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Introduction

- As portfolio managers, how can we evaluate the performance of our portfolio?
- We know that there are 2 major requirements of a portfolio manager’s performance:

1. The ability to derive above-average returns conditioned on risk taken, either through superior market timing or superior security selection.

2. The ability to diversify the portfolio and eliminate non-systematic risk, relative to a benchmark portfolio.

Today

- Performance Measurement Risk Adjusted Performance MeasuresMeasures of Sharpe, Treynor and JensenMeasures of Skill and Timing
- Attribution de performance

Concept de mesures ajustées pour le risque

Mesures de Sharpe, Treynor et Jensen

Mesure des habilités de timing

Averaging Returns

Arithmetic Mean:

Example:

(.10 + .0566) / 2 = 7.83%

Geometric Mean:

Example:

[ (1.1) (1.0566) ]1/2 - 1

= 7.808%

17-3

Geometric Average

The arithmetic average provides unbiased estimates of the expected return of the stock. Use this to forecast returns in the next period.

The fixed rate of return over the sample period that would yield the terminal value is know as the geometric average.

The geometric average is less than the arithmetic average and this difference increases with the volatility of returns.

The geometric average is also called the time-weighted average (as opposed to the dollar weighted average), because it puts equal weights on each return.

17-4

Dollar- and Time-Weighted Returns

Dollar-weighted returns

- Internal rate of return.
- Returns are weighted by the amount invested in each stock.

Time-weighted returns

- Not weighted by investment amount.
- Equal weighting
- Geometric average

17-5

Example: Multiperiod Returns

PeriodAction

0 Purchase 1 share of Eggbert’s Egg Co. at $50

1 Purchase 1 share of Eggbert’s Egg Co. at $53

Eggbert pays a dividend of $2 per share

2 Eggbert pays a dividend of $2 per share

Sell both shares for $108

17-6

Dollar-Weighted Return

PeriodCash Flow

0 -50 share purchase

1 +2 dividend -53 share purchase

2 +4 dividend + 108 shares sold

Internal Rate of Return:

Dollar Weighted: The stocks performance in the second year,

when we own 2 shares, has a greater influence on the overall return.

Time-Weighted Return

Geometric Mean:

[ (1.1) (1.0566) ]1/2 - 1

= 7.808%

Time Weighted: Each return has equal weight in the geometric

average.

17-8

Early Performance Measure Techniques

- Portfolio evaluation before 1960
- Once upon a time, investors evaluated a portfolio’s performance based purely on the basis of the rate of return.
- Research in the 1960’s showed investors how to quantify and measure risk.
- Grouped portfolios into similar risk classes and compared rates of return within risk classes.

Peer Group Comparisons

- This is the most common manner of evaluating portfolio managers.
- Collects returns of a representative universe of investors over a period of time and displays them in a box plot format.
- Example: “US Equity with Cash” relative to peer universe of US domestic equity managers.
- Issue: There is no explicit adjustment for risk. Risk is only considered implicitly.

Treynor (1965)

- Treynor (1965) developed the first composite measure of portfolio performance that included risk.
- He introduced the portfolio characteristic line, which defines a relation between the rate of return on a specific portfolio and the rate of return on the market portfolio.
- The beta is the slope that measures the volatility of the portfolio’s returns relative to the market.
- Alpha represents unique returns for the portfolio.
- As the portfolio becomes diversified, unique risk diminishes.

Treynor Measure

A risk-adjusted measure of return that divides a portfolio's

excess return by its beta.

The Treynor Measure is given by

The Treynor Measure is defined using the average rate of return for portfolio p and the risk-free asset.

Treynor Measure

A larger Tp is better for all investors, regardless of their risk preferences.

Because it adjusts returns based on systematic risk, it is the relevant performance measure when evaluating diversified portfolios held in separately or in combination with other portfolios.

Treynor Measure

- Beta measures systematic risk, yet if the portfolio is not fully diversified then this measure is not a complete characterization of the portfolio risk.
- Hence, it implicitly assumes a completely diversified portfolio.
- Portfolios with identical systematic risk, but different total risk, will have the same Treynor ratio!
- Higher idiosyncratic risk should not matter in a diversified portfolio and hence is not reflected in the Treynor measure.
- A portfolio negative Beta will have a negative Treynor measure.
- Also known as the Treynor Ratio.

Sharpe Measure

- Similar to the Treynor measure, but uses the total risk of the portfolio, not just the systematic risk.
- The Sharpe Ratio is given by
- The larger the measure the better, as the portfolio earned a higher excess return per unit of total risk.

