Basis = -

S1: 2.50

If the asset to be hedged and the asset underlying the futures contract are the same, the basis should be zero

Prior to expiration, the basis may be positive or negative

When the underlying asset is a stock index or low-interest-rate currency or gold or silver, the futures price is greater than the spot price; the basis is negative

For high-interest-rate currencies and many commodities, the reverse is true and the basis is positive

If spot price increases by more than the futures price; this is referred to as a strengthening of the basis

If future price increases by more than the spot price; this is referred to as a weakening of the basis

Basis Risk

In practice, hedging is often not quite as straightforward as our earlier examples indicate; reasons are as follows:

The asset whose price is to be hedged may not be exactly the same as the asset underlying the futures contract

The hedger may be uncertain as to the exact date when the asset will be bought or sold

Example: Examinimg the Basis Risk:spot at time t1

The hedge may require the futures contract to be closed out well before its expiration date

These problems give rise to what is termed basis risk:

The Basis:

Spot price of asset to be hedged Futures price of contract used

S2: 2.00

F1: 2.20

F2: 1.90

b1: (S1 - F1) 0.30

b2: (S2 - F2) 0.10

2.30 = 2.30

Profit (F1 - F2) 0.30

2.30 = 2.30Profit (F1 - F2) -0.30

spot at time t2

futures price at time t1

The effective price that is obtained for the asset with short hedging is therefore:

S2 + F1 - F2 = F1 + b2

Long Hedge: Price realizedThe effective price that is paid for the asset with long hedging is therefore:

futures price at time t2

basis at time t1

basis at time t2

Short Hedge: Price realized

S2 + F1 - F2 = F1 + b2

The hedging risk is the uncertainty associated with b2 (the basis risk)

The hedging risk is the uncertainty associated with b2 (the basis risk)

For investment assets such as currencies, stock indices, gold and silver, the basis risk tends to be less than for consumption commodities; because arbitrage arguments lead to well-defined relationship between the future price and the spot price of an investment asset

S1*: 2.50S2*: 2.00S2: 2.20F1: 2.20F2: 1.90b1: (S1 - F1) 0.30b2: (S2 - F2) 0.10

2.50 ≠ 2.30Profit = (F1 - F2) 0.30 (+) (S2 - S2*) 0.20Profit =

S1:S2: 0.7200spot at time t2

2.50This can be written as:

Example: Examinimg the Basis Risk in a Short Hedge (Receive 50 million Yen in July; enter Short September Futures)

spot at time t1

Short four (4) September yen futures contracts on March 1Close out contract when yen arrive at end of JulyBasis Risk: uncertainty as to the difference between the spot price and September futures price of the yen at the end of July

F1 + (S2* - F2) + (S2 - S2*)

0.50

The basis risk for an investment asset arises mainly from uncertainty as to the level of the risk-free interest rate in the future.

The asset that gives rise to the hedger's exposure is sometimes different from the asset underlying the hedge; the basis risk is usually greater.

Example: Examining the Basis Risk:spot at time t1spot at time t2

futures price at time t1

The effective price that is obtained for the asset with hedging is therefore:S2 + F1 - F2 = F1 + b2

spot of asset being hedged

futures price at time t2basis at time t1basis at time t2

Short Hedge: Price realized

F1: 0.7800F2: 0.7250b1: (S1 - F1)b2: (S2 - F2) -0.0050

0.7750 = 0.7750Profit (F1 - F2) 0.0550

S1:S2: 20.0000F1: 18.0000F2: 19.1000b1: (S1 - F1)b2: (S2 - F2) 0.9000

18.90 = 18.90Profit (F2 - F1) 1.1000

The hedging risk is the uncertainty associated with b2 (the basis risk)

futures price at time t2basis at time t1basis at time t2

Long Hedge: Price realizedThe effective price that is obtained for the asset with short hedging is therefore:S2 + F1 - F2 = F1 + b2

Closes out contract when it is ready to purchase the oilBasis Risk: uncertainty as to the difference between the spot price and December futures price of oil when the oil is needed

spot at time t1spot at time t2

futures price at time t1

basis at time t2

Short Hedge: Price realizedThe effective price that is obtained for the asset with short hedging is therefore:S2 + F1 - F2 = F1 + b2

Takes a long position in 20 NYM December oil futures contracts in June

basis at time t1

The hedging risk is the uncertainty associated with b2 (the basis risk)

