1. Hedging Strategies Using Futures

2. Long & Short Hedges
A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price
A short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price

3. Arguments in Favor of Hedging
Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables

4. Arguments against Hedging
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

5. Basis Risk Basis is the difference between spot & futures.
Basis = Spot price of asset to be hedged – Futures price of a contract used
As time passes, the spot price and futures price for a particular month do not necessarily change by the same amount.
An increase (decrease) in the basis is referred as strengthening (weakening) of the basis.
Basis risk is the risk that the futures contract may not be able to remove all the risk arising from the price of the asset on that date. It arises because of the uncertainty about the basis when the hedge is closed out

6. Basis Convergence Basis = Cash price – futures price
End Basis , b2 = S2 – F2 or
Initial Basis , b1 = S1 – F1
As times goes by, the difference between the cash index and the futures price will narrow.
At the end of the futures contract, when T = 0, the futures price will equal the index, a phenomenon known as basis convergence.

7. Convergence of Futures to Spot
Price
Spot Price
Futures
Price
Spot Price
Time
Time
(a)
(b)

8. Long Hedge S1 : Initial Spot Price at time t1
Suppose that
S1 : Initial Spot Price at time t1
S2 : Final Spot Price at time t2
F1 : Initial Futures Price at time t1
F2 : Final Futures Price at time t2
b1 : Initial basis at time t1
b2 : Final basis at time t2
You hedge the future purchase of an asset by entering into a long futures contract
The price realized for the asset is S2 and the loss/gain from the futures hedge (F2 – F1).-Price is expected to
Effective Price obtained by hedging is ;
=S2 –(F2 – F1) = F1 + Basis

9. Short Hedge F1 : Initial Futures Price at time t1
Suppose that
F1 : Initial Futures Price at time t1
F2 : Final Futures Price at time t2
S2 : Final Asset Price at time t2
You hedge the future sale of an asset by entering into a short futures contract
The price realized for the asset is S2 and the profit from the futures hedge (F1 – F2).- Price is expected to
Effective Price obtained by hedging is ;
Price Realized=S2+ (F1 –F2) = F1 + Basis

10. Choice of Contract
One of the key factors affecting basis risk is the choice of the future contract to be used in hedging. It had two components;
Choice of the asset underlying the futures contract
When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. There are then 2 components to basis
Choice of delivery month
Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge

11. Effective price
The effective price is the price at which a commodity is sold or bought after the hedge has been lifted (liquidated).
It can be calculated either by adding/substracting the basis change to the original cash price, or by adding/substracting the hedge of the futures (the original futures price at the time the hedge was placed minus the futures price at the time the hedged was lifted) to the final cash price.
If a short hedger has made a profit, the effective cash price will be higher than the original cash price being hedged. If a long hedger has made a profit, the effective cash price will be lower than the original cash price being hedged.

12. Choice of Contract: Example 1
On 1st march, a US company expects to receive 50 million Japanese yen at the end of July. Yen futures at the CME Group have delivery months of March, June, Sept. and December. One contract is for delivery of 12.5 million yen. Suppose that the futures price on March 1 in cents per yen is and that the spot and futures prices when the contract is closed out are and , respectively.
Determine
The effective price and
The total amount received by the company

13. Choice of Contract: Example 1
The company therefore shorts 4 (50M / 12.5M) September yen futures contracts in March 1. When the yen are received in July, the company closes out its position.
Gain on the futures contract : F1 – F2 = – = cents per yen.
Basis = S2 – F2 = – = cents per yen.
Effective price = F1 + basis = ( ) = cents per yen.
Effective price = S2 + gain = (0.0550) = cents per yen.
Total amount received by US company = 50M *
= $ 387, 500.

14. Choice of Contract: Example 2
On June 8, a company knows it will need to purchase 20,000 barrels of crude oil at some time in October or November. Oil contracts are currently traded every month on the NYMEX and contract size is 1,000 barrels. Suppose that the futures price on June 8 is $68.00 per barrel. The company finds that it is ready to purchase the oil in November 10 and hence decides to close out on November 10 when the spot and futures prices are $70.00 and $ per barrel, respectively.
Determine
The effective price and
The total amount paid by the company

15. Choice of Contract: Example 2
The company therefore longs 20 contracts (20,000 barrels / 1,000 barrels) December futures contracts on June 8. The company closes out its position on November 10.
Gain on the futures contract : F2 – F1 = $69.10 – $68.00 = $1.10 per barrel.
Basis = S2 – F2 = $ $ = $0.90 per barrel.
Effective price = F1 + basis = $ $0.90 = $68.90.
Effective price = S2 - gain = $ $ 1.10 = $ 68.90
Total price paid = $68.90 * 20,000 barrels = $ 1,378,000.

