There are two questions about the minimum-variance hedge ratio that are of interest: 1. By how much does optimal hedging reduce uncertainty over the alternative of not hedging
(i.e., using h= 0)?
2. How much larger is the variance of cash flows if we hedge one-for-one (i.e., set h= 1) rather than using h∗?
(A) The Alternative of Not Hedging
If we do not put on a hedge, then h = 0. Substituting h = 0 in (5.9), the variance of the unhedged cash flow is
Q2σ2(S) (5.15)
Comparing (5.14) and (5.15), we see that optimal hedging reduces cash-flow variance by a factor ofρ2. For instance, ifρ = 0.90, then ρ2= 0.81, so optimal hedging removes 81%
of the unhedged cash-flow variance, i.e., the variance of the hedged position is only 19% of the variance of the unhedged position. On the other hand, ifρ = 0.30, then ρ2= 0.09, so
even optimal hedging removes only 9% of the unhedged cash-flow variance.
(B) The Alternative of Hedging One-for-One
If we use a hedge ratio of h= 1, the cash-flow variance in (5.9) becomes
Q2σ2(
S)+ σ2(F)− 2cov (S,F)
(5.16)
which can be rewritten as
Q2 σ2(S) (1− ρ2)
+ Q2[σ (
F)− ρσ(S)]2 (5.17)
Comparing this to the variance (5.14) under h∗, we see that using a hedge ratio of unity
quantity. This is intuitive: a lower correlation implies a lower minimum-variance hedge ratio h∗, so the greater is the error we are making by using a hedge ratio of unity.
Indeed, hedging one-for-one may even be worse than not hedging at all! Compare (5.17) and (5.15). The difference between these quantities is
Q2 σ2(F)− 2cov (S,F) = Q2σ2 (F)− 2ρσ(S)σ(F) (5.18)
IfσF > 2ρσS, this difference is positive, which means the variance of the cash flow with a
hedge ratio of unity is higher than the variance of the unhedged cash flow.
5.6
Examples
In this section, we present two examples to illustrate minimum-variance hedging. Both examples involve basis risk arising from commodity mismatches. The first example looks at cross-hedging in currencies. The second example concerns hedging an equity portfolio using futures on another portfolio.
Example 5.2
Cross-Hedging with Currencies
Suppose that a US exporter will receive 25 million Norwegian kroner (NOK) in three months and wishes to hedge against fluctuations in the US dollar (USD)-NOK exchange rate. Assume there is no active forward market in NOK, so the company decides to use a forward contract on the euro (EUR) instead. The company has gathered the following data:
1. The standard deviation of quarterly changes in the USD/NOK exchange rate is 0.005. 2. The standard deviation of quarterly changes in the USD/EUR forward rate is 0.025. 3. The correlation between these changes is 0.85.
What should be the company’s minimum-variance hedging strategy?
The spot asset in this example is the NOK, so one “unit” of the spot asset is one NOK. The company will receive 25 million NOK in three months, which must be converted to USD. Thus, it is effectively as if the company has a commitment to sell Q= 25 million NOK in three months, i.e., it has a short spot exposure.
The forward contract used to hedge this exposure has the euro as its underlying asset, so one “unit” of the forward contract is a forward calling for delivery of one euro at maturity. There is commodity basis risk since the asset underlying the forward contract and the asset being hedged are not the same.
We are givenσ(S)= 0.005, σ(F)= 0.025, and ρ = 0.85. From (5.12), the variance-
minimizing hedge ratio is given by
h∗ = ρ σ (S)
σ (F) = 0.85 ×
0.005
0.025 = 0.17
In words, the optimal hedge position is to take 0.17 units of forwards per unit of spot exposure. Why only 0.17, i.e., why is the hedge position so “small”? Loosely speaking, the euro trades roughly on par with the dollar (at the time of writing, around USD 1.45/EUR), while the Norwegian kroner costs only a fraction of that (at the time of writing in September 2009, around USD 0.17/NOK). Reflecting these price differentials, the quarterly standard deviation of the USD/EUR forward rate in the example is five times larger than the 0.005 quarterly standard deviation of the USD/NOK exchange rate.
