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RISK EXPOSURE AND HEDGING Everyone faces uncertainties, but only some of those uncertainties matter to your business. For example, a farmer might see that future exchange rates, employment levels, crop yields, and commodity prices are all uncertain. Of these uncertainties, crop yields and prices are likely the only ones that really affect the farmer’s business. These types of uncertainties are true risks; the others are uncertain, but are not truly risks to the farmer. It is common to say that people have “exposure” to risks. Exposure then relates to uncertainties that “matter.”1 While this explanation of exposure may seem straightforward, analyzing and managing exposure requires careful consideration of the degree of exposure. To manage exposure, one must be able to determine exactly which uncertainties are important and measure their effects. This requires two key steps: •
Determine whether a change in a given variable will make you better or worse off. If you are unaffected by changes in the variable, there is no exposure. If the variable matters to you, you have exposure to it.
•
Once you have identified the variables to which you have exposure, quantify the relationship between an incremental change in each variable and the resulting change in your NPV (or something else you care about—quarterly profits, project costs, etc.). Exposure is often described in incremental terms—“an incremental increase in variable X will result in an adverse impact on NPV of Y.”2
Consider an airline. It has to buy a lot of jet fuel to keep its planes running. What happens to the airline if the price of jet fuel increases by 10 cents a gallon—will the airline be 1
This section and definition draw heavily on David E. Bell and Arthur Schleifer, Jr., chap. 2 in Risk Management (Cambridge, Mass.: Course Technology, Inc., 1995). 2 Describing exposures this way implies a linear relationship between the risk factor X and the variable of concern Y. This convention is used even when the relationship is not linear over an extended range. For example, modified duration is used in bond trading to describe how much the price of a bond will change for a given change in yields—even though the relationship between yield and a bond’s price is not linear over a big range. However, the approximation should be satisfactory for relatively small changes. This technical note was written by Lee Fiedler, MBA ’02, and Professor Samuel Bodily, John Tyler Professor of Business Administration, Darden Graduate School of Business Administration, University of Virginia. Copyright 2003 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to
[email protected]. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without the permission of the Darden School Foundation. ◊
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better or worse off? If the airline immediately recovers the increased cost with a fuel surcharge to its passengers (and that the same number of people fly even with this surcharge), the airline has no exposure. On the other hand, the airline may not be able to pass on the higher costs of jet fuel. If this is true, the airline has exposure, and it is useful to quantify the exposure in terms of incremental effects on the business. For example, if the airline buys 2,000,000 gallons of jet fuel every month, it is exposed to the extent of $20,000 per 1 cent increase in jet fuel price per month. Quantifying exposure may be relatively straightforward if only one variable affects economic performance. For example, the airline may be able to enter into a contract today that specifies a price it will pay for jet fuel at the time of next month’s jet fuel purchase. It quickly becomes more complicated if more than one variable is involved. For example, the amount of jet fuel needed next month may be uncertain. If the airline is exposed to both the quantity and the price of jet fuel, it gets trickier to develop effective policies for managing the combined exposure. Simply put, multiple exposures make it more difficult to describe exposure in terms of incremental changes in the company’s interests.3 Hedging a Single Exposure Once a firm identifies its exposure, it may seek to reduce or eliminate it. This activity is called hedging. In its most simple terms, hedging is taking a long position in one asset that moves in the opposite direction to the exposure. A common situation is that there is no uncertainty about the quantity of an asset that needs to be hedged—only about what the value of that asset will be in the future. In this situation, a manager can use forward contracts to hedge. A forward contract is simply an agreement to exchange an asset in the future for a price that is determined today. When two parties initially enter into a forward contract, no cash changes hands; payment is made when the asset is delivered. For example, a farmer who knows how much corn his land will produce can “lock in” his revenues by selling corn forward. The farmer delivers the corn at the specified time and receives the price specified in the contract at that time. If a manager is concerned about hedging the price of a known quantity of an asset and can enter into forward contracts for that asset, the manager should be able to construct a hedge that eliminates virtually all risk. In many situations, however, there may be no forwards (or other hedging instruments) available for the relevant asset.
