hedging using futures contracts finance (derivative securities) 312 tuesday, 22 august 2006...

Hedging Using Hedging Using Futures ContractsFutures Contracts

Finance (Derivative Securities) 312

Tuesday, 22 August 2006

Readings: Chapters 3 & 6

Long v ShortLong v Short

Long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price

Short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price

Arguments For HedgingArguments For Hedging

Companies should focus on the main business they are in and take steps to minimise risks arising from interest rates, exchange rates, and other market variables

Arguments Against Arguments Against HedgingHedging

Shareholders are usually well diversified and can make their own hedging decisions

It may increase risk to hedge when competitors do not

Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult

ConvergenceConvergence

Time

Spot Price

FuturesPrice

t1 t2

Basis RiskBasis Risk

Basis is the difference between spot & futures

Basis risk arises because of uncertainty about the basis when the hedge is closed out

Long HedgeLong Hedge

Suppose that

F1 : Initial Futures Price

F2 : Final Futures Price

S2 : Final Asset Price

Hedge future purchase of an asset by entering into a long futures contract

Cost of Asset = S2 – (F2 – F1) = F1 + Basis

Short HedgeShort Hedge

Suppose that

F1 : Initial Futures Price

F2 : Final Futures Price

S2 : Final Asset Price

Hedge future sale of an asset by entering into a short futures contract

Price Realised = S2 + (F1 – F2) = F1 + Basis

Choice of ContractChoice of Contract

Choose delivery month as close as possible to, but later than, the end of the life of the hedge

When no futures contract available on asset being hedged, choose contract whose futures price is most highly correlated with the asset price• Creates two components to basis

Optimal Hedge RatioOptimal Hedge Ratio Proportion of exposure that should optimally be

hedged is

where S is standard deviation of S, the change in the

spot price during the hedging period F is standard deviation of F, the change in the futures price during the hedging period is correlation coefficient between S and F

h S

F

Index FuturesIndex Futures

To hedge risk in a portfolio, the number of contracts that should be shorted is

where P is value of the portfolio, βis its beta, and A is value of the assets underlying one futures contract

P

A

P

A

Reasons for Hedging Equity Reasons for Hedging Equity PortfoliosPortfolios

Desire to be out of the market for a short period of time• Hedging may be cheaper than selling portfolio

and buying it back

Desire to hedge systematic risk• Appropriate when you feel that you have

picked stocks that will outperform the market

Hedging a PortfolioHedging a Portfolio

Suppose that:• Value of S&P 500 is 1,000• Value of Portfolio is $5 million• Beta of portfolio is 1.5• Risk-free rate is 4%, dividend yield is 1% p.a.

What position in 4-month futures contracts on the S&P 500 is necessary to hedge the portfolio?

N* = 1.5 x 5,000,000/250,000 = 30


Suppose index is 900 in three months’ time, futures price is 902• Futures gain is 30 x (1,010 – 902) x 250 =

$810,000• Index loss 10%, dividend 0.25% per 3

months, overall loss –9.75%

• E(RP) = Rf + β[E(RM) – Rf]

= 1 + 1.5(–9.75 – 1) = –15.25


Expected value of portfolio after 3 months• 5,000,000 x (1 – 0.15125) = $4,243,750

Hedger’s position• 4,243,750 + 810,000 = $5,053,750• Value has increased by the risk-free rate (1%)

Rolling Hedges ForwardRolling Hedges Forward

Can use a series of futures contracts to increase the life of a hedge

Each switch from one futures contract to another incurs an element of basis risk

Interest Rate FuturesInterest Rate Futures

Treasury Bonds: Actual/Actual (in period)

Corporate Bonds: 30/360

Money Market Instruments: Actual/360

PricingPricing

T-bond Cash price = Quoted price + Accrued Interest

T-bill Quoted price:

where Y is the cash price of a Treasury bill that has n days to maturity

360100

nY( )

T-bond FuturesT-bond Futures

Cash price received (short position) =

Quoted futures price × Conversion factor + Accrued interest

Conversion factor for a bond is approximately equal to the value of the bond on the assumption that the yield curve is flat at 6% with semi-annual compounding


Suppose that:• Cheapest-to-deliver bond is 12% bond with

conversion factor of 1.4000• Delivery in 270 days• Last coupon 60 days ago, next in 122 days,

then 305 days• Yield curve flat at 10%• Quoted price $120

What is the quoted futures price?


Cash price:• 120 + (6 x 60/[60 + 122]) = 121.978

PV of next coupon:• 6e-0.1(0.3342) = 5.803

Cash futures price:• (121.978 – 5.803)e0.1(0.7397) = 125.094

Quoted futures price:• 125.094 – 6 x (148/[148 + 35]) = 120.242• 120.242/1.4000 = 85.887


Factors that affect the futures price:

• Delivery can be made any time during the delivery month

• Any of a range of eligible bonds can be delivered

• The wild card play

Eurodollar FuturesEurodollar Futures If Q is the quoted price of a Eurodollar futures

contract, the value of one contract is 10,000[100-0.25(100-Q)]

A change of one basis point or 0.01 in a Eurodollar futures quote corresponds to a contract price change of $25

Settled in cash Expires on third Wednesday of the delivery

month and all contracts are closed out• Z is set equal to 100 minus the 90 day Eurodollar

interest rate (actual/360)

Eurodollar Forward Eurodollar Forward RatesRates

Eurodollar futures contracts last as long as 10 years

For Eurodollar futures lasting beyond two years we cannot assume that the forward rate equals the futures rate

Eurodollar Forward Eurodollar Forward RatesRates

A "convexity adjustment" often made is

Forward rate = Futures rate

where is the time to maturity of the

futures contract, is the maturity of

the rate underlying the futures contract

(90 days later than ) and is the

standard deviation of the short rate changes

per year (typically is about

1

2

0 012

21 2

1

2

1

t t

t

t

t

. )

DurationDuration

Duration of a bond that provides cash flow ci at time ti is:

where B is its price and y is its yield (continuously compounded)

tceBi

i

ni

yti

1

DurationDuration

This leads to:

When the yield y is expressed with compounding m timesper year

This expression is referred to as “modified duration”

yDB

B

my

yBDB

1

D

y m1

Duration MatchingDuration Matching

This involves hedging against interest rate risk by matching the durations of assets and liabilities

It provides protection against small parallel shifts in the zero curve

Duration HedgingDuration Hedging

Suppose that:• On 2 August, fund manager has $10m

invested in govt bonds• Hedges porftolio with December T-bond

futures, priced at 93.0625• Duration of portfolio in 3 months is 6.8• CTD bond is 20yr 12% bond, yield = 8.8%,

duration will be 9.2 at maturity of futures

Duration HedgingDuration Hedging

No. of contracts:• (10m x 6.8) / (93,062.50 x 9.20) = 79.42

If portfolio rises to 10.45m on 2 Nov, futures price = 98.50, loss on contracts:• 79 x (98,500 – 93,062.50) = $429,562.50

Net change in manager’s position:• $450,000 - $429,562.50 = $20,437.50

Duration-based Hedge Duration-based Hedge RatioRatio

FC = Contract price for Interest Rate Futures

DF = Duration of asset underlying futures at maturity

P = Value of portfolio being hedged

DF = Duration of portfolio at hedge maturity

FC

P

DF

PD

hedging using futures contracts finance (derivative securities) 312 tuesday, 22 august 2006...

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