Hull Chapter3 2013(2)

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Copyright @ Suman Banerjee 2013

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Cost of assetGain on Future

Net amount paid

2

2 1

S2 (F2 F1) = F1 + b2

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Price of assetGain on Futures

Price of asset

S2

F1 F2

S2 + (F1 F2 = F1 + b2

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Suppose we expect to sell NA units of asset at time t2 and choose to hedge at time t1 by shortingfutures contracts on NF units of a similar asset. The hedge ratio, which we denote by h, is

h =NF

NA

We will denote the total amount realized for the asset when the profit or loss on the hedge istaken into account byY, so that Y= S2NA (F2 F1)NFor

Y = S1NA + (S2 S1)NA (F2 F1)NF

Where S1 and S2 are the asset prices at times t1 and t2, and F1 and F2 are the futures prices at

times t1 and t2. From equation 1, the expression for Y in equation 2 can be written as

(1)

(2)

Y= S1NA +NA(S hF) (3)

Where S2 S1 and F = F2 F1

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Because S1 and NA are known at time t1, the variance of Y in equation (3) isminimized when the variance ofS - hF is minimized. The variance ofS-hF is

Where S, F, and are as defined so that

Setting this equal to zero, and noting that d2v/dh2 is positive, we see thatthe value of h that minimizes the variance is

dv

dh= 2h

2

F 2SF

v = 2

S+ h22

F 2hSF

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Table 1: Data to calcis used to hedge pur late minimum variance hedge ratiohase of jet fuel. hen heating oil futures contract

Monthi

Change in future priceper gallon (=xi)

Change in fuel priceper gallon (=yi)

1 0.021 0.029

2 0.035 0.0203 -0.046 -0.044

4 0.001 0.008

5 0.044 0.026

6 -0.029 -0.019

7 -0.026 -0.010

8 -0.029 -0.007

9 0.048 0.043

10 -0.006 0.01111 -0.036 -0.036

12 -0.011 -0.018

13 0.019 0.009

14 -0.027 -0.032

15 0.029 0.023

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xi =

0.013

x2i

= 0.0138

y2

i= 0.0097

yi = 0.003

xiyi = 0.0107

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Standard formulas for statistics give the estimate of F as

The estimate of S is

The estimate of is

From equation (1), the minimum variance hedge ratio, h*, is therefore

Each heating oil contract traded on NYMEX is on 42,000 gallons of heating oil. From equation(2), the optimal number of contracts is

or, rounding to the nearest whole number, 37.

x2

i

n 1

(

xi)2

n(n 1)= 0.0313

y2

i

n

1

(

yi)2

n(n

1)

= 0.0263

n

xiyi

xi

yi

[n

x2

i (

xi)2][n

y2

i (

yi)2]= 0.928

0.9280.0263

0.0313= 0.78

0.78 2, 000, 000

42, 000= 37.14

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Optimal number of contracts ifno tailing adjustment

Optimal number of contractsafter tailing adjustment to allow

or daily settlement of futures

(= SA QA)

(= FA QF)

=

hQA

QFhVA

VF

Spot Price

Futures priceSize of position being hedged (units)

Size of one futures contract (units)

Value of position being hedged

Value of one futures contract

QA

QF

VA

F

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h

= 0.928 0.0263

0.0313

= 0.7777

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The size of one heating oil contract is 42,000 gallons The spot price is 1.94 and the futures price is 1.99 (both dollars per

gallon) so that

VA = 1.94 2, 000, 000 = 3, 880, 000

VF = 1.99 42, 000 = 83, 580

Optimal number of contracts assuming no daily settlement =

Optimal number of contracts after tailing =

0.7777 2, 000, 000/42, 000 = 37.03

0.7777 3, 880, 000/83, 580 = 36.10

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To hedge the risk in a portfolio the number of contracts thatshould be shorted is

VA

VF

where VA is the value of the portfolio, is its beta, and VF is the

value of one futures contract

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N=

P

A

For example, a portfolio worth $1 million mirrors the S&P 500. The current value of the index is1,000, and each futures contract is on $250 times the index. In this case P = 1,000,000 and A =250,000, so that 4 contracts should be shorted to hedge the portfolio.

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N

= P

A

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Suppose the index turns out to be 900 in 3 months and the futures price 902. The

gain from the short futures position is then

The loss on the index is 10%. The index pays a dividend of 1% per annum, or 0.25% per3 months. When dividends are taken into account, an investor in the index would

therefore earn -9.75% in the 3-month period. The risk-free interest rate is approximately1% per 3 months. Because the portfolio has a of 1.5, the capital asset pricing modelgives

Expected return on portfolio Risk-free interest rate = 1.5 x (Return on index Risk- freeinterest rate)

It follows that the expected return (%) on the portfolio during the 3 months is

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Time t1 : Short futures contract 1

Time t2 :Close out futures contract 1Short futures contract 2

Time t3 :Close out futures contract 2Short futures contract 3

:

Time tn :Close out futures contract n-1Short futures contract n

Time T: Close out futures contract n

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Data for the exam le on rolling oil hedge forward

Date Apr. 2004 Sept. 2004 Feb. 2005 June 2005

Oct. 2004 futures price 18.20 17.40

Mar. 2005 futures price 17.00 16.50

July. 2005 futures price 16.30 15.90

Spot Price 19.00 16.00

(18.20 17.40) + (17.00 16.50) + (16.30 15.90) = 1.70

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Hull Chapter3 2013(2)

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