Introduction to Futures Hedging

Hedging Linear Risk

The traditional approach to market risk management is

hedging

.

Hedging

consist of tacking position that lower the risk profile of the portfolio. We distinguish between two hedging method:

Static hedging

– consists of putting on the position, and leaving it until the hedging horizon.

Dynamic hedging

– consists of continuously rebalancing the portfolio to the hedging horizon.

Hedging Linear Risk

In general, hedging will create a

basis risk

.

Basis risk

arises when changes in payoffs on the hedging instruments do not perfectly offset changes in the value of the underlying asset. generally, hedging eliminates the downside risk but its also reduces upside in the position Thus, the profitability of hedging should be examined in the context of a risk-return tradeoff.

Futures Hedging

Numerical Example

Consider the situation of a U.S. exporter who has been promised a payment of Y125M in seven months. Two hedging alternatives: 1.

Forward hedging

– enter a 7-month forward contract in the OTC market –

perfect hedge

but with

low liquidity

.

2.

Futures hedging

– The CME lists yen contracts with face amount of Y12.5M that expire in 9 month –

creates

a

Basis Risk

but with

high liquidity

.

The exporter places an order to sell 10 contracts, with the intention of reversing the position in 7 months – when the contract will still have 2 months to maturity.

The P&L of the Hedged Position

Maturity r US r YEN Spot ($/Y) Futures ($/Y) Basis ($/Y) Initial Time

9 6% 5% 0.008

0.00806

-0.00006

Exit Time

2 6% 2% 0.006667

0.006711

-0.000045

Gain/Loss

-$166,667 $168,621 $1,954 Cash  Y 125 M  ( 0 .

00667  0 .

00800 )   $ 166 , 667 Futures  (  10 )  Y 12 .

5 M  ( 0 .

006711  0 .

00806 )  $ 168 , 621

The P&L on unhedged position: Q  S 2  S 1  The P&L on hedged position: Q Q     S 2 S 2   S 1 F 2     F 2 S 1   F 1 F 1       Q[b 2  b 1 ] where b=S-F which called the

basis

: The hedger objective is to minimize the basis risk

The Optimal Hedge Ratio

Definitions

 S – the change in the

dollar

value of the unhedged position.

 F – the change in the

dollar

value of the one futures contract.

N – the number of futures contracts.

 V – the total change in the

dollar

value of the hedged portfolio: Δ V  Δ S  N Δ F

The Optimal Hedge Ratio

The variance of the portfolio’s profits is: σ 2 Δ V  σ 2 Δ S  N 2 σ 2 Δ F  2 N ρ Δ S , Δ F σ Δ S σ Δ F Minimizing the variance of the portfolio’s profits with respect to N, we obtain that optimal number of futures contracts is: N *   ρ Δ S , Δ F σ Δ S σ Δ F Plugging N* in the variance equation, we obtain:  * 2  V   2  S   2  F ,  S  2  F

The Optimal Hedge Ratio-Rates of Changes

The OHR also can be expressed in terms of rates of changes in unit prices.

Definitions: Q

s

– number of units (shares, bonds…) in the cash position Q

f

– number of units in one futures contract

s

– unit spot prices

f

- unit forward prices S = Q

s *s

- the cash position F = Q

f * f

- the notional amount of one futures contract

The Optimal Hedge Ratio-Rates of Changes

R s – the rate of change in the spot price R

f

– the rate of change in the futures price We than can write:   F  Q

f f

 (R

f

)   S  Q

s

 (R

s

) N

*

  

f, s

  (R

s

)  (R

f

)  Q Q

f s s f

    Q Q

f s s f

Where  is the coefficient in the following regression R

s

    R

f

 

Numerical Example

Consider the pervious example of a U.S exporter and assume the following data: σ(R

s

)  8 % σ(R f )  10 % 

s, f

 0

.

85

What are the standard deviations in dollars?

What is the optimal hedge ratio?

What is the SD ($) of the full hedged portfolio?

What is the SD ($) of the optimal portfolio?

Numerical Example

The standard deviation in dollars:   S  

(

R

s )

 Q

s



s

 0

.

