Appendix I.A.1.B: Utility Functions
X. The portfolio’s expected return is
µπ= w ⊤
µ.
Let Σ denote the variance–covariance matrix of the returns X1, . . . , Xd. This is the d × d matrix with entries
Σij= σiσjρij, i, j = 1, . . . , d.
Along the diagonal i = j, we have ρii = 1 (each asset is perfectly correlated with itself), so this reduces to
Σii= σ2i, the variance of the ith asset return.
The portfolio variance can be expressed compactly as σ2
Π= w ⊤
Σw using Σ and the portfolio weights w.
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15% 20% 25% 30% 35% 8% 10% 12% 14% 16% 18%
Portfolio Standard Deviation
Portfolio Mean
Figure I.A.2.7: The shaded region shows achievable pairs of mean and standard deviation for a portfolio invested in four assets. The thick line marking the upper boundary of the region is the efficient frontier.
I.A.2.4.3 Efficient Frontier
By varying the portfolio weights w1, . . . , wd, we get different combinations of the portfolio mean µΠ and standard deviation σΠ. As in the case of two assets, different investors may choose different combinations, but there are some combinations that no rational investor should choose because they are dominated by other portfolios.
To illustrate this point, we consider a four-asset example. The individual asset means and standard deviations are given by
(µ1, µ2, µ3, µ4) = (8%, 12%, 15%, 18%), (σ1, σ2, σ3, σ4) = (16%, 24%, 30%, 36%). For simplicity, we take all the correlations ρij, j 6= i, equal to 0.33. These parameters determine the set of pairs (σΠ, µΠ) that can be achieved by varying portfolio weights w1, w2, w3and w4. Achievable pairs of portfolio mean and standard deviation for this example are shown in Figure I.A.2.7. The four kink points on the right side of the region correspond to the four underlying assets: the risk–return combinations that can be attained by fully investing the portfolio in a single asset are (16%,8%), (24%,12%), (30%, 15%) and (36%,18%).
The curves connecting the kink points are the combinations that can be achieved by combining just those two assets. For example, the curve that connects the two middle kink points reproduces the curve in Figure I.A.2.4, because the second and third assets in this example have the same parameters as the assets XXX and YYY used in Figure I.A.2.4.
By investing the portfolio in all four assets, we can achieve all the combinations in the shaded region in Figure I.A.2.7. This shows that the possibilities are much richer with four assets than they were with just two.2
2The shaded region shows only those combinations that are at least as attractive as combinations that can
be achieved with just two of the four assets. One can construct inferior combinations — lying to the right of the shaded region — using the four assets, but these have been omitted from the figure.
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The thick line marking the upper boundary of the shaded region indicates the efficient frontier. These are the only combinations a rational investor should consider. Any combination not on the efficient frontier is dominated, meaning that there are other portfolios that achieve higher expected returns for the same risk or lower risk for the same expected returns.
The point marked by a dot inside the shaded region of Figure I.A.2.7 is an example of a dominated portfolio. Indeed, all combinations to the northwest (i.e., in the region bounded by the dotted lines) dominate because they offer a higher mean and a lower standard deviation. The portfolios on the efficient frontier correspond to solutions to the problem
min w w⊤Σw subject to w⊤µ= µ∗ 0 ≤ wi≤ 1, i = 1, . . . , d w1+ w2+ · · · + wd= 1,
as µ∗varies between the lowest and highest means of the underlying assets. This is a quadratic programming problem. It asks for the minimum variance that can be achieved with an expected return of µ∗.
The efficient frontier can also be characterized as the set of solutions to the utility maximization problem with utility function (I.A.2.25), as the risk aversion parameter γ varies.
I.A.2.5 A Hedging Example
As a further application of the ideas developed in the previous sections, we now consider a problem of hedging commodity price risk.
