- Differentiate between time-weighted and dollar weighted returns of a portfolio, and describe their appropriate uses.
- Describe and distinguish between risk-adjusted performance measures, such as Sharpe's measure, Treynor's measure, Jensen's measure (Jensen's alpha), and information ratio.
- Describe the uses for the Modigliani-squared and Treynor's measure in comparing two portfolios, and the graphical representation of these measures.
- Determine the statistical significance of a performance measure using standard error and the t-statistic.
- Describe style analysis.
- Explain the difficulties in measuring the performance of hedge funds.
- Explain how changes in portfolio risk levels can affect the use of the Sharpe ratio to measure performance.
- Describe techniques to measure the market timing ability of fund managers with a regression and with a call option model, and compute return due to market timing.
- Describe and apply performance attribution procedures, including the asset allocation decision, sector and security selection decision, and the aggregate contribution.

- Video Lecture
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- PDFs
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- List of chapters

- Time – Weighted Returns (TWR)
- Time – Weighted Returns (TWR) – Example
- Dollar – Weighted Returns (DWR)
- TWR Versus DWR
- Adjusting Returns For Risk
- The M
- The M
- The Treynor Ratio – Representation
- The Treynor Ratio
- The Treynor Ratio And T
- The Role Of Alpha In Performance Measures
- Style Analysis
- Performance Measurement With Changing Portfolio Composition
- Market Timing
- Market Timing Example
- Valuing Market Timing As A Call Option
- Performance Attribution Procedures
- Practice Questions

- The time-weighted rate of return (TWR) is the geometric mean return which measures the compound growth rate of a portfolio. It is that constant rate of return over the relevant time horizon which when applied, would give the same cumulative return as obtained when the effect of all individual returns are multiplied over that time horizon. Hence, if TWR over 𝑛 periods is given by 𝑟
_{g}, then

- TWR breaks up a portfolio’s returns into separate intervals (known as sub-periods) based on incoming and outgoing cash flows. Hence, this measure eliminates the distortions on rates of growth created by money inflows and outflows which makes it appropriate for comparing the returns of investment managers.

- The dollar-weighted rate of return (DWR) calculates the rate of return of a portfolio, taking into account the timing of cash flows. It is same as the internal rate of return (IRR). It is calculated by finding the rate of return at which the discounted values of all cash outflows is equal to the discounted value of all cash inflows.
- The DWR return is also time-weighted, because cash inflows and outflows at different time intervals affect the overall DWR.
- DWR is more suitable to demonstrate the performance of their own funds to clients.

- If funds are contributed to an investment portfolio just before a poor return period of relatively poor portfolio performance, DWR will be less. But, if funds are contributed to a portfolio just before a high return period, DWR will be more. As discussed earlier, use of the time-weighted return removes these distortions, providing a better measure of a manager’s ability to select investments over the period.
- DWR is more appropriate to judge performance for an investor who has complete control over cash flows into and out of an investment account. But in the investment management industry, the timing of cash inflows and outflows which may not be in the manager’s control. Hence TWR is preferable to judge performance for a portfolio manager.

- Sharpe ratio

\( \frac{\bar{r}_P – \bar{r}_f}{\sigma_P} \)

𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 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.

- Treynor measure

\( \frac{\bar{r}_P – \bar{r}_f}{\beta_P} \)

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

- Jensen’s alpha

\( \alpha_{\rho} = \bar{r}_{\rho} – \left[ \bar{r}_f + \beta_{\rho} \left( \bar{r}_M – \bar{r}_f \right) \right] \)

𝐽𝑒𝑛𝑠𝑒𝑛’𝑠 𝑎𝑙𝑝ℎ𝑎 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.

- Information ratio

\( \frac{\alpha_{\rho}}{\sigma(e_{\rho})} \)

The 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 divides the alpha of the portfolio by the nonsystematic risk of the portfolio, called “tracking error” in the industry. It measures abnormal return per unit of risk that in principle could be diversified away by holding a market index portfolio.

