Futures Hedging Strategies

1
Futures Hedging Strategies

• FIN 653 Lecture Notes
• Yea-Mow Chen
• Department of Finance
• San Francisco State University

2
I. Interest Rate Futures as a Hedging Device

• An interest rate futures has an interest-bearing
  discount security as the underlying commodity,
  its value depends on the market value of the
  underlying asset the price of an interest rate
  futures contract also changes inversely with
  interest rates. Thus a financial institution can
  use futures to reduce its exposure to adverse
  rate changes.
•  

3
I. Interest Rate Futures as a Hedging Device

• I. Long Hedge
• A long hedge is chosen in anticipation of
  interest rate declines and requires the purchase
  of interest rate futures contract. If the
  forecast is correct, the profit on the hedge
  helps to offset losses in the cash market.
• Example In June 2002, the manager of a money
  market portfolio expects interest rates to
  decline. New funds, to be received invested in
  90 days, will suffer from the drop in yields. The
  manager expects an inflow of 10m in September.
  The discount yield currently available on 91-day
  T-bills is 10, and the goal is to establish a
  yield of 10 on the anticipated funds.

4
I. Interest Rate Futures as a Hedging Device
5
I. Interest Rate Futures as a Hedging Device

• Effective Discount Yield with the Hedge
•   10,000,000- (9,797,778- 50,000) 360
• --------------------------------------------
  - ----------
• 10,000,000
  91
•   9.978
• Note 1. At a discount yield of 10, the price of
  a 91-day T-bill is
  91
• P 10,000,000 1 - 10 -----------
  9,747,222
• 360
• 2. T-bill futures are standardized at 90-day
  maturity, resulting in a price different from the
  one calculated in the cash market.

6
I. Interest Rate Futures as a Hedging Device

• Even if the expectation on future interest rates
  for the cash market is incorrect, the position is
  still hedged. The cost is that the potential
  profitable opportunities in the cash market is
  foregone.
• EX Assume the T-bill discount yield rises to 12
  , instead of declining to 8 as expected.
•  

7
I. Interest Rate Futures as a Hedging Device

• Cash Market Futures Market
• __________________________________________________
  __June T-bill discount yield at 10 June buy
  10 T-bill Contracts
• Price of 91-day T-bills, for September delivery
  at
• 10m par 9,747,222 10 discount yield.
  Value
• of contracts 9,750,000
•  
• Sept T-bill discount yield at 12 Sept Sell 10
  Sept. T-bill
• Price of 91-day T-bills, contracts at 12
  discount
• 10m par 9,696,667 yield.
• Value of contracts
• 9,700,000
• __________________________________________________
  _______
• Opportunity gain 50,555 Loss 50,000

8
I. Interest Rate Futures as a Hedging Device
9
I. Interest Rate Futures as a Hedging Device

• Effective Discount Yield with the Hedge
•  
• 10,000,000- (9,696,667 50,000) 360
• -----------------------------------------------
  - ----- --
  10,000,000 91
• 10.022
• The investor is still making about 10, the
  target rate of return.

10
I. Interest Rate Futures as a Hedging Device

• Long speculation Instead of expecting new funds
  to arrive invest in September, the manager
  could speculate on the direction of interest
  rates.

11
I. Interest Rate Futures as a Hedging Device

• If he/she speculates on a declining interest
  rate, but market rate rises in September instead

12
I. Interest Rate Futures as a Hedging Device

• II. Short Hedge
• A short hedge is chosen in anticipation of
  interest rate increases and requires the sale of
  interest rate futures. If the forecast is correct
  the profit on the hedge helps to offset losses in
  the cash market.
• Example A saving institution in September 1999
  wants to hedge 5m in short-term CDs whose owners
  are expected to roll them over in 90 days. If
  market yields go up, the thrift must offer a
  higher rate on its CDs to remain competitive,
  reducing the net interest margin. If the CD rare
  rises from 7 to 9, the interest cost will
  increase by 25,000 for the 3-month period. The
  asset/liability manager can reduce these by the
  sale of T-bill futures contracts.

