drake drake university fin 288 hedging strategies in futures markets fin 288 fixed income analysis

DrakeDRAKE UNIVERSITY

Fin 288

Hedging Strategies in Futures Markets

Fin 288Fixed Income Analysis

DrakeDrake University

Fin 288Hedge Terminology

Short HedgeA short hedge occurs when the hedger already owns an asset or will own an asset soon and expects to sell it at some date in the future. In this case the hedger will take a short position in the futures market, guaranteeing the price in the future at which the asset can be sold.


Fin 288Short Hedge Example

You have agreed to sell 10,000 bushels of corn on July 1 at the spot price on that day.You are afraid that the price of corn may decrease between now and July 1. The current futures price for delivery of corn in July is $2.10.The current spot price of corn is approximately $1.79 a bushel.

www.agriculture.com/ag/markets


Fin 288Recent Corn Spot Prices*


Fin 288Short Hedge

By taking agreeing to take a short position 20 futures contracts you decrease the impact of a price decline.Assume that on July 1 the spot price for corn is $1.60. You will sell your corn for

(1.6)(10,000)=$16,000Assume that the contract expires on July1 so the futures price equals the spot price. You can close out the futures contract making (2.10-1.6) (10,000) = $5,000


Fin 288The two positions combined

You made $16,000 in the spot and $5,000 in the futures market for a total of $21,000.Given that you still sold 10,000 bushels of corn You have effectively received $21,000/10,000 = $2.10 per bushel (ignoring transaction costs)


Fin 288Short Hedge

What if the spot price of corn is $2.40 on July 1?You sell your corn for (2.4)(10,000) = $24,000In the futures market when you close out the contract you loose(2.1-2.4)10,000 = -$3,000The total amount you receive is $21,000


Fin 288Impact of Hedge

Regardless of the changes in the spot price the result of the hedge is that you have received $21,000 for 10,000 bushels of corn.Note: If you had not hedged you would have been better off when the price increased without the hedge.


Fin 288Hedge Terminology

Long HedgeA long hedge occurs when the hedger knows that it will be necessary to purchase a given asset at a point in the future and wants to lock in the future price today. The alternatives to the hedge are buying the asset in the future at the market price or purchasing it today and holding onto it until the asset is needed in the future.


Fin 288Long Hedge

Similar to the short hedge by simultaneously entering into a long position and the spot market you can fix the price to be paid in the future.


Fin 288Assumptions

The hedge worked because of three assumptions:

The underlying asset in the futures market is the same as the asset in the spot market. The end of the exposure matches the delivery date exactlyThe contract was closed out at the futures price prior to delivery


Fin 288Basis Risk

The basis is a hedging situation is defined as the Spot price of the asset to be hedged minus the futures price of the contract used. When the asset that is being hedged is the same as the asset underlying the futures contract the basis should be zero at the expiration of the contract.

Basis = Spot - Futures


Fin 288Basis Risk

The easiest way to illustrate the basis risk is with an example:

Let: St represent the spot price at time tFt represent the futures price at time tbt represent the basis at time t


Fin 288Basis Risk Illustration

Assume we enter into a short hedge at time t = 1 and close out the hedge at time t = 2.

The profit on the futures position will equal F1- F2

The total price paid received from the hedge is then

S2 + F1 - F2

By definition:b1 = S1-F1 and b2 = S2-F2


Fin 288Basis Risk

By rearranging the price equation:S2 + F1 - F2 = F1 + (S2- F2) = F1 + b2

When the hedge is entered into F1 is known but b2 is unknown.

The fact that b2 is not known represents the basis risk.


Fin 288Basis Risk Long Hedge

The same expression holds for a hedger undertaking a long hedge.Loss on Hedge = F1-F2 price paid is S2 +F1-F2

Again the effective price paid is F1+b2 where b2 is unknown when the hedge is taken out.


Fin 288Mismatch of Maturities 1

Assume that the maturity of the contract does not match the timing of the underlying commitment.Assume that our short hedger for Corn has agreed to sell corn on the spot market on October 15. However, the months that corn delivery are available are March, May, July, September and December.


Fin 288New Short Hedge

To hedge the position you now need to take out a short position for the September futures contract. The current futures price for September is $2.28 a bushel. The contract will now need to be closed out on October 15, prior to when the futures price and spot price converge.


Fin 288New Short Hedge

What if the Spot price on October 15 is $1.90 and the futures price for December Delivery is $2.10?You sell 10,000 bushels for $1.90 each or $19,000You close out the futures position and profit:(2.28-2.10)(10,000) = $1,800The total price received is $20,800 or $2.08 a bushel.


