The objective of this chapter is to provide the difference between a portfolio’s time-weighted returns and its dollar-weighted returns. The chapter will also give a description of the appropriate application of the time- and dollar-weighted returns.

Furthermore, risk-adjusted performance measures will also be described and distinguished. In the comparison of two portfolios, the applications of the Modigliani-squared and Treynor’s measure will be described and the uses of their graphical representation explained.

A performance measure’s statistical significance will be described applying the standard error and the t-statistic. The chapter will go ahead and offer an explanation of the challenges associated with the evaluation of hedge funds’ performance.

The effect of portfolio risk level changes on the application of Sharpe ratio in performance measurement will also be explained. The chapter will also describe the techniques used in the measure of fund managers’ ability to time the market via a call option model, and calculate returns due to the market timing.

Finally, the style analysis and performance attribution procedures will be described in this chapter.

# The Conventional Theory of Performance Evaluation

## The Conventional Evaluation

Assuming that a portfolio’s performance within 5 years from 20 quarterly rates of return is evaluated. The best estimates of the portfolio’s expected rate of return for the next quarter are given by this sample’s arithmetic average.

The constant quarterly return over the 20 quarters yielding similar total cumulative return, contrary to that, is the geometric average, \({ r }_{ G }\), defined by:

$$ { \left( 1+{ r }_{ G } \right) }^{ 20 }=\left( 1+{ r }_{ 1 } \right) \left( 1+{ r }_{ 2 } \right) \dots \left( 1+{ r }_{ 20 } \right) $$

A $1 investment that is to earn the 20 quarterly return rates over an observation period of 5 years has a compounded final value given by the right side of the equation. For a $1 investment earning \({ r }_{ G }\) quarterly, the left side is its compounded value. Therefore:

$$ 1+{ r }_{ G }={ \left[ \left( 1+{ r }_{ 1 } \right) \left( 1+{ r }_{ 2 } \right) \dots \left( 1+{ r }_{ 20 } \right) \right] }^{ \frac { 1 }{ 20 } } $$

The geometric average is called the time-weighted average since it contains all returns with equal weight.

Let a stock pay an annual dividend of $3.5 selling for $65 at present. Assuming that first, the stock is bought, then the $3.5 dividend collected, and at year end the stock is finally sold at $70. Then the rate of return is:

$$ \frac { Total\quad Proceeds }{ Initial\quad Investment } =\frac { Income+Capital\quad Gain }{ 65 } =\frac { 3.5+5 }{ 65 } =0.1308\quad or\quad 13.08\% $$

Setting up the investment as a discounted cash flow problem is another useful way in the more challenging multi-period case. In the above case, we have that a stock bought for $65 generates cash flows of $3.5 annually and $70 upon selling.

Therefore:

$$ 65=\frac { \left( 3.5+70 \right) }{ 1+r } $$

$$ \Rightarrow r=0.1308\quad or\quad 13.08\% $$

## Time Weighted Returns versus Dollar-Weighted Returns

Assuming that an investor is to buy a second share, of a similar stock described in the example above, at the closure of the first year and held each of the shares until year 2 comes to an end and then sells each of the shares at $72. Then we have that:

$$ \begin{array}{|ll|} \hline Time & Outlay \\ \hline 0 & $65 \quad to \quad buy \quad the \quad first \quad share \\ 1 & $70 \quad to \quad buy \quad the \quad second \quad share \quad one \quad year \quad later \\ Proceeds & {} \\ 1 & $3.5 \quad is \quad the \quad dividend \quad earned \quad from \quad the \quad initially \quad bought \quad share \\ 2 & $7 \quad is \quad the \quad dividend \quad for \quad holding \quad the \quad two \quad shares \quad in \quad year \quad 2, \\ {} & with \quad a \quad $144 \quad received \quad by \quad selling \quad them \quad at \quad $72 \quad each \\ \hline \hline \end{array} $$

When we apply the discounted cash flow (DCF) approach, the two years’ average return can be computed if we equate the present values of the cash inflows and outflows:

$$ 65+\frac { 70 }{ 1+r } =\frac { 3.5 }{ 1+r } +\frac { 151 }{ { \left( 1+r \right) }^{ 2 } } $$

$$ \Rightarrow 65\left( 1+r \right) +70=3.5+\frac { 151 }{ 1+r } $$

$$ r=9.61\% $$

\(r\) is referred to as the internal rate of return, or the dollar-weighted rate of return on the investment.