16. Stock index futures Similar to other indexes but are settled on cash.
Features of a futures contract (S & P 500) include:
Contract size: $250 * Index level
Minimum price change = 0.10 ($25)
Performance margins:
Speculator:
Initial = $ 20,000
Maintenance = $ 16,000
Hedger:
Initial = $ 16,000
Contracts are marked to market throughout the trading day

17. Stock index futures
Features of a futures contract (S & P 500) include:
The contract does not earn dividends.
Trading hours: 9.30 a.m. – 4.15p.m EST
Settlement Months: March, June, September, and December. The last trading day is the Thursday prior to the third Friday of the contract month.
Ticker symbols: Futures (SP), Cash (SPX)
The futures are nicknamed SPUs (Pronounced SPOOZ) in the trading pit.

18. Pricing of Stock index futures
Since an index futures is a function of the price of the underlying cash index, the same factors affect it and the stock market.
Futures value depends on four elements;
Level of spot index itself
Dividend yield on the stocks in the index (e.g. on 500 stocks in the index)
Current levels of interest rates
Time until final contract cash settlement.

19. Pricing of Stock index futures
Underlying index
T-bill rate
Dividend yield
Time to maturity/settlement

20. Pricing of Stock index futures
F = S е(R-D)T
F – Futures value; S – Current level of cash index; R – T-bill yield; D – Underlying stocks dividend yield; T – Time to maturity in years.
Example 3: Stock Index Futures Information. Determine the futures price
August 2011
Current level of the cash index (S) =
T – bill yield (R) = 6.07%
S & P 500 Dividend yield (D) = 1.06%
Days until December settlement = 121 days = 0.33yrs
F = е( – )121/365
F =

21. Synthetic Index Portfolio
An investor can replicate a well-diversified portfolio of common stock by simply holding a long position in the stock index futures contract and satisfying the good-faith deposit, or margining, the position with T-bills.
Long Treasury Bills + Long Stock Index Futures = Long index portfolio
Added benefits of the futures approach are;
Transaction costs will be much lower on the futures contracts than on 500 separate stock issues
The portfolio will be much easier to follow and manage

22. Effective Hedging with Stock Index Futures
A hedger seeks to eliminate risk. Unsystematic risk is eliminated by diversification but systematic risk is not.
Beta is the measure of relative riskiness of a portfolio compared to a benchmark portfolio like the S & P index.

23. Reasons for Hedging an Equity Portfolio
Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.)
Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will outperform the market.)

24. Cross Hedging
Cross hedging is when you hedge a position by investing in two positively correlated securities or securities that have similar price movements.
The investor takes opposing positions in each investment in an attempt to reduce the risk of holding just one of the securities.
The success of cross-hedging depends completely on how strongly correlated the instrument being hedged is with the instrument underlying the derivatives contract.
When cross hedging, the maturity of the two securities has to be equal. In other words, you cannot hedge a long-term instrument with a short-term security. Both financial instruments have to have the same maturity.

25. Hedge Ratio
Portfolios are of different sizes and different risk levels.
Hedge ratios incorporates the relative value of the stocks and futures and accounts for relative riskiness of the two “portfolios”.
It is the ratio of the size of the position taken in futures contracts to the size of the exposure.
The index has a beta of 1.0 while the stocks maybe less risky (beta less than 1, thus fewer contracts) or more risky (beta more than 1, hence more futures contracts are necessary.)

26. Hedge Ratio
To hedge a long position, the manager needs to go short the futures contracts.
Because futures contracts are not available in fractional amounts, the portfolio manager must also round to a whole number
To determine the hedge ratio, you require;
Value of the chosen futures contract
Value of the portfolio to be hedged
Beta of the portfolio

27. Hedge Ratio HR = Value of the portfolio to be hedged * Beta Example 4:
Value of the chosen futures contract
Example 4:
The size of an S& P 500 futures contract is established as $250 times the value of the S& P 500 index. Suppose, in August, the manager of a $75 million stock portfolio (beta of 0.9 and dividend yield = 1%) studies a possible hedge using the December S&P 500 futures (for four months hedge). The previous day’s closing value for the S& P 500 index was December S&P 500 futures contract closed at
Determine the hedge ratio.