3
A lot of work has gone into developing models that distill the multiple dimensions of option exposure into a single metric. The most common of these is called Value at Risk (VAR). Unlike the definition of exposure used in this note (e.g., a change in risk factor X will lead to a given change in the measure of concern Y), VAR is based on percentiles of a risk profile of the potential financial effects of a set of risks. VAR reports the best and worst outcomes—within a given confidence level.
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Cross-Hedging In many situations, hedging instruments are not available for the specific asset a firm is concerned about, but they may be for other related assets. For example, there may be forward contracts available on gasoline, fuel oil, and other petroleum products, but not on aviation fuel. If you are interested in hedging the price of aviation fuel, this creates the need for a type of hedging strategy called cross-hedging. In a cross hedge, you use a forward contract (or some other hedging instrument) based on one asset to hedge the price of another asset. Conceptually, cross hedging is like using the price of apples to hedge the price of oranges. For something to be useful as a cross hedge, it must have a correlation to the exposure of concern. Let’s return to the airline example; assume you are the operations manager. At the end of the month, you must pay for the 2,000,000 gallons of aviation fuel that the airline will use in the following month, which at this point we assume to be a known quantity. You hate uncertainty about the price of aviation fuel, since fluctuations in it can dramatically affect your bottom line. Your goal is to reduce the uncertainty about the cost of the aviation fuel for next month. There are no forward contracts on aviation fuel,4 so you will have to use another type of asset for the hedge. Over the past two years, you have observed the following prices for aviation fuel and finished gasoline:5
4 5
A similar example—aviation fuel and heating oil—is used in Hull, Options, Futures, & Other Derivatives. The data in this example were taken from The CRB Commodity Yearbook, 2001.
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-4Table 1 Price History for Gasoline and Aviation Fuel
Jan-99 Feb-99 Mar-99 Apr-99 May-99 Jun-99 Jul-99 Aug-99 Sep-99 Oct-99 Nov-99 Dec-99 Jan-00 Feb-00 Mar-00 Apr-00 May-00 Jun-00 Jul-00 Aug-00 Sep-00 Oct-00 Nov-00 Mean Standard deviation
Average refiner price of finished gasoline (cents per gallon) 59.20 56.80 65.10 79.00 78.20 75.60 80.60 86.50 88.80 87.10 88.40 90.30 91.70 98.70 113.10 108.70 110.30 121.30 116.20 109.30 116.70 114.80 113.40 93.47
Average price of Aviation Fuel (cents per gallon) 87.00 85.00 89.70 101.30 103.50 103.30 110.00 114.80 117.70 118.40 117.40 120.70 119.60 123.80 133.80 130.70 133.60 140.80 142.10 140.15 138.20 134.90 134.90 119.19
19.38
17.67
By plotting the price of aviation fuel against the refiner price of finished gasoline, you notice the following pattern:
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Average monthly price of aviation fuel
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135.00 125.00 115.00 105.00 95.00 85.00 56.00
66.00
76.00
86.00
96.00
106.00
116.00
Average monthly price of finished gasoline
Let’s assume that you can enter into forward contracts on finished gasoline. Since the prices of aviation fuel and finished gasoline tend to move closely together, you should be able to use forward contracts for finished gasoline to hedge the cost of the aviation fuel. To set up the hedge, you will want to enter into a contract to buy some amount of finished gas at the end of the month. Entering the contract does not cost you anything today, but at the end of the month, you will pay for the gasoline and immediately sell it at the spot price, without taking delivery. You will use the proceeds from selling the gasoline to offset the cost of buying your aviation fuel. If you do this right, you should be able to eliminate a lot of the exposure you have to changes in the price of aviation fuel. (In this example we will ignore transaction costs and assume that all exchange of money takes place at the same time.) The question is: How