For the time-weighted (geometric average) return, we have:

$$ { r }_{ 1 }=\frac { 70+3.5-65 }{ 65 } =13.08\% $$

$$ { r }_{ 2 }=\frac { 72+3.5-70 }{ 65 } =7.86\% $$

$$ { r }_{ G }={ \left( 1.1308\times 1.0786 \right) }^{ \frac { 1 }{ 2 } }-1=0.1044=10.44\% $$

Since the money invested was more leading to lower the return in year 2, the time-weighted average becomes more than the dollar-weighted average.

## Dollar-Weighted Return and Investment Performance

As far as the choice of a venue of investment is concerned, households have a considerable latitude and they, therefore, will regularly check results. To do this, a spreadsheet of time-dated cash inflows and outflows must be maintained by the household, by keeping a record of the current value of investment account. Therefore, over an investment period, the effective return rate earned in the period will be yielded by the dollar-weighted average.

## Adjusting Returns for Risk

Before a meaningful comparison is carried out, returns must be adjusted for risks. A comparison of return rates with those of other investment funds having risk characteristics that are similar is the simplest and most popular approach of adjusting a portfolio’s risk return.

However, one can be misled by such rankings. Some managers may decide to focus on particular subgroups, within a particular universe, making the comparison of portfolio characteristics to be difficult. For instance, a manager may concentrate on high beta or aggressive growth stocks within the equity universe. In a similar fashion, there may be variations in durations across managers within the fixed income universes. Therefore, there is a need for a means of risk adjustment that is precisely based on these considerations.

Risk-adjusted performance evaluation methods by means of mean-variance criteria occur simultaneously with the capital asset pricing model. However, there are limitations of widely applying each risk-adjusted performance measures. Furthermore, a long history of consistent management, including a level performance field that is a representative sample of investment environments, is a requirement for the risk-adjusted returns to be relied upon. Here are the most common performance measures:

**Sharpe Ratio:**\({ \left( { \bar { r } }_{ P }-{ \bar { r } }_{ f } \right) }/{ { \sigma }_{ P } }\)The average portfolio excess return over sample period is divided by the returns’ standard deviation over the said period, via the Sharpe Ratio which is a measurement of the reward to volatility trade-off.**Treynor Measure:**\({ \left( { \bar { r } }_{ P }-{ \bar { r } }_{ f } \right) }/{ { \beta }_{ P } }\)Treynor’s measure is similar to the Sharpe ratio in the sense that excess return per unit of risk is given, but in the place of total risk, systemic risk is applied instead.**Jensen’s Alpha:**\({ \alpha }_{ P }={ \bar { r } }_{ P }-\left[ { \bar { r } }_{ f }+{ \beta }_{ P }\left( { \bar { r } }_{ M }-{ \bar { r } }_{ f } \right) \right] \)Given the beta of the portfolio and the average market return, this measure is the average return on the portfolio over and above the CAPM prediction.**Information Ratio:**\({ { \alpha }_{ P } }/{ \sigma \left( { e }_{ P } \right) }\)The portfolio’s alpha is divided by the portfolio’s nonsystematic risk (tracking error), via the information ratio. It is a measure of the abnormal return per unit of risk, which could be diversified away in principle by holding a market index portfolio.**Morningstar Risk-Adjusted Return:**\(MRAR\left( \gamma \right) ={ \left[ \frac { 1 }{ T } { \Sigma }_{ t=1 }^{ T }{ \left( \frac { 1+{ r }_{ t } }{ 1+{ r }_{ ft } } \right) }^{ -\gamma } \right] }^{ \frac { 12 }{ \gamma } }-1, \)Where, \(t=1\dots ,\) \(T\) are monthly observations, and \(\gamma\) is the risk aversion measure and usually equal to 2 for an average retail client in mutual funds. Morningstar rating is somehow a harmonic average of excess returns and can be interpreted as the portfolio’s risk-free equivalent return for an investor whose risk aversion is measured by \(\gamma\).

## The \({ M }^{ 2 }\) Performance Measure

It is not easy to interpret the numerical value of the Sharpe ratio despite it being applied to rank portfolio performance. The M2 approach is an equivalent representation of the Sharpe ratio, which similarly focuses on the total volatility as a risk measure. However, an easy-to-interpret differential return in relation to the benchmark index comes about due to its risk adjustment.

In the calculation of \({ M }^{ 2 }\), there is the assumption that for the complete or adjusted portfolio to match a market index’s volatility, a managed portfolio, \(P\), is mixed with a position in T-bills. The adjusted portfolio will be two-thirds invested in the portfolio and a third in bills, in the event that the managed portfolio has 1.5 times the index’s standard deviation.

