1. Performance Evaluation
Portfolio Management
Performance Evaluation
How to evaluate the performance of a portfolio manager? The question is not simple and several are the issues that one has to deal with when doing this evaluation. Even the seemingly simple task of computing average returns is not trivial. Also, after having done this, an adjustment for risk is required. This entails discussing several criteria and how and when to apply them. Finally, more sophisticated techniques which are designed to analyze in detail performance will be discussed.

2. One period returns Gross return Net return
Also, average returns can be computed in two different ways. One can use either arithmetic or geometric means. An arithmetic mean corresponds to the most familiar concept of average. We add up the value of the return each period and we divide this summation by the number of observations. The geometric mean is different and it corresponds to the return value that after being compounded a number of times equal to the number of observations is equal to the compounded value of the different realizations. Its formula is given above and in our case is equal to...

3. Average Returns Arithmetic Mean Geometric Mean
Also, average returns can be computed in two different ways. One can use either arithmetic or geometric means. An arithmetic mean corresponds to the most familiar concept of average. We add up the value of the return each period and we divide this summation by the number of observations. The geometric mean is different and it corresponds to the return value that after being compounded a number of times equal to the number of observations is equal to the compounded value of the different realizations. Its formula is given above.

4. Example Period Price Dividend 50 1 53 2 5450 1 53 2 54
Our first stop is thus some of the issues which can be brought up when computing average returns. The task is simple when there we look at one period investments. In the case of a multiperiod setup things can be a little more complicated. Let us illustrate everything with an example. Suppose we look at a stock with price 50 today and price 53 one year from now and 54 two years from now. At the end of the year it delivers a 2 SEK dividend. An investor buys today one share, buys another one year later and sells the two at the end of this two year period.

5. Example
Arithmetic Mean
Geometric Mean

6. Arithmetic vs Geometric
Past Performance - generally the geometric mean is preferable to arithmetic
Predicting Future Returns- generally the arithmetic average is preferable to geometric
Their usage depends on the goal of the calculation. Geometric averages are used when one wants to look at past performance. If we want to extract conclusion of past historical realizations in order to predict future returns the arithmetic average should be used instead. Let us illustrate this with an example. Suppose I invested in a mutual fund two years ago 100 SEK and the return the first year was 50% whereas the second year was -50%. An average return would be equal to 0% but is this really telling me what I need to know. NO! I gave 100 SEK to the fund manager and he or she gives me back two years after 75 SEK. I am loosing money, a zero-return measure would be misleading. What really happened should be measured with a geometric average. However, if one wants to predict returns based on those two observations the best prediction would be based on the arithmetic average: 0%. For a one-period forecast this is the measure. However, if the forecast is over longer periods and for future cumulative performance, this introduces an upward bias (see BKM for details).

7. Example A stock price doubles or halves Same probability We observe
Period
Price 10 1 20 2
The price of a stock can either double or halve and both outcomes have the same probability

8. Example (True) Average mean (Observed) Geometric mean
The price of a stock can either double or halve and both outcomes have the same probability

9. Example Period Price Return 10 1 20 100% 2 -50%10 1 20
100% 2
-50%
Time weighted return = arithmetic average return
(100-50) = 25%

10. Example Period Price Number of shares bought 10 100 1 20 2 -20010
100 1 20 2
-200
Dollar weighted return = Internal Rate of Return

11. Measuring Returns Dollar-weighted returns
Internal rate of return considering the cash flow from or to investment
Returns are weighted by the amount invested in each stock
Time-weighted returns
Not weighted by investment amount
Equal weighting
The first measure takes into account the cash flows. This weighting depends on the amount invested each period. Time weighted returns considers an investment in one stock as an investment in one share, the weighting each year is the same. This will be clear with our example.

12. Adjusting for risk
Mean returns are not enough and one must also adjust for risk
Find the appropriate comparison universe
Mean-variance risk adjustments
Like we said before, our task is not done for quite understandable reasons. If we are trying to evaluate a portfolio, the average return alone will not be very informative. I need to know how risky each portfolio could be. That is, I need to adjust my predictions of future expected return with an estimate of the risk I might be bearing. The usual way to proceed is to compare that portfolio with other investments with the same risk characteristics. This entails establishing the appropriate comparison universe which is in most cases done by choosing a benchmark portfolio. This benchmark portfolio can be S and Poor if the portfolio contains only a representative sample of listed US stocks. But it might have to be more specific if the stocks of my portfolio focuses only on a particular type of stocks withing the above type: for example low beta stocks. With the arrival of the CAPM, a whole battery of risk adjustment methods based on the mean-variance criteria were proposed. We will review the most important ones.

13. The Sharpe Ratio
Sharpe’s measure: expected excess return per unit of risk (measured as total volatility)
Apropriate scenario: Evaluate a portfolio which represents the entire investor’s initial wealth
Slope of the CAL
The mean-variance criteria delivers in a natural way the measure named after his creator Willian Sharpe: the Sharpe ratio. Risk premium divided by the standard deviation. Simple. A natural reward-to-volatility ratio which happens to be the slope of the capital allocation line of the portfolio. It should be use when the portfolio will represent the entire investor’s initial wealth.