13
I. Interest Rate Futures as a Hedging Device

• Cash Market Futures Market
• __________________________________________________
  _______
• Sept. CD rate 7 Sept.
  Sell 5 Dec. T-bill Interest cost on 5m
  3-month contracts at 7 discount yield
• Interest costs 87,500 Value of
  contract
• 4,912,500
• Dec. CD rate 9 Dec. Buy 5 Dec. T-bill
  contracts
• at 9 discount
• Interest cost on 5m 3-month interest Value of
  contracts
• 112,500 4,887,500
• __________________________________________________
  __Opportunity Loss 25,000 Gain 25,000
• Net result of hedge 0

14
I. Interest Rate Futures as a Hedging Device

• 112,500 -25,000 360
• Effective CD rate -------------------------
  ------ 7
• 5,000,000
  90

15
I. Interest Rate Futures as a Hedging Device

• Basis Risk Using the Long Hedge Example
•  
• Example In June 1993, the manager of a money
  market portfolio expects interest rates to
  decline. New funds, to be received invested in
  90 days, will suffer from the drop in yields. The
  manager expects an inflow of 10m in September.
  The discount yield currently available on 91-day
  T-bills is 10, and the goal is to establish a
  yield of 10 on the anticipated funds.

16
I. Interest Rate Futures as a Hedging Device

• Cash Market Futures Market
• __________________________________________________
  _______June T-bill discount yield at 10
  June buy 10 T-bill Contracts
• Price of 91-day T-bills, for September delivery
  at
• 10m par 9,747,222 10 discount yield.
• Value of contracts
• 9,750,000
• Sept. T-bill discount yield at 8 Sept Sell 10
  Sept. T-bill
• Price of 91-day T-bills, contracts at 8
  discount
• 10m par 9,797,778 yield.
• Value of contracts 9,800,000
• __________________________________________________
  ______ 
• Opportunity Loss 50,556 Gain 50,000

17
I. Interest Rate Futures as a Hedging Device
18
I. Interest Rate Futures as a Hedging Device

• Revised Example Rather than using T-bill
  contract for hedging, a long-term T-bond futures
  contract is used for hedging which is price at
  96-12. If the T-bill rate drops to 8 in
  September as expected, the T-bond futures will
  have it price increased to 98-16.

19
I. Interest Rate Futures as a Hedging Device
20
I. Interest Rate Futures as a Hedging Device

• The how many contracts to buy to make it a
  perfect hedge?
•  

21
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• When the futures price changes by less than the
  cash price, a larger futures position than cash
  position is optimal. On the other hand, if the
  futures price changes by more than the cash
  price, a smaller futures position than cash
  position is optimal.
• The first step in structuring a perfect hedge is
  to identify the assets and/or liabilities to be
  protected. The volume and interest rate
  characteristics of the instruments to be hedged
  are the foundation for the futures decision.
• Once the size and cash market position has been
  chosen, the hedge ratio, or the number of
  contracts to be traded must be determined.

22
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• a. Optimal Hedging - Mean-Variance Approach
• The uncertain gain, pn, of the hedger who holds
  NA, units of the asset (commodity) and hedges
  using NF futures contracts is
•  
• pn (ST S0)NA (FT-F0)NF
•  
• The term (ST S0) is the random price change per
  unit of asset over the life of the hedge, and the
  term (FT-F0) is the random price change of the
  futures contract over the life of the hedge.

23
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• To derive the optimal number of futures contracts
  to sell, first rewrite the above equation in
  terms of price change per unit of the underlying
  asset
• pn /NA (ST S0) (FT-F0)NF / NA
• ?s h?F
•  When price change terms are ?S (ST S0) and ?F
  (FT-F0) and the hedge ratio, h, is the number
  of future contracts per unit of the underlying
  asset. Since the hedger is concerned with
  minimizing risk, the variance of the hedge
  portfolio profit
• sn2 ss2 h2sF2 2hsSF

24
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• The value of h that minimizes s2 is found by
•   dsn2 / dh 2hsF2 2sSF 0
•  solve for h, the optimal hedge ration is
•   h -sSF / sF2
• The optimal hedge ratio thus depends on the
  covariance between the cash and future prices
  changes relative to the variance of the future
  price changes.
• It is interesting to note that the expression for
  the optimal hedge ratio, sSF /sF2 is the slope
  coefficient in an ordinary least squares (OLS)
  regression of the cash price changes, ?S, on the
  futures price change, ?F.