Therefore, the standard deviation of the adjusted portfolio, \({ P }^{ \ast }\), would be similar to the index. The performance of the market index and portfolio \({ P }^{ \ast }\) may simply be compared by comparing results since their standard deviation is similar.

$$ { M }_{ P }^{ 2 }={ r }_{ P }-{ r }_{ M } $$

## Sharpe’s Ratio Is the Criterion for Overall Portfolios

Assuming we are to evaluate the performance of a portfolio that has been constructed, held for a considerable period of time without variations in its composition in the course of the said period, and having constant means, variances, and covariance for the daily rates of return on all securities.

The following is a utility function that has been applied:

$$ U=E\left( { r }_{ P } \right) -\frac { 1 }{ 2 } A{ \sigma }_{ P }^{ 2 } $$

The risk aversion coefficient is denoted as \(A\). The Sharpe ratio \({ \left[ E\left( { r }_{ P } \right) -{ r }_{ f } \right] }/{ { \sigma }_{ P } }\) is to be maximized, with mean-variance preferences.

## Appropriate Performance Measures in Two Scenarios

For the portfolio’s choice to be evaluated, the first concern is if the portfolio is the client’s exclusive investment vehicle. The question as to whether the portfolio is the entire investment fund or simply a fraction of the overall wealth of the investor will critically determine the appropriate portfolio performance measure.

## The Investor’s Portfolio represents his Entire Risky Investment Fund

Here, it is necessary to only ascertain if the Sharpe measure of the portfolio is the highest. The following three steps can be applied:

- The past security performance is assumed to be a representative of expected performance. This implies that averages and covariance that are the same as those anticipated by the investor will be exhibited by the realized security returns over the investor’s holding period.
- Had he chosen a passive strategy, the held benchmark portfolio should be determined.
- His Sharpe measure or \({ M }^{ 2 }\) should be compared to that of the best portfolio.

The benchmark is the market index or another specific portfolio in case the portfolio is a representation of his entire investment fund. This performance criterion is the actual portfolio’s Sharpe measure versus the benchmark.

## The Investor’s Choice Portfolio is one of the Many Portfolios Combined into a Large Investment Fund

The situation described in this case is where the corporate pension fund is managed by the investor, as a corporate financial manager. The entire fund will be parceled out to various portfolio managers. The individual managers’ performance will then be evaluated for reallocation of the fund and consequent improvement of performance in future.

The basis of the Sharpe ratio is the average excess return against total standard deviation/portfolio risk. Nonsystematic risk will be largely diversified away in the event that various managers are employed, hence systematic risk becomes the risk measure that is relevant. Treynor’s performance metric, taking the ratio of average excess return to the beta, will now be the appropriate performance metric.

Supposing there are two portfolios, \(P\) and \(Q\), as indicated in the following table. The nonsystematic risk will largely be diversified away since is assumed that \(P\) and \(Q\) are two of the many sub-portfolios in the fund. The distance of \(P\) and \(Q\) above the security market line (SML) indicates the value of \({ \alpha }_{ P }\) and \({ \alpha }_{ Q }\) by the SML.

$$ \begin{array}{|lccc|} \hline {} & Portfolio \quad P & Portfolio \quad Q & Market \\ \hline Beta & .90 & 1.6 & 1.0\% \\ Excess\quad Return\left( \bar { r } -{ \bar { r } }_{ f } \right) & 11\% & 19\% & 10\% \\ { Alpha }^{ \ast } & 2\% & 3\% & 0 \\ \hline \end{array} $$

$$ { Alpha }=Excess\quad return-\left( Beta\times Market\quad excess\quad return \right) $$

$$ =\left( \bar { r } -{ \bar { r } }_{ f } \right) -\beta \left( { \bar { r } }_{ M }-{ \bar { r } }_{ f } \right) =\bar { r } -\left[ { \bar { r } }_{ f }+\beta \left( { \bar { r } }_{ M }-{ \bar { r } }_{ f } \right) \right] $$

In this case, the T-line slope gives the appropriate performance criterion and is the trade-off between excess return and beta. The slope for \(P\), \({ T }_{ P }\), is given as:

$$ { T }_{ P }=\frac { { \bar { r } }_{ P }-{ \bar { r } }_{ f } }{ { \beta }_{ P } } $$

Treynor’s measure is similar to \({ M }^{ 2 }\) in the sense that they are both percentages.