14. The Sharpe Ratio and M2
Equates the volatility of the managed portfolio with the market by creating a hypothetical portfolio made up of T-bills and the managed portfolio
If the risk is lower than the market, leverage is used and
The hypothetical portfolio is compared to the market
Like we said, the evaluation also requires a comparison universe. In many cases a comparison based on the Sharpe ratio might be difficult to interpret. Is .59 as opposed to .64 a meaninful difference? Franco and Leah Modigilian, grandfather and grandson, developed the M2 measures which translates a sharpe ratio difference into a difference in expected excess returns in the following way.
We want to equate the volatility of the portfolio’s return under consideration with the volatility of the benchmark by building a portfolio of T-bills combined with the managed portfolio. We compare then the expected excess return of this new portfolio with the expected excess return of the benchmark.

15. The Sharpe Ratio and M2 Find the value where a is the value for which
Then:

16. The Sharpe Ratio and M2 E(r) M M2 P* P rf σP σM
So a graphic illustration goes as follows suppose my managed portfolio is P and consider its CAL. We move upwards on the line until we get to a point with the same volatility of the market M. The difference in expected return between this point and the market is the M2. Obviously, a higher sharpe ratio than the market will always give a positive M2. This measure translate the sharpe ratio difference into an adjusted by risk difference of expected returns. It can thus be easily interpreted as a differential return relative to the benchmark index. rf σP σM

17. Jensen’s alpha & AP
Jensen’s measure: the expected return of the portfolio above its CAPM counterpart
The appraisal ratio: alpha divided by the portfolio’s nonsystematic risk
A quite natural measure of performance based on CAPM is Jensen’s measure. The idea is simple. We did an example when presenting CAPM where we illustrated how to declare a stock a good buy or a bad buy. The difference between the portfolio’s actual expected return and its CAPM prediction. This is the managed portfolio’s alpha.
A companion of this measure is the appraisal ratio. It is designed for situations where as opposed as the previous case the managed portfolio is not the full investment but part of a portfolio of itself mixed with the market portfolio. The idea is that a positive alpha in this case for the managed portfolio will obviously increase the expected return of the total portfolio but this comes with a cost: the nonsystematic risk. The ratio gives a measure of the increase in the reward to the increase in volatility in the total portfolio after including a given amount of the managed portfolio.

18. Jensen’s alpha & AP
The AP is used in situations where the portfolio to be evaluated will be mixed with the market
Why? For the optimal mix, the complete portfolio’s sharpe ratio is
It measures improvement in the Sharpe ratio
Another way to interpret the appraisal ratio is a little less intuitive. If we try to find the optimal portfolio of the two as the one that maximizes the Sharpe ratio, we have that the maximized sharpe ratio of the complete portfolio satisfies the following equality: its squared value is equal to the sum of the Sharpe ratio of the market (squared) plus the square of the appraisal ratio. The appraisal ratio is a measure that capture the improvement in the S after including the managed portfolio.

19. Treynor’s measure
Treynor’s measure: excess expected return per unit of systematic risk (measured as beta)
Appropriate when the portfolio is part of a large investment portfolio
The slope of the T-line
The T-line is to the SML the same as the CAL to the CML(check this, cause it is inconsistent with BKM)!!!. It gives as the resulting line where the returns in expected return-beta space will be of any portfolio of P and the treasury bill. The difference is only about the particular measure of risk used in each case. The slope of this line is the appropriate measure in this case given that the portfolio will be part of a large investment portfolio its non systematic risk will be mostly diversified away when included in the larger portfolio. Thus, the relevant measure of performance should be Treynor measure since it uses the relevant measure of risk in this case.

20. Treynor’s measure TP E(r) TQ Q SML P M E(rM) Slope(SML)=TM=E(rM)- rf
In arguing why this is the relevant measure of performance when comparing across different portfolios that will be part of larger portfolio, we can do it in the following way, if we construct a portfolio of Q and treasury bills we can do in such a way that we match P’s Beta, at that point the two investments P and Q* are comparable now by lookin only at their expected return, since they hold the same systematic risk and since the idea is to forget about idyosincratic risk, P will have a higher expected return than Q* whenever the slope of its SCL (its Treynor measure) is higher. By similarity with M2, the difference of the treynor measure of an asset with the treynor measure of the market (the expected return of the market in excess of the riskless rate (the beta of M is one)) is referred to as the T2 measure. rf b
bM= 1.0 bP

21. Some Issues Assumptions underlying measures limit their usefulness
Constant distributions
Preferences
When the portfolio is being actively managed, basic stability requirements are not met
An example: market timing
All these measures are consistent in the sense that superior performance (with the benchmark/market) requires a positive alpha and this is probably why alpha is the most wildly used measure of performance.