*Note – The industry jargon tends to be a little loose concerning this topic. Some define the information ratio as excess return‐rather than alpha‐per unit of nonsystematic risk, using appraisal ratio to refer to the ratio of alpha to nonsystematic risk.*

Each performance measure has some appeal. But these competing measures do not necessarily provide consistent assessments of performance, because the risk measures used to adjust returns differ substantially. Hence, it is important to consider the circumstances in which each of these measures is appropriate. For example, consider the following data for a particular

\( \text{Sharpe Ratio} = \frac{\bar{r}_P – \bar{r}_f}{\sigma_P} = \frac{0.35 – 0.06}{0.42} = 0.690 \)

\( \text{Treynor Measure} = \frac{\bar{r}_P – \bar{r}_f}{\beta_P} = \frac{0.35 – 0.06}{1.2} = 0.2417 \)

\( \text{Jensen’s Alpha} = \alpha_{\rho} = \bar{r}_{\rho} – \left[ \bar{r}_{f} + \beta_{\rho} \left( \bar{r}_{M} – \bar{r}_{f} \right) \right] = 0.35 – \left[ 0.06 + 1.2 \times (0.28 – 0.06) \right] = 0.26 \)

\( \text{Information Ratio} = \frac{\alpha_{\rho}}{\sigma(e_{\rho})} = \frac{0.26}{0.18} = 0.144 \)

- For the market portfolio, if calculations are done, then Sharpe ratio will be 𝑆
_{M}= 0.733 and - Treynor measure will be 𝑇
_{M}= 0.220 - Hence based on Sharpe ratio of 0.690 versus 0.733, portfolio P underperforms the market, and based on Treynor measure of 0.2417 versus 0.220, portfolio P outperforms the market Jensen’s alpha will of course be 0 for the market and 𝐼𝑅 (information ratio) can’t be defined as denominator (tracking error) is 0.
- While the Sharpe ratio can be used to rank portfolio performance, its numerical value is not easy to interpret. For this example, after performing the calculations, it was found that 𝑆
_{P}= 0.690 and 𝑆_{M}= 0.733 which suggests that portfolio P underperformed the market index. But is a difference of 0.04 in the Sharpe ratio economically meaningful ? Rates of return are easy to compare, but these values are difficult to interpret. - This table shows the definition of the various performance measures and the situations in which each is the relevant performance measure.

- An equivalent representation of Sharpe’s ratio is known as the 𝑀
^{2}measure (for Modigliani- squared). Like the Sharpe ratio, 𝑀^{2}focuses on total volatility as a measure of risk, but its risk adjustment leads to an easy-to-interpret differential return relative to the benchmark index.

- To compute 𝑀
^{2}, it is assumed that an active portfolio, 𝑃, is mixed with a position in T-bills so that the resulting “adjusted” portfolio matches the volatility of a passive market index such as the 𝑆&𝑃 500. - For example, if the active portfolio has 1.5 times the standard deviation of the index, it would be mixed with T-bills using proportions of 1/3 in bills and 2/3 in the active portfolio. The resulting adjusted portfolio, which can be termed as 𝑃
^{∗}, would then have the same standard deviation as the index. If the active portfolio had lower standard deviation than the index, it would be leveraged by borrowing money and investing the proceeds in the portfolio, i.e. weight of T-bills would be negative. - Because the market index and portfolio 𝑃
^{∗}have the same standard deviation, their performance can be compared simply by comparing returns. This is the 𝑀^{2}measure for portfolio 𝑃

\( M_{P}^2 = r_{P}^{\star} – r_{M} \)

- Consider the data from the previous table again –

P has a standard deviation of 42% versus a market standard deviation of 30%. Therefore, the adjusted portfolio 𝑃^{∗} would be formed by mixing bills and portfolio P with the following weights –

\( \frac{30}{42} = 0.714 \text{ in } P \)

\text{and}

\( 1 – 0.714 = 0.286 \text{ in bills.} \)

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

Thus portfolio 𝑃 has an 𝑀^{2} measure of

\( M_P^2 = r_P^\star – r_M = (26.7\% – 28\%) = -1.3\% \)

- A graphical representation of 𝑀
^{2}appears in this figure.

A downward movement is done on the capital allocation line corresponding to portfolio 𝑃 (by mixing 𝑃 with T-bills) until the standard deviation of the adjusted portfolio is reduced to match that of the market index.

𝑀^{2} is then the vertical distance (the difference in expected returns) between portfolios 𝑃^{∗} and 𝑀. It can be seen from the figure that 𝑃 will have a negative 𝑀^{2} when its capital allocation line is less steep than the capital market line, that is, when its Sharpe ratio is less than that of the market index.

- When employing a number of managers, nonsystematic risk will be largely diversified away, so systematic risk becomes the relevant measure of risk. The appropriate performance metric when evaluating potential components of the full risky portfolio is the Treynor measure.