I.A.2.5.1 Problem Formulation
FlyFreight, a cargo company, has won a contract for a major shipping job that will begin in three months. As part of this job, FlyFreight will need to buy two million gallons of jet fuel. In order to win the contract, FlyFreight bid aggressively, and it is concerned that an increase in jet fuel prices could wipe out its profit on the deal. It therefore decides to try to hedge its price risk with futures contracts. As there are no exchange-traded contracts on jet fuel, it follows the industry practice of hedging with heating oil contracts traded on the New York Mercantile Exchange. Heating oil and jet fuel are chemically quite similar and their price fluctuations have thus historically shown a high degree of correlation.
The spot prices of heating oil and jet fuel are currently at 65 cents and 68 cents per gallon, respectively. The heating oil futures price for delivery in three months is 67 cents per gallon. Each heating oil futures contract is for 42,000 gallons (1,000 barrels).
Key to the hedging strategy is the observation that if the price of heating oil rises, then so does the value of a heating oil futures contract. When FlyFreight enters in a futures contract at 67 cents per gallon, the contract has zero value. If the price of heating oil rises to 70 cents, the contract — which allows the holder to buy at 67 cents — acquires positive value. Moreover, FlyFreight can collect this positive value by closing out its position, without ever buying or selling heating oil. If the price of heating oil drops to 65 cents, then the contract — which still commits the holder to buying at 67 cents — acquires negative value, and FlyFreight must pay to close out its position. Thus, the value of the contract moves in the same direction as the futures price of heating oil, and this will usually be the same direction as the price of jet fuel. We would like to address the following questions:
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(i) What is the price risk (as measured by standard deviation) faced by FlyFreight if it does no hedging?
(ii) What is the price risk if FlyFreight hedges jet fuel with heating oil on a gallon-for-gallon basis?
(iii) What is the risk–minimizing hedge? How effective is the best hedge?
To address these questions, we need more information. Suppose the percentage change (the “return”) in the price of jet fuel over three months has a standard deviation of 18%. The standard deviation of the percentage change in the futures price of heating oil over three months is 23%. The correlation between the two is 0.82.
In order to quantify the risk faced by FlyFreight, we will work with dollar amounts rather than percentage changes. So, let
X = change in heating oil futures price per gallon over three months, and Y = change in jet fuel price per gallon over three months.
Then, based on the information above, we find that X has a standard deviation of σX = 23% × $0.67 = $0.1541
and Y has a standard deviation of
σY = 18% × $0.68 = $0.1224.
We further posit that X and Y have zero expected value. This is particularly appropriate in the case of X, because a futures price already reflects expectations about price changes over the life of the contract. In contrast, the expected value for a change in the spot price of jet fuel could reasonably be different from zero. In this case, we could reinterpret Y as the difference between the spot price in three months and the expected price (which would imply E[Y ] = 0). For example, if the expected change in the price of jet fuel is 2 cents per gallon, we could assume that this expected price increase was reflected in FlyFreight’s bid; the risk to which FlyFreight is exposed lies, then, in deviations from this expected increase, not in the increase itself.
With this formulation of the problem, we can answer question (i), above. The change in FlyFreight’s unhedged position over the next three months is
−2,000,000Y,
because FlyFreight will need to buy 2,000,000 gallons and each gallon will cost Y more than expected. This position has a standard deviation (see (I.A.2.22)) of
2,000,000σY = $244,800.
I.A.2.5.2 Gallon-for-Gallon Hedge
Consider, next, a simple hedging strategy in which FlyFreight hedges the 2 million gallons of jet fuel it needs with 2 million gallons of heating oil. As each heating oil contract is for 42,000 gallons, this means that FlyFreight goes long
2,000,000
42,000 = 47.6 contracts,
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or simply 48 contracts.
FlyFreight will not take delivery of heating oil. Instead, it will close out its positions before the futures expire. If the prices of both heating oil and jet fuel go up, FlyFreight’s costs will go up but this will be offset, at least partly, by a profit from the heating oil contracts. If both prices go down, FlyFreight will lose money on the futures, but its cargo contract will be more profitable because its fuel costs will be lower.