## The role of Alpha in Performance Measures

The relationship between the three performance measures discussed can be derived with some algebra as shown in the following table:

$$ \begin{array}{|l|c|c|} \hline {} & Treynor\quad \left( { T }_{ P } \right) & { Sharpe }^{ \ast }\quad \left( { S }_{ P } \right) \\ \hline Relation\quad to\quad alpha & \frac { E\left( { r }_{ P } \right) -\bar { r } _{ f } }{ { \beta }_{ P } } =\frac { { \alpha }_{ P } }{ { { \beta }_{ P } } } +{ T }_{ M } & \frac { E\left( { r }_{ P } \right) -\bar { r } _{ f } }{ { \sigma }_{ P } } =\frac { { \alpha }_{ P } }{ { { \sigma }_{ P } } } +\rho { S }_{ M } \\ Deviation \quad from \quad market & { T }_{ P }^{ 2 }={ T }_{ P }-{ T }_{ M }=\frac { { \alpha }_{ P } }{ { \beta }_{ P } } & { S }_{ P }-{ S }_{ M }=\frac { { \alpha }_{ P } }{ { \beta }_{ P } } -\left( 1-\rho \right) { S }_{ M } \\ performance & {} & {} \\ \hline \end{array} $$

Since superior performance requires a positive alpha, all these measures are consistent. Therefore, this makes alpha the most widely used performance measure.

## Performance Manipulation and the Morningstar Risk-Adjusted Rating

The assumption so far used as the basis of performance evaluation is that: in each period, rates of return are drawn from similar distributions and are independent. In case the managers try to game the system, the crumbling by this assumption will be in an insidious way since their compensation depends on performance.

Since managers observe the unfolding of returns over the course of the period and can accordingly adjust portfolios, they can evaluate the performance over the said period. In the latter part of the period, the rate of return will then come to depend on the rates at the start of the period.

With the exception of the morning star RAR, all performance measures in this chapter can be manipulated, as shown by Ingersoll, Speigel, Goetzmann, and Welch. The Morningstar is a manipulation-proof performance measure (MPPM).

The following requirements must be fulfilled by the manipulation-proof performance measure (MPPM):

- To rank a portfolio, a single-value score should be produced by the measure;
- There should be no reliance on the portfolio’s dollar value by the score;
- Deviating from the benchmark portfolio should not be used by an uninformed investor to improve the expected score; and
- There should be consistency in the measure and the standard financial market equilibrium conditions.

All these necessities are fulfilled by the Morningstar RAR making it an MPPM.

## Realized Returns versus Expected Returns

Neither the original expectations of the portfolio manager nor whether the expectations are sensible is known by the evaluator during portfolio evaluation.

A significant level of a performance measure to determine whether the ability is reliably indicated should, therefore, be determined so as not to make mistakes.

Take for example John Mitt who is a portfolio manager whose portfolio has an alpha of 20 bps per month, accounting for a hefty 2.4% per annum, prior to compounding. Assume that there is a constant mean, beta, and alpha for his portfolio’s return distribution, in line with the usual treatment of performance measure.

Let the portfolio beta be 1.2 with a residual monthly deviation of 2%. The following equation represents the portfolio systematic variance having a market index standard deviation of 6.5% per month:

$$ { \beta }^{ 2 }{ \sigma }_{ M }^{ 2 }={ 1.2 }^{ 2 }\times { 6.5 }^{ 2 }=60.84 $$

Therefore, between his portfolio and the market index, the correlation coefficient is:

$$ \rho ={ \left[ \frac { { \beta }^{ 2 }{ \sigma }_{ M }^{ 2 } }{ { \beta }^{ 2 }{ \sigma }_{ M }^{ 2 }+{ \sigma }^{ 2 }\left( e \right) } \right] }^{ \cfrac { 1 }{ 2 } }=\left[ \frac { 60.84 }{ 60.84+4 } \right] =0.97 $$

This indicates that the portfolio is well diversified.

The portfolio excess returns are regressed on the market index so as to estimate Mitt’s portfolio alpha from the security characteristic line (SCL). Assuming ideally that precisely true parameters are yielded by the regression, then from the \(N\) months our SCL estimates are:

$$ \hat { \alpha } =0.2\%,\quad \quad \quad \hat { \beta } =1.2,\quad \quad \quad \hat { \sigma } \left( e \right) =2\% $$

In the SCL regression, the following is a standard error of the alpha estimate:

$$ \hat { \alpha } \left( \alpha \right) =\frac { \hat { \sigma } \left( e \right) }{ \sqrt { N } } $$

