CHAPTER 26: DERIVATIVES AND HEDGING RISK

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CHAPTER 26 DERIVATIVES AND HEDGING RISK

• TOPICS
• 26.1 Forward Contracts
• 26.2 Futures
• 26.3 Hedging
• 26.4 Interest Rate Futures Contracts
• 26.5 Duration Hedging

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Overview

• Risks to be managed, and the methods used to
  finance them.
• Commodity price risk (futures)
• Interest rate exposure (duration hedging/swaps)
• FX exposure (derivatives)
• Hedging
• Find two closely related assets
• Buy one and sell the other in proportions to
  minimize the risk of your net position
• If the assets are perfectly correlated, your net
  position is risk free

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How risk is managed

• Production costs 1.50/bu
• Selling price in Sept. Unknown
• What can the farmer do to reduce risk?
• 1. Do nothing
• 2. Buy Crop Insurance
• 3. Buy a put option
• 4. Enter a Forward/futures contract to sell

Plant
Harvest
May
Sept.
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26.1 Forward Contracts

• A forward contract is an agreement to buy / sell
  an asset at a particular future time for a
  specified price called the delivery price
• Forward contracts are customized and not usually
  traded on an exchange
• The long (short) position agrees to buy (sell)
  the asset on the specified date for the delivery
  price
• When the contract is entered into, the delivery
  price is chosen so that the value of the contract
  is zero to each party.
• the forward price is the delivery price which
  makes the contract value zero (so the forward
  price is equal to the delivery price at the
  inception of the contract)

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Examples of a forward

• Pizza forward contract.
• Order pizza by phone. Specify topping (type),
  size, delivery time and location and price -
  fixed when contract is established. Pay on
  delivery.
• Energy forward
• You buy 50,000 cubic feet (50 Mcf) of heating gas
  in summer from your heating company for 10 per
  thousand cubic feet (Mcf), deliverable from Jan.
  March.
• Long in forward You
• Short Heating Co.

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Payoffs From Forward Contracts

• let ST denote the spot price of the asset at the
  delivery date T and let Ft be the delivery price
  (price set at t payable at T)
• The payoffs on the delivery date are
• long position payoff ST - Ft
• short position payoff Ft - ST

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Problems with hedging with forwards

• Hedged with forwards is imperfect, since you do
  not know the quantity you will have to trade.
• There is credit risk with forward contracts.
• Bipartisan arrangement
• In the previous example, if heating gas price
  increase sharply in the winter, your heating
  company will lose, and it might default.

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26.2 Futures

• Very similar to forwards in payoff profile, but
  addresses credit risk problem by
  marking-to-market every day.
• Highly standardized contracts (delivery
  location, contract size etc.), which permit
  exchange trading.
• The futures price is analogous to the forward
  price it is the delivery price for a futures
  contract
• the futures price will converge to the spot price
  of the underlying asset when the contract matures
  
• More institutional details
• The exact delivery date is usually not specified
  in a futures contract rather it is some time
  interval within the delivery month
• Actual delivery rarely occurs, instead parties
  close out positions by taking offsetting
  transactions prior to maturity. Cash settlement.
• There are commodities futures and financial
  futures (stocks, bonds and currencies).

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Example Corn Futures at CBOT
4 Digit Price Quote Fourth digit is 1/8 cent/bu
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Futures/forward price can change everyday

• In options, X does not change
• In futures, your profit/loss is based on
  fluctuation of futures price
• Your daily profit/loss for long futures is New
  futures price the futures price that you agreed
  
• If an offsetting position is taken before the
  expiry for a long futures, then profit/loss is
  Fnew - Ft ,where Fnew is the new futures price
  at the offsetting time.
• Likewise, if an offsetting position is taken
  before the expiry for a short futures, then
  profit/loss is Ft Fnew.

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Example

• Consider an investor who enters a futures
  contract expiring one month from now to purchase
  100 oz. of gold at the futures price of 275 per
  ounce.
• If the spot price of gold is 290 on the expiry
  date, the profit/loss is _____.
• If the investor closes out her position two week
  from now with a futures price of 280 on the same
  contract. The spot price is 290. Her profit/loss
  is ______.

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Marking to market/Margin

• Profit/loss is settled every day on a margin
  account
• Minimize default risk
• Details
• Initial margin
• If the value of the margin account falls below
  the maintenance margin, the contract holder
  receives a margin call.
• You need to add to bring margin balance back to
  initial margin level (otherwise contract will be
  forced to close out.)

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Example Marking-to-market Consider an investor
who enters a futures contract to purchase 100 oz.
of gold at the futures price of 875 per ounce.
Suppose that the initial margin is set at 6,000
and the maintenance margin is set at 4,500. The
contract is closed out after 6 days.
Day Futures Price Cash Flow Starting Margin Cash added to Margin Ending Margin
1 875 0 0 6,000 6,000
2 872
3 869
4 874
5 875 1,100 7,100 -1,100 6,000
6 884 900 6,900 -6,900 0

• What is the profit and loss to the investor,
  e.g. at days 2 and 3?
• (2) When does he receive a margin call? What to
  do when receiving a margin call?
• (3) Whats his ultimate gain/loss?