- As an example, consider the properties of portfolios 𝑃 and 𝑄 as given in this table.

In this figure, 𝑃 and 𝑄 are plotted in the expected return-beta plane (rather than the expected return standard deviation), because it is assumed that 𝑃 and 𝑄 are two of many subportfolios in the fund, and thus that nonsystematic risk will be largely diversified away.

The security market line (𝑆𝑀𝐿) shows the value of α_{P} and a α_{Q} as the distance of 𝑃 and 𝑄 above the 𝑆𝑀𝐿.

- If 𝑤
_{Q}is invested in 𝑄 and 𝑤_{F}= 1 — 𝑤_{Q}is invested in T-bills, the resulting portfolio, 𝑄^{∗}, will have alpha and beta values proportional to 𝑄’𝑠 alpha and beta scaled down by 𝑤_{Q}:

α_{Q}* = 𝑤_{Q}α_{Q}

𝛽_{Q}* = 𝑤_{Q}𝛽_{Q}

- Thus, all portfolios such as 𝑄
^{∗}, generated by mixing 𝑄 with T-bills, plot on a straight line from the origin through 𝑄. It is known as the T-line for the Treynor measure, which is the slope of this line. - The previous figure shows the T-line for portfolio 𝑃 as well. 𝑃 has a steeper T-bills despite its lower alpha. 𝑃 is a better portfolio after all. For any given beta, a mixture of 𝑃 with will give a better alpha than a mixture of 𝑄 with T-bills.
- Equalizing Beta – Suppose 𝑄 is mixed with T-bills to create a portfolio 𝑄
^{∗}with a beta equal to that of 𝑃. 𝑤_{Q}can be obtained by solving

\( \beta_Q^\star = w_Q \beta_Q \Rightarrow 0.9 = w_Q \times 1.5 \Rightarrow w_Q = \frac{9}{16} \)

Portfolio 𝑄^{∗} therefore has an alpha of α_{Q} = 9⁄_{16} × 3% = 1.69%

which is less than that of 𝑃

- The slope of the 𝑇-line, giving the trade-off between excess return and beta, is the appropriate performance criterion in this case. The slope for 𝑃, denoted by 𝑇
_{P}, is given by

\( T_P = \frac{\bar{r}_P – \bar{r}_f}{\beta_P} \)

- Like 𝑀
^{2}, Treynor’s measure is a percentage. If the market’s excess return is subtracted from Treynor’s measure, it will be same as the difference between the return on the 𝑇_{P}line in the previous figure and the 𝑆𝑀𝐿, at the point where 𝛽 = 1. - Analogous to 𝑀
^{2}, this difference can be called 𝑇^{2}. - 𝑀
^{2}and 𝑇^{2}are as different as Sharpe’s measure is from Treynor’s. They may well rank portfolios differently

- Because superior performance requires positive alpha, it is the most widely used performance measure. However, while positive alpha is necessary, it is not sufficient to guarantee that a portfolio will outperform the index. Taking advantage of mispricing means departing from full diversification, which entails a cost in terms of nonsystematic risk. A mutual fund can achieve a positive alpha, yet, increase its volatility enough that its Sharpe ratio will actually fall.

- The correlation coefficient between portfolio and the market index is given by

\( \rho = \left[ \frac{\beta^2 \sigma_M^2}{\beta^2 \sigma_M^2 + \sigma^2(e)} \right] \)

If this measure comes out to be close to 1, then the portfolio is quite well diversified.

- To estimate the portfolio alpha from the security characteristic line (𝑆𝐶𝐿), portfolio excess returns is regressed on the market index. The standard error of the alpha estimate in the SCL regression is approximately.

where

𝑁 is the number of observations, and

\( \hat{\sigma}(e) \) is the sample estimate of nonsystematic risk.

The t-statistic for the estimate of alpha is then

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

EXAMPLE – For a significance level of 5%, with a large number of observations, 𝑡 α^ should be greater than or equal to 1.96 to reject the null. Assume that α^ = 0.5% and 𝜎^ 𝑒 = 2% and that there are 180 monthly observations, Then

\( t(\hat{\alpha}) = \frac{\hat{\alpha} \sqrt{N}}{\hat{\sigma}(e)} = \frac{0.005 \sqrt{180}}{0.04} \approx 1.68 \)

Since 1.68 is not greater than or equal to 1.96, so null can’t be rejected, which means that α is not statistically significant, and it can’t be concluded that portfolio manager has superior return generating ability.