The number of observations is denoted as \(N\), the nonsystematic risk’s sample estimate is \(\hat { \sigma } \left( e \right) \), and then the \(t\)-statistic alpha estimate is:

$$ t\left( \hat { \alpha } \right) =\frac { \hat { \alpha } }{ \hat { \alpha } \left( \alpha \right) } =\frac { \hat { \alpha } \sqrt { N } }{ \hat { \sigma } \left( e \right) } $$

If a significant level of 5% is required, then we need a \(t\left( \hat { \alpha } \right)\) value of 1.96 in case \(N\) is large. \(\hat { \alpha } =0.2\) and \(\hat { \sigma } \left( e \right)=2\). Therefore:

$$ 1.96=\frac { 0.2\sqrt { N } }{ 2 } $$

$$ N=384 \quad months $$

# Performance Measurement for Hedge Funds

Supposing an investor, whose portfolio was earlier described, is satisfied with his well-diversified mutual fund. Supposing that he stumbles upon information on hedge funds. The funds have a tendency of concentrating on opportunities offered by temporarily misplaced securities, with less concern shown for broad diversification.

The funds are therefore alpha driven and are considered best as possible additions to core positions in more traditional portfolios established for diversification. The actively managed portfolio’s information ratio is the crucial data for this mixture. Therefore, this ratio is made from the performance of the active fund.

Assuming that \(H\) is the portfolio established by the hedge fund, and \(M\) is the investor’s baseline passive portfolio. Therefore, the following relationship gives the optimal position \(H\) in the overall portfolio, denoted as \({ P }^{ \ast }\):

$$ { w }_{ H }=\frac { { w }_{ H }^{ 0 } }{ 1+\left( 1-{ \beta }_{ H } \right) { w }_{ H }^{ 0 } } ;{ w }_{ H }^{ 0 }=\frac { \frac { { \alpha }_{ H } }{ { \sigma }^{ 2 }\left( { e }_{ H } \right) } }{ \frac { E\left( { R }_{ H } \right) }{ { \sigma }_{ M }^{ 2 } } } \quad \quad \quad \quad \quad I $$

The information ratio, \(\frac { { \alpha }_{ H } }{ { \sigma }\left( { e }_{ H } \right) } \), determines the improvement in the Sharpe measure in the event that the hedge fund is optimally combined with the baseline portfolio using equation \(I\), in accordance with:

$$ { S }_{ P\ast }^{ 2 }={ S }_{ M }^{ 2 }+{ \left[ \frac { { \alpha }_{ H } }{ { \sigma }\left( { e }_{ H } \right) } \right] }^{ 2 }\quad \quad \quad \quad II $$

The appropriate performance measure for the hedge fund is its Information Ratio (IR), according to equation \(II\) above. The IR of portfolios \(P\) and \(Q\) is computed as:

$$ { IR }_{ P }=\frac { { \alpha }_{ P } }{ \sigma \left( { e }_{ P } \right) } =\frac { 1.63 }{ 2.02 } =0.81;{ IR }_{ Q }=\frac { 5.38 }{ 9.81 } =0.54 $$

The following are some of the challenges posed by a practical evaluation of hedge funds:

- A rapid change may occur in a hedge fund’s risk profile. It, therefore, becomes hard for exposure to be evaluated at a given time due to this lack of stability;
- There is a tendency of investing in illiquid assets in the hedge funds;
- Strategies capable of providing apparent profits over long time horizons are often pursued by most hedge funds; however, a specific hedge fund is exposed to losses that may be infrequent but severe;
- Hedge funds have ample latitude to change their risk profiles and thus considerable ability of conventional performance measures manipulation; and
- The survivorship bias becomes a major consideration in the event that hedge funds are evaluated as a group. This can be attributed to the fact that, in this industry, turnover is far higher as compared to investment companies.

# Market Timing

Depending on whether the market index is expected to outperform the risk-free asset, market timing in its pure form involves shifting funds between a market index portfolio and the risk-free asset.

Assuming that an investor only holds the market index portfolio and T-bills, then the beta of the portfolio would be constant if we had a constant market weight, hence making the SCL plotted as a straight line.

On the other hand, the market could be correctly timed by the investor, and funds shifted into it in periods where the market does well. When the \({ r }_{ M }\) is higher, the portfolio beta and the SCL slope will be higher. This results in a curved line which can be estimated by adding a squared term to the usual linear index model:

$$ { r }_{ P }-{ r }_{ f }=a+b\left( { r }_{ M }-{ r }_{ f } \right) +c{ \left( { r }_{ M }-{ r }_{ f } \right) }^{ 2 }+{ e }_{ P } $$

Where \({ r }_{ P }\) is portfolio return, and \(a\), \(b\), and \(c\), are regression analysis estimates. The evidence of timing ability becomes obvious in case \(c\) turns out to be positive. This is due to the fact that this term will make the characteristic line steeper.

