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Topic 10 (Ch. 24) Portfolio Performance Evaluation

Measuring investment returns The conventional theory of performance

evaluation Market timing Performance attribution procedures

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 Measuring Investment Returns

One period:

Find the rate of return (r) that equates the present value of all cash flows from the investment with the initial outlay.

investmentinitialgaincapitalincome

investmentintialproceedstotalreturnofrate

3

Example:

Consider a stock paying a dividend of $2 annually that currently sells for $50.

You purchase the stock today and collect the $2 dividend, and then you sell the stock for $53 at year-end.

%10)5053(2 50

returnofrate

r1 53250 %10 r

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Multiperiod:

Arithmetic versus geometric averages:

Arithmetic averages:

021.0

50491.01088.02869.0221.01189.0

Ar

5

Geometric averages:

The compound average growth rate, rG, is calculated as the solution to the following equation:

In general:

where rt is the return in each time period.

)0491.01)(1088.01)(2869.01)(221.01)(1189.01(

5)1( Gr

0275.1

)1)...(1)(1()1( 21 nn

G rrrr

0054.0 Gr

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Geometric averages never exceed arithmetic ones:

Consider a stock that doubles in price in period 1 (r1 = 100%) and halves in price in period 2 (r2 = -50%).

The arithmetic average is: rA = [100 + (-50)]/2 = 25%

The geometric average is: rG = [(1 + 1)(1 - 0.5)]1/2 – 1 = 0

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The effect of the -50% return in period 2 fully offsets the 100% return in period 1 in the calculation of the geometric average, resulting in an average return of zero.

This is not true of the arithmetic average.

In general, the bad returns have a greater influence on the averaging process in the geometric technique. Therefore, geometric averages are lower.

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Generally, the geometric average is preferable for calculation of historical returns (i.e. measure of past performance), whereas the arithmetic average is more appropriate for forecasting future returns:

Example 1:

Consider a stock that will either double in value (r = 100%) with probability of 0.5, or halve in value (r = -50%) with probability 0.5.

Investment Outcome

Final Value of Each Dollar Invested

One-Year Rate of Return

Double $2.00 100% Halve $0.50 -50%

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Suppose that the stock’s performance over a 2-year period is characteristic of the probability distribution, doubling in one year and halving in the other.

The stock’s price ends up exactly where it started, and the geometric average annual return is zero:

which confirms that a zero year-by-year return would have replicated the total return earned on the stock.

1%)501%)(1001()1( 2 Gr

0 Gr

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However, the expected annual future rate of return on the stock is not zero.

It is the arithmetic average of 100% and -50%: (100 - 50)/2 = 25%.

There are two equally likely outcomes per dollar invested: either a gain of $1 (when r = 100%) or a loss of $0.50 (when r = -50%).

The expected profit is ($1 - $0.50)/2 = $0.25, for a 25% expected rate of return.

The profit in the good year more than offsets the loss in the bad year, despite the fact that the geometric return is zero. The arithmetic average return thus provides the best guide to expected future returns.

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Example 2:

Consider all the possible outcomes over a two-year period:

Investment Outcome Final Value of Each

Dollar Invested Total Return

over Two Years Double, double $4.00 300% Double, halve $1.00 0% Halve, double $1.00 0% Halve, halve $0.25 -75%

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The expected final value of each dollar invested is:

(4 + 1 + 1 + 0.25)/4 = $1.5625

for two years, again indicating an average rate of return of 25% per year, equal to the arithmetic average.

Note that an investment yielding 25% per year with certainty will yield the same final compounded value as the expected final value of this investment:

(1 + 0.25)2 = 1.5625.

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The arithmetic average return on the stock is:

[300 + 0 + 0 + (-75)]/4 = 56.25%

per two years, for an effective annual return of 25% since:

(1 + 25%)(1 + 25%) – 1 = 56.25%.

In contrast, the geometric mean return is zero since:

[(1 + 3)(1 + 0)(1 + 0)(1 – 0.75)]1/4 = 1.0

Again, the arithmetic average is the better guide to future performance.

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Dollar-weighted returns versus time-weighted returns:

Example: Time Outlay

0 $50 to purchase first share 1 $53 to purchase second share a year later Proceeds

1 $2 dividend from initially purchased share 2 $4 dividend from the 2 shares held in the

second year, plus $108 received from selling both shares at $54 each

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Dollar-weighted returns:

Using the discounted cash flow (DCF) approach, we can solve for the average return over the two years by equating the present values of the cash inflows and outflows:

%117.7 r

2)1(112

12

15350

rrr

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This value is called the internal rate of return, or the dollar-weighted rate of return on the investment.