It can be seen that seen that a large number of observations will be required to demonstrate statistical significance, which will in turn demonstrate ability rather than luck. For example, if 𝑁 = 250 in this example, then 𝑡 α^ would be

- Style analysis was introduced by Nobel laureate William Sharpe as a tool to systematically measure the exposures of managed portfolios. The popularity of the concept was aided by a well-known study concluding that 91.5% of the variation in returns of 82 mutual funds could be explained by the funds’ asset allocation to bills, bonds, and stocks. While later studies have taken issue with the exact interpretation of these results, there is widespread agreement that asset allocation is responsible for a high proportion of the variation across funds in investment performance.
- Sharpe’s idea was to regress fund returns on indexes representing a range of asset classes. The regression coefficient on each index would then measure the fund’s implicit allocation to that “style”. Because funds are barred from short positions, the regression coefficients are constrained to be either zero or positive and to sum to 100%, so as to represent a complete asset allocation. The R-squared of the regression would then measure the percentage of return variability attributable to style choice rather than security selection. The intercept measures the average return from security selection of the fund portfolio. It therefore tracks the average success of security selection over the sample period.

- As seen earlier, the volatility of stock returns requires a very long observation period to determine performance levels with any precision, even if portfolio returns are distributed with constant mean and variance. This problem is compounded when portfolio return distributions are constantly changing. It may be acceptable to assume that the return distributions of passive strategies have constant mean and variance when the measurement interval is not too long. However, return distributions of active strategies change by design, as the portfolio manager updates the portfolio in accordance with ongoing financial analysis. In such a case, 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. In the first year, the portfolio manager executes a low-risk strategy and realizes an (annualized) mean excess return of 1% and standard deviation of 2%. This makes for a Sharpe ratio of 0.5, which beats the passive strategy. Over the next year, the manager decides that a high-risk strategy is optimal and achieves an annual mean excess return of 9% and standard deviation of 18%. Here, again, the Sharpe ratio is 0.5. Over the 2-year period, our manager consistently maintains a better-than- passive Sharpe measure.

- This figure shows a pattern of (annualized) quarterly returns that are consistent with our description of the manager’s strategy of two years. In the first four quarters the excess returns are —1%, 3%, —1 %, and 3%, making for an average of 1 % and standard deviation of 2%. In the next four quarters the excess returns are —9%, 27%, —9%, and 27%, making for an average of 9% and standard deviation of 18%. Thus both years exhibit a Sharpe measure of 0.5. However, the mean and standard deviation of the eight quarterly returns are 5% and 13.42%, respectively, making for a Sharpe measure of only 0.37, apparently inferior to the passive strategy!
- Here, the problem was that the shift of the mean from the first four 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. It can be concluded that for actively managed portfolios it is necessary to keep track of portfolio composition and changes in portfolio mean and risk.

- In its pure form, market timing involves shifting funds between a market-index portfolio and a safe asset, depending on whether the market index is expected to outperform the safe asset. In practice, most managers do not shift fully between T- bills and the market. There are partial shifts into the market when it is expected to perform well.
- To simplify, 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 SCL would plot as a straight line with slope .6, as in Panel A of this figure. If, however, the investor could correctly time the market and shift funds into it in periods when the market does well, the SCL would plot as in Panel B of this figure. If bull and bear markets can be predicted, the investor will shift more into the market when the market is more likely to go up.
- The portfolio beta and the slope of the SCL will be higher when 𝑟
_{M}is higher, resulting in the curved line that appears in Panel B of this figure.

- The curved line can be represented by adding a squared term to the usual linear index model, as proposed by Treynor and Mazuy –

\( r_P – r_f = a + b (r_M – r_f) + c (r_M – r_f)^2 + e_P \)

where

𝑟_{P} 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 𝑟_{M} — 𝑟_{ƒ}, is larger.

Treynor and Mazuy estimated this equation for a number of mutual funds, but found little evidence of timing ability.

- A similar but simpler methodology was proposed by Henriksson and Merton. These authors suggested that the beta of the portfolio takes only two values: a large value if the market is expected to do well and a small value otherwise. Under this scheme the portfolio characteristic line appears as shown in Panel C here.