25
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• Example Suppose that a securities Portfolio
  manager, anticipating a decline in interest rates
  over the next 3 months, wishes to protect the
  yield on an investment of 15m T-bills and that
  a T-bill futures contract is now selling far
  989,500. If the hedge ratio between price
  changes in T-bills and T-bill futures contract
  has been estimated through regression to be 0.93,
  the number of contracts to be used in the hedged
  can be determined by
• NF (V/F)h
• The number of contracts to be purchased is
• (15,000,000/989,500) 0.93 14.098

26
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• b. Optimal Hedging- OLS Regression Approach
• Consider the following regression equation
• ?S a0 a1?F e
• The intercept term, a0, captures any expected
  change in the cash price unaccompanied by an
  expected change in the futures price. We know
  that, if E(?F) 0, the expected cash price
  change equal the basis, i.e., E(?F) E(ST) - S0
  (F0 - S0). The regression model states that the
  expected cash price change equals a, under the
  assumption that E(?F) 0, then the intercept
  represents the basis.

27
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• The basis in turn, reflects the storage costs
  which a store of the assets must recover by price
  appreciation. The term a1?F, reflects the fact
  that the random changes in the futures price will
  be reflected in the cash price according to the
  slope coefficient, a1.a1, the slope coefficient,
  is equal to
•   Cov(?S,?F)
• a1 -----------------
• Var(?F)

28
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• Moreover, the value of a1 has theoretical meaning
  as the hedge ratio (h) that minimizes the risk of
  a portfolio of spot assets and futures contracts.
  That is, we can use the estimated of a1 from the
  regression model as the appropriate measure of
  the hedge ratio h to be used by the FI manager.
• The term e reflects basis risk which arises from
  the fact that certain random changes in ?S are
  unique to the cash asset and uncorrelated with
  the futures price change.

29
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• The hedged portfolio profit can be written
•   pn /ns a0 a1?F e h?F
•   a0 (a1 h) ??F e
• This equation shows clearly that the profit on
  the hedge portfolio, ?h ,can be made independent
  of movements in cash and futures prices by
  setting h -a1,

30
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• Ifa1 1.0, a one-dollar change in the cash price
  is matched by a one-dollar change in the futures
  prices. In this case, the optimal hedge is h -1
  or 100 percent hedge.
• If a1 0.0, futures and cash prices are
  unrelated and there is no point in hedging and
  the optimal hedge ratio, h, is zero.

31
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• The degree of confidence the FI manager may have
  in using such a method to determine the
  appropriate hedge ratio depends on how well the
  regression line fits the scatter of observations.
  
• The standard measure of the goodness of fit of a
  regression line is the R2 of the equation, which
  is the square of the correlation coefficient
  between S and F
•   
• R2 p2 Cov(?S,?F)/ (s?s s?F)

32
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• A low R2 would mean that we might have little
  confidence that the slope coefficienta1 , from
  the regression is actually the true hedge ratio.
• As R2 approaches 1, our degree of confidence
  increases in the use of futures contracts, with a
  given hedge ration estimate, to hedge our cash
  asset-risk position. R2 therefore measures
  hedging effectiveness.

33
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• Optimal Hedge Ratio An Application to Cross
  hedges
• Example Ajax expects at the beginning of 1991
  that it has to borrow 36m on June 1 by issuing
  one month commercial paper. On January 2, the
  3-month LIBOR is 9.25, the CP rate is 8.75, and
  the price of June Eurodollar Futures is 90.45.
•  

34
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• 1. Selecting the Appropriate Futures Contracts
• The appropriate futures contracts for instituting
  a cross-hedge is normally selected on that
  futures contracts most highly correlated with the
  underlying exposure.