Sharpe Measure

It adjusts returns for total portfolio risk, as opposed to only systematic risk as in the Treynor Measure.

Thus, an implicit assumption of the Sharpe ratio is that the portfolio is not fully diversified, nor will it be combined with other diversified portfolios.

It is relevant for performance evaluation when comparing mutually exclusive portfolios.

Sharpe originally called it the "reward-to-variability" ratio, before others startedcalling it the Sharpe Ratio.

SML vs. CML

- Treynor’s measure uses Beta and hence examines portfolio return performance in relation to the SML.
- Sharpe’s measure uses total risk and hence examines portfolio return performance in relation to the CML.
- For a totally diversified portfolio, both measures give equal rankings.
- If it is not a diversified portfolio, the Sharpe measure could give lower rankings than the Treynor measure.
- Thus, the Sharpe measure evaluates the portfolio manager in terms of both return performance and diversification.

Price of Risk

- Both the Treynor and Sharp measures, indicate the risk premium per unit of risk, either systematic risk (Treynor) or total risk (Sharpe).
- They measure the price of risk in units of excess returns per each unit of risk (measured either by beta or the standard deviation of the portfolio).

Jensen’s Alpha

- Alpha is a risk-adjusted measure of superior performance
- This measure adjusts for the systematic risk of the portfolio.
- Positive alpha signals superior risk-adjusted returns, and that the manager is good at selecting stocks or predicting market turning points.
- Unlike the Sharpe Ratio, Jensen’s method does not consider the ability of the manager to diversify, as it is only accounts for systematic risk.

Multifactor Jensen’s Measure

Measure can be extended to a multi-factor setting, for example:

Information Ratio 1

- Using a historical regression, the IR takes on the form

where the numerator is Jensen’s alpha and the denominator is the standard error of the regression. Recalling that

Note that the risk here is nonsystematic risk, that could, in theory,

be eliminated by diversification.

Information Ratio 2

Measures excess returns relative to a benchmark portfolio.

Sharpe Ratio is the special case where the benchmark equals the risk-free asset.

Risk is measured as the standard deviation of the excess return (Recall that this is the Tracking Error)

For an actively managed portfolio, we may want to maximize the excess return per unit of nonsystematic risk we are bearing.

Portfolio Tracking Error

Excess Return relative

to benchmark portfolio b

Average Excess Return

Variance in Excess Difference

Tracking Error

Information Ratio

- Excess return represents manager’s ability to use information and talent to generate excess returns.
- Fluctuations in excess returns represent random noise that is interpreted as unsystematic risk.

Information to noise ratio.

- Annualized IR

M2 Measure

Developed by Leah and her grandfather Franco Modigliani.

M2 = rp*- rm

rp* is return of the adjusted portfolio that matches the volatility of the market index rm. It is mixed with a position in T-bills.

If the risk of the portfolio is lower than that of the market, one has to increase the volatility by using leverage.

Because the market index and the adjusted portfolio have the same standard deviation, we may compare their performances by comparing returns.

M2 Measure: Example

Managed Portfolio: return = 35% st dev = 42%

Market Portfolio: return = 28% st dev = 30%

T-bill return = 6%

Hypothetical Portfolio:

30/42 = .714 in P (1-.714) or .286 in T-bills

Return = (.714) (.35) + (.286) (.06) = 26.7%

Since the return of the portfolio is less than the market, M2 is negative, and the managed portfolio underperformed the market.

17-35

Which Portfolio is Best?

- It depends.
- If P or Q represent the entire portfolio, Q would be preferable based on having higher sharp ratio and a better M2.
- If P or Q represents a sub-portfolio, the Q would be preferable because it has a higher Treynor ratio.
- For an actively managed portfolio, P may be preferred because it’s information ratio is larger (that is it maximizes return relative to nonsystematic risk, or the tracking error).

Style Analysis

Introduced by William Sharpe

1992 study of mutual fund performance

- 91.5% of variation in return could be explained by the funds’ allocations to bills, bonds and stocks

Later studies show that 97% of the variation in return could be explained by the funds’ allocation to set of different asset classes.

17-41

Sharpe’s Style Portfolios for the Magellan Fund

Monthly returns on Magellan Fund over five year period.

Regression coefficient only positive for 3.

They explain 97.5% of Magellan’s returns.

2.5 percent attributed to security selection within asset classes.

17-42

Fidelity Magellan Fund Returns vs Benchmarks

Fund vs Style and Fund vs SML

19.19%

Impact of positive alpha on abnormal returns.

