Evaluación de la integración ambiental en la programación

We recall some basic definitions and properties of random variables. Let X and Y be random variables with variancesσ2

X andσY2, respectively. Let E(·) denote expectation. Then, the

covariance of X and Y is defined as

cov ( X, Y ) = E(XY ) − E(X)E(Y )

The correlationρ(X, Y ) between X and Y and cov (X, Y ) are related via

ρ(X, Y ) = cov ( X, Y )_σ

XσY

If a is any constant, then

Variance (a X ) = a2_{Variance ( X ) = a}2_σ2

X

Finally, if a and b are any constants, then

Variance (a X− bY ) = a2Variance ( X )+ b2Variance (Y )− 2ab cov (X, Y )

5.3

The Cash Flow from a Hedged Position

Suppose that a specific futures contract has been chosen for hedging purposes. (We formalize later the criterion that should guide this choice.) Let F denote the current price of the contract and S the current spot price of the asset being hedged. Let FT and ST denote, respectively,

We stress that one or both kinds of basis risk may be present: the asset underlying the futures contract may not be the same as the asset being hedged, and the date T may not be the maturity date of the futures contract. Thus, we may not have ST = FT.

We treat the futures contract as if it is a forward contract that is marked-to-market once at termination. That is, the resettlement profits (or losses) from taking a long futures position at inception and closing it out at time T are given by FT− F. The impact of daily

marking-to-market on the optimal hedge position is considered in Section 5.8 (see “Tailing the Hedge”).

Consider first an investor with a commitment to buy Q units of the asset on date T . To hedge this position, the investor

1. Takes a long futures position of size H at inception at the futures price F .

2. Closes out the futures position at time T by taking a short futures position of size H . 3. Buys the required quantity Q on the spot market at time T .

To handle the possibility that the initial futures position may be a short one, we will allow

H to take on negative values also and interpret a long position of (say)−10 units as a short

position of 10 units. Under this strategy, there is a cash outflow of Q ST at time T towards

the spot purchase. There are also resettlement profits from the futures position at this time of H ( FT − F). Thus, the net cash outflow is

Q ST− H (FT− F) (5.4)

The investor must choose H to minimize the variance of the cash flow (5.4).

Now consider an investor with a commitment to sell Q units of the asset on date T . To hedge this, the investor

1. Takes a short futures position of size H at inception at the futures price F .

2. Closes out the futures position at time T by taking a long futures position of size H . 3. Sells the quantity Q on the spot market at time T .

Once again, we allow H to be negative to allow for the possibility that the initial futures position is a long one. Under this strategy, there is a cash inflow of Q STat time T from the

spot market sale. There are also resettlement profits from the futures position of H ( F− FT).

Thus, the net cash inflow is

Q ST+ H(F − FT) (5.5)

which is identical to (5.4). Thus, both a long and short investor want to choose H to minimize the variance of the cash flow (5.4).

5.4

The Case of No Basis Risk

If there is no basis risk, identifying the minimum variance hedge ratio is a simple matter. In this case, we must have ST = FT, so (5.4) becomes

Q ST− H (FT− F) = QST − H (ST − F)

= (Q − H) ST+ H F (5.6)

At the time the hedging strategy is initiated, Q and F are known quantities, so the only unknown here is ST. If we set H = Q, the term involving ST drops out of (5.6) and the

cash flow reduces to the known quantity H F = QF. The variance of this cash flow is zero. Since variance cannot be negative, we cannot improve upon this situation. Thus, if there is

no basis risk, it is optimal to hedge completely, i.e., the minimum-variance hedge ratio is

h∗= 1, and this eliminates all risk.

The important question is, of course, what if basis risk is present? The next section provides the answer.

5.5

The Minimum-Variance Hedge Ratio

To identify the minimum-variance hedge ratio, we first rewrite the cash flow (5.4) from a hedged futures position in terms of price changes. LetS = ST − S and F = FT − F

denote the changes in spot and futures prices, respectively, over the hedging horizon. Add and subtract the quantity Q S to (5.4) to obtain

Q ST− QS + QS − H(FT− F) = Q(ST− S) − H(FT− F) + QS

= QS− HF+ QS (5.7)

Now, let h= H/Q denote the hedge ratio. The cash flow (5.7) can be expressed in terms of the hedge ratio as

Q [S− hF]+ QS (5.8)

We want to pick h to minimize the variance of this quantity. Note that the last term Q S is a known quantity at the time the hedge is put on, so contributes nothing to the variance. From (5.8), the variance of hedged cash flows comes from three sources:

• The variance of spot price changesS. Denote this quantity byσ2(S).

• The variance of futures price changesF. Denote this quantity byσ2(F).

• The covariance between these quantities, denoted cov (S,F).

Using this notation, the variance of hedged cash flows (5.8) is Var [Q (S− hF)]= Q2Var (S− hF)

= Q2 _σ2

(S)+ h2σ2(F)− 2h cov (S,F)



(5.9)

The presence of the h2_{term ensures that the last term is U-shaped as a function of h (see}

Figure 5.1). To identify the point of minimum variance, we take the derivative of (5.9) with respect to h and set it equal to zero. This yields

2hσ2(F)− 2cov (S,F) = 0

or hσ2_(

F)= cov (S,F). Thus, the variance-minimizing value of h is

h∗ = cov (S,F) σ2

F

(5.10)

To express h∗in terms of the correlationρ between SandF, note that by definition

ρ = cov (S,F)

σ (S)σ (F)

(5.11)

Thus, cov (S,F)= ρσ(S)σ (F), so h∗can also be written as

h∗ = ρ σ(S) σ(F)

(5.12)

Expression (5.12) is the main result of this chapter. In words, as mentioned earlier, it says that the optimal hedge ratio is the correlationρ between price changes adjusted by a “scaling factor”σ (S)/σ (F).

FIGURE 5.1

The Minimum- Variance Hedge Ratio

Hedge Ratio

h*LC h*HC h*PC

0

Cash-Flow Variance

Var(CF): Low Correlation Var(CF): High Correlation Var(CF): Perfect Correlation

The minimum-variance hedge ratio is illustrated graphically in Figure 5.1. The figure considers a low level of correlation, a high level of correlation, and the limiting case of perfect correlation. It highlights two points. First, the minimum-variance hedge ratio increases as correlation increases. Second, the minimized cash-flow variance (i.e., the variance of cash flows under the minimum-variance hedge ratio) is lower as correlation is higher, which is intuitive: higher correlation implies a superior ability to offset cash-flow risk by hedging. In the limit, when correlation is perfect, the minimized cash-flow variance is also zero.