08  125 M  0

.

008  $ 0

.

08 M   F   (R

f

)  Q

f



f

 0

.

1  12

.

5 M  0

.

00806  $ 0

.

01 M The optimal hedge ratio:    F

,

S  (R S )  (R F )  0

.

85 0

.

08 0

.

1  0

.

68 N

*

   Q Q

f s

 

s f

  0

.

68 125 M 12

.

5 M   0

.

008 0

.

00806   6

.

75

Numerical Example

The SD($) of a full hedged portfolio (N=-10):   V   2  S  N 2  2  F  2 N   S

,

 F   S   F  0

.

08 M 2  (  10 ) 2  0

.

01 M 2  2  (  10 )  0

.

85  0

.

08 M  0

.

01  $ 0

.

053 M The SD($) of the optimal portfolio (N=-7):   V   2  S  N 2  2  F  2 N   S

,

 F   S   F  0

.

08 M 2  (  7 ) 2  0

.

01 M 2  2  (  7 )   0

.

85  0

.

08 M  0

.

01  $ 0

.

042 M

The Optimal Hedge Ratio

0.08

0.07

0.06

0.05

0.04

0.03

0 2 4 6 Number of Contracts 8 10

OHR

It can be shown from the OHR equation that: ρ 2 s, f  (σ 2 ΔS  σ 2 ΔS σ * 2 ΔV )  0 .

08 2  0 .

042 2 0 .

08 2  72 .

25 % which is the effectiveness of the hedge –the proportion of variance eliminated by the OHR. σ * 2 ΔV  σ ΔS ( 1  ρ 2 s,f )

Beta Hedging

Beta is a measure of the exposure of the rate of return on a portfolio to movement in the market portfolio rate of return: R it  α  β R mt  ε t Where  is the residual component and it is assumed to be uncorrelated with the market.

Ignoring the residual component: Δ S  β ( Δ M / M ) S

Beta Hedging

Assume that there is a stock-index futures contract, which has a beta of unity (the stock-index represent the market portfolio). Thus: Δ F  1  ( Δ M / M ) F Thus, the total change in the

dollar

value of the hedged portfolio composed of some portfolio and a short of

N

stock-index futures contracts : Δ V  Δ S  N Δ F  S β ( Δ M / M )  NF ( Δ M / M )  ( Δ M / M )  ( S β  NF )

Beta Hedging

Δ V  ( Δ M / M )  ( S β  NF )  V is set to zero when:  S   NF Thus, the optimal number of contracts to short is: N   β S F The quality of the hedge depends on the size of the residual risk –  2 (  ) The larger the portfolio the smaller the residual risk, as the individual stocks’ residual risk cancel each other.

Numerical Example

A portfolio manager holds a stock portfolio worth $10M, with a beta of 1.5 relative to the S&P 500. The current S&P index futures price is 1400, with a multiplier of $250. What is the number of contracts to sell short for optimal protection?

N

*

   S F   1

.

5 10 M 250  1400   43

Duration Hedging

Modified duration can be viewed as a measure of the exposure of rates of changes in prices to movement in yields:  p   D

*

P  y We can rewrite this expression for the cash and the futures position  S   D

*

S S  y  F   D

*

F F  y

Duration Hedging

The variances and the covariance are:  S 2 

 

S S 2  2

(

 y

)

 2 F 

 

F 2  2

(

 y

)

 S

,

F 

  

F D

*

S S  2

(

 y

)

Therefore the OHR: N

*

   S

,

F  2 F  

  

F D

 

F 2

*

S S  

 

S

 

F

Duration Hedging

In the case we have a target duration of Dv, this can be achieved by: D

*

S S  ND

*

F F  D

*

V V N

*





D

*

V V  D

*

F F D

*

S S



Numerical Example

A portfolio manager has a bond portfolio worth $10M with modified duration of 6.8 years. The current futures price is 93% with a notional amount of $100K and modified duration of 9.2 years. What is the number of contracts to sell for optimal protection N

*

 

 

S

 

F  6

.

8  10 M 9

.

2  0

.

93  100 K  79

.

5