investment characteristics as the security in question Their characteristic selectivity (CS) performance statistic is given by: Where: RBjt = the Period t return to a passive portfolio whose investment characteristics are matched at the beginning of Period t with those of Stock j

18.5.2 Characteristic Selectivity Performance Measure

18.6 The Decomposition of Portfolio Returns

18.6 The Decomposition of Portfolio Returns
The preceding risk-adjusted and holding-based measures were designed to answer the first question of performance measures: how did the portfolio manager actually perform? The answer to the question: why did the manager perform as he or she did? This requires an additional decomposition of portfolio returns

18.6.1 Performance Attribution Analysis
Attribution analysis attempts to distinguish the source of the portfolio’s overall performance Method compares the manager’s total return to the return for a predetermined benchmark policy portfolio and decomposes the difference into an allocation effect and a selection effect where: wpi, wbi = the investment proportions of the ith market segment the manager’s portfolio and the policy portfolio, respectively Rpi, Rbi = the investment return to the ith market segment in the manager’s portfolio and the policy portfolio, respectively Rbi = total return to the benchmark portfolio

An Example Consider an investor whose top-down portfolio strategy consists of two dimensions: He decides on a broad allocation across three asset classes: U.S. stocks, U.S. long-term bonds, and cash equivalents, such as Treasury bills The investor’s second general decision is choosing which specific stocks, bonds, and cash instruments to buy As a policy benchmark, he selects a hypothetical portfolio with a 60 percent allocation to the Standard & Poor’s 500 Index, a 30 percent investment in the Barclays Aggregate Bond Index, and a 10 percent allocation to three-month T-bills

An Example (continued) Suppose that at the start of the investment period, the investor believes equity values are inflated relative to the fixed-income market Compared to the benchmark, he decides to underweight stocks and overweight bonds and cash with 50 percent in equity, 38 percent in bonds, and 12 percent in cash. Further, he decides to concentrate on equities in the interest rate–sensitive sectors, such as utilities and financial companies, while deemphasizing the technology and consumer durables sectors Finally, he resolves to buy shorter-duration bonds of a higher credit quality than are contained in the benchmark bond index, and to buy commercial paper rather than Treasury bills

Performance Attribution Extension The attribution methodology can also be used to distinguish security selection skills from other decisions that an investor might make For instance, the manager of an all-equity portfolio must decide which economic sectors (for example, basic materials, consumer nondurables, transportation) to under- and overweight before selecting her preferred companies in those sectors

Measuring Market Timing Skills Tactical asset allocation (TAA) attempts to produce active value-added returns solely by adjusting their asset class exposures based on perceived changes in the relative valuations of those classes Thus, the relevant performance measurement criterion for a TAA manager is how well he is able to time broad market movements Two reasons why attribution analysis is ill-suited for this task: First, by design, a TAA manager indexes his actual asset class investments, and so the selection effect is not relevant Second, TAA might entail dozens of changes to asset class weightings during an investment period, which could render meaningless an attribution effect computed on the average holdings Because of these problems, many analysts consider a regression-based method for measuring timing skills to be a superior approach

18.6.2 Fama Selectivity Performance Measure
Fama suggested overall performance, in excess of the risk-free rate, consists of two components: Overall Performance = Excess return = Portfolio Risk + Selectivity The selectivity component represents the portion of the portfolio’s actual return beyond that available to an unmanaged portfolio with identical systematic risk and is used to assess the manager’s investment prowess

Evaluating Selectivity Formally, you can measure the return due to selectivity as: where: Ra = the actual return on the portfolio being evaluated Rx(βa) = the return on the combination of the riskless asset and the market portfolio that has risk βx equal to βa Overall performance can be written: Overall Performance = Selectivity + Risk

Evaluating Diversification If a portfolio manager attempts to select undervalued stocks and in the process gives up some diversification, then it is possible to measure the added return necessary to justify this decision

The portfolio’s gross selectivity is made up of net selectivity plus diversification: Or: Where: Rx(σ (Ra)) = return on the combination of the riskless asset and the market portfolio that has return volatility equivalent to that of the portfolio being evaluated

