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ExposureThis page describes the concept of exposure, otherwise known as exposed market value. Before reading this page, is will be a big help if you first read The Concept of Exposure on the Measurement Basics page. What is Exposure?Simply put, exposure measures "what is at stake" in a holding in an investment portfolio. The "amount at stake" is used for several purposes:
The section on The Concept of Exposure introduces a simple example of these calculations (using an S&P 500 futures contract). For simple instruments such as equities, exposure is a redundant concept, since the exposure is always equal to the market value. However, for instruments such as futures, options, and swaps, exposure can be completely different from market value. We refer to these instruments, where exposure and market value can be different, as leveraged instruments. Leveraged instruments entail an exposure to something (for example, S&P 500 futures entail an exposure to equities). This "something" to which exposure is created is known as the "underlying". However, leveraged instruments also modify the portfolio's exposure to cash. Specifically, the difference between market value and exposure brings with it a leveraging effect (i.e. it is equivalent to borrowing cash). This is one of the more complicated aspects of exposure calculations, because it entails indirect modification of the effective cash position in a portfolio. We discuss this further in the next section (and beyond). The Fundamental Rule of ExposureWhen we are looking at a single position in a portfolio, a salient feature to notice is the difference between a position's exposure and market value. We refer to this difference as an "exposure adjustment". Definition of the Term: The exposure adjustment for a position is the difference between its exposure and its market value. Exposures and market values can increase and decrease over time, in accordance with market fluctuations. However, at any fixed moment in time, we can observe that a fundamental rule (just like Newton's laws of motion) applies. The Fundamental Rule of Exposure states: Definition of the Rule: At any fixed moment in time, the exposure adjustment for cash is by definition equal to the sum of the exposure adjustments for non-cash positions in the portfolio. We discuss below how to use this rule in practice during performance calculations. A corollary of this rule is: Corollary: At any fixed moment in time, the sum of exposures in a portfolio is by definition equal to the sum of the market values in a portfolio. It is quite easy to prove that this corollary follows from the rule. This rule is not one that can be empirically tested. It is essentially a mathematical convention. Examples Using the Fundamental Rule of ExposureHow does one apply this rule in practical terms? Consider our example portfolio, which holds a million dollars in cash, and which buys an S&P 500 futures contract at 1400 points. For the sake of simplicity, assume that the brokerage on the futures contract was zero. The market value of cash in the portfolio would still be a million dollars, and the market value of the futures contract would be zero. The total market value remains at a million dollars. However, in terms of exposures, the exposure to S&P 500 is $350,000. This implies that the exposure to cash is only $650,000. We draw this inference about the cash exposure because of the other data we have (the total market value and the fact that the S&P 500 futures has an exposure that exceeds its market value by $350,000). To reiterate, the reduced cash exposure is not something that one empirically observes: it is true as a matter of convention. Notionally reducing the cash exposure by $350,000 preserves the fundamental rule of exposure, that the sum of exposures in a portfolio equals the sum of market values. Taking the example a bit further, think what would happen if one added to the portfolio a MSCI World futures contract with exposure of $350,000. Once again, the market value of this futures contract at the time it is purchased will be (approximately, i.e. neglecting brokerage) zero. The net cash exposure for the portfolio in this case will be reduced to $300,000 (even though the portfolio is still actually holding a million dollars of cash). In this case, the portfolio weights would be 35% S&P 500, 35% MSCI World, and 30% cash. To keep the data consistent with the fundamental rule of exposure, whenever a non-cash security has an exposure greater than its market value, there must be an equal-and-opposite exposure adjustment in the cash sector. There are two different strategies that one could use for keeping track of the adjustments to cash exposure that are required for each leveraged instrument:
For example, in the first strategy, the cash sector holdings would consist of $1 million cash, minus $350,000 cash exposure generated by the S&P 500 futures, and minus $350,000 cash exposure generated by the MSCI World futures. The sum of these exposures is $300,000. The essence of this calculation strategy is that the exposure adjustments for each leveraged position outside the cash sector are provided explicitly. As an example of the second strategy, the only explicit holding loaded into a performance measurement system for the cash sector might be a million dollars. But because the exposure of the non-cash sectors exceeds the market value of the non-cash sectors by $600,000, the performance measurement system would know that, simply in order to make the numbers reconcile with the fundamental law of exposure, the exposure of the cash sector would have to be adjusted downwards by $600,000. The system would then make this adjustment automatically, before calculating the weights and returns for performance measurement purposes. It is a sound "belt and braces" approach for a performance measurement system to implement the second strategy automatically, regardless of whether the investment manager is attempting to explicitly load exposure adjustments for the cash sector (using the first calculation strategy). The rationale for this is that there is simply no point in doing performance calculations using data that clearly doesn't reconcile. Provided that the performance system clearly discloses the value of its exposure adjustment for the cash sector, there is nothing to lose (and plenty to gain) by having a system that automatically introduces a balancing item into the exposure calculations. Changes Over TimeOn every day for which performance calculations are done, one needs to carry out the