With a gallon-for-gallon hedge, FlyFreight’s position thus becomes
(48)(42,000)X − 2,000,000Y.
This is actually a slight simplification in that it ignores the difference between futures and forwards.
Using (I.A.2.15), we find that the standard deviation of this position is
q
(48 × 42,000)2σ2
X+ (2,000,000)2σ2Y − 2(48)(42,000)(2,000,000)σXσYρXY. With σX = 0.1541, σY = 0.1224, and ρXY = 0.82, we get
StdDev[(48)(42,000)X − 2,000,000Y ] = $178, 092, which is lower than the unhedged standard deviation of $244,800.
I.A.2.5.3 Minimum-Variance Hedge
Can FlyFreight do better? Instead of assuming a gallon-for-gallon hedge, we now treat the number of contracts c as a variable over which we will optimize.
With c contracts, FlyFreight’s position becomes
42,000cX − 2,000,000Y. The variance of the position is therefore
(42,000c)2σ2X+ (2,000,000)2σ2Y − 2(42,000c)(2,000,000)σXσYρXY (I.A.2.27) and the standard deviation is the square root of this expression.
The variance of the position is a quadratic function of c. We can minimize the variance by differentiating with respect to c, setting the derivative equal to zero, and solving for c. In other words, we need to solve
(42,000)2σ2 Xc − (42,000)(2,000,000)σXσYρXY = 0. The solution is c∗ = 2,000,000 42,000 σY σX ρXY. (I.A.2.28) This evaluates to c∗ = 2,000,000 42,000 0.1224 0.1541 0.82 = 31.0
This is the number of contracts that minimizes the variance (equivalently, the standard devi- ation) of the hedged position.
c
0 10 20 30 40 50 60 0 50 100 150 200 250 Number of Contracts
Standard Deviation ($thousands)
Figure I.A.2.8: Standard deviation as a function of the number of contracts used to hedge.
Figure I.A.2.8 plots the standard deviation against the number of contracts. The figure con- firms that hedging with 48 contracts reduces risk compared to no hedge and that the optimal hedge is 31 contracts.
The shape of the curve in Figure I.A.2.8 follows from the fact that the variance (I.A.2.27) is quadratic in the number of contracts, but it can also be explained more intuitively as follows. If we start at c = 0 and increase the number of contracts, the risk initially decreases because fluctuations in jet fuel prices are partly offset by fluctuations in heating oil prices; this is the effect of the last term in (I.A.2.27), which is negative. However, beyond a certain point, the additional risk resulting from a long position in heating oil begins to overwhelm the hedging effect — the first term in (I.A.2.27) becomes greater than the third. In this “overhedging” region, the variance and standard deviation increase as we increase the number of contracts. The optimal point c∗
= 31 is the point that balances these two effects.
It is useful to decompose the optimal number of contracts in (I.A.2.28) into two pieces. The most important part is
β = σY σXρXY (I.A.2.29) which evaluates to β = 0.1224 0.1541 0.82 = 0.65.
This is the minimum-variance hedge ratio for hedging a gallon of jet fuel with gallons of heating oil. If we had exactly one gallon of jet fuel to hedge, we would hedge it with β gallons of heating oil. For 2,000,000 gallons of jet fuel we need 2,000,000β gallons of heating oil. To convert this from gallons to contracts, we divide by 42,000, the size of each contract. That gives c∗
in (I.A.2.28).
Notice that β does not equal 1, the coefficient implicit in the gallon-for-gallon hedging strategy. We would have β = 1 if the two price changes had the same standard deviation and were perfectly correlated.
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I.A.2.5.4 Effectiveness of the Optimal Hedge
We have seen that the risk-minimizing hedge for FlyFreight requires 31 contracts. But how effective is this optimal hedge? By how much does it reduce FlyFreight’s risk?