According to another similar methodology introduced by Henriksson and Merton, only two values are taken by the portfolio beta, namely:

- If the market is expected to perform well, a large value;
- Otherwise, a small value.

The regression form of the portfolio characteristic line under this scheme is given as:

$$ { r }_{ P }-{ r }_{ f }=a+b\left( { r }_{ M }-{ r }_{ f } \right) +c{ \left( { r }_{ M }-{ r }_{ f } \right) }D+{ e }_{ P } $$

Where \(D\) is dummy variable equal to 1 for \({ r }_{ M }>{ r }_{ f }\), 0 otherwise.

A regression of the excess returns of portfolios \(P\) and \(Q\) on the excess returns of \(M\) and the square of these returns gives:

$$ { r }_{ P }-{ r }_{ f }={ a }_{ P }+{ b }_{ P }\left( { r }_{ M }-{ r }_{ f } \right) +{ c }_{ P }{ { \left( { r }_{ M }-{ r }_{ f } \right) } }^{ 2 }+{ e }_{ P } $$

$$ { r }_{ Q }-{ r }_{ f }={ a }_{ Q }+{ b }_{ Q }\left( { r }_{ M }-{ r }_{ f } \right) +{ c }_{ Q }{ { \left( { r }_{ M }-{ r }_{ f } \right) } }^{ 2 }+{ e }_{ Q } $$

It is important to note that for performance evaluation, most effects of portfolio composition change to confound the more conventional mean-variance measures that can be captured by expanded regressions.

## Potential Value of Market Timing

Let perfect market timing be defined as the ability to tell if the S&P 500 at the beginning of each year will outperform the rolling over 1-month T-bills all year-round strategy. All the funds are shifted by the market timer, at the start of each year, either into cash equivalents or equities, depending on whichever is to perform better.

## Valuing Market Timing as a Call Option

Being cognizant of the fact that perfect foresight is equivalent to holding a call option on the equity portfolio is the key to market timing ability valuation. The investment by the perfect timer, either in the risk-free asset or equity portfolio, is 100% depending on the one capable of providing higher returns.

Assuming that currently, the market index is at \({ S }_{ 0 }\) having a call option with an excise price of \(X={ S }_{ 0 }\left( 1+{ r }_{ f } \right) \). If over the coming periods, bills are outperformed by the market, \({ S }_{ T }\) will surpass \(X\), whereas otherwise, it will be less than \(X\).

Consider a payoff to a portfolio made up of this option and \({ S }_{ 0 }\) dollars invested in bills:

$$ \begin{array}{|c|c|c|} \hline {} & { S }_{ T }<X & { S }_{ T }\ge X \\ \hline Bills & { S }_{ 0 }\left( 1+{ r }_{ f } \right) & { S }_{ 0 }\left( 1+{ r }_{ f } \right) \\ Call & 0 & { S }_{ T }-X \\ Total & { S }_{ 0 }\left( 1+{ r }_{ f } \right) & { S }_{ T } \\ \hline \end{array} $$

When the market is bearish, the risk-free returns are paid by the portfolio, and when bullish, it pays the market return and beats bills. Such is a perfect market timer portfolio.

The final value of the T-bill investment is the exercise price of the perfect-timer call option on $1. This becomes \($1\times { e }^{ rt }\) when continuous compounding is applied. The Black-Scholes formula reduces to the following expression when this exercise price is applied to it for the call option value:

$$ MV\left( Perfect\quad timer\quad per\quad $\quad of\quad assets \right) = $$

$$ C=2N\left( \frac { 1 }{ 2 } { \sigma }_{ M }\sqrt { T } \right) -1 $$

So far, annual forecasts have been assumed, i.e.,\(T = 1\) year. There would be a dramatic increment on the value of the timer if every month the correct choice could be made by the timer rather than yearly. There will be an increment in the value of the services without bound, as the frequency of such perfect predictions increases without bound.

## The Value of Imperfect Forecasting

The overall proportion of the correct forecasts is rarely the appropriate measure of market forecasting ability. It is necessary to examine the proportion of bull markets \(\left( { r }_{ M }<{ r }_{ f } \right) \) correctly forecasted and the proportion of bear markets \(\left( { r }_{ M }>{ r }_{ f } \right) \) correctly forecasted.