Fin 288Result 2

What if the futures price for delivery in December is $2.35 and the spot price is $2.20You sell corn in the spot market and receive:$2.20(10,000) = $22,000You close out the futures position and loose(2.28-2.35)(10,000)=-$700The total you receive is $21,300 (less than you would have received in the spot market alone)


Fin 288Additional Risk

In our examples we assumed that the timing of the spot position was fixed. It may be the case that the timing of the spot position is not known with certainty. This is especially the case of a long hedger who knows that s/he will need to acquire an asset in the future, but not know the exact date.


Fin 288Minimizing Basis Risk

Given that the actual timing of the spot asset may also be uncertain the standard practice is to use a futures contract slightly longer than the anticipated spot position. The futures price is often more volatile during the delivery month also increasing the uncertainty of the hedge Also the long hedger could be forced to accept delivery instead of closing out.


Fin 288Mismatch in Maturities 2

Assume that instead of our original problem there are a string of future dates over which corn will be needed.Anticipated corn demand

Date AmountMay 1 15,000 BushelsJuly 1 10,000

BushelsSeptember 1 20,000 Bushels


Fin 288Strip Hedge

To hedge this risk, it is possible to hedge each position individually.On Feb 1 the firm could:

enter into short May contracts for 15,000 bushels enter into short July contracts for 10,000 bushelsenter into short Sept contracts for 20,000 bushels


Fin 288Strip Hedge continued

On each date the respective hedge should be closed out. The effectiveness of the hedge will depend upon the basis at the time each contract is closed out.(Note in this example each hedge again coordinated with the maturity of a contract)


Fin 288Rolling Hedge

Another possibility is to Roll the Hedge:Feb 1 enter into short May contracts for 45,000 BushelsMay 1 enter into long March contracts for 45,000 Bushels enter into short July contracts for 30,000

BushelsJuly 1 enter into long July contracts for 30,000 Bushels

enter into short Sept contracts for 20,000 Bushels

Sept 1 enter into long Sept contracts for 20,000 Bushels


Fin 288Rolling the Hedge

Again the effectiveness of the hedge will depend upon the basis at each point in time that the contracts are rolled over.This opens the from to risk from the resulting rollover basis. When the contract is closed out there is a cost if there has been a loss on the position. Therefore there may be a dollar cost to rolling over the hedge (basically a margin call).


Fin 288Hedging

So far we have assumed that the underlying asset is an exact match for the spot position to be hedged. Often this is not the case. Even if the asset underlying the futures contract is identical to the spot asset, the prices of the two will not always move together. Two questions

What futures contract should be used?How many contracts should be taken out?


Fin 288Hedge Ratio

The hedge ratio is the ratio of the size of the position in the futures market to the size of the spot exposure being hedged. In our examples so far we have utilized a hedge ratio equal to one. In other words the size of the futures position was the same as the size of the position in the underlying asset.


Fin 288

Minimum Variance Hedge Ratio

The ideal hedge ratio should be the one that minimizes the variance of the value of the hedged position.


Fin 288


S be the change in the spot price S during a period of time equal to the life of the project

F be the change in the futures price F during a period of time equal to the life of the project

S be the standard deviation of S

F be the standard deviation of F be the coefficient of correlation between S and

Fh be the hedge ratio


Fin 288The hedge ratio

The hedge ratio is the ratio of the amount of futures positions undertaken in the futures market to the number of positions held in the spot market. Let NA= the units of asset A needed at time 2

Let NF= the number of futures contracts held to offset the price variation in the spot asset.

The hedge ration is then:

A

F

N

Nh


Fin 288

Determining the Hedge Ratio

Assume that you are holding an NA units of an asset which can be stored for free and you plan on selling it in the future.To hedge the risk of a price decline you want to undertake a short hedge using NF futures contracts.


Fin 288Total value of portfolio

When you sell the asset and close the futures position the total change in the value of your two positions will equal:

AA

F12A12

A

AF12A12

F12A12

NN

N)F(F)NS-S(

grearragnin

N

N)NF(F)NS-S(

or

)NF(F)NS-S(

Y

Y

Y


Fin 288

Given that

You can substitute:A

F

N

Nh

A1212

AA

F12A12

N)hF(F)S-S(

NN

N)F(F)NS-S(

Y

Y

AA

F12A12 N

N

N)F(F)NS-S(

Y


Fin 288

Given our earlier definitions this can be written as

A1212 N)hF(F)S-S( Y

ANFS hY


Fin 288Hedger’s Objective

The objective of the hedger is to minimize the change in the value of the two positions NA is known at the beginning of the period and will not change. Therefore if the hedger can minimize the changes to

FS h


Fin 288Hedge positions

We just showed the change in the short hedgers position is

Likewise, the change in the long hedgers position is

FS h

S-F h


Fin 288


We want to find the hedge ratio that minimizes the variance of the change in the position held by the hedger. This will depend upon the covariance between the spot price and futures price and the variance of each variable.