One way to answer this question is to look at the ratio StdDev[Optimal hedge]
StdDev[No hedge]
of the standard deviation with the optimal hedge and with no hedge. We have already found that the denominator is $244,080. To find the numerator, we can substitute c = 31 in (I.A.2.27) and then take the square root of the resulting value, which gives $140,115. The reduction in standard deviation is thus
140, 115
244, 080= 57.24%. (I.A.2.30) Equivalently, we may say that the optimal hedge eliminates 42.76% of the standard deviation. It is instructive to look at this calculation in greater generality. In calculating the optimal number of contracts c∗
, we minimized an expression of the form
σ2(c) = a2c2σ2X+ b2σ2Y − 2abcσXσYρXY, with a, b > 0. The optimal value of c is
c∗ = b a σY σX ρXY.
It follows by substituting the values c = c∗
and c = 0 that σ(c∗ ) σ(0) = q 1 − ρ2 XY. In other words,
StdDev[Optimal hedge]
StdDev[No hedge] =p1 − ρ 2 XY (I.A.2.31) At ρXY = 0.82, we get q 1 − ρ2 XY = 0.5724, consistent with what we obtained previously in (I.A.2.30).
The interesting feature of the risk reduction in (I.A.2.31) is that it is completely determined by the correlation coefficient ρXY. This means that from the outset, FlyFreight could have looked at the correlation ρXY = 0.82 between heating oil and jet fuel and known immediately that the optimal hedge would reduce risk to 57.24% of the unhedged risk.
Equation (I.A.2.31) also indicates that the effectiveness of the optimal hedge depends on the magnitude of the correlation, but not its sign. If the two commodities had a correlation of −0.82, we would achieve exactly the same hedge effectiveness, though of course in this case the optimal hedge would sell rather than buy heating oil contracts.
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-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.6 -0.4 -0.2 0 0.2 0.4 0.6
Change in Heating Oil
C h a n g e in J e t F u e l
Figure I.A.2.9: Scatter plot of price changes, with regression line
I.A.2.5.5 Connection with Regression
The minimum-variance hedge ratio β in (I.A.2.29) is the slope of the regression line in regressing Y against X. This leads to a graphical interpretation of minimum-variance hedging.
Figure I.A.2.9 shows a (hypothetical) scatter plot of changes in jet fuel prices and changes in the futures price of heating oil over three-month intervals. The standard deviations and correlation of the points in the figure match the values given above for σX, σY and ρXY. The line through the figure is the least-squares regression line.
The slope of the line, β, measures the average change in the price per gallon of jet fuel for each unit change in the future price per gallon of heating oil. The slope is thus an indication of how one asset moves with the other.
Consider, for example, an extreme case in which the changes fall exactly on a straight line:
∆Jet Fuel Price = β∆Heating Oil Futures Price;
i.e.,
Y = βX.
In this case, a position of the form Y − βX would be riskless, as would a position of the form −2,000,000Y + 2,000,000βX. (I.A.2.32) This is, in fact, the minimum-variance hedge in (I.A.2.28), because
2,000,000β = 2,000,000 × 0.65 = 1,300,000 gallons which is 31 contracts.
In Figure I.A.2.9, the linear relation does not hold exactly. Instead we have
Y = βX + ǫ
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for some residual ǫ. The optimal hedge (I.A.2.32) now leaves an unhedged variability repre- sented by
−2,000,000Y + 2,000,000βX = 2,000,000ǫ.
The residual in the regression is the portion of the variability in Y that cannot be removed by hedging with X. The optimal risk reduction (I.A.2.31) is
StdDev[ǫ] StdDev[Y ] =
q 1 − ρ2
XY.
The connection between minimum-variance hedging and regression is particularly useful in hedging with multiple assets. Let Y denote the price change to be hedged and let X1, . . . , Xd denote the changes in prices of the hedging instruments. We can use regression to estimate coefficients in the equation
Y = β0+ β1X1+ · · · + βdXd+ ǫ. The variance-minimizing hedge is then
Y − β1X1− · · · − βdXd.