exposure adjustments we have just described. Consider again a portfolio that starts off (at Time 0) with $1 million cash, and an S&P 500 futures contract whose exposure is $250 x 1400 = $350,000. At the end of the first performance measurement period (Time 1), let us suppose that the price of the S&P 500 futures contract has gone up to 1500 points. For the sake of simplicity, let us also suppose that the cash return has been zero. We would like to calculate the portfolio value, and the asset class weights, at Time 1. The key to both of these is understanding what has happened to the S&P 500 futures contract. The exposure of this contract at Time 1 will have gone up by $250 x 100 = $25,000. Thus, the exposure at Time 1 will be $375,000. The market value of this contract is identical with the unrealized profit or loss. This will have moved from a starting position of zero to a position of $25,000. The return on the S&P 500 sector will be $25,000 / $350,000 = 7.14%. The return of the cash sector will be 0 / $650,000 = zero. The overall portfolio return will be $1,025,000 / $1,000,000 = 2.50% exactly. We can check this with a contribution analysis. The weighted sum of the sector returns should agree with the portfolio return. 65% x 0 + 35% x 7.14% = 2.50% exactly. The foregoing shows how we can calculate sensible weights, returns, and contributions, by having regard to exposure. For the sake of completeness, we may like to consider what the sector weights are at Time 1. The exposure of S&P 500 futures is $375,000. The market value of the futures contract is $25,000. The leverage entailed in the S&P 500 Futures Contract is $375,0000 - $25,000 = $350,000. Thus, the cash exposure has not changed at all: it is $1 million less $350,000 = $650,000. The weights at Time 1 will therefore be:
Astute readers will have noticed that we glossed-over a couple of tricky details in this section. In particular, we assumed a cash return of zero, and this allowed us to gloss-over issues concerning the relationship between cash rates and futures valuation. Readers who are troubled by this can rest assured that we will dig into this extra detail very soon. Using Exposures in Return Calculations In the preceding section, we did some return calculations. These followed the same approach that we always recommend:
Note that, if one doesn't have exposure data, one can calculate perfectly meaningful returns at the total portfolio level (because, in line with the fundamental rule of exposure, the sum of exposures and sum of market values in a portfolio will always be the same). However, if one doesn't have exposure data, one will be unable to calculate correct returns below the total portfolio level whenever derivatives are involved. Equitization of CashThis is a handy place to discuss a common investment management technique: equitization of cash. A classic example of this would be an investment manager who believes that always being fully-invested in the market is essential to good management. Every afternoon, the manager gathers data about all of the cash and receivables in the fund. Before the close of trade, he buys as many S&P 500 futures contracts as he is able to without actually driving the cash exposure below zero (i.e. leveraging the fund). By simply purchasing the right number of futures contracts, the manager is able to transfer virtually all the portfolio's cash exposure into equities exposure every evening. In the event that there had been a substantial redemption from the fund, and the fund was in danger of effectively holding negative cash (i.e. being leveraged), the portfolio manager could sell S&P 500 futures contracts (whether or not he already owned any), in order to increase the cash exposure and decrease the S&P 500 exposure. This highlights a key property of futures and other derivatives: they do not create exposure. Rather, they move exposure between cash and the underlying. To properly understand the performance effect of a derivatives position, one always needs to look both at the effect on the underlying exposure, and the effect on the cash exposure. Going Deeper into the S&P 500 Futures ExampleThe example we have been using to explore exposure calculations is a portfolio that starts out with $1 million cash, then buys an S&P futures contract at 1400 points, creating an equity exposure of $250 per point = $350,000. As we have seen, the increased equity exposure is matched by an "equal and opposite" decrease in cash exposure (i.e. the change in cash exposure is equal in sign and opposite in magnitude). Accordingly, the fundamental economic effect of the futures contract is to transfer exposure from cash to equities. To understand some aspects of this, it is useful to understand exactly how a futures contract is priced. A useful explanation of futures pricing is Determining the Relevant Fair Value(s) of S&P 500 Futures by Ira G. Kawaller. This document explains the relationship between the fair value and spot value of the futures contract, using the following equation:  In this equation:
The fair value of the futures price is the price we would normally expect for an S&P 500 futures contract, based on information such as the spot value, interest rates, and dividend yields. On the other hand, the spot value of the index is simply the current value of the index (for example, as one would see it reported in the newspaper or on a financial web site). The fair value for the futures contract differs from the spot value, because the futures contract is for delivery on a future date. An easy way to understand this difference is by working through an example. To make the numbers easy, suppose that the cash interest rate is 8% per year, while the anticipated dividend yield on the S&P 500 futures index is 2% per year. Somebody who holds S&P 500 futures instead of actually owning the physical shares will be sacrificing the 2% dividend yield on the shares. On the other hand, they will be earning 8% interest on the cash that they otherwise would have needed to outlay in order to buy the shares. The net difference is 6% per year. Using the above equation, we can see that if one is holding a futures contract whose maturity date is 60 days in the future, the fair value for the futures contract will be (8% - 2%) x (60/360) = 1% higher than the spot value of the S&P 500 index. Accordingly, if the spot value of the index was 1200, one might expect to see the futures contract whose expiry is 60 days away trading at something like 1212. The extra 12 points constitutes a time premium that will decay away day-by-day as the contract approaches expiry.