- Such a two sloped line appears in regression form as

𝑟_{P} — 𝑟_{ƒ}, = 𝑎 + 𝑏(𝑟_{M} — 𝑟_{ƒ}, ) + 𝑐 (𝑟_{M} — 𝑟_{ƒ}) 𝐷+ 𝑒_{P}

where 𝐷 is a dummy variable that equals 1 when 𝑟_{M} > 𝑟_{ƒ}, 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. They also found little evidence of market timing ability.

- The example in the next page illustrates the inadequacy of conventional performance evaluation techniques that assume constant mean returns and constant risk. The market timer constantly shifts beta and mean return, moving into and out of the market. So market timing presents another instance in which portfolio composition and risk change over time, complicating the effort to evaluate performance. Whereas the expanded regression captures this possibility, the simple SCL does not.
- The important point for performance evaluation is that expanded regressions can capture many of the effects of portfolio composition change that would confound more conventional mean-variance measures.

- Consider the regression estimates from the single variable regression reported in this table for two portfolios 𝑃 and 𝑄–

- Portfolio 𝑃 shows no evidence of attempted timing: Its timing coefficient, 𝑐, is estimated to be zero. It is not clear whether this is because no attempt was made at timing or because any effort to time the market was in vain and served only to increase portfolio variance unnecessarily.

- The results for portfolio 𝑄, however, reveal that timing has, in all likelihood, been attempted. Here the coefficient, 𝑐, is positive, with an estimated value of .10. The evidence thus suggests successful timing, offset by unsuccessful stock selection (negative 𝑎). Note that the estimate of alpha, 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.

- Let’s define perfect market timing as the ability to tell (with certainty) at the beginning of each year whether the 𝑆&𝑃 500 portfolio will outperform the strategy of rolling over 1-month 𝑇- bills throughout the year. Accordingly, at the beginning of each year, the market timer shifts all funds into either cash equivalents (𝑇-bills) or equities (the 𝑆&𝑃 portfolio), whichever is predicted to do better.
- The key to valuing market timing ability is to recognize that perfect foresight is equivalent to holding a call option on the equity portfolio-but without having to pay for it. The perfect timer invests 100% in either the safe asset or the equity portfolio, whichever will provide the higher return. The rate of return is at least the risk-free rate.
- To see the value of information as an option, suppose that the market index currently is at 𝑆
_{0}and that a call option on the index has an exercise price of 𝑋 = 𝑆_{0}(1 + 𝑟_{ƒ}). At the maturity of the option, assume that the value of the market index is 𝑆_{T}. If the market outperforms bills over the coming period, 𝑆_{T}will exceed 𝑋; otherwise it will be less than 𝑋. The following table gives the payoff to a portfolio consisting of this option and 𝑆_{0}dollars invested in bills – - The portfolio pays the risk-free return when the market is bearish (i.e., the market return is less than the risk-free rate), and it pays the market return when the market is bullish and outperforms bills. Such a portfolio is a perfect market timer.
- The market timing ability is thus equivalent to a position in acquiring a (free) call option on the market and adding it to a position in bills. Hence it can be assigned a dollar value using option- pricing models. This value would constitute the fair fee that a perfect timer could charge investors for its services. Placing a value on perfect timing also helps to assign value to less- than-perfect timers.
- Using Black-Scholes formula for the value of the call option, the formula simplifies considerably to

\( MV(Perfect\ timer\ per\ \$\ of\ assets) = C = 2 \times N \left( \frac{1}{2} \sigma_M \sqrt{T} \right) – 1 \)

- Superior investment performance depends on an ability to be in the “right” securities at the right time. Such timing and selection ability may be considered either
- broadly, such as being in equities as opposed to fixed-income securities when the stock market is performing well, or
- at a more detailed level, such as choosing the relatively better-performing stocks within a particular industry.

- Portfolio managers continually make broad-brush asset allocation decisions as well as more detailed sector and security allocation decisions within asset classes. Performance attribution studies attempt to decompose overall performance into discrete components that may be identified with a particular level of the portfolio selection process.
- Attribution analysis starts from the broadest asset allocation choices and progressively focuses on ever-finer details of portfolio choice. 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 three components –
- broad asset market allocation choices across equity, fixed-income, and money markets
- industry (sector) choice within each market; and
- security choice within each sector.

- This attribution method explains the difference in returns between a managed portfolio, 𝑃, and a selected benchmark portfolio, 𝐵, called the
**bogey**. The bogey is designed to measure the returns the portfolio manager would earn if he or she were to follow a completely passive strategy. “Passive” in this context has two attributes –- First, it means that the allocation of funds across broad asset classes is set in accord with a notion of “usual”, or neutral, allocation across sectors. This would be considered a passive asset-market allocation.
- Second, it means that within each asset class, the portfolio manager holds an indexed portfolio, such as the S&P 500 index for the equity sector.