Let the proportion of the correct forecasts of bull markets be denoted as \({ P }_{ 1 }\) and the proportion of bear markets be denoted as \({ P }_{ 2 }\). Therefore, the correct measure of timing ability is given as:

$$ { P }_{ 1 }+{ P }_{ 2 }-1 $$

According to Merton, the following equation gives the market value of the services of an imperfect timer, \(MV\), in case the overall accuracy is measured by the statistic \( { P }_{ 1 }+{ P }_{ 2 }-1\), and if the timing is imperfect:

$$ MV\left( Imperfect\quad timer \right) =\left( { P }_{ 1 }+{ P }_{ 2 }-1 \right) \times C=\left( { P }_{ 1 }+{ P }_{ 2 }-1 \right) =\left[ 2N\left( \frac { 1 }{ 2 } { \sigma }_{ M }\sqrt { T } \right) -1 \right] \quad \quad III $$

The performance of an imperfect timer can be stimulated by drawing random numbers to capture the likelihood that an incorrect forecast will sometimes be issued by the timer.

A case whereby the timer fails to fully shift from asset to asset is a further variation on the valuation of market timing. Assuming that a fraction \(\omega \) of the portfolio between T-bills and equities is shifted by the timer, then equation \(III\) can be generalized as:

$$ MV\left( Imperfect\quad timer \right) =\omega \left( { P }_{ 1 }+{ P }_{ 2 }-1 \right) \left[ 2N\left( { \sigma }_{ M }\sqrt { T } \right) -1 \right] $$

# Style Analysis

According to William Sharpe who introduced style analysis, funds returns are regressed on indexes that represent a range of asset classes. Therefore, the implicit allocation of the fund to that style is measured by the regression coefficient on each index.

The percentage of return variability attributable to style or asset allocation is measured by the R-Square of the regression. The remainder of return variability will either be attributable to security selection or to the market timing by periodically changing the weights of the asset class.

We can attribute the proportion of return variability, not explained by asset allocation, to security selection within asset classes, and timing showing up as periodic changes in allocation. Based on CAPM’s SML, an alternative to performance evaluation is provided by style analysis.

Only one comparison portfolio is applied by the SML, namely the broad market index. For style analysis, a tracking portfolio is constructed more freely from various specialized indexes.

The higher R-Square of the regression with only a single factor can be explained relative to the style regression, deploying a number of stock indexes, say 6. Extra constraints are imposed by the style analysis of the regression coefficients, forcing them to be positive and sum to 1.

Relative to the theoretically prescribed passive portfolio, a better representation of performance is the SML benchmark, as it is the broadest available market index. Conversely, the strategy that tracks the activity of the fund and evaluates performance relative to this strategy is revealed by style analysis.

## Style analysis and Multifactor Benchmarks

Assuming that a superior performance is exhibited by growth-index portfolios relative to a mutual fund benchmarks over some period of measurement, then the superior performance is eliminated from the estimated alpha value of the portfolio by adding this growth-index in a style analysis.

Today, the conventional performance benchmark is a four-factor employing the three Fama-French factors augmented by a momentum factor. The application of any benchmark other than the single-index benchmark of the fund is only legitimate under the assumption that part of the alternative passive strategy of the fund is the factor portfolios in question.

The attempt of portfolio managers to uncover the decisions contributing to superior performance is shown in the section on performance attribution. The starting point of this performance attribution procedure is the benchmark allocation to various indexes attributing performance to asset allocation based on deviation from actual benchmark allocations.

## Style Analysis in Excel

Excel’s solver can be applied in conducting style analysis. In this strategy, the rate of returns of funds is regressed on those of a number of style portfolios. Assuming that three style portfolios labeled 1-3 are chosen, then alpha and some three slope coefficients corresponding to each style index will be the coefficients in our style regression.

The sensitivity of the fund’s performance in following the return of each passive style portfolio is revealed by the slope coefficients. The noise is represented by the residuals from this regression, \(e\left( t \right) \).

To conduct a style analysis applying the solver, the arbitrary coefficients are first used to calculate the time series of residuals from the style regression in accordance to:

$$ e\left( t \right) =R\left( t \right) -\left[ \alpha +{ \beta }_{ 1 }{ R }_{ 1 }\left( t \right) +{ \beta }_{ 2 }{ R }_{ 2 }\left( t \right) +{ \beta }_{ 3 }{ R }_{ 3 }\left( t \right) \right] $$

We have:

\(R\left( t \right)\)=Excess return on the evaluated fund for date \(t\);

\({ R }_{ i }\left( t \right) \)=Excess return on the \(i\)th style portfolio (\(i=1,2,3\));

\(\alpha\)=Abnormal performance of the fund over the sample period;and

\({ \beta }_{ 1 }\)=Beta of the fund on the \(i\)th style portfolio.