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Futures vs. Options

• Similarities
• Deferred delivery markets
• Limited number of contracts
• Standardized contracts
• Exchange is middleman

• Differences
• Options
• Longs have right, not obligation to buy/sell
• Frequent exercise
• Futures
• Both longs and shorts have obligation to buy/sell
• Daily price limits
• Marked-to-market
• Delivery seldom occurs

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Hedging with futuresLocking in price

• There are two types of investors who use
  futures/forward
• Speculators try to profit from price movements
• Hedgers try to protect against price movement
  and to reduce risk by making outcome less
  variable
• Short hedge (take a short position in futures) is
  used when you have asset to sell in the future
• Conversely, long hedge (take a long position in
  futures) is used when you have asset to purchase
  in the future

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Short Hedge

• Consider a firm which will be selling an asset at
  some future date T, and suppose there is a
  futures contract on that asset for delivery at T
• The firm is exposed to the risk that the price of
  the asset might fall between now and T
• If the firm takes a short position in a futures
  contract, its overall payoff is
• futures payoff -(FT - F0) -(ST - F0)
• payoff from selling asset at T ST
• total F0
• i.e. the price of F0 is locked in today
• If the asset price falls, the firm loses on the
  asset sale but gains on the futures contract
• If the asset price rises, the firm gains on the
  sale but loses on the futures contract

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Example Short hedge

• It is November 2003. The canola farmer is
  worried about the price of his crop (output). He
  sells canola futures say 50 tonnes Feb 2004 at
  300 per tonne. In February 2004, when the farmer
  harvests his crop, the market price of canola is
  250 per tonne.
• The farmer's profits from futures _____________
  per tonne
• The farmer's proceeds from sale of canola
  ___________ per tonne
• Total _______ per tonne
• Suppose in February 2004, when the farmer
  harvests his crop, the market price of canola is
  450 per tonne.
• The farmer's profits from futures _____________
  per tonne
• The farmer's proceeds from sale of canola
  ___________ per tonne
• Total ___________ per tonne

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Long Hedge

• Suppose instead a firm wants to purchase an asset
  at some future date T
• The firm is exposed to the risk that the price of
  the asset might rise between now and T
• If the firm takes a long position in a futures
  contract, its overall payoff is
• futures payoff FT - F0 ST - F0
• payment from purchasing asset at T -ST
• total F0
• i.e. the price of F0 is locked in today
• if the asset price falls, the firm gains on the
  asset purchase but loses on the futures contract
• And vice versa
• Note that futures hedging does not necessarily
  improve the overall outcome you can expect to
  lose on the futures contract roughly half of the
  time gt the objective of hedging is to reduce
  risk by making the outcome less variable

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Interest Rate Futures Contracts

• Futures contract whose underlying security is a
  debt obligation.
• Well consider interest rate futures
• Interest rate futures are used to lock into the
  forward term structure (lock into future interest
  rates).

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Term Structure of Interest Rates

• The text coverage of this material is in Appendix
  6A
• Although in almost all cases in this course we
  consider a flat term structure ( interest rates
  of different maturities), it is important to keep
  in mind that this is a simplification
• With a flat term structure, discount rates are
  the same for all maturities, but this is rarely
  (if ever) the case
• For Oct. 31, 2007, the Bank of Canada reported
  government zero coupon government bond yields as
  follows
• Maturity 1 yr 3 yr 5 yr 7 yr 10
  yr 15 yr
• Yield 4.18 4.16 4.18 4.21 4.28
  4.37
• This means, for example, that the price on Oct.
  31 of a one year zero coupon government bond
  paying 1,000 at maturity was 1,000/1.0418
  959.88, while the price of a ten year zero
  coupon government bond paying 1,000 at maturity
  was 1,000/1.042810 657.64
• The rates above, which can be used to determine
  prices at which bonds may be currently traded,
  are known as spot rates

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Pricing of Government Bonds

• Consider a Government of Canada bond that pays a
  semi-annual coupon of C for the next T/2 years
  (Note that there is a total of T2(T/2) payments)

If the term structure is flat, i.e. r1 r2
rT r , then the above formula simplifies to
the familiar C ATr F/(1 r )T
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Pricing of Interest Rate Forward Contracts

• An N-period forward contract on that Government
  Bond

Can be valued as the present value of the forward
price

• In the above, PV is the current value of the
  forward contract.
• Pforward is the forward contract price (the
  price youll pay in the future). One implies the
  other.

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Example

• Consider a 5-year forward contract on a 20-year
  Government of Canada bond. The coupon rate is 6
  percent per annum and payments are made
  semiannually on a par value of 1,000. The quoted
  yield to maturity is 5. Assume that the term
  structure is flat. What is the value of the bond
  today? What is the forward price?