17-43

Perfect Market Timing

- A manager with perfect market timing, that shifts assets efficiently across stocks, bonds and cash would have a return equal to

With Perfect Forecasting Ability

- Switch to T-Bills in 90 and 94
- Mean = 18.94%,
- Standard Deviation = 12.04%
- Invested in large stocks for the entire period:
- Mean = 17.41%
- Standard Deviation = 14.11

17-48

Performance of Bills, Equities and Timers

Beginning with $1 dollar in 1926, and ending in 2005....

Value of Imperfect Forecasting

- Suppose you are forecasting rain in Seattle. If you predict rain, you would be correct most of the time.
- Does this make you a good forecaster? Certainly not.
- We need to examine the proportion of correct forecasts for rain (P1) and the proportion of correct forecasts for sun (P2).
- The correct measure of timing ability is

P = P1 + P2 – 1

An forecaster who always guesses correctly will show P1 = P2 = P =1, whereas on who always predicts rain will have P1 = 1, P2 = P = 0.

Identifying Market Timing

If an investor holds only the market and the risk free security, and the weights remained constant, the portfolio characteristic line would be a straight line.

Adjusting portfolio weights for up and down movements in market returns, we would have:

- Low Market Return - low weight on the market - low ßeta
- High Market Return – high weight on the market - high ßeta

Henriksson (1984) showed little evidence of market timing. Evidence of market efficiency.

17-51

Characteristic Lines: Market Timing

No Market Timing

Beta Increases with Return

Two Values of Beta

17-52

Testing Market Timing

- The following regression equation, controls for the movements in bond and stock markets, and captures the superior market timing of managers
- Gamma was found to be equal to .3 and statistically significant, suggesting that TAA managers were able to time the markets.
- However, the study also found a negative alpha of -.5.

Selectivity

- The basic premise of the Fama method is that overall performance of a portfolio can be decomposed into a portfolio risk premium component and a selectivity component.
- Selectivity is the portion of excess returns that exceeds that which can be attained by an unmanaged benchmark portfolio.
- Overall performance = Portfolio Risk Premium + Selectivity

Overall performance = Portfolio Risk Premium+ Selectivity

Overall

Performance

Selectivity

Portfolio Risk Premium

Selectivity measures the distance between the return on portfolio p and the return on a benchmark portfolio with beta equal to the beta of portfolio p.

Attribution Analysis

Portfolio managers add value to their investors by

1) selecting superior securities

2) demonstrating superior market timing skills by allocating funds to different asset classes or market segments.

Attribution analysis attempts to distinguish is the source of the portfolio’s overall performance.

Total value added performance is the sum of selection and allocation effects.

Formula for Attribution

Set up a ‘Benchmark’ or ‘Bogey’ portfolio

Where B is the bogey portfolio and p is the managed portfolio.

17-58

Allocation Effect

- Asset Allocation Effect
- Captures the manager’s decision to over or underweight a particular market segment i.
- Overweighting a segment i when the benchmark yield is high is rewarded.

Selection Effect

- Security Selection Effect
- Captures the stock picking ability of the manager, and rewards the ability to form specific market segment portfolios. Rewards the manger for placing larger weights on those segments where his portfolio outperforms the benchmark portfolio in that particular segment.

Benchmark Error

- Market portfolio is difficult to approximate
- Benchmark error
- can effect slope of SML
- can effect calculation of Beta
- greater concern with global investing
- problem is one of measurement
- Note: Sharpe measure not as dependent on market portfolio as the Treynor measure and others relying on Beta.

Differences in Betas

- Two major differences in the various beta statistics:
- For any particular stock, the beta estimates change a great deal over time.
- There are substantial differences in betas estimated for the same stock over the same time period when two different definitions of the benchmark portfolio are employed.

Bond Portfolio Measures

- Returns-Based Bond Performance Measurement
- Early attempts to analyze fixed-income performance involved peer group comparisons
- Peer group comparisons are potentially flawed because they do not account for investment risk directly.
- How did the performance levels of portfolio managers compare to the overall bond market?
- What factors lead to superior or inferior bond-portfolio performance?

Fama-French Measure

- Fama and French extended their 3-factor equity pricing model with 2 additional factors to account for the return characteristics of bonds
- TERM – captures the term premium in the slope of the yield curve.
- DEF – captures the default premium in the credit spread between corporate bonds and treasuries.
- These two bond factors are the dominate drivers of bond portfolio returns.

Bond Performance Attribution

- A Bond Market Line
- Need a measure of risk such as beta coefficient for equities
- Difficult to achieve due to bond maturity and coupon effect on volatility of prices
- Composite risk measure is the bond’s duration
- Duration replaces beta as risk measure in a bond market line

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