The time series of residuals from the regression equation is yielded with those arbitrary coefficients. We then square each residual and add up the squares. Four constraints should also be added to the optimization.

The three style coefficients and the estimate of the fund’s unique abnormal performance will be given by the solver’s output.

# Performance Attribution Procedures

Asset allocation decisions made by portfolio managers are constantly broad-brush while making security allocation decisions that are more detailed, within asset classes. The starting point of the attribution studies is the broadest asset allocation choices and then focuses progressively on the ever-finer details of portfolio choice.

The sum of contributions to the performance of a series of decisions created at various levels of the constriction process of the portfolio is the difference between a managed portfolio’s performance and the benchmark.

The difference in returns between a managed portfolio, \(P\), and a selection benchmark portfolio, \(B\), called the Bogey, is explained by the attribution method. The rate of return for a Bogey portfolio set to have fixed weights in each asset class is given by:

$$ { r }_{ B }=\sum _{ i=1 }^{ n }{ { w }_{ Bi } } { r }_{ Bi } $$

Where the weight of the Bogey in asset class \(i\) is given as \({ w }_{ Bi }\), and the return of benchmark portfolio of that class over the evaluation period is given as \({ r }_{ Bi }\). The basis of choosing the weights,\({ w }_{ Pi }\), by the managers in each class is their capital expectations.

The portfolio of the securities within each class is chosen on the basis of their security analysis returns,\({ r }_{ Pi }\), over the period of the evaluation. The managed portfolio’s return is, therefore:

$$ { r }_{ P }=\sum _{ i=1 }^{ n }{ { w }_{ Pi } } { r }_{ Pi } $$

Therefore:

$$ { r }_{ P }-{ r }_{ B }=\sum _{ i=1 }^{ n }{ { w }_{ Pi } } { r }_{ Pi }-\sum _{ i=1 }^{ n }{ { w }_{ Bi } } { r }_{ Bi }=\sum _{ i=1 }^{ n }{ \left( w_{ Pi }{ r }_{ Pi }-w_{ Bi }{ r }_{ Bi } \right) } $$

We also have that:

$$ \begin{array}{c} Contribution\quad from\quad asset\quad allocation\quad \left( { w }_{ Pi }-{ w }_{ Bi } \right) { r }_{ Bi } \\ +Contribution\quad from\quad security\quad selection\quad { w }_{ Pi }\left( { r }_{ Pi }-{ r }_{ Bi } \right) \\ \hline =Total\quad contribution\quad from\quad asset\quad class\quad i\quad \left( w_{ Pi }{ r }_{ Pi }-w_{ Bi }{ r }_{ Bi } \right) \end{array} $$

## Asset Allocation Decisions

For the impact of asset allocation choice of a manager to be isolated, the performance of a hypothetical portfolio invested in the indexes for each market having particular weights should be evaluated. The impact of the shift away from the benchmark, say 70/40/20, weights are measured without allowing for any effects that could be attributed to active management of the selected securities within each market.

Overweighting investments in markets that turn out to perform well will achieve superior performance relative to the Bogey. This asset allocation contribution is equated to the sum of all markets of the excess weight in each market multiplied by the return of the market index.

# Practice Questions

1) James Harrington is retail investor. Assume that he is interested in a stock that pays an annual dividend of $10 selling for $135 at the present moment. He decides to first buy the stock, then collect the $10 dividend, and at the end of the year sell the stock in exchange for $150. However, instead of selling his share, Harrington wishes to purchase a second share when the first year elapses and hold each of the shares until the end of the second year and then sell each of the shares at $170. Compute the geometric average/time-weighted return on the investment.

- 19.26%
- 24.07%
- 38.52%
- 48.15%

The correct answer is **A**.

$$ { r }_{ 1 }=\frac { 150+10-135 }{ 135 } =0.1852 $$

$$ { r }_{ 2 }=\frac { 170+10-150 }{ 150 } =0.2 $$

Then:

$$ { r }_{ G }={ \left( 1+{ r }_{ 1 }\times 1+{ r }_{ 2 } \right) }^{ \cfrac { 1 }{ 2 } }-1 $$

$$ { r }_{ G }={ \left( 1.1852\times 1.2 \right) }^{ \cfrac { 1 }{ 2 } }-1=19.26\% $$