It is “dollar weighted” because the stock’s performance in the second year, when two shares of stock are held, has a greater influence on the average overall return than the first-year return, when only one share is held.

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Time-weighted returns:

Ignore the number of shares of stock held in each period.

The stock return in the 1st year:

The stock return in the 2nd year:

%1050

)5053(21

r

%66.553

)5354(22

r

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The time-weighted (geometric average) return is:

This average return considers only the period-by-period returns without regard to the amounts invested in the stock in each period.

Note that the dollar-weighted average is less than the time-weighted average in this example because the return in the second year, when more money is invested, is lower.

%81.71%)]66.51%)(101[( 2/1 Gr

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Note: For an investor that has control over

contributions to the investment portfolio, the dollar-weighted return is more comprehensive measure.

Time-weighted returns are more likely appropriate to judge the performance of an investor that does not control the timing or the amount of contributions.

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Several risk-adjusted performance measures:

Sharpe’s measure:

Sharpe’s measure divides average portfolio excess return over the sample period by the standard deviation of returns over that period.

It measures the reward to (total) volatility trade-off.

Note: The risk-free rate may not be constant over the measurement period, so we are taking a sample average, just as we do for rP.

 The Conventional Theory of Performance Evaluation

P

frr P

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Treynor’s measure:

Like Sharpe’s, Treynor’s measure gives excess return per unit of risk, but it uses systematic risk instead of total risk.

Jensen’s measure:

Jensen’s measure is the average return on the portfolio over and above that predicted by the CAPM, given the portfolio’s beta and the average market return.

Jensen’s measure is the portfolio’s alpha value.

P

frr P

)]([ fMPfP rrrr P

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Information ratio:

The information ratio divides the alpha of the portfolio by the nonsystematic risk of the portfolio.

It measures abnormal return per unit of risk that in principle could be diversified away by holding a market index portfolio.

Note: Each measure has some appeal. But each does not necessarily provide consistent

assessments of performance, since the risk measures used to adjust returns differ substantially.

)( P

Pe

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Example: Consider the following data for a particular sample period:

The T-bill rate during the period was 6%.

　 Portfolio P Market M

Average return 35% 28%

Beta 1.20 1.00

Standard deviation 42% 30%

Nonsystematic risk, σ(e) 18% 0

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Sharpe’s measure:

Treynor’s measure:

69.042

635

P

fP

rrS P

73.030

628 MS

2.242.1

635

P

fP

rrT P

221

628 MT

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Jensen’s measure:

Information ratio:

%6.2)]628(2.16[35)]([ fMPfP rrrr P

0)](1[ fMfM rrrr M

144.018

6.2)(

: P

Pe

PPortfolio

0:Market

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While the Sharpe ratio can be used to rank portfolio performance, its numerical value is not easy to interpret.

We have found that SP = 0.69 and SM = 0.73. This suggests that portfolio P under-performed the

market index.

But is a difference of 0.04 in the Sharpe ratio economically meaningful?

We often compare rates of return, but these ratios are difficult to interpret.

The M2 measure of performance

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To compute the M2 measure, we imagine that a managed portfolio, P, is mixed with a position in T-bills so that the complete, or “adjusted,” portfolio (P*) matches the volatility of a market index (such as the S&P500).

Because the market index and portfolio P* have the same standard deviation, we may compare their performance simply by comparing returns.

This is the M2 measure: MP rrM *2

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Example:

P has a standard deviation of 42% versus a market standard deviation of 30%.

The adjusted portfolio P* would be formed by mixing portfolio P and T-bills and :

weight in P: 30/42 = 0.714 weight in T-bills: (1 - 0.714) = 0.286.

The return on this portfolio P* would be: (0.286 6%) + (0.714 35%) = 26.7%

Thus, portfolio P has an M2 measure: 26.7 – 28 = -1.3%.

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We move down the capital allocation line corresponding to portfolio P (by mixing P with T-bills) until we reduce the standard deviation of the adjusted portfolio to match that of the market index.

The M2 measure is then the vertical distance (i.e., the difference in expected returns) between portfolios P* and M.

P will have a negative M2 measure when its capital allocation line is less steep than the capital market line (i.e., when its Sharpe ratio is less than that of the market index).

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Suppose that Jane constructs a portfolio (P) and holds it for a considerable period of time.

She makes no changes in portfolio composition during the period.

