|
|
|
|
Investment Performance MeasurementInvestment performance measurement provides some of the most basic and useful information that one could possibly know about an investment portfolio. Specifically, performance measurement entails calculating the returns for portfolios and their benchmarks. By comparing the portfolio return with the benchmark return, one can determine whether the portfolio has generated a better return than would have been possible from a passive (benchmark) alternative. Performance measurement also can entail specialized return calculations for indices, derivatives, unit-priced products, fixed income, and common stocks. In all cases, the fundamental principles still hold. However, there are usually a few key points that need to be observed in order to ensure that the calculation remains sensible. What is a Return?All studies of investment performance measurement must start with the question "What is a Return?". If we can devise a robust answer to this simple question, we will be able to rely upon it again and again in tricky situations where it's not clear how one should go about calculating returns. In general terms, the equation used to calculate a return is: Return = Gain / Amount at Stake So, for example, if I placed $100 into a fixed-term deposit with a bank, and received $105 back when the deposit matured, the gain would be $5, and the amount at stake would be $100. The return calculation would be 5% = 5 / 100. In more complicated cases (such as calculating the performance of derivatives or leveraged positions), one sometimes needs to think clearly about what the gain is, and what the amount at stake is. A return corresponds quite naturally to the idea of dollars and cents (or pounds and pennies, or ...). If I start off with $100, and earn a return of 5%, my gain will be exactly $5. In this way, return calculations should always make sense at an "accounting level", where one treats the calculation mainly from an accounting perspective. Indeed, even when we consider performance attribution, it often makes the calculations much more intuitive if we carefully explain their relationship to dollars-and-cents calculations. Calculating ReturnsFor detailed information about how to calculate returns for shares, fixed income, unit-priced products, portfolios, and derivatives, see the Measurement Basics page. CompoundingTo combine returns over time, one uses the principle of compounding. For example, if a portfolio earned a return of 10% per year over two consecutive years, the total return would be (1+10%) x (1+10%) - 1 = 21%. This corresponds naturally to the fact that an initial portfolio value of $100 would have grown to $121 over the two-year period. The intuition behind this sort of calculation is that, in the first period, one earns interest only on the principal. In subsequent periods, one starts earning interest on principal plus the accrued interest. To state it another way, if returns combined additively over time, two periods with a 10% return might yield 20% when combined. However, the compounding calculation shows that there is a payoff for patient investing: The investor who holds for two years gets two lots of 10%, plus a bonus percent, which represents the earnings on the first period's interest over the second period. In general, the equation for compounding arbitrarily many periods runs as follows: Rtotal = (1 + R1) x (1 + R2) x ..... (1 + RN) - 1 Compounding is not always a very intuitive calculation. For example, returns of +10% and -10% do not cancel out to a net return of zero (instead the net return is negative 1%). Compounding is one of the fundamental skills that one needs to grasp in order to do performance measurement and attribution. We will see later that it is central to the question of creating high-quality attribution analysis. Benchmark ReturnsFor information about how to calculate benchmark returns, see the Benchmarks page. Active ReturnsThe active return over any period is simply the difference between the (gross) portfolio return and the benchmark return. Since the benchmark return represents the "do nothing" scenario, the active return indicates the effect of management decisions by the portfolio manager. Broadly speaking, a positive active return indicates that the investment manager has added value (and vice versa). For example, if the portfolio return was 12%, while the benchmark return was only 10%, this would indicate that the portfolio manager added 200 basis points of value. An obvious question is, "What about risk -- could the outperformance be due to extra risk (rather than manager skill)?". The answer to this question is that one can definitely generate outperformance by incurring extra risk (at least in a rising market). This leads to the whole area of risk analytics, and risk-adjusted performance. The extent of a portfolio's outperformance after risk adjustment is known technically as the excess return. If the portfolio is no more risky than the benchmark, excess return and active return will be identical. However, if the portfolio risk deviates from the benchmark risk, the excess return measures value added after allowing for