many gallons of finished gasoline should you contract to buy at the end of the month? The number of gallons of finished gasoline you buy forward for every gallon of aviation fuel is called the hedge ratio, which we will denote as H in the following calculations. Since your goal is to reduce the uncertainty about the cost of the aviation fuel, your focus should be on achieving the lowest standard deviation in your net cost. By using the historic price data, we can examine the standard deviation of the net cost for different hedge ratios. Every month the company buys one gallon of aviation fuel and sells H gallons of gasoline. The net cost is thus: Net cost = Price per gallon of aviation fuel - (H * Price of per gallon of finished gasoline) Trying the calculations for different values of H should tell us about the relative merits of different hedge ratios. Calculating the Net Cost for the price data of Table 1 gives the following:
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-6Table 2 Net Cost for Various Hedges Using Price Data H= Jan-99 Feb-99 Mar-99 Apr-99 May-99 Jun-99 Jul-99 Aug-99 Sep-99 Oct-99 Nov-99 Dec-99 Jan-00 Feb-00 Mar-00 Apr-00 May-00 Jun-00 Jul-00 Aug-00 Sep-00 Oct-00 Nov-00 Standard deviation
0.25 72.20 70.80 73.43 81.55 83.95 84.40 89.85 93.18 95.50 96.63 95.30 98.13 96.68 99.13 105.53 103.53 106.03 110.48 113.05 112.83 109.03 106.20 106.55 12.93
0.50 57.40 56.60 57.15 61.80 64.40 65.50 69.70 71.55 73.30 74.85 73.20 75.55 73.75 74.45 77.25 76.35 78.45 80.15 84.00 85.50 79.85 77.50 78.20 8.31
0.75 42.60 42.40 40.88 42.05 44.85 46.60 49.55 49.93 51.10 53.08 51.10 52.98 50.83 49.78 48.98 49.18 50.88 49.83 54.95 58.18 50.68 48.80 49.85 4.24
1.00 27.80 28.20 24.60 22.30 25.30 27.70 29.40 28.30 28.90 31.30 29.00 30.40 27.90 25.10 20.70 22.00 23.30 19.50 25.90 30.85 21.50 20.10 21.50 3.72
In this example, a hedge ratio of 1.00 offers the lowest standard deviation of net cost for the four different hedge ratios considered. There may however be an even better hedge amount. Using regression analysis to find the optimal hedge ratio Finding the hedge ratio by plugging in different numbers and calculating standard deviations is not very efficient. Fortunately, regression analysis can be used to quickly find the optimal hedge ratio. Regression also provides information about the amount of exposure reduction in this hedge. To use regression for hedging, regress the price of the variable that is important to you (Y variable) against the price of the variable you are considering to use as a hedge (X variable). The regression coefficient between the two variables is the optimal hedge ratio. For example, in the aviation fuel and finished gasoline example from above, regression returns the following information:
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-7Price of aviation fuel versus price of finished gasoline Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.9841 0.9685 0.9670 3.2106 23.0000
ANOVA
df Regression Residual Total
Intercept Average refiner price of finished gasoline (cents per gallon)
MS 6,650.29 10.31
F 645.16
Significance F 0.00
1.00 21.00 22.00
SS 6,650.29 216.47 6,866.76
Coefficients 35.33
Standard Error 3.37
t Stat 10.49
P-value 0.00
Lower 95% 28.33
Upper 95% 42.34
Lower 95.0% 28.33
Upper 95.0% 42.34
0.8971
0.04
25.40
0.00
0.82
0.97
0.82
0.97
The regression thus tells us that the optimal hedge ratio is .8971. This answers the question of how many gallons of gasoline the airline manager should buy forward. The best hedge would be a forward contract to buy 1,794,200 gallons of finished gasoline (2M gallons aviation fuel * .8971). Intuition may have told you that the best thing to do was to hedge on a gallon-for-gallon basis, which means buying 2,000,000 gallons of gasoline forward—a hedge ratio of 1. However, this results in a higher standard deviation than the optimal hedge ratio will be (3.72 versus 3.14). In addition, note how much lower the hedged standard deviation is than the standard deviation of jet fuel alone (3.14 versus 17.67); even without an ideal hedging instrument (e.g., a forward contract for aviation fuel), you can significantly reduce risk by crosshedging.6 You may be curious about why the regression coefficient works as the optimal hedge ratio. The answer is that regression, by definition, fits a line to the data so that total variance is minimized. Since variance equals standard deviation squared, the regression coefficient will produce the lowest standard deviation for Y – (H * X). You can also calculate the hedge ratio directly by solving for: H = CorrelationX & Y * (Standard DeviationY / Standard DeviationX) 6