In addition, suppose that the daily rates of return on all securities have constant means, variances, and covariances. This assures that the portfolio rate of return also has a constant mean and variance.

We want to evaluate the performance of Jane’s portfolio.

Appropriate performance measures in 3 scenarios

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Jane's portfolio P represents her entire risky investment fund:

We need to ascertain only whether Jane’s portfolio has the highest Sharpe measure.

We can proceed in 3 steps:

Assume that past security performance is representative of expected performance, meaning that realized security returns over Jane’s holding period exhibit averages and covariances similar to those that Jane had anticipated.

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Determine the benchmark (alternative) portfolio that Jane would have held if she had chosen a passive strategy, such as the S&P 500.

Compare Jane’s Sharpe measure to that of the best portfolio.

In sum: When Jane’s portfolio represents her entire

investment fund, the benchmark is the market index or another specific portfolio.

The performance criterion is the Sharpe measure of the actual portfolio versus the benchmark.

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Jane’s portfolio P is an active portfolio and is mixed with the market-index portfolio M:

When the two portfolios are mixed optimally, the square of the Sharpe measure of the complete portfolio, C, is given by:

where P is the abnormal return of the active portfolio relative to the market-index, and (eP) is the diversifiable risk.

222 ])(

[P

PMC e

SS

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The ratio P/(eP) is thus the correct performance measure for P in this case, since it gives the improvement in the Sharpe measure of the overall portfolio.

To see this result intuitively, recall the single-index model:

If P is fairly priced, then P = 0, and eP is just

diversifiable risk that can be avoided.

PfMPPfP errrr )(

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However, if P is mispriced, P no longer equals zero. Instead, it represents the expected abnormal return.

Holding P in addition to the market portfolio thus brings a reward of P against the nonsystematic risk voluntarily incurred, (eP).

Therefore, the ratio of P /(eP) is the natural benefit-to-cost ratio for portfolio P.

This performance measurement is the information ratio.

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Jane’s choice portfolio P is one of many portfolios combined into a large investment fund:

The Treynor measure is the appropriate criterion.

E.g.:

][* returnexcessmarketbetareturnexcessalpha

)]([)()( fMffMf rrrrrrrr

　 Portfolio P Portfolio Q Market

Beta 0.90 1.60 1.00 Excess return 11% 19% 10%

Alpha* 2% 3% 0 )( frr

38

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Note:

We plot P and Q in the expected return-beta (rather than the expected return-standard deviation) plane, because we assume that P and Q are two of many sub-portfolios in the fund, and thus that nonsystematic risk will be largely diversified away, leaving beta as the appropriate risk measure.

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Suppose portfolio Q can be mixed with T-bills.

Specifically, if we invest wQ in Q and wF = 1 - wQ in T-bills, the resulting portfolio, Q*, will have alpha and beta values proportional to Q’s alpha and beta scaled down by wQ:

Thus, all portfolios Q* generated from mixing Q with T-bills plot on a straight line from the origin through Q.

We call it the T-line for the Treynor measure, which is the slope of this line.

QQQ w * QQQ w *

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P has a steeper T-line.

Despite its lower alpha, P is a better portfolio after all.

For any given beta, a mixture of P with T-bills will give a better alpha than a mixture of Q with T-bills.

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Suppose that we choose to mix Q with T-bills to create a portfolio Q* with a beta equal to that of P.

We find the necessary proportion by solving for wQ:

Portfolio Q* has an alpha of:

which is less than that of P.

9.06.1* PQQQQ ww

16/9 Qw

%69.13)16/9(* QQQ w

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In other words, the slope of the T-line is the appropriate performance criterion for this case.

The slope of the T-line for P, denoted by TP, is:

Treynor’s performance measure is appealing because when an asset is part of a large investment portfolio, one should weigh its mean excess return against its systematic risk rather than against total risk to evaluate contribution to performance.

P

fP

rrT P

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An example:

Excess returns for portfolios P & Q and the benchmark M over 12 months:

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Performance statistics:

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Portfolio Q is more aggressive than P, in the sense that its beta is significantly higher (1.40 vs. 0.70).

On the other hand, from its residual standard deviation P appears better diversified (2.02% vs. 9.81%).

Both portfolios outperformed the benchmark market index, as is evident from their larger Sharpe measures (and thus positive M2) and their positive alphas.

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Which portfolio is more attractive based on reported performance?

If P or Q represents the entire investment fund, Q would be preferable on the basis of its higher Sharpe measure (0.49 vs. 0.43) and better M2 (2.66% vs. 2.16%).