Example of Fama Performance Measure Suppose that over a recent five-year investment period, you observed that the average annual return on the market portfolio and the risk-free security were percent and 5.28 percent, respectively Thus, an investment portfolio with a beta of would be expected to deliver a return of percent Suppose further that this portfolio actually returned percent per annum The return for selectivity is the difference between the actual excess performance ( ) = and the required excess return for risk of (= ) or 0.02, indicating the manager fell slightly short of matching expectations consistent with the actual risk level of the portfolio What if the manager also did not fully diversify the portfolio? Assume that the standard deviations on the market and the manager’s portfolio were percent and percent, respectively

Example of Fama Performance Measure (continued) The ratio of total risk in the portfolio per unit of market total risk is = (13.41/14.95), but because the manager’s beta (0.815) is less than this, it appears that the portfolio contained elements of unsystematic risk Thus, the selectivity measure of 0.02 understates the true performance shortfall To adjust the selectivity measure for the lack of complete diversification, notice that the fund’s required return given its standard deviation is [= ( )] The difference of 1.45 (= ) between the required returns using total versus systematic risk is the added return required because of less-than-perfect diversification This is subtracted from the selectivity measure to create the manager’s net selectivity performance of (=-0.02 – 1.45) After accounting for the added cost of incomplete diversification, this manager’s performance would plot substantially below the market line

18.7 Factors That Affect Use of Performance Measures

18.7 Factors That Affect Use of Performance Measures
The problem arises in finding a realistic proxy for the theoretical market portfolio Benchmark portfolios Performance evaluation standard Usually a passive index or portfolio May need benchmark for entire portfolio and separate benchmarks for segments to evaluate individual managers Benchmark error Can effect slope of SML Can effect calculation of beta Greater concern with global investing Problem is one of measurement

18.7.1 Demonstration of the Global Benchmark Problem
As an illustration of the benchmark problem in global capital markets, consider how individual measures of risk change when the world equity market is employed as the market portfolio proxy There are two major differences in the various beta statistics: For many stocks, the beta estimates change a great deal over time Although the mean and median values for the U.S. and world beta estimates appear to be somewhat similar during both time periods, the “% Diff” columns show that there are some substantial differences in betas estimated for the same stock over the same time period when two different definitions of the benchmark portfolio are employed

18.7.1 Demonstration of the Global Benchmark Problem

18.7.2 Implications of the Benchmark Problems
Benchmark problems do not negate the value of the CAPM as a normative model of equilibrium pricing There is a need to find a better proxy for the market portfolio or to adjust measured performance for benchmark errors Multiple markets index (MMI) is major step toward a truly comprehensive world market portfolio

18.7.3 Required Characteristics of Benchmarks
Bailey, Richards, and Tierney (2007) contend that any useful benchmark should have the following characteristics: Unambiguous Investable Measurable Appropriate Specified in advance Owned

18.8 Reporting Investment Performance

18.8 Reporting Investment Performance
The performance measures described represent the essential elements of how any investor’s performance should be evaluated How should the returns used in the evaluation process be reported to the investor? Two dimensions: Consider the issue of how returns should be computed for a portfolio that experiences infusions and withdrawals of cash during the investment period The performance presentation standards created by the CFA Institute

18.8.1 Time-Weighted and Money-Weighted Returns
The holding period yield (HPY) for any investment position was determined by that position’s market value at the end of the period divided by its initial value: The dollar-weighted and time-weighted returns are the same when there are no interim investment contributions within the evaluation period When there are contributions, a method for adjusting holding period yields is: Where: DW = factor represents the portion of the period that the contribution is actually held in the account

18.8.2 Performance Presentation Standards
CFA Institute introduced PPS in 1987 In 1999 adopted the companion Global Investment Performance Standards (GIPS), which were intended to accomplish the following goals: To establish investment industry best practices for calculating and presenting investment performance To obtain worldwide acceptance of a single standard for the calculation and presentation of investment performance To promote the use of accurate and consistent investment performance data To encourage fair, global competition among investment firms without creating barriers to entry To foster the notion of industry “self-regulation” on a global basis

Fundamental Principles of GIPS Total return must be used Time-weighted rates of return must be used Portfolios must be valued at least monthly, and periodic returns must be geometrically linked Composite return performance (if presented) must contain all actual fee-paying accounts Performance must be calculated after deduction of trading expenses Taxes must be recognized when incurred Annual returns for all years must be presented Disclosure requirements must be met

In addition to these requirements, the CFA Institute also encourages managers to disclose the volatility of the composite return and to identify benchmarks that parallel the risk or investment style the composite tracks

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