In fact, the standard deviation of the portfolio Y – (H * X) is exactly the same as the standard deviation of the regression residuals. You may also notice that the standard deviation of Y – (H * X) is very close to the standard error of the regression model. The two measures of dispersion are very similar, but the standard error makes a slight adjustment. To be more precise: Standard deviation = SQRT((Σ(x – mean)2) / (n – 1)) Standard error = SQRT((Σ(x – mean)2) / (n – 2)) The only difference between the two measures is that standard deviation divides by (n – 1) and standard error divides by (n – 2). You can convert the standard deviation of the residuals into the standard error by multiplying the standard deviation by the ratio of the square root of the number of observations – 1 and the square root of the number of observations – 2. In our case, the calculation looks like 3.1368 * sqrt(23 – 1)/sqrt(23 – 2) = 3.2106, which is the standard error.
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In our example, this looks like: H = .9841* (17.6671 / 19.3799) = .8971 Although you can calculate the hedge ratio directly using the formula above, the regression output tells you important things about the desirability of the hedge. The R-squared effectively tells you what percentage of the variance in Y can be eliminated with the cross hedge.7 In looking for potential vehicles for cross-hedging, it is thus best to find the one that results in the highest R-squared and lowest standard error. This should help you to eliminate the most variability. In the aviation fuel example, the R-squared is .9685, which means that we have eliminated about 97 percent of the uncertainty about the net cost of next month’s aviation fuel. When using regression for forecasting, analysts generally like to start from an assumed cause and effect relationship; if X causes Y, using regression to forecast Y makes sense. In using regression for hedging, however, it is not necessary for X to cause Y; all that you need is a correlation between the two variables. One caveat—using regression for hedging assumes that the standard deviation is the appropriate measure of risk. In many cases this may be acceptable, but if the risk profile is not normally distributed, standard deviation is not synonymous with risk. Hedging Multiple Uncertainties In many situations there is only one source of exposure. For example, an airline may know exactly how much fuel it will use in a given period of time but be uncertain what that fuel will cost. However, managers often face multiple uncertainties. For example, a manager hedging revenues in a foreign currency may not know today what those revenues will turn out to be. Nor does a farmer hedging corn prices today know exactly how many bushels of corn his land will produce. As you might expect, deciding how to hedge multiple exposures, in these examples both quantity and price, is more complex than hedging a single exposure. Hedging multiple exposures requires a manager to understand all of the uncertainties she faces and how they interrelate. Let’s look at an aviation fuel example to consider what happens when we add uncertainty about the quantity of fuel the airline will use. Fuel usage is expected to be 2,000,000 gallons next month, but it is uncertain, a normal distribution with standard deviation of 100,000 gallons. 7
The standard deviation of the price of aviation fuel itself is 17.6671; with a hedge, we can reduce the standard deviation to 3.1368. Since the variance equals the standard deviation squared, you can check this by calculating the variance without the hedge (17.6671 ^ 2 = 312.1264) and comparing it to the variance with the hedge set at .8971 (3.1368 ^ 2 = 9.8395). The percentage change in the variance equals R-squared: R-squared = (312.1264 – 9.8395) / 312.1264 = .9685).
