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Performance Measurement BasicsThis page will cover performance measurement basics. Specifically, how to calculate returns for various instruments:
We start, however, by summarizing some fundamental principles of investment performance measurement. Market ValueA basic principle of performance analysis is that returns should be calculated using market values (rather than historical cost). There are a number of operational challenges in conducting valuation of investment portfolios. Traditionally, a lot of performance measurement and attribution has been based on calculating monthly returns using monthly valuations. However, when cashflows are present, monthly calculations give rise to approximation errors. These errors are discussed on the Measurement Errors page. Accordingly, it is best practice to calculate daily returns using daily valuations. The Concept of ExposureHaving accurate market values is very important for high-quality investment performance measurement. But it is not always enough. In particular, for derivatives, one needs to know the exposed market value (often known simply as the Exposure) of each holding. Futures contracts are a great example of the kind of instrument for which one needs exposure data. Consider an S&P 500 futures contract, as traded on the Chicago Mercantile Exchange. The pricing for these contracts is $250 per index point. Suppose that one bought an S&P 500 futures contract at 1400, then sold it at 1500. The gain is 100 index points, which at $250 per point amounts to $25,000. This indicates what the market value would have been just before the contract was sold: approximately $25,000. In calculating the return for this holding, the numerator would be $25,000. But what would the denominator be? Cutting to the chase, it would be 1400 x $250 = $350,000. This is the exposure of this holding. The exposure is used for several purposes:
The concept of exposure is not really necessary for instruments such as stocks and bonds. For these non-leveraged instruments, the exposed market value is identical to the unexposed market value. For example, if a portfolio holds $10 million of IBM stock, the market value and the exposure are both $10 million. Some software systems for investment performance measurement and attribution do not incorporate the notion of exposure. These systems are fine, except for the fact that they give crazy results when futures or options or other kinds of derivatives are involved. In other words, they are best avoided. For a detailed treatment of the role of exposure in investment performance measurement and attribution, see the Exposure page. Calculating the Return on Short PositionsA short position exists when one holds a negative exposure to an instrument. In normal (long) holdings, one holds a position hoping to make a profit if the instrument increases in value. However, for a short holding, the payoff is exactly reversed. For example, consider an ordinary investment in one share of IBM. If one purchases this share at $100, then sells it at $120, one makes a profit of $20. However, short selling works differently. To open the short position, you sell one share of IBM (even though you don't own it) at $100. To close the position, you need to buy back one share of IBM. Since the price in the meantime has moved by $20, you will have lost $20 on the short position. The arithmetic for calculating the returns for these positions in IBM is: Return on long position = (EMV - BMV) / BMV = (120 - 100) / 100 = 20% Return on short position = (EMV - BMV) / BMV = (-120 - -100) / -100 = -20 / -100 = 20% Can that be right? Should the return for the short position the same as for the long position? Specifically, should both of these returns be of the same sign? Indeed they should. We will go into a full justification further below, but at this stage it will help you to avoid confusion if you know the golden rule for returns on short positions: The return on a short position is no different from the return on a long position. The different contribution to overall portfolio return arising from a short position results from the portfolio weights being of opposite signs for long and short positions. It seems, however, that agreement on this golden rule is not universal. A number of influential people still advocate approaches where one takes the absolute value of the denominator in the return calculation in order to get returns where the signs are opposite for long and short positions. In a letter published in the Journal of Performance Measurement (Spring 2002), Damien Laker tried to make it clear why this is indeed a big mistake, and a source of great confusion. That letter explains some key points:
Another problem for the method that uses an absolute value on the denominator is that the returns it produces simply do not compound properly! This is a fairly grievous problem. For a worked example, please refer to the spreadsheet Crazy method for short returns does not compound properly.xls. For those who wonder how the effect of their short positions can come out in performance reporting, the answer is that short positions have a negative weight in the portfolio. Therefore, the weights and the returns need to be considered together. Perhaps your portfolio held a security last month whose return was 55%. This would be wonderful news if the portfolio was holding that security at a weight of 10%, but on the other hand, if the portfolio was holding that security at a weight of -20% (i.e. it was a large short holding), this news is disastrous. Contribution reports and attribution reports always take weights into account, so it is important to educate people that if they want to know what made the portfolio go up or down last month, they should not look at the holding returns: rather, they should look at a contribution report. Market Exposures and Currency ExposuresTo understand a portfolio's position with respect to market risks, it is vital to have a clear picture of the market exposures. For example, a diversified global portfolio might hold 20% domestic equities, 60% international equities, 15% global bonds, and 5% domestic cash. In most different kinds