Fin 288Min Variance Hedge

The variance of either hedge position is

Taking the first derivative of the variance and setting it to zero produces the hedge ratio

FSFS hhv 2222

2,

2

),(

022

FF

SFS

FSF

SFCovh

hh

v


Fin 288Estimating the Hedge Ratio

The hedge ratio can be rewritten to allow easy estimation via regression analysis


Fin 288Regression Review

Equation of a line: Y = a + bXGraphing combinations of X and Y form a line.X is the independent variable and placed on the horizontal axis. Y the dependent variable and placed on the vertical axis (The value of Y depends upon X)a is the Y intercept and b the slope of the line.


Fin 288

We can observe observations of X,Y and plot

them


Fin 288

Regression Estimates the line that best explains the

relationship between the variables


Fin 288

The goal is to minimize the sum of the squared residuals


Fin 288Estimating the Regression

Y = a + bXThe slope of the line is then equal to

The Intercept is:

XVariance

y)Cov(x,b

)( XY AverageslopeAverage


Fin 288

Confidence in the ResultsR-Squared (R2)

R2 will range up to one. It is the portion of the relationship explained by the regressionR-Squared (R2) = correlationYX

2=b2x2/Y

2

Examples: An R2 of one implies all the points are on

the line An R2 of 0.5 would mean that half of the

relationship is explained by the line.


Fin 288

Confidence in the ResultsT-statistic

The t-statistic tells us whether or not we can reject the hypothesis that the variable is equal to zero.The higher the t-statistic the higher the confidence that we can reject the hypothesis that the slope is zero.If you cannot reject the hypothesis -- It implies that the dependent variable has no impact on the independent variable.


Fin 288T-Statistic

A Rule of Thumb:The confidence levels are based upon the number of observations, but in general:If you have a t-statistic above 2.0 you can reject the null hypothesis at the 95% level.(With 120 observations a t-statistic of 2.36 allows rejection at the 99% level)


Fin 288Standard Error

Provides a measure of “spread” around each variable. Provides a confidence band “similar to standard deviation)We can use standard error to estimate the T- Statistic (Assuming a normal distribution)T-Statintercept=A/SEA T-Statslope = B/SEB


Fin 288Quick Review

Linear Regression - Provides line the best describes the relationship between two variablesR2 - Portion of relationship explained by the estimated lineT-Statistic - Confidence in the estimate of the variable (Is is statistically significant?)Standard Error - Confidence Interval


Fin 288

Applying the Regression to the Hedge Ratio

The minimum variance hedge ratio could be estimated by in the regression.

(St) = + (Ft) + t

tF

tt

Variance

)S,Cov(F Where

2,

2

),(

022

FF

SFS

FSF

SFCovh

hh

v


Fin 288

Hedging using the hedge ratio

Assume that the airline you are working for wants to hedge against a possible increase in the price of jet fuel. There are not futures contracts available for jet fuel so a contract on a different asset must be used.

What contract should be used?What is the associated hedge ratio?

http://corporate.bmo.com/rm/commodity/images/Hedging_JetKero_Prices.pdf


Fin 288Contract options

NYMEX futures contracts trade on Unleaded Gasoline, Light Sweet Crude Oil, Brent Crude Oil, Heating Oil, Natural Gas, and PropaneHigh correlation of spot prices for Heating Oil and Jet Fuel indicate it might be a good candidate for the contract (Correlation = .994 from Jan 1995 to October 2004).

http://corporate.bmo.com/rm/commodity/images/Hedging_JetKero_Prices.pdf


Fin 288A Hypothetical Hedge

Assume you know that the airline has average consumption of 100 Million gallons each month and you want to hedge the price of Jet Fuel for June.The Heating Oil contract calls for trading to stop on the last business day prior to the beginning of the delivery month. Assume you plan to close out your contract during June at the same time you make a spot market purchase for the month.http://www.eia.doe.gov/oil_gas/petroleum/info_glance/prices.html


Fin 288A Hypothetical Hedge

You would need to use the July contract so you had the month of June to close out your position.The Price for July delivery on 2/3/05 is $1.2277 per gallonThere are 42,000 gallons (1,000 barrels) per contract.