22. Market Timing
Adjusting portfolio for up and down movements in the market
Low Market Return - low ßeta
High Market Return - high ßeta
Regression:
Interpretration. If manager is capable of timing the market, then the beta with it will increase when the market goes up (bullish) and it will go down when the market is bearish, giving a pattern that could conform with the following structure. Consider an extreme example to simplify, one portfolio of treasury bill and the market P. The manager has market timing capabilities and when the market is bullish it will leveraged its portfolio to buy the market and viceversa. In this picture one sees that excess returns are usually higher than those of the market when the market is bullish (the beta is higher than one) and lower than the market when the market is bearish (the beta is lower than one).

23. An Example of Market Timing*
rp - rf
rm - rf
Steadily Increasing the Beta
Market timing is an example of the inadequacy of conventional performance evaluation techniques that assume constant mean returns and constant risk. The market timer constantly shifts beta and mean return by moving in and out of the market. Whereas the expanded regression captures this phenomenon, the simple SCL does not. Note that when the market is bearish, that is, when rm-rf is negative the loss of the timing strategy is less that that of the market and viceversa. A perfect timer

24. Market Timing A simple alternative: Regression
Beta is large if the market does well
Beta is small otherwise
Regression

25. Market timing
rp - rf *
rm - rf
the Beta takes only two values

26. Performance Attribution
Decomposing overall performance into components
Components are related to specific elements of performance
Example components
Broad Allocation
Industry
Security Choice
It is also important when evaluating performance to be able to determine its sources: is it due to capital allocation or is it due to security selection. There are two main procedures to do so and they can be identified with the kind of information it is required. The first method works as follows and it requires knowledged of internal information, it requires knowledge of the portfolio composition at each time the returns are observed. It also requires to determine a relevant reference return also known as bogey.

27. Performance Attribution
Set up a ‘Benchmark’ or ‘Bogey’ portfolio
Use indexes for each component: depends on the asset class
Use target weight structure: neutral, depend on preferences of the client
BKM give the example:
10% cash, 15% bonds and 75% equity for risk-tolerant client.
45% cash, 20% bonds and 35% equity for risk-averse.
This bogey must be chosen so that it is a good description of what the manager would earn if she or he were to follow a completely passive strategy. Passive in this context means two things. One is that the allocation across broad classes of assets is done in accordance with a notion of usual or neutral. Second, within each asset class the portfolio manager holds an indexed portfolio like SP 500 for equity. This notion of neutrality of the capital allocation decision depends on the investor’s preferences and it should be done in consultation with the client. BKM come up with an example that I consider striking.

28. A question The bond-to-equity ratio is
15/75 = 0.2 (low risk aversion)
20/35 = 0.57 (high risk aversion)
If cash is riskless, does it make sense according to standard assumptions?

30. Asset allocation puzzle
Canner, Mankiew and Weil:
”Popular financial advisors appear not to follow the mutual-fund separation theorem. When these advisors are asked to allocate portfolios among stocks, bonds, and cash, they recommend more complicated strategies than indicated by the theorem”
And so do BKM!!!!

31. Performance Attribution
Calculate the return on the ‘Bogey’ and on the managed portfolio
Explain the difference in return based on component weights or selection
Summarize the performance differences into appropriate categories

32. Performance Attribution
Asset Allocation
Security Selection

33. Performance Attribution
Contribution for asset allocation (wpi - wBi) rBi
+ Contribution for security selection wpi (rpi - rBi)
= Total Contribution from asset class wpirpi -wBirBi

34. Style analysis
Regress the returns under evaluation on a sufficiently representative set of asset classes
This allows identification of the capital allocation decision
The proportion not explained: security selection
It was presented by W. Sharpe (Nobel prize winner). A passive fund manager provides an investor with an investment style, while an active manager provides both style and selection. The idea is that performance contribution like we have previously presented requires information that might not be easily attainable: the weights of the manager’s portfolio, internal information that for many reasons might not be easy to reach or even kept secret. This approach is thus external in as much it uses only the returns of the portfolio, as oposed to the previous method which might be referred to as internal.

35. This tables illustrate the exercise
This tables illustrate the exercise. Here there is a regression that uses observations from 1985 to 1989 (montly). The explanatory variable corresponds to a US mutual fund. The unscontrained regression shows a high R2, but it has two problems, it does not add up to one and it shows the presence of negative weights.
Note that the addiction of constraints reflecting the fund’s actual investment policy causes a slight reduction in the fit of the resulting equation to the data at hand. Now, however, the coefficients conform far more closely to the reality of the fund’s investment style, making the resulting characterization more likely to provide meaningful results with out-of-sample data.

36. Style analysis Magellan Fund Growth stocks 47% Medium cap 31%
Small stocks %
European stocks 4%
An investor in Magellan Fund should know that its style favors growth stocks. The choice to expose his portfolio to this risk is the investor’s. Results of this choice (good or bad) should be atributed to the investor, not to the manager of a fund following this style.

37. Results of this choice (good or bad) should be atributed to the investor, not to the manager of a fund following this style.

38. The manager’s performance should be evaluated with respect to its style. In this example, one can see that the performance of this fund is outstanding. It beats its style consistently.

39. Some Complications Two major problems To measure well
Need many observations even when portfolio mean and variance are constant
Active management leads to shifts in parameters making measurement more difficult
To measure well
You need a lot of short intervals
For each period you need to specify the makeup of the portfolio