See Exposure.xls for a spreadsheet showing the daily decay of the time premium right through to expiry. This example assumes that "everything else is equal" as the contract rolls toward expiry. Specifically, it assumes that the S&P500 index remains constant, and that the dividend yield and cash yield are completely smooth. In reality, of course, these other variables do fluctuate over time. But it is useful to understand how the pricing of leveraged instruments such as futures contracts accurately reflects the fundamental economic leverage involved in these instruments. Summarizing this section, the fair value of a futures contract reflects the underlying economic reality that the instrument transfers exposure from the case sector to the equity sector. One component of the return from a futures contract is the "time decay" corresponding to the implicit cash borrowing behind the instrument). We will see some specific implications arising from this in the section on Synthetic Income. However, the general point to understand is that exposure calculations are essential not just for understanding the performance effects in the sector where the leveraged instrument is being held, but also for understanding the performance effects on a portfolio's cash holdings. Synthetic IncomeAs we saw in Table 1, the time decay on an S&P 500 futures contract might cause its value to drop from $303,000 to $300,000 over two months, even though the underlying S&P 500 index had not changed in value at all (recall that this is on the assumption of an 8% cash rate and a 2% dividend yield on the S&P 500 index). One might consider the slight performance difference between the futures contract and the underlying index to be a distortion. This is one of the issues in measuring the performance of a leveraged instrument. But a another (closely-related) issue is how one measures the performance of cash. Let us continue working with the example of a portfolio that buys a single S&P 500 futures contract at 1212 (exposure = $303,000), which then decays in value to $300,000 over two months. Suppose that the portfolio started out with $1 million cash. Even after buying the futures contract, the portfolio would still actually be holding $1 million cash. According to the assumptions we introduced in the preceding section, the cash return is 8% per year. Hence, over two months, the cash return would be one-sixth of 8%, which is 1.33%. The initial amount of cash is exactly a million dollars. After two months, this would grow to $1,013,333.33. In other words, the interest earned by the cash sector would have been a bit over thirteen thousand dollars. This all seems very simple and straightforward. But remember the Fundamental Rule of Exposure. This tells us that purchasing the S&P500 futures contract reduced the cash exposure by exactly the same amount as it increased the equity sector exposure. Hence the cash sector exposure at the start of the calculation period was $1 million less $303,000, which equals $697,000. If we calculate the return of the cash sector on that basis, we would have to say that the cash sector return was $13,333.33 / $697,000 = 1.91%. This is way higher than the index return for cash, purely because the exposure calculation in this case reduces the denominator. This is quite arguably an even bigger distortion than the $3,000 time decay in the equity sector. Essentially, the problem is that the denominator in the return calculations has been adjusted (to take account of exposure), but the numerator has not been adjusted at all. One possible remedy would be to abandon the idea of exposure, but this would lead to much larger problems (for example, one needs to understand that the futures contract takes the portfolio's equity weight to over 30%). A less drastic remedy, which has been proven by many years of use, is to use a concept called "synthetic income". Synthetic income is simply an adjustment to the numerator in the return calculation. This adjustment simply transfers some interest between leveraged sectors and the cash sector. For example, TODO What is the Sound of One Hand Clapping?A useful guideline when doing performance measurement on derivatives is the Zen question "What is the sound of one hand clapping?". If one always considers this principle, one will look for the other things that are affected when a derivatives instrument is introduced to the portfolio. For example, an S&P 500 futures contract, as we have seen, changes the performance of the sector holding the futures, as well as the performance of the sector holding the cash to back the S&P 500 futures contract. Similarly, a swap can create exposure to asset classes such as equities or bonds, but just like a futures contract, it entails a financing cost, which needs to be treated as borrowing cash. Given the almost infinite complexity in derivative instruments, it may seem impossibly demanding to require that performance calculations for every instrument must always consider the impact on cash as well as the impact on the underlying sector. TODO: Always need to consider cash as well as the non-cash sector in leverage calculations: Equal and opposite exposure adjustments Equal and opposite synthetic income.
TODO: Examples for futures and options. Examples should show weights and returns. Also they should illustrate the effect on cash backing. TODO NB Exposure is important for calculating the correct security-level return. But it is also important for calculating the correct security-level weight. TODO: One can have multiple cash sectors in a portfolio. By choosing how to arrange these, one can influence where the financing costs for derivatives will appear. Options Example TODO Margin Payments - Essentially Not a Performance Issue TODO: basic explanation of margin payments. Investors A and B both have portfolios whose starting position is one million dollars cash. They each purchase one S&P 500 futures contract at 1400 points (an exposure to equities of $350,000). So their asset allocation is 65% cash, 35% equities.
TODO Total Return Swap Example TODO: Provide an example showing how a TR swap works. Two important differences are the regular margining, and the fact that one pays fees by swapping for LIBOR+Ybps. Show a diagram for how the cost of the swap can be contained within the equities sector (if it's an equities swap) by having a sub-sector.
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13th November 2006.