As an active portfolio to be mixed with the market index, P is preferable to Q, as is evident from its information ratio (0.81 vs. 0.54).

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When P and Q are competing for a role as one of a number of subportfolios, Q dominates again because its Treynor measure is higher (5.38 versus 3.97).

Thus, the example illustrates that the right way to evaluate a portfolio depends in large part how the portfolio fits into the investor’s overall wealth.

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Relationships among the various performance measures

The relation between Treynor’s measure and Jensen’s :

P

fPP

rrET

)(

P

fMpfMpfP rrErrErrE

))(()])(([)(

MP

PfM

P

P TrrE

])([

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The relation between Sharpe’s measure and Jensen’s :

P

fPP

rrES

)(

P

fMpfMpfP rrErrErrE

))(()])(([)(

P

MP

MP

MP

MP

MPPM

rr

2),cov(

M

PM

P

P

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P

fMp

P

PP

rrES

))((

])([ fMM

PM

P

P rrE

MPMP

P S

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Estimating various statistics from a sample period assuming a constant mean and variance may lead to substantial errors.

Example:

Suppose that the Sharpe measure of the market index is 0.4.

Over an initial period of 52 weeks, the portfolio manager executes a low-risk strategy with an annualized mean excess return of 1% and standard deviation of 2%.

Performance measurement with changing portfolio composition

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This makes for a Sharpe measure of 0.5, which beats the passive strategy.

Over the next 52-week period this manager finds that a high-risk strategy is optimal, with an annual mean excess return of 9% and standard deviation of 18%.

Here, again, the Sharpe measure is 0.5.

Over the two-year period our manager maintains a better-than-passive Sharpe measure.

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Portfolio returns in last four quarters are more variable than in the first four:

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In the first 4 quarters, the excess returns are -1%, 3%, -1%, and 3%, making for an average of 1% and standard deviation of 2%.

In the next 4 quarters the returns are -9%, 27%, -9%, 27%, making for an average of 9% and standard deviation of 18%.

Thus both years exhibit a Sharpe measure of 0.5.

However, over the 8-quarter sequence the mean and standard deviation are 5% and 13.42%, respectively, making for a Sharpe measure of only 0.37, apparently inferior to the passive strategy.

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The shift of the mean from the first 4 quarters to the next was not recognized as a shift in strategy.

Instead, the difference in mean returns in the two years added to the appearance of volatility in portfolio returns.

The active strategy with shifting means appears riskier than it really is and biases the estimate of the Sharpe measure downward.

We conclude that for actively managed portfolios it is helpful to keep track of portfolio composition and changes in portfolio mean and risk.

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Market timing involves shifting funds between a

market-index portfolio and a safe asset (such as T-bills or a money market fund), depending on whether the market as a whole is expected to outperform the safe asset.

In practice, most managers do not shift fully, but partially, between T-bills and the market.

 Market Timing

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Suppose that an investor holds only the market-index portfolio and T-bills.

If the weight of the market were constant, say, 0.6, then portfolio beta would also be constant, and the security characteristic line (SCL) would plot as a straight line with slope 0.6.

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No market timing, beta is constant:

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If the investor could correctly time the market and shift funds into it in periods when the market does well.

If bull and bear markets can be predicted, the investor will shift more into the market when the market is about to go up.

The portfolio beta and the slope of the SCL will be higher when rM is higher, resulting in the curved line.

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Market timing, beta increases with expected market excess return:

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Such a line can be estimated by adding a squared term to the usual linear index model:

where rP is the portfolio return, and a, b, and c are estimated by regression analysis.

If c turns out to be positive, we have evidence of timing ability, because this last term will make the characteristic line steeper as (rM - rf) is larger.

PfMfMfP errcrrbarr 2)()(

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A similar and simpler methodology suggests that the beta of the portfolio take only two values: a large value if the market is expected to do well and a small value otherwise.

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Such a line appears in regression form as:

where D is a dummy variable that equals 1 for rM > rf and zero otherwise.

Hence, the beta of the portfolio is b in bear markets and b + c in bull markets.

Again, a positive value of c implies market timing ability.

PfMfMfP eDrrcrrbarr )()(

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Example:

Regressing the excess returns of portfolios P and Q on the excess returns of M and the square of these returns:

we derive the following statistics:

PfMPfMPPfP errcrrbarr 2)()(

QfMQfMQQfQ errcrrbarr 2)()(

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The numbers in parentheses are the regression estimates from the single variable regression (reported in Table 24.3).