In other words, to hedge a long position in Y , we should sell βi units of the ith asset for each unit of Y (or buy |βi| units if βi < 0). The effectiveness of this hedge (in the sense of the risk-reduction ratio (I.A.2.31)) is given by √1 − R2, where R2 is the usual coefficient of determination in the regression.
I.A.2.6 Serial Correlation
In Section I.A.2.4, we explained how correlations between pairs of assets affect the variability of portfolio returns. We now show how similar ideas can be used to relate the variability in returns in a single asset over different time horizons. For example, we can address the following question:
• How is the standard deviation of annual returns of a single asset related to the standard deviation of monthly returns for the same asset?
Throughout this section, we use X1, . . ., Xn to denote the returns of a single asset over con- secutive time periods, such as consecutive months. Notice that this differs from the notation in Section I.A.2.4, where the Xi were returns on different assets in the same period.
Also, throughout this section we will interpret the Xi as continuously compounded returns, in the sense of (I.A.2.2). With this convention, the return X over the entire period is the sum of the returns over the individual periods,
X = X1+ X2+ · · · + Xn. (I.A.2.33) To see this, suppose the returns are calculated using (I.A.2.2) from asset prices S0, S1, . . . , Sn; then X1+ X2+ · · · + Xn= ln S1 S0 + lnS2 S1 + · · · + ln Sn Sn−1 = ln S1 S0 S2 S1· · · Sn Sn−1 = ln Sn S0 .
From (I.A.2.33) we see that the effect of serial correlation in a single asset — i.e., correlations between the Xi — can be calculated in the same way as for a portfolio of correlated assets.
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Although (I.A.2.33) does not hold exactly for the simple returns in (I.A.2.1), it is sufficiently close that the qualitative conclusions of this section apply to simple returns as well.
Consider the case of uncorrelated returns: the correlation between Xi and Xj is zero, for all i 6= j. In this case, it follows from (I.A.2.15) or (I.A.2.26) that
Var[X] = Var[X1] + Var[X2] + · · · + Var[Xn].
The return variance over the full horizon is the sum of the variances over the individual periods. If we further assume that the Xi all have the same variance σ2(e.g., all monthly returns have the same variance), we get
Var[X] = nσ2, and then
StdDev[X] = StdDev[X1+ · · · + Xn] =√nσ. (I.A.2.34) In the case of n = 12 monthly returns, this says that the standard deviation of annual returns is √
12 times larger than the standard deviation of monthly returns. To annualize a daily standard deviation one would similarly use√250 because there are approximately 250 business days in a year.
The fact that the annual standard deviation is only√12 times larger than the monthly number and not 12 times larger is similar to the diversification effect we observed in our discussion of portfolio standard deviation. In a portfolio, losses in some assets will tend to be partly offset by gains in others. Similarly, losses in some months will tend to be offset by gains in others. The “square root of time” formula (I.A.2.34) is often used, for example, to convert a value- at-risk figure from one time horizon to another. However, it is important to stress the two assumptions that underlie this formula:
• the monthly returns have a common standard deviation σ; • the monthly returns are uncorrelated with each other.
What happens if we drop the second assumption? From (I.A.2.26) we see that if all the serial correlations ρij are positive, then
StdDev[X] = StdDev[X1+ · · · + Xn] > √
nσ,
and the simple rule (I.A.2.34) understates the true variability. Positive serial correlations in returns suggest an asset with a price trend or “momentum”.
If, on the other hand, many of the largest correlations are negative — for example, the con- secutive correlations ρi,i+1 — then we may have
StdDev[X] = StdDev[X1+ · · · + Xn] <√nσ,
in which case (I.A.2.34) overstates the true variability. Negative serial correlations may suggest a mean-reverting price process, in which losses are often followed by gains, and vice versa.