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And for the price of next month’s aviation fuel, let’s assume that since you know this month’s price, you have built a regression model that forecasts next month’s price for aviation fuel using the price in the current month. Your forecast price is 135.6 cents per gallon with a standard error of 3.9 cents (normally distributed). Independent Uncertainties In introducing another uncertainty about quantity, we also have to consider whether or not the quantity uncertainty is related to the uncertainty in price. Let’s assume first that the quantity of fuel used is independent of price. This is reasonable if fuel usage is driven by factors like the number of cancelled flights and wind speeds, which should be independent of fuel prices. Making that assumption means that the influence diagram for total fuel costs looks like: ~ Quantity
~ Price
Total Cost
Introducing an independent uncertainty about quantity means that there should be more total exposure for the manager. A Crystal Ball model of this scenario returns the graph below. As you would expect, the exposure here (as measured by the standard deviation) is greater than it is if there is no uncertainty about the fuel requirements. If there was no quantity exposure, the forecast mean cost of fuel would be $2,712,000 with a standard deviation of $78,000 (3.9 cent standard deviation in price * 2M gallons). Simulating both quantity and price exposure returned a standard deviation in total costs of about $157,600. Because the manager has much greater exposure in this case, his interest in hedging should be greater. While it is common to assume no quantity exposure, quantity risk can make a real difference to your total exposure and your strategy for hedging it.
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Forecast: Total costs no correlation 10,000 Trials
Frequency Chart
9,932 Displayed
.022
223
.017
167.2
.011
111.5
.006
55.75 Mean = 2,709,191
.000 2,291,309
2,497,565
2,703,822
2,910,078
0 3,116,334
Dependency Now let’s assume that the quantity of fuel used depends somehow on price. If the two exposures are dependent, the analysis gets even more complex. With dependent exposures, the influence diagram looks like: ~ Price
~ Quantity
Total Cost
The dotted line here is meant to indicate that price and quantity are not independent. To model the dependency, you need some way of describing the relationship mathematically. A typical measure of dependency is correlation. The sign (i.e., positive or negative) and magnitude of correlation between different risks affects both the overall exposure and the appropriate hedging strategy to use.
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Crystal Ball has a feature that facilitates correlating assumptions. First, define the assumptions in your model (to use this feature, you have to define assumptions with the Crystal Bar toolbar—it will not let you use the typed functions). Then go to the CB Tools menu and select “Correlation Matrix.” In the menus that follow, select the assumptions you want to correlate and tell Crystal Ball where to put the matrix in your model. The correlation matrix tool will put a matrix in your worksheet, after which you can enter correlations between different variables. Crystal Ball then automatically correlates the assumptions in your model. To explore what dependency does, we will use the same Crystal Ball model that generated the risk profile above. Below are the results of three trial runs in Crystal Ball and the same case with no uncertainty about the quantity of fuel used. As you can see, the standard deviation and variance with positively correlated risks are higher than they are with independent risks. The fact that a positive correlation between risks results in a higher standard deviation suggests that hedging should become more valuable.
Statistics Mean Standard Deviation Variance Percentage change in variance
Total costs with no quantity uncertainty
Total costs with two independent uncertainties
Total costs with two dependent uncertainties +.99 correlation
Total costs with two dependent uncertainties -.99 correlation
2,712,000
2,709,191
2,715,236
2,708,045
78,000
157,599
212,449
61,971
6,084,000,000
24,837,311,576
45,134,635,795
3,840,443,226
0%
308%
642%
-37%
By contrast, the standard deviation with negatively correlated risks is lower than it is with independent risks and is even lower than the standard deviation with no uncertainty in quantity. This situation is called a natural hedge. Depending on the strength of a natural hedge, managers may want to avoid using forward contracts or other derivatives, since doing so could actually result in greater exposure!