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Portfolio P shows no timing (cP = 0).

The results for portfolio Q reveal that timing has, in all likelihood, successfully been attempted (cQ = 0.10).

The evidence thus suggests successful timing (positive c) offset by unsuccessful stock selection (negative a).

Note that the alpha estimate, a, is now -2.29% as opposed to the 5.26% estimate derived from the regression equation that did not allow for the possibility of timing activity.

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Portfolio managers constantly make broad-brush asset allocation decisions as well as more detailed sector and security allocation decisions within asset class.

Performance attribution studies attempt to decompose overall performance into discrete components that may be identified with a particular level of the portfolio selection process.

 Performance Attribution Procedures

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The difference between a managed portfolio’s performance and that of a benchmark portfolio then may be expressed as the sum of the contributions to performance of a series of decisions made at the various levels of the portfolio construction process.

For example, one common attribution system decomposes performance into 3 components:

• broad asset market allocation choices across equity, fixed-income, and money markets.

• industry (sector) choice within each market.• security choice within each sector.

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The attribution method explains the difference in returns between a managed portfolio, P, and a selected benchmark portfolio, B (called the bogey).

Suppose that the universe of assets for P and B includes n asset classes such as equities, bonds, and bills.

For each asset class, a benchmark index portfolio is determined.

For example, the S&P 500 may be chosen as benchmark for equities.

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The bogey portfolio is set to have fixed weights in each asset class, and its rate of return is given by:

where wBi: weight of the bogey in asset class i. rBi: return on the benchmark portfolio of that class over the evaluation period.

n

iBiBiB rwr

1

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The portfolio managers choose weights in each class (wPi) based on their capital market expectations, and they choose a portfolio of the securities within each class based on their security analysis, which earns rPi over the evaluation period.

Thus, the return of the managed portfolio will be:

n

iPiPiP rwr

1

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The difference between the two rates of return is:

We can decompose each term of the summation into a sum of two terms as follows:

Contribution from asset allocation:

+ Contribution from selection:

= Total contribution from asset class i:

n

iBiBiPiPi

n

i

n

iBiBiPiPiBP rwrwrwrwrr

11 1)(

BiBiPi rww )(

)( BiPiPi rrw

BiBiPiPi rwrw

74

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Example:

Consider the attribution results for a portfolio which invests in stocks, bonds, and money market securities.

The managed portfolio is invested in the equity, fixed-income, and money markets with weights of 70%, 7%, and 23%, respectively.

The portfolio return over the month is 5.34%.

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　 Bogey Performance and Excess Return

Component Benchmark Weight Return of Index during Month (%)

Equity (S&P 500) 0.60 5.81

Bonds (Barclays Aggregate Bond Index)

0.30 1.45

Cash (money market ) 0.10 0.48

Bogey = (0.60 5.81 ) + (0.30 1.45) + (0.10 0.48) = 3.97%

　 Return of managed portfolio 5.34%

- Return of bogey portfolio 3.97%

　 Excess return of managed portfolio 1.37%

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Note:

The bogey portfolio is comprised of investments in each index with the following weights:

60%: equity 30%: fixed income 10%: cash (money market securities).

These weights are designated as “neutral” or “usual”.

They depend on the risk tolerance of the investor and must be determined in consultation with the client.

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This would be considered a passive asset-market allocation.

Any deviation from these weights must be justified by a belief that one or another market will either over- or underperform its usual risk-return profile.

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　 A. Contribution of Asset Allocation to Performance 　

　 (1) (2) (3) (4) (5) = (3) (4)

　 Actual Benchmark Contribution to

　 Weight in Weight in Excess Market Performance

Market Market Market Weight Return (%) (%)

Equity 0.70 0.60 0.10 5.81 0.5810

Fixed-income 0.07 0.30 -0.23 1.45 -0.3335

Cash 0.23 0.10 0.13 0.48 0.0624

Contribution of asset allocation 　 　 0.3099

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　 B. Contribution of Selection to Total Performance 　　 (1) (2) (3) (4) (5) = (3)

(4)

　 Portfolio Index Excess

　 Performance Performance Performance Portfolio Contribution

Market (%) (%) (%) Weight (%)

Equity 7.28 5.81 1.47 0.70 1.03

Fixed- income 1.89 1.45 0.44 0.07 0.03

Contribution of selection within markets

　 　 1.06

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Sector selection within the equity market:

82

Portfolio attribution summary: