Financial Engineering 5

References: Villalobos, Luenberger, Faerber

Lecture 16

Introduction to Hedging (Risk Mitigation)

Lecture Topics • Introduction • The perfect hedge • Minimum variance hedge • Examples • Assignments

Risk • Risk comes from uncertainty

– Weather – Economy – Demand – Market

• How does uncertainty affect companies: – Cost of materials – Sales performance – Profits

• How can people protect themselves from risk? – Insurance – Contracts – Futures

Hedging • Making an investment to reduce the risk of adverse price movements

in an asset. – Normally, a hedge consists of protecting a position in a related

security (or securities). • If you own a stock, then short selling an equal amount would be a

hedge. • Hedging can be achieved through related securities or commodities.

– For example, using corn futures to reduce the risk of sorghum prices.

– Sorghum does not have its own futures market, but prices tend to track closely with corn.

• Investors hedge when they are uncertain of what the market will do. • A perfect hedge reduces your risk to nothing (except for the cost of

the hedge). • The perfect hedge is achieved by taking an equal and opposite

position in the security or the futures market.

Real Examples • Southwest Airlines hedged 80% of fuel prices for 2004 with

prices below $24 per barrel, 80% for 2005 at $25 per barrel and 45 percent hedged for 2006 with prices capped at $28 per barrel.

• Northwest Airlines – only 7% of its 2004 fuel needs are hedged at $37 per barrel and none of its 2005 fuel needs are hedged.

• American Airlines – no hedging. • United Airlines – no hedging.

• Southwest’s net income for the second quarter of 2004, was $113 million, $90 million of which was attributable to the lower jet fuel prices afforded by their hedging program.

• American and United paid around $700 million and $750 million respectively in additional fuel costs during 2004.

The Perfect Hedge • An example of a perfect hedge is a corn farmer sells future or

forward contracts for an amount identical to the expected corn harvest from the farmer’s corn field.

– In this case the farmer will have a short position in futures and a long position in the commodity (corn).

• Another example is when a buyer of the corn such as a producer of cattle feed or ethanol, hedges the risk of unfavorable corn prices by buying a futures or forward contract on corn based on the amount corn that will be actually purchased at a future date.

– In this case the buyer has a long position on futures and short position on the commodity (corn).

• Note, it might be considered a perfect hedge if the farmer used a forward contract with a default free party.

• It might be considered an almost perfect hedge if the farmer used a futures contract due to the standardized contracts and process of the market.

Not Quite a Perfect Hedge • There are different reasons why perfect hedging is usually infeasible.

– Such as the uncertainty of production, or the date of production might not correspond to availability of contracts, etc.

• Let’s take the example of a corn farmer who expects to harvest her crop on the middle of July.

– In theory the farmer can hedge her investment by taking a short position in the futures market at the same level as the expected harvest (long position).

– Once the farmer takes a short position she has protection against reductions in price.

– However, she will not benefit from price increases. • Once the harvest date approaches the farmer would “close” her short

position and the proceedings from the futures account plus the proceedings of selling the crop at the prevailing spot market prices would in theory be equivalent to the expected total proceedings when the futures contract was acquired.

• However, because of factors such as the location of the actual harvest, timing of harvest relative to the acquisition and closing of contracts positions, the resulting hedge is less than perfect.

Definition of Basis • Basis is defined as the difference between the Future market price

and the corresponding future spot price of the commodity. Basis = S – F or Basis = cash price – closest futures price

• The basis represent the spread between the future’s price for a contract and the corresponding spot price at the location of delivery.

• Basis is most often calculated as the difference between the cash price and the closest to expiration futures contract.

– For example, in June the wheat basis would be calculated using the current cash price minus the July futures contract price.

• The basis are affected by the carrying cost for the commodity. – So, it is affected by interest rates and their expectation, and

storage costs and their dynamics. • As the expiration of the contracts approach, the spread tends to

reduce until it is zero at expiration date. • For holding periods less than the expiration date we need to be

concerned with the movement of basis. • For current spot prices see http://www.ams.usda.gov

Historical Spread

Sources of Risk for the Basis • When the contract will not be held to maturity:

– Hedger benefits from a narrow spread of the basis. – Speculator benefits from a wider spread of the basis.

• Changes in the convergence of the futures prices to cash prices.

• Changes in factors that affect the cost of carry (storage) such as interest rates.

• Mismatches between the exposure being hedged and the futures contracts being used as the hedge.

• Random deviations from cost-of-carry relation.

Expectations • From the previous discussion we can say that the price of the futures

contract is the same as the expected spot price at the time of the contract expiration.

• This implies that the return made over the investment is only the return provided by the risk free rate.

• Some people have argued that to entice people to invest in futures they expect the prices to behave in a different manner.

– They argue that the commodities market transfers risk from the hedgers to the speculators.

• Thus the speculators expect a rate higher than the risk free rate investment.

– Assuming the hedgers are the SELLERS or farmers, the spectators have an interest in the market if the expected spot prices are higher than the futures’ prices and rise as the contract maturity approaches.

– This is called Normal Backwardation. • If the hedgers are the BUYERS of the commodity, then for the speculators

to enter the market, the expected spot price must be lower than the future’s prices and decrease as the contract maturity approaches.

– This is called Normal Contango.

Minimum Variance Hedge • A perfect hedge is possible only if there is a futures contract that

exactly matches, in terms of the nature of the asset and delivery, the obligation being hedged.

• Because this is not usually possible, and completely eliminating the risk is not an option, an alternative security or commodity is contracted to minimize the risk.

• A measure of the “perfection” of the hedge is given by the Basis where:

Basis = spot price of asset being hedged – futures contract price

• In general the final basis may not be zero but a random variable. • The variance of this random variable is an estimation of the risk

associated with the hedge.

Minimum Variance Hedge • Suppose that at time zero the situation to be hedged is

described by a cash flow x to occur at time T.

where S is the spot price at time T and W is the position or quantity taken in the “hedged” asset.

• By assuming that the account is settled at time T, and assuming no interest paid in the margin account or its payments, the cash flow (y) at time T is given by:

• Where h is the hedge position, and FT – F0 is the payments in Futures account.

• What should h be such that we minimize the variability of our cash flow y.

( )hFFxy T 0−+=

x WS=

Minimum Variance Hedge • The variance of the cash flow is:

( ) ( ) ( ) ( ) 2var,cov2varvar hFhFxxy TT ++=

( ) ( )T

T

F Fxh

var ,cov

−=

( ) ( )T

T

F Fxxy

var ,cov)var()var(

2

−=

• By taking the derivative with respect to h and setting it to zero we can solve for the minimum-variance hedge:

• Thus the variance of the total cash flow is given by:

How much do we hedge? • When we are trying to determine the number of units or amount

W of the commodity to hedge, we use:

Wh β−=

( ) ( )T

TT

F FS

var ,cov

=β

Where

Looks like the Capital Asset Pricing Model β from our earlier lectures!

Luenberger Exercise • Farmer D. Jones has a crop of grapefruit that will be ready for

harvest and sale as 150,000 pounds of grapefruit juice in three months.

• Jones is worried about possible price changes, so he is considering hedging.

• There is no futures contract for grapefruit juice, but there is a futures contract for orange juice.

• His son, Gavin, recently studied minimum variance hedging and suggests it is a possible approach.

• Currently, the spot prices are $1.20 per pound for orange juice and $1.50 per pound for grapefruit juice.

• The standard deviation of the prices of orange and grapefruit juice is about 20% per year, and the correlation coefficient between the two is about 0.7.

• What is the minimum variance hedge for farmer Jones, and how effective is this hedge compare to no hedge?

Solution • So, starting with:

( ) ( ) ( )( ) ( )

( ) 22

22 2 2

cov , var var

var T

T

x FT x x x

T F

x F y x

F

ρσ σ σ σ ρσ

σ = − = − = −

( ) ( ) 2

cov , var

T

T T

x FT x

T F F

x F h W W W W

F ρσ σ ρσβ σ σ

= − = − = − = −

0.2 1.50 0.30 0.2 1.20 0.24

0.70 150,000 lbs

T

x

F

W

σ σ

ρ

= × = = × =

= =

we need to determine:

Solution • The minimum variance hedge is given by:

2 0.875 150,000 131,250 lbs of orange juice futures T

T

x F

F

h W ρσ σ σ

= − = − × = −

( ) ( )

( )( )( )[ ] ( )

0459.0 24.

24.3.7.00.3)( var

,cov)var()var( 2 2

2 2

=−=−= T

T

F Fxxy

0.214 0.714 0.30

y

x

σ σ

= = 21 0.714y x xσ ρ σ σ= − =

• And the variance of this position is:

• Assuming independence, the StDev of y is 0.214, and given the StDev of x for grape fruit, the improvement of the hedge is:

Products Traded • Commodities

– Soybeans – Gold

• Economic Derivatives – Consumer Price Index (CPI)

• Equities – S&P 500

• Foreign Exchange • Interest Rates • Real Estate • Weather Derivatives

Another Example • Agricultural commodities traded:

Corn Oates Soybeans Ethanol Wheat Rough Rice

• What if we wanted to hedge the risk for sorghum?

• Let’s assume that a farmer has an uncertain price for his 10,000 tons of sorghum, but there is not a future or option available to perform a perfect hedge.

• So he hires a professional to hedge his crops using futures.

Correlation of Crops • Using the historical monthly prices for different traded and not

traded future commodities we get the following correlation table.

• The highest correlation of Sorghum is with corn, so we will use this commodity to make the hedge.

• What happens if we use other commodities?

Corn Sorghum Alfalfa Rice Oats Wheat Soybeans Avg. Price 74.75074 35.9656 96.27188 65.4104 96.15234 107.4406 182.9018 Std 10.42879 7.0072 10.26378 17.1934 19.1519 18.93611 39.80732 Var 108.7597 49.1008 105.3452 295.614 366.7955 358.5762 1584.623 Current Price (dec. 2006) $107.50 $ 60.90 $112.00 $98.70 $123.13 $150.32 $ 205.98

Corn Sorghum Alfalfa Rice Oats Wheat Soybeans Corn 1.00 Sorghum 0.88 1.00 Alfalfa 0.27 0.43 1.00 Rice 0.38 0.28 0.00 1.00 Oats 0.49 0.50 0.65 -0.15 1.00 Wheat 0.73 0.78 0.56 0.33 0.67 1.00 Soybeans 0.76 0.62 0.14 0.51 0.25 0.56 1.00

21y xσ ρ σ= −

Solution

• We need an equivalent hedge of 5,900 tons of corn futures

• The equivalent number of tons in sorghum is: 634250/60.90 = 10,415. • In theory the variance reduction provided by the hedge is the following:

• A ton of corn is equivalent to 35.7 bushels; since the standard corn contract is for 5000 bushels (140 tons), we need 43 ( round up 42.126) futures contracts.

• 43 futures of corn = -6022 tons = -$647,409

0.88(7.01)10,000 0.59 10,000 5,900 10.43

S

C

h W Wρσβ σ − −

= − = = = − × = −

( ) ( ) ( )( ) ( )( )( )

( )

22 0.88 10.43 7.01cov , var var 49.10 11.04

var 108.76 T

T

x F y x

F   = − = − =

3.32 0.47y sσ σ= =

5,900 107.50 $634,250= − × = −

Other Cases? • What happens if we use other commodities?

Corr Std Var Std New Var New Std Compare Corn 0.88 10.43 49.10 7.01 11.03 3.32 0.47 Rice 0.28 17.19 49.10 7.01 45.18 6.72 0.96 Oats 0.50 19.15 49.10 7.01 36.67 6.06 0.86 Wheat 0.78 18.94 49.10 7.01 19.19 4.38 0.63 Soybeans 0.62 39.81 49.10 7.01 29.94 5.47 0.78

Sorghum

Some Practical Issues with Hedging • There are some practical issues related to hedging with futures

including: – Difference between the expiration date of a contract and

when the hedged commodity will be used or available. – Which of the potentially many outstanding future contracts

to use? – Spread of the basis? – What volatility to use?

Example • Consider the data given in the file:

– “Lecture 16 Feed Grains Excel Villalobos.xlsx” – “Spot Price” spreadsheet

• Based on the first 55 points (oldest) of data we get the following information:

• Considering the last 55 points (newest) of data we have:

• The estimation of parameters are quite different!

Corn Sorghum Alfalfa Oats Average 2.49036 3.93927 58.75091 1.31849 Std. Dev. 0.29117 0.51240 6.60552 0.18587 Variance 0.08478 0.26256 43.63292 0.03455 Coef. Corr. 0.11692 0.13008 0.11243 0.14097

Corn Sorghum Alfalfa Oats Corn 1 Sorghum 0.94737 1 Alfalfa 0.50432 0.35367 1 Oats 0.71944 0.63459 0.67225 1

Corn Sorghum Alfalfa Oats Average 4.13700 6.74101 130.4000 2.54745 Std. Dev. 1.01959 1.91134 20.79583 0.52372 Variance 1.03956 3.65320 432.4666 0.27428 Coef. Var. 0.24646 0.28354 0.15948 0.20559

Corn Sorghum Alfalfa Oats Corn 1 Sorghum 0.97864 1 Alfalfa 0.48222 0.41514 1 Oats 0.80768 0.77006 0.80605 1

Example Month

Average (Corn)

Average (Sorghum)

StdDev (Corn)

StdDev (Sorghum) Correlation Beta

The Hedged Position “h”

Current Position

Difference with Hedge

+5mo

Difference without Hedge

+5mo Var.

Reduction 1 0.0479 0.0429 0.0033 0.0034 0.8568 -0.8779 -17557383.0382 -26352.7307 -12725.6291 -140000.0000 0.5156 2 0.0474 0.0424 0.0031 0.0035 0.8493 -0.9612 -19224109.4237 -115084.9241 -65052.0495 -204000.0000 0.5279 3 0.0472 0.0419 0.0032 0.0037 0.8454 -0.9993 -19986949.3606 -168274.5250 -86437.6580 -168000.0000 0.5341 4 0.0470 0.0414 0.0033 0.0040 0.8771 -1.0667 -21334624.6973 -212150.8129 -90994.8115 -96000.0000 0.4803 5 0.0467 0.0409 0.0036 0.0045 0.9119 -1.1194 -22387425.2823 -251490.2748 -175519.4661 24000.0000 0.4105 6 0.0461 0.0402 0.0043 0.0049 0.9339 -1.0558 -21116562.7748 -143017.0301 -153808.4371 42000.0000 0.3574 7 0.0452 0.0392 0.0055 0.0056 0.9531 -0.9817 -19633543.7709 -78066.2127 -93811.8010 128000.0000 0.3026 8 0.0444 0.0382 0.0062 0.0059 0.9749 -0.9316 -18631358.4045 -42865.0916 -37273.2513 142000.0000 0.2226 9 0.0435 0.0374 0.0067 0.0060 0.9787 -0.8710 -17420637.4382 37607.6270 21844.5651 146000.0000 0.2052

10 0.0428 0.0368 0.0066 0.0057 0.9767 -0.8522 -17043866.7102 28680.8079 20069.5457 104000.0000 0.2146

-1500000.0000

-1000000.0000

-500000.0000

0.0000

500000.0000

1000000.0000

Fe b-

77 De

c- 77

O ct

-7 8

Au g-

79 Ju

n- 80

Ap r-

81 Fe

b- 82

De c-

82 O

ct -8

3 Au

g- 84

Ju n-

85 Ap

r- 86

Fe b-

87 De

c- 87

O ct

-8 8

Au g-

89 Ju

n- 90

Ap r-

91 Fe

b- 92

De c-

92 O

ct -9

3 Au

g- 94

Ju n-

95 Ap

r- 96

Fe b-

97 De

c- 97

O ct

-9 8

Au g-

99 Ju

n- 00

Ap r-

01 Fe

b- 02

De c-

02 O

ct -0

3 Au

g- 04

Ju n-

05 Ap

r- 06

Fe b-

07 De

c- 07

O ct

-0 8

Au g-

09 Ju

n- 10

Difference with Hedge +5mo Difference without Hedge +5mo

Avg StDev Max Profit Most Lost Hedged 40139.73 188132.58 686906.45 -623419.60

No Hedge 16148.96 215672.01 806820.00 -1228900.00

Assignments • Using the date in “Lecture 16 Commodities Villalobo.xlsx”,

perform a hedging exercise with alfalfa. – Determine the best commodity and perform a hedge. – Determine the variance reduction of the hedge.

• Luenberger 2nd edition Chapter 12 problems 12.1, 12.3, 12.7, 12.13.

• Luenberger 1st edition Chapter 10 problems 10.1, 10.3, 10.7, 10.11.

• Begin reading Luenberger 2nd edition Chapter 13; 1st edition Chapter 12.