35
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• 2. Determining the Appropriate Number of Futures
  Contracts
• (1)  the relation between movements in the
  underlying exposure and the price of the futures
  contract being used as a hedge
• The estimate of relative sensitivity tells how
  the interest rate to which the firm is exposed
  moves in relation to the interest rate imbedded
  in the futures contract. If the hedge ratio
  (beta) is estimated to be .75, then the treasurer
  knows that to hedge 36m exposure, he would need
  to sell 36 contracts .75 27 contracts June
  Eurodollar contracts.

36
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• (2)  If a given change in the interest rate has a
  larger impact on the underlying exposure than on
  the value of the futures contract, fewer futures
  contracts will be needed to hedge the position,
  and vice versa.
•  
• If the response of the futures contract is three
  times that of the underlying exposure, then to
  hedge the 36m exposure, it only need
•   36 contracts .75 .33 9 contracts

37
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• Spot Future
• __________________________________________________
  __________
• Jan 2 Jan 2 Sell 9 June Eurodollar
• contracts at 90.45 to yield
• 9.55
• June 1 Borrow 36m at June 1 Close out (buy
  back)
• 1-month CP rate of 10.20 the futures position
  at 89.25
• to yield 10.75
• __________________________________________________
  __________
• Additional Cost Gain
• 36m (10.20-8.75) 30/360
  9m(10.75-9.55) 90/360
• 43,500 27,000

38
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• C. Duration-Based Hedging Strategies
• For interest rate sensitive assets, assume
• F interest rate futures contract price
• DF duration of the asset underlying the
  futures
• contract at the maturity of the futures
  contract
• A asset portfolio value to be hedged
• DA duration of the asset portfolio at the
  maturity
• of the hedge

39
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• C. Duration-Based Hedging Strategies
• If assume that the change in yield, I, is the
  same for all maturities, i.e., only parallel
  shifts in the yield curve can occur, it is
  approximately true that
• ? A -A DA ? I
• To a reasonable approximation, it is also true
  that
• ? F -F DF ? I
• The number of contracts required to hedge against
  an uncertain is therefore given by
• N ADA/FDF

40
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• Ex Hedging a bond portfolio
• On August 2, a fund manager has 10m invested in
  government bonds and is concerned that interest
  rates are expected to be highly volatile over the
  next 3 months. The fund manager decides to use
  the December T-bond futures contract to hedge the
  value of the portfolio.
• The current futures price is 93-02 or 93.0625 or
  93,062.50 per contract. The duration of the
  bond portfolio in 3 months is 6.8 years. The
  cheapest-to-deliver T-bond is a 20-year 12 per
  annum coupon bond. The yield on this bond is
  currently 8.8 per annum, and the duration will
  be 9.2 years at maturity of the futures contract.

41
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• Ex Hedging a bond portfolio
• The fund manager requires a short position in
  T-bond futures to hedge the bond portfolio. The
  number of bond futures that should be shorted
• N (10,000,000 6.8)/(93,062.50 9.20
  79.42

42
II. Choosing the Optimal Numbers of Contracts
The Hedge Ratio

• The result During the period Aug. 2 to Nov. 2,
  interest rate declined rapidly. The value of the
  bond portfolio increased from 410m to
  10,450,000.
• On Nov. 2, the T-bond futures price was 98-16 or
  98,500 per contract. A loss of
  7998,500-93,062.50) 429,562.50 was
  therefore made on the contracts.
• Overall the value of the portfolio changed by
  only
• 450,000 - 429,562.50 20,437.50

43
III. Macrohedging with Futures for a Financial
Institution

• Suppose a FI's balance sheet structure is as
  follows Assets 100m, Liabilities 90m, and
  equity 10m. The average duration of assets and
  liabilities is 5 and 3 years, respectively. If
  interest rates are expected to rise from 10 to
  11, then
• ?E - (DA - kDL) A (?R/1R)
• - (5 - .9 3) 100m (.01/1.1)
• - 2.09m
• The manager's objective is to fully hedge the
  balance sheet exposure by constructing a futures
  position to make a gain to just offset the loss
  of 2.09m on equity.