risk differences. Despite this technical difference in terminology, you will sometimes see people use the term "excess return" when, strictly speaking, they should have said "active return". Most people consider this imprecise use of terminology to be acceptable in many contexts. Performance attribution is the process of analyzing how the portfolio manager generated the active return. After one has measured a portfolio's returns and compared them with the benchmark, performance attribution is clearly one of the most interesting forms of analysis that one can perform in order to understand what was happening in a portfolio. Geometric vs. Arithmetic Active ReturnSo, if the portfolio return is 21%, and the benchmark return is 10%, the active return is therefore 11%, right? Well, from an arithmetic perspective, it is. Many (probably most) people adopt an arithmetic perspective. An alternative approach is the geometric paradigm. From this perspective, if the fund return was 21% and the benchmark return was 10%, the active return would be 10% (rather than the 11% answer which seems obvious from the arithmetic perspective). This is because the 10% active return compounds with the 10% benchmark return to produce the 21% portfolio return. A good overview of this topic is provided by the paper Measuring and Attributing Value Added Using Arithmetic and Geometric Approaches, by Owen Davies. Owen explains how each approach to dealing with the active return has pros and cons. Geometric active return can be a useful concept in multicurrency settings, for pretty much the same reason as continuously compounding returns are (see below). In both cases, it becomes a bit easier to translate any active performance effect from one currency into another. This can be useful, and it is probably a good reason to consider adopting geometric active return. On the other hand, some people say that one should use geometric active returns so that one can use geometric multiperiod performance attribution (based on the premise that arithmetic multiperiod attribution is a problem that's too hard to solve properly). This is a case of "the tail wagging the dog". We recommend that you should decide which approach you want to take to active returns: arithmetic or geometric. Once you've done this, you should then adopt a kind of attribution that is consistent with the kind of active return you've chosen (e.g. geometric attribution to explain a geometric active return). In broad terms, Europe tends to favor geometric active returns, while the rest of the world mostly favors arithmetic active returns. This web site mostly uses the arithmetic paradigm, but most of the ideas found here could easily be translated into a geometric paradigm if one so desired. Continuously Compounding ReturnsOne useful way of manipulating returns is to express them in a continuously compounding form. Before exploring how this works, let's cut to the chase and examine why continuously compounding returns are useful. A bit of terminology. The distinction is between continuously compounding returns (a.k.a. continuous returns) on one hand, and periodic returns (a.k.a. discrete returns) on the other hand. "Periodic returns" are simply the usual garden variety of returns that one normally uses in everyday life (and in everyday performance analysis also). In a nutshell, continuously compounding returns are logarithms. As we all learned in high school, a convenient feature of logarithms is that one can add logarithms together instead of multiplying. Also, one can subtract logarithms instead of dividing. Accordingly, if one wishes to compound together continuous returns of 10.5% and -3.0%, one simply adds them to obtain 7.5%. For simple examples, this keeps everything in round figures. This is especially useful when trying to explain complicated ideas concerning (say) multicurrency attribution. To see how continuous returns make things simpler, consider what is involved if one compounds periodic returns of 10.5% and -3.0%. The arithmetic required for this calculation is (1 + 10.5%) x (1 - 3.0%) - 1 = 7.185%. Continuous returns are also much easier to manipulate in spreadsheets. If you are not yet convinced that continuous returns can simplify many calculations, think about how one could obtain a mean monthly return given the two monthly returns we just considered. With continuous returns, the mean monthly return would simply be (10.5% - 3.0%) / 2 = 3.75%. This is simple mental arithmetic that anybody can follow. However, with periodic returns, one would have to calculate ((1 + 10.5%) x (1 - 3.0%))^(1/2) - 1 = 3.53%. This is way beyond the mental arithmetic powers of most educated people. We conclude that continuous returns can bring greatly improved simplicity to some common calculations. To calculate a continuous return from market values, one takes the natural logarithm (usually denoted as ln) of the "wealth relative". The wealth relative is simply the ratio of two market values. So, if an initial investment of $100 increased to a value of $111.0711, the continuous return would be ln(1.110711) = 10.5%. Note that the periodic return for these values would be 11.0711%.