28. Hedge Ratio HR = $75,000,000 * 0.9 = 177.96 = 178 contracts
F1 =
S1 =
S2 = F2 =
Gain = F1 – F2
HR = $75,000,000 * 0.9 = = 178 contracts
* 250
If the beta was same as market = 1.0, then the number of contracts required for a hedge will be given by;
HR = $75,000,000 * 1.0 = = 198 contracts
We are hedging 20 contracts less as the asset has lower sensitivity (beta) to market movements/ beta

29. Hedge Ratio: Consequences of different market scenarios at final delivery day of contract
Market falls
S & P 500 index falls by 5% from to
Portfolio: -5% * 0.9 * $75M = $ 3,375,000 loss
Dividends: 1% * 0.333(4mths) * $75M = $ 250,000 gain
Futures: ( – ) * 250 * 178 = $4,761,500 gain
Net effect: $ 1,636,500 gain

30. Hedge Ratio: Consequences of different market scenarios at final delivery day of contract
2. Market rises
S & P 500 index rises by 5% from to
Portfolio: 5% * 0.9 * $75M = $ 3,375,000 gain
Dividends: 1% * 0.333(4mths) * $75M = $ 250,000 gain
Futures: ( – ) * 250 * 178 = $1,846,750 loss
Net effect: $ 1,778,250 gain

31. Hedge Ratio: Consequences of different market scenarios at final delivery day of contract
3. Market is unchanged
S & P 500 index remains unchanged at
Portfolio: 0% * 0.9 * $75M = $ 0
Dividends: 1% * 0.333(4mths) * $75M = $ 250,000 gain
Futures: ( – ) * 250 * 178 = $1,455,150 gain
Net effect: $ 1,705,150 gain

32. Example 5 Value of SP500 index = 1,000
A portfolio is worth: P = $5,000,000
r = 10% (risk-free); D = 4%;  = 1.5 (all rates are continuously compounded)
Futures on SP500 matures in 4 months and is used to hedge the portfolio over the next 3 months.
The value of SP500 index in 3 months is = 900
Required:
How many contracts are required to hedge the position?
What are the gains (losses) on futures?
Determine the net gain or loss.

33. Example 5
F0 = 1,000e(.1-.04)x4/12 = 1,020.20
HR = $5,000,000 * 1.5 = 29.4 = 30 contracts
* 250
St=3m= 900; Ft = 900e(.1-.04)1/12 =
S & P 500 index falls by 10% from 1000 to 900
Gains on the short futures:
h(F1-F2) = (1, ) x $250 x 30= $867,675 gain
Net effect:
Portfolio: -10% * 1.5 * $5M = $ 750,000 loss
Dividends: 4% * 0.25(3mths) * $5M = $ 50,000 gain
Net effect: $ 167,675 gain

34. Adjusting Market Risk - Changing Beta
Futures can be used to adjust the level of market risk in a portfolio. e.g. maybe to increase or reduce market exposure…
# contracts = Portfolio value * (βdesired - βcurrent )
Value of the chosen futures contract
What position is necessary to raise our previous beta from 0.9 to 1.4?
# contracts = $75* (1.40 – 0.90) = 99 contracts
* $250
Should add extra 99 contracts
While this does not put more dollars in the market, it increases the sensitivity of the aggregate portfolio to market changes so that to mimic those of a large portfolio

35. Example 6
A company would like to hedge its $20 mil portfolio ( = 1.2) with SP 500 futures. 1 contract is for $250* Index. Index is at now.
Required:
What is the optimal hedge ratio?
What should the company do to reduce  of the portfolio to 0.6?

36. Example 6 What is the optimal hedge ratio?
h = 1.2*20/(250*1080) ≈ 89 contracts
What should the company do to reduce  of the portfolio to 0.6?
Short ( )*20/(250*1080) ≈ 44 contracts (should short / reduce by 44 contracts)

37. In conclusion
It is important to not that in practice, it is not possible to acquire a perfect hedge. This is because;
Future contracts are only available in integer quantities, hence portfolio manager must round to the nearest whole number.
Stock portfolio rarely behaves exactly as their beta says they should…betas are estimated
Futures prices does not move in lockstep with the underlying index….this is referred as basis risk.
Dividends do not occur uniformly over time as suggested in the formulae.