In hedging NOK price risk with the euro, we are trying to compensate for losses from NOK price movements with gains from euro price movements and vice versa. Since the typical euro price move is five times as large as the typical NOK price move, we want to use far fewer euros in the hedge position than the number of NOK in the spot exposure.
Returning to the computations, since Q is given to be 25 million and we have estimated
h∗= 0.17, the optimal forward position calls for the delivery of H∗ = h∗Q = 4.25 million euros
Finally, note that since the hedge ratio is positive and the company has a short spot exposure, this forward position must be a short one.
To summarize: the company’s optimal hedge is to take a short forward position calling for the delivery of 4.25 million euros in three months. If the company’s data is correct, this optimal hedge will removeρ2= (0.85)2= 0.7225, or about 72% of the variance associated
with the unhedged position. ■
Example 5.3
Cross-Hedging with Equities
Consider the problem of hedging a portfolio consisting of S&P 100 stocks using S&P 500 index futures.1Suppose that:
1. The value of the portfolio is $80,000,000. 2. The current level of the S&P 100 index is 800. 3. The current level of the S&P 500 index futures is 960.
4. One S&P 500 index futures contract is for 250 times the index.
The underlying asset in this problem is the S&P 100 index. That is, one “unit” of the underlying asset is the basket of stocks used to construct the S&P 100 index. The current price per unit S of this asset is simply the current level of the index, so S = 800. Since the portfolio value is given to be $80 million, the number of “units” in the portfolio is [80,000,000/800] = 100,000. Therefore, Q = 100,000.
The asset underlying the futures contract is the S&P 500 index, i.e., one “unit” of the asset underlying the futures contract is the basket of securities used to construct the S&P 500 index. The current futures price per unit is simply the current level of the S&P 500 index futures, which gives us F = 960. Note that the futures contracts are standardized in size: one futures contract calls for delivery of 250 units of the S&P 500 index.
There is evidently basis risk in this problem since we are hedging one asset (the S&P 100 index) with futures written on another asset (the S&P 500 index). To determine the optimal hedging scheme, therefore, we need information on variances of spot and futures price changes over the hedging horizon, and the covariance of these price changes. Suppose we are given the following information:
1. σ(S)= 60. 2. σ(F)= 75. 3. ρ = 0.90.
Then, the optimal hedge ratio is
h∗ = ρ σS
σF = 0.90 ×
40
50 = 0.72
i.e., to take 0.72 units of futures positions per unit of spot exposure. Since Q = 100,000, the size of the optimal futures position is
H∗ = h∗· Q = (0.72)(100,000) = 72,000
That is, the optimal futures position calls for the delivery of 72,000 units of the S&P 500 index. One unit of the futures contract is for 250 units of the index. Therefore, we should take a futures position in (72,000)/250 = 288 contracts.
Should this be a long or short futures position? By hedging, we are trying to protect the value over the hedging horizon of the S&P 100 portfolio that we hold. Thus, it is as if we have a short spot exposure in three months and want to lock-in a value for this. Since the
hedge ratio is positive, our futures position should also be a short one. ■
5.7
Implementation
To implement a minimum-variance hedging scheme in practice we must identify h∗. There are two equivalent ways in which this may be accomplished, both using historical data on spot and futures price changes. The first is to estimate each of the three parameters (σ(S),
σ (F), andρ) that go into the computation of h∗. The second, and easier, method is to
estimate h∗directly from the data using regression analysis. We describe both approaches below.
In each case, we rely on the use of data on spot and futures prices at specified sampling intervals. For specificity, we take the sampling interval to be daily, though, of course, data of different frequency could also be used.
So, suppose that we have data on daily spot and futures price changes. Assume that price changes across different days are independent and identically distributed. LetδSdenote the
random daily spot price change andδFthe random daily futures price change. Further, let
• σ2(δ
S) denote the variance of daily spot price changesδS.
• σ2(δ
F) denote the variance of daily futures price changesδF.
• cov (δS,δF) denote the covariance ofδSandδF.
• ρ(δS,δF) denote the correlation ofδSandδF.
Each of these quantities may be estimated easily from historical time-series data on daily spot and futures prices.