Example: Examinimg the Basis Risk in a Long Hedge (Airline needs to purchase 20,000 barrels of crude oil in October or November; enter Long December Futures)

futures price at time t1futures price at time t2

∆S∆FδSδFph*

δSδF

Hedge Ratio: hh*

0.5

δ∆Fδ∆S

Variance of Position

Dependence of variance of hedger's position on hedge ratio

1 = 1

Minimum Variance Hedge RatioThe hedge ratio is the ratio of the size of the position taken in futures contracts to the size of the exposureIf the objective of the hedger is to minimize risk, setting the hedge ratio equal to one (1.0) is not necessarily optimal

h* = p

The Optimal Hedge Ratio

correlation of coefficient betw: ∆S & ∆FThe Optimal Hedge Ratio

0.5

0.51

Month ∆F ∆Si xi Xi^2 yi Yi^21 0.021 0.000441 0.0290 0.000841 0.0006092 0.035 0.001225 0.0200 0.000400 0.00073 -0.046 0.002116 -0.0440 0.001936 0.0020244 0.001 0.000001 0.0080 0.000064 0.0000085 0.044 0.001936 0.0260 0.000676 0.0011446 -0.029 0.000841 -0.0190 0.000361 0.0005517 -0.026 0.000676 -0.0100 0.000100 0.000268 -0.029 0.000841 -0.0070 0.000049 0.0002039 0.048 0.002304 0.0430 0.001849 0.002064

10 -0.006 0.000036 0.0110 0.000121 -0.00006611 -0.036 0.001296 -0.0360 0.001296 0.00129612 -0.011 0.000121 -0.0180 0.000324 0.00019813 0.019 0.000361 0.0090 0.000081 0.00017114 -0.027 0.000729 -0.0320 0.001024 0.00086415 0.029 0.000841 0.0230 0.000529 0.000667

∑Xi^2 ∑Yi^2 ∑XiYi∑Xi -0.0130 0.0138 ∑Yi 0.0030 0.0097 0.0107

Mean = -0.00086667

δF 0.03134341 δS 0.026254795

p 0.928

0.02630.0313

= 1

Data to calculate Minimum Varaiance Hedge

0.778 = 0.928

0.5

1 = 1

NA 20,000.00 QF 1,000.00N*

h*NA 15,553.01

h*NAQF

0.778*20,0001,000

S = NA * SF = QF * FδS =δF =p =

δSδF

Futures contract priceStd. Dev New SStd. Dev. New FCoefficient of correlation betw: new S and new F

The Optimal Hedge Ratio

N* = p

N* =

15.55 =

Notation Transition:value of position being hedged

This means that the futures contracts bought should have 77.8% of the face value of the asset being hedged

Optimal Number of Contractssize of position being hedged (units)size of one futures contract (units)Optimal number of future contracts for hedging

Futures contracts used should have a face value of = h*NA

SF

= 200.00= #########= 0.10= 0.04= 1.50= 0.33

500.00$ 204.04

################

Spot Index . = 180 180180.90

Gain = #########

30.0 = 1.50

Current One mos. Fo=

Value of S&P indexValue of portfolioRisk-free interest rateDividend yield on S&P 500

Future contract is for delivery of Multiplier:Current Fo=

Maturity - Four mos.

Stock Index Futures - Hedging

ß=(beta); this is the slope of the best fitted line obtained when the excess return on the portfolio over the risk-free rate is regressed against the excess return on the market over the risk-free rate

N* = ß

Beta of Portfolio

Example: Futures contract on SP 500 with 4 mos. to maturity, over the next three

Loss on Index = 10.0%Dividend 4% p.a. 1.0%Net Loss o/ 3 mos 9.0%

= 1.5 X (Return on Index - Risk-free interest rate)= -0.1475

Portfolio Value = #########Gain = #########Hedged Port. Value = #########

From 1.5 to .75 Short 15 contractsFrom 1.5 to 2.0 Long poistion in 10 contracts

SF

15.00 0.75 20.00

SF

10.00 0.50 20.00

10.00 (ß* - ß)

To change beta of the portfolio brom ß to ß*; where ß > ß*; pursue a short position in:

To change beta of the portfolio brom ß to ß*; where ß < ß*; pursue a Long position in:

Expected Return on the portfolio - Risk-free interest rate

Changing Beta (using the above example of 30 contracts

Beta: Has been reduced to zero

To reduce the beta of a portfolio to some value other than zero:

(ß - ß*)15.00