44
III. Macrohedging with Futures for a Financial
Institution

• When interest rates rise, the price of futures
  contracts falls. The sensitivity of the price of
  a futures contracts depends on the duration of
  the deliverable bond underlying the contract, or
•  
• ?F/F - DF (?R/1R), or
• ?F - DF F (?R/1R)
• - DF (NF PF) (?R/1R)

45
III. Macrohedging with Futures for a Financial
Institution

• Fully hedging can be defined as selling
  sufficient number of futures contracts so that
  the loss of net worth on the balance sheet is
  just offset by the gain from off-balance-sheet
  selling of futures
• ?F ?E
• which implies
•   N F (DA - kDL) A / DF PF
• (5-.93)100m/(9.597,000)
• 249.59 contracts

46
III. Macrohedging with Futures for a Financial
Institution

• If a T-bond futures contract is used for hedging.
  The futures is quoted 97 per 100 of face value
  for the benchmark 20-yr., 8 coupon bond that has
  a duration of 9.5 yrs.
•  
• Suppose instead of using the 20-yr. T-bond
  futures to hedge it had used the 3-month T-bill
  futures that has a price of 97 per 100 par
  value and a duration of .25 yrs. Then
•  NF (5 - .93)100m/.2597,000 948.45
  contracts

47
III. Macrohedging with Futures for a Financial
Institution

• The Problem of Basis Risk
• Because spot bonds and futures on bonds are
  traded in different markets, the shift in yields
  (?R/1R) affecting the value of the
  on-balance-sheet cash portfolio may differ from
  the shift (?RF/1RF) in yields affecting the
  value of the underlying bond in the futures
  contracts i.e., spot and futures prices or
  values are not perfectly correlated. To take this
  basis risk into account
•   ?E -(DA - kDL) A (?R/1R)
• ?F - DF (N FP F ) (?RF/1RF)

48
III. Macrohedging with Futures for a Financial
Institution

• Setting ?E ?F , we have
• N F (DA - kDL) A / DF PF b,
•  
• Where b (? RF /1 RF)/ (?R/1R) which measures
  the degree to which the futures price yields move
  more or less than spot price yields.
• For example, if b 1.1, this implies that for
  every 1 change in discounted spot rate (?R/1R),
  the implied rate on the deliverable bond in the
  futures market moves by 1.1.
•   NF (5 -.93) 100m/9.597,000 1.1
• 226.9 contracts

49
IV. Hedging Credit Risk with Futures

• For well-diversified FIs their credit risk
  exposure may be largely nondiversifiable
  systematic risk. In particular, the return on the
  loan portfolio may come to be uniquely reliant on
  general macro-factors relating to the state of
  the economy. Under this circumstance, a FI might
  consider the use of stock index futures to hedge
  the systematic credit risk of its portfolio.
• The reason that stock index futures may be useful
  is that stock prices could reflect the underlying
  current and expected present values of the
  earnings and dividends of the firms, which are
  positively correlated with the performance of the
  economy.

50
IV. Hedging Credit Risk with Futures

• Suppose that a FI manager attempts to use SP 500
  index futures to hedge its loan portfolio the
  value of which will be adversely affected by
  economic recession. If the settlement price of
  the index futures dropped from 507.30 to 450, the
  FI would make
• (507.30 - 450) 500 28,650
• for each contract shorted. The cash flow profits
  the FI manager gets from selling futures when bad
  economic states arise can offset losses from the
  loan portfolio due to increased systematic risk.

51
IV. Hedging Credit Risk with Futures

• There might be two problems in using futures to
  hedge credit risk.
• 1. Selling futures may produce sufficient cash
  flows to offset credit losses in bad economic
  states. In good states, the seller of futures
  contracts loses as the index rises and the
  marking-to-market cash flows favor the contract
  buyer. Such losses are not likely to be
  compensated fully by gains on the loan portfolio
  since the return on this portfolio has a limited
  upside return potential that is the full payment
  of loan interest and principal. Thus a manager
  would prefer a derivative securities product that
  limits the risk of losses in good economic states
  while still producing profits in bad states.

52
IV. Hedging Credit Risk with Futures

• 2. Managers must decide how close and stable the
  correlation is between stock index futures prices
  and the general macroeconomic conditions
  affecting the systematic credit risk exposure to
  losses of the FI's loan portfolio. The stronger
  this correlation, the smaller the basis risk and
  the better the hedging effectiveness.