Fin 288Hedge Alternatives

Without using the hedge ratio you would need to enter into 100 Million / 42,0000 = 2380.95 or approximately 2381 long contracts By running a regression using the spot price and futures price assume that you discover that your hedge ratio is 1.07 futures positions for each spot position. This implies a need to enter into 107 million / 42,000 = 2547.62 or apporximately 2548 long contracts


Fin 288Hedge Results

The current spot price of jet fuel is $1.4345 per gallon. Assume that on June 15 you decide to close out the contract and the price of Jet Fuel is 1.5345 per gallon.The effectiveness of the hedge depends upon the futures price for delivery of Heating oil in July. Assume that the futures price is $1.3077


Fin 288Hedge Results

Assuming a 1 to 1 hedge ratioSpot Price of Fuel =1.5345 per gallonGain on Hedge = 1.3077-1.2277=.09 per gallonEffective cost of jet fuel = 1.5345-.09 =$1.4445

Assuming a 1.07 hedge ratioSpot Price of Fuel =1.5345 per gallonGain on Hedge = 1.3077-1.2277(1.07) =.0963 per gallonEffective cost of jet fuel = 1.5345-.0963 =$1.4382


Fin 288Effective Cost

Total cost with 1 to 1 hedge =$144,450,000Total cost with 1.07 Hedge ratio = $143,820,000A difference of $630,000 for the month!


Fin 288Problems

Current Open Interest for July 2005 is 8532 contracts, there may not be enough liquidity in the market to cover the hedge (will there be enough short participants willing to take a short position?It might be difficult to close out the futures position, however current open interest for the March 2005 contract is 69970 contracts (there is some seasonal variation to also worry about).


Fin 288Tailing the Hedge

Adjustments to the margin account will also impact the hedge and need to be made.The idea is to make the PV of the hedge equal the underlying exposure to adjust for any interest and reinvestment in the margin account.

Some of the discsussion and examples for these topics are from Kolb Futures Options and Swaps 3rd edition


Fin 288Should a firm Hedge?*

Tax incentives for HedgingCosts of Financial Distress as an IncentivePrincipal –Agent Conflicts as an IncentivePrincipal-Agent Conflicts as a DisincnetiveLack of Owner Diversification as an IncentiveTransaction Costs as a DisincentiveCompetitive Environment


Fin 288

Tax incentive for Hedging

Consider a mining firm that expects to mine 1,000 ounces of gold bullion this year at a cost of $300 per ounce.Assume that there are two possible prices for gold, $300 or $500 and both are equally possible.If the firm has positive income it can use a 20,000 tax credit to offset taxes and it expects to pay a 20% tax rate.


Fin 288Unhedged Firm

Sale Price 300 500Gold Revenue 300,000 500,000Futures Result 0 0Less Production Costs -300,000 -300,000Pretax Profit 0 200,000Tax Obligation 0 -40,000Add Tax Credit 0 20,000Net Income 0 180,000 Expected After tax Revenue 90,000


Fin 288Hedged Firm

Sale Price 300 500Gold Revenue 300,000 500,000Futures Result 100,000 -100,000

Less Production Costs -300,000 -300,000Pretax Profit 100,000 100,000Tax Obligation -20,000 -20,000Add Tax Credit 20,000 20,000Net Income 100,000 100,000 Expected After tax Revenue 100,000


Fin 288Cost of Financial Distress

In the previous example the pretax expected income was the same for cases, but the after tax net income differed.In a perfect market, investors could diversify with out any costs and eliminate any risk associated with the change in expected profits.However, if a firm pursues a high risk strategy in the real world and goes bankrupt, there are high transaction costs.


Fin 288Cost of financial distress

By Hedging the firm can minimize the cost associated with possible bad outcomes and therefore increase the value of the firm.


Fin 288

Principal Agent Conflicts as an Incentive

In efficient markets manger (agents) act in the best interest of the shareholders (principals). Shareholders, in theory, can diversify by holding a portfolio of securities, if the firms fails the loss is limited. The Manager has a much larger stake in the firm succeeding and may be more risk averse than the shareholder – therefore hedging when the shareholder would prefer not to hedge.


Fin 288

Principal Agent Conflicts as an Disincentive

The manger may also run into internal conflict if the hedge looses money. It is difficult to explain a loss in derivative markets to the board of directors and shareholders, even if the loss was associated with a hedging strategy. Therefore the manager may resist hedging for fear of perceived poor performance or even job loss.


Fin 288

Lack of Owner Diversification

It may be that the owners are not really diversified as assumed in efficient markets they would then have an incentive to pressure the manger to hedge to decrease risk.


Fin 288Transaction Costs

In the long run gains and losses on hedging should offset each other ignoring transaction costs. However regardless of a loss or gain there is a transaction cost to hedging, therefore in the long run there may be a cost to hedging that decreases the value of the firm.


Fin 288Competitive Environment

If the retail price fluctuates with the wholesale price of inputs then the profit margin for the firm stays relatively constant without hedging. In this case hedging may actually increase the volatility of income compared to competitors.

drake drake university fin 288 hedging strategies in futures markets fin 288 fixed income analysis

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