of investment performance analysis, an essential piece of information is the percentage of total assets allocated to each different market sector. "Market sectors" might be broad asset classes (as in the example we just mentioned). For a domestic equities portfolio, "market sectors" might be different industries. Or, in global bonds, "market sectors" might be different countries or regions of the world. Whichever classification scheme one chooses, this paradigm of classifying holdings into market sectors is a mainstay of investment performance analysis. The correct way to calculate a portfolio's weights in different market sectors is by using exposures (rather than market values). This is why it is really quite essential to know the exposure of each asset. The Exposure page provides details of how to do these calculations. In multicurrency portfolios, the currency weights comprise a completely separate dimension. This currency dimension is just as important as the market exposure dimension. Without the use of currency derivatives, a portfolio's currency exposures will simply be determined by its market exposures. For example, if an equities portfolio was 60% Japan, 30% Korea, and 10% USA, then the currency exposures would accordingly be 60% Japanese Yen, 30% Korean Won, and 10% US Dollar. However, by using currency derivatives (principally Forward Rate Agreements), one could alter the currency exposure of this portfolio to anything one chose. For example, if the portfolio was being managed on behalf of Japanese investors, the currency exposure might be hedged back into 100% Japanese Yen. Or, if the investment manager had a bullish view on the US Dollar, they might adopt a position where the currency allocation was mostly in US dollars. Indeed, it would be equally possible to completely hedge this portfolio across into Euro or Australian Dollars. Going further, one could even take a currency position of +300% Russian Ruble, -200% US Dollar. Something this extreme is more likely to be adopted by a hedge fund rather than a run-of-the-mill investment portfolio. From the perspective of investment performance analysis, the important thing to grasp is that, in general, a portfolio's currency exposures can be analyzed quite independently of the market exposures. The term "currency overlay" has often been used in the past to describe the way that currency positions can be managed quite separately from market positions. Even in a passively managed (indexed) portfolio, currency is a very important topic. For example, an investment manager might offer two global equities products: one fully hedged for currency, the other unhedged. It would not be particularly unusual for a fully-hedged benchmark to outperform (or underperform) an unhedged benchmark by 10% per year over several consecutive years. There is a very big difference between calculating a global equity index (unhedged) in US dollars, and a global equity index hedged into US dollars. This difference arises solely from the currency positioning of the benchmark.
American Depository Receipts (ADRs) and Exchange Traded Funds (ETFs)Some instruments can introduce unexpected difficulties in understanding a portfolio's market exposures and currency exposures. For example, many companies that are listed on exchanges outside the USA choose to access the US capital markets by issuing American Depository Receipts (ADRs) on a US exchange. Without going into all the implementation details, each ADR simply gives exposure to a specified number of underlying shares. For example, XYZ Bank in Australia might issue ADRs where each ADR represents 5 ordinary shares in XYZ Bank. The ADR is an instrument traded in US dollars on a US exchange. The XYZ share is an instrument traded in Australian dollars on an Australian exchange. At first glance, one might think that a US investor buying the ADR was not assuming foreign currency risk. However, you would be wrong. Just because the ADR is priced in US dollars, this does not mean that it gives a currency exposure to US dollars. Rather, it gives a currency exposure to Australian dollars. A simple way of understanding this is to consider the potential for arbitraging between one position and the other. For example, the data for day 0 may be:
Note that AUD means Australian Dollars, and USD means US Dollars. Suppose that the XYZ share price does not change on day 1. However, the exchange rate moves dramatically. The data for day 1 might then be:
The return on XYZ shares in Australian Dollars is exactly zero. But the return on the XYZ American Depository Receipts (in US Dollars) is $157.89 / $166.67 - 1 = -5.26%. This is exactly the same return that a US investor would have obtained by purchasing XYZ shares on the Australian Securities Exchange, with no currency hedging in place. This demonstrates clearly that the ADR, while priced in US Dollars, actually provides an exposure to Australian Dollars. This is a very subtle issue for performance analysts to deal with: the ADR is priced in one currency (US Dollars), but gives economic exposure to a different currency (Australian Dollars). Another interesting instrument is Exchange-Traded Funds (ETFs). Like ADRs, ETFs can also have an economic currency exposure that differs materially from their pricing exposure. For example, an ETF traded in US Dollars on a US Exchange might be managed to closely track a particular global equity index. The currency exposures of this ETF will therefore be very close to those of the benchmark global equity index (assuming that the ETF is unhedged). Some ETFs are offered in different countries, where different currencies may apply. For example, the global equity index ETF might have originally been developed as a US product, but it might now be sold in Australia, Singapore, Japan, and the UK. This would mean that it was possible to trade the ETF in numerous different currencies. The key point to remember is that none of these currencies is particularly important for understanding how this ETF really works. Rather, the ETF gives you an exposure to the various currencies of the world index, regardless of the currency you use to purchase the ETF.
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