- In such a manner, the passive strategy used as a performance benchmark rules out asset allocation as well as security selection decisions. Any departure of the manager’s return from the passive benchmark must be due to either asset allocation bets or security selection bets
- The return on the bogey portfolio (B) is be given by \( r_B = \sum_{i=1}^{n} w_{B_i} r_{B_i} \)
- The return on the managed portfolio (P) is be given by \( r_P = \sum_{i=1}^{n} w_{P_i} r_{P_i} \)
- The difference in the two returns is be given by –

\( r_P – r_B = \sum_{i=1}^{n} w_{P_i} r_{P_i} – \sum_{i=1}^{n} w_{B_i} r_{B_i} = \sum_{i=1}^{n} (w_{P_i} r_{P_i} – w_{B_i} r_{B_i}) \)

To decompose each term, a little algebraic manipulation is done where 𝑤_{Pi}𝑟_{Bi} is added and subtracted to give

- As an example, consider the attribution results for a hypothetical portfolio. The portfolio invests in stocks, bonds, and money market securities. The portfolio return over the month is given as 5.34%.

In this table, the usual or neutral weights have been set at 60% equity, 30% fixed income, and 10% cash (money market securities). The bogey portfolio, comprised of investments in each index with the 60/30/10 weights, returned 3.97%. The managed portfolio’s measure of performance is positive and equal to its actual return less the return of the bogey is 5.34 – 3.97 = 1.37%. The next step is to allocate the 1.37% excess return to the separate decisions that contributed to it.

- This table shows that in this month, the manager established asset allocation weights of 70% in equity, 7% in fixed income, and 23% in cash equivalents.

- Superior performance relative to the bogey is achieved by overweighting investments in market that turn out to perform well and by underweighting those in poorly performing markets. The contribution of asset allocation to superior performance equals the sum over all markets of the excess weight (or active weight) in each market times the return of the index for that market.
- Panel A of the previous table demonstrates that asset allocation contributed 31 basis points to the portfolio’s overall excess return of 137 basis points. The major factor contributing to superior performance in this month is the heavy weighting of the equity market in a month when the equity market has an excellent return of 5.81%.
- Panel B of the previous figure details the contribution of the managed portfolio’s sector and security selection to total performance. Panel B shows that the equity component of the managed portfolio has a return of 7.28% versus a return of 5.81% for the S&P 500. The fixed-income return is 1.89% versus 1.45% for the Barclays Aggregate Bond Index. The superior performance in both equity and fixed-income markets weighted by the portfolio proportions invested in each market sums to the 1.06% contribution to performan attributable to sector and security selection.
- This table documents the decisions that led to the superior equity market performance. The first three columns detail the allocation of funds within the equity market compared to their representation in the S&P 500. Column (4) shows the rate of return of each sector. The contribution of each sector’s allocation presented in column (5) equals the product of the difference in the sector weight and the sector’s performance.

- Good performance (a positive contribution) derives from overweighting well performing sectors such as consumer non-cyclicals and underweighting poorly performing sectors such as technology. The excess return of the equity component of the portfolio attributable to sector allocation alone is 1.2898% ≈ 1.29%
- The circled cell in an earlier table showed that the equity component of the portfolio outperformed the S&P 500 by 1.47%. It can be conclude that the effect of security selection within sectors must have contributed an additional 1.47% — 1.29%, or 0.18%, to the performance of the equity component of the portfolio.

**Summary of Component Contributions**

- Asset allocation across the major security markets contributes 31 basis points. Sector and security allocation within those markets contributes 106 basis points, for total excess portfolio performance of 137 basis points.
- The sector and security allocation of 106 basis points can be partitioned further. Sector allocation within the equity market results in excess performance of 129 basis points, and security selection within sectors contributes 18 basis points. (The total equity excess performance of 147 basis points is multiplied by the 70% weight in equity to obtain contribution to portfolio performance.)
- A similar analysis can be applied to the fixed-income portion of the portfolio.

- Suppose the benchmark weights had been set at 70% equity, 25% fixed-income, and 5% cash equivalents. What would have been the breakup of contributions ?
- Suppose the S&P 500 return is 5%. Compute the new value of the manager’s security selection choices.