In one sense, continuously compounding returns are extremely intuitive: continuous returns of plus 10% and minus 10% do indeed cancel out to exactly zero. However, continuous returns sacrifice the intuitive notion that returns correspond to the cumulative value of an initial investment. For example, if a portfolio starts out with $100 and earns a continuously compounding return of 10% for one period, the cumulative value is approximately $110.52 (rather than exactly $110). Similarly, if a portfolio starts out with a value of $100 and earns a continuously compounding return of negative 10% for one period, the cumulative value is approximately $90.48 (rather than exactly $90). So, when is it best to use continuous returns, and when is it best to use discrete returns? Continuous returns are really useful for all sorts of quantitative analysis. They are also tremendously useful when discussing Multicurrency Attribution. This is because one often needs to convert between local currency and base currency when doing multicurrency analysis. The convention of continuous returns makes it possible to do this conversion simply by adding the currency return to the local market return. On the other hand, virtually all of the everyday work in a performance department is done using periodic returns. This is because periodic returns correspond naturally to market values. The context normally dictates which convention one uses for returns. One could liken this to the decision of when to speak in English and when to speak in French: for those who are fortunate enough to be familiar with both languages, one normally chooses one or the other based on what will be most useful in any particular context. Extending this analogy, it doesn't make much sense to try asserting that one convention is somehow objectively "better" than the other (although the Academie Francaise may beg to differ!)
Time Weighted Returns and Money Weighted ReturnsSuppose that a portfolio manager is bullish about domestic equities. At the start of the month, the manager has $25 million in the domestic equities sector of the portfolio. In the first 10 days of the month, the domestic equities sector provides a return of 10%. Unable to contain his bullish urges any further, the investment manager allocates another $25 million into domestic equities. Unfortunately, the trend reverses, and the value of the domestic equities sector drops by 8%. This means that the return for the domestic equities sector over the whole month has been 10% compounded with minus 8%, i.e. (1+10%) * (1 - 8%) -1 = 1.20%. This seems a fairly intuitive solution, and perhaps there is nothing more to consider in this trivial problem. Well actually, there's a lot more to consider. The 1.20% result is a time-weighted return. It accurately measures the return of the domestic equities sector over the month. But there is another way to calculate the domestic equities performance. This other approach is called a money-weighted return. Money-weighted returns measure both:
In this case, there are a number of different money weighted returns that one could choose, depending on the assumptions that you made. These are illustrated in the spreadsheet MoneyWeightedorTimeWeighted.xls Because the portfolio manager allocated more money into domestic equities just before it fell by 8%, the portfolio actually lost money on domestic equities over the month. The final value of domestic equities was $48.3 million, when the total amount invested had been $50 million. Thus the dollar loss was $1.7 million. Some ways of calculating a money-weighted return are:
These money-weighted returns all reflect the fact that the portfolio lost money on domestic equities over the month. But the important thing to remember is that the portfolio didn't lose money because domestic equities had a negative return. Rather, the reason for the loss was that the portfolio manager made a bad market-timing decision by only having $25 million in domestic equities for the period where it went up, but $52.5 million in domestic equities for the period it went down. The money-weighted return can be a good method to use if one wants a single number that reflects both the return of the asset and the effect of timing cashflows into the asset. In this sense, money weighted returns provide a rough and ready method for performance attribution. However, the only way to get a true measure of the return of the asset (independently of cashflow timing) is by using a time weighted return. So, for example, if the benchmark return on domestic equities had been exactly zero during the month, the sector manager responsible for domestic equities would have done a good job by outperforming the benchmark by 1.20% during the month. It is not the sector manager's fault that the overall portfolio manager made a poor market timing decision. Similar reasoning applies when calculating total-level returns. A money-weighted return for a portfolio reflects both the manager's skill and the client's skill or luck in timing cashflows. Money-weighted returns can serve a useful purpose, but they are a poor substitute for time-weighted returns if one's purpose is to measure investment performance independently of cashflows. This is why the Global Investment Performance Standards (GIPS®) recommend time-weighted returns as the best practice. However, the minimum requirement under GIPS is monthly money-weighted returns using day-weighted cashflows (so long as the cashflows fall within certain limits). This certainly means that compliant returns can still be 10 basis points in error every single month if one is using the monthly money-weighted return method. Ponder that next time somebody tells you that GIPS is "best practice". Of course, for the example we have been using, the cashflow is so large that nobody could reasonably consider it to be a "relatively small" cashflow. In extreme cases such as this, GIPS requires valuation of the portfolio on the date the cashflow takes place, so that you can calculate the two sub-period returns and then compound them together. For a detailed treatment of the errors that arise from using money-weighted returns as a substitute for time-weighted returns, see the Measurement Errors page. It also outlines some other sources of performance measurement errors. GIPS is a registered trademark of the CFA Institute. |
|
Send mail to
webmaster@compoundinghappens.com with
questions or comments about this web site.
|