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Interest rate futures contracts and hedging

• In practice, futures contracts on bonds are
  typically used rather than forward contracts
• Futures contracts on bonds are referred to as
  interest rate futures contracts
• The pricing relationships derived above for
  forward contracts will only be an approximation
  in this context
• The exact delivery date is determined by the
  short party in a futures contract

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,84-165 equals 84 16.5/32
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Use interest rate futures to lock into future
interest rate

• Example You own 10 million worth of 20 year 10
  coupon bond (semiannual coupon payments). The
  term structure is flat at 5 (semi-annual).
  These bonds are therefore selling at 1,000.
• If the term structure shifts up uniformly to
  5.5, the new price per bond is
• Since you have 10,000 of these bonds, you have
  lost
• You want to lock into the interest rates to
  prevent the loss. What should you do?

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Example contd Opposite position in futures

• Suppose government bond futures contract
  specifies 6-month delivery of 100,000 par value
  of 20 year 8 coupon bond. The current price
  (value) for this futures contract is
• After the term structure shift, it is
• Each short futures contract gains
• Suppose you hedge by shorting K futures
  contracts
• K Size of exposure/size of futures contract
• Gain on futures
• Overall Approximately you lock into the 5
  interest rate.

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Reasons in practice why interest hedging using
futures may not work perfectly

• Different maturities (bonds in portfolio vs.
  futures contract)
• Different coupon rates
• Different risk (e.g. corporate bonds in
  portfolio, government bonds in futures contract)

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Interest rate risk

• Interest rate risk impact of changing market
  yields on price
• Assume for simplicity a flat term structure.
  Consider these four bonds, each with 1,000 par
  value and coupons paid annually
• Note percentage price changes are calculated
  relative to the price when r 10, e.g.
  (877.93-875.66)/875.66 .2592.

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Interest rate risk contd

• Observations from the previous slide
• Comparing A, B, and C low coupon bond prices are
  more sensitive (i.e. higher percentage price
  change) to changes in r , given the same T
• Comparing C and D longer maturity bond prices
  are more sensitive to changes in r, given the
  same coupon
• Rank bonds by their interest rate risk

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Duration

• How do we measure the sensitivity of bond prices
  to changes in interest rates?
• This means that the percentage change in price
  for a given change in r is

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Duration contd

• Duration is defined as
• Or
• Duration measures how long, on average, a
  bondholder must wait to receive cash payments (a
  measure of the effective maturity of the bond
  given when its cash flows occur)

CFt cash flow at t
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Example
Calculate the duration for a 3 year bond, P
1,026.25, 8 annual coupon, r 7
1 2 3
Payment or cash flow 80 80 1,080
PV of cash Flow 74.77 69.88 881.60
Relative value (Wt) 0.0729 0.0681 .8591
Weighted maturity (tWt)
Duration
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Duration and Interest Rate Risk
A bit of algebra yields

• Duration measures the sensitivity of bond
  prices to changes in interest rates
• It is the first-order approximation of price
  sensitivity to interest rate
• For a small change in interest rate, duration is
  quite an accurate estimate for percent price
  change
• For a given change in yield, the larger a bond's
  duration the greater the impact on price
  (interest rate risk/sensitivity)

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Example

• Which bond has the higher duration (treat each
  column separately, assume everything else being
  equal)?

Bond Coupon Maturity Yield
A 10 10 years 10
B 5 5 years 5
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Portfolio Duration

• The duration of a portfolio P containing M bonds
  is
• where wi is the percentage weight of bond i in P.

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Examples

• D 3.3, P1,000, r 10, if r drops to 9, what
  is the price change as measured by duration?
• Portfolio duration
• A bond mutual fund holds the following two
  zero-coupon bonds (1) 5-year maturity and 5
  yield with 40 of portfolio investment (2)
  10-year maturity and 6 yield with 60 portfolio
  investment. Whats the duration for the funds
  portfolio?

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Immunization Balance sheet hedging based on
duration

• Immunization is a hedging strategy based on
  duration
• designed to protect against interest rate risk.
• Match the value changes in both sides of balance
  sheet
• The drop in the value of assets can be
  (partially) offset by the drop in the value of
  liabilities.
• Immunization is accomplished by equating the
  interest rate exposure of assets and liabilities
• Asset Duration assets Liability Duration
  liabilities

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Example

• You have just learned that your firm has a future
  liability of 1 million due at the end of two
  years. Suppose there are two different bonds
  available and r 10.
• Duration of liability 2 years
• Bond 1 7 annual coupon, T 1 year, 1,000 par
  value
• P1
• Duration (D1) 1 year
• Bond 2 8 annual coupon, T 3 years, 1,000 par
  value
• P2
• duration (D2) 2.78 years after some calculation

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Example contd Immunization strategies

• If
• Buy bond 1 and then another 1-year bond after a
  year - runs risk of lower rates available for
  second year - reinvestment risk
• Buy bond 2 and sell after 2 years - If rates rise
  before then, bond prices fall, so investment may
  not be enough to cover liability - price risk
• Invest in a combination of bonds 1 and 2 so that
  the exposure to interest rate risk will be the
  same between assets (your investment) and
  liability
• Basic idea if rates rise, the portfolios losses
  on the 3 year bonds will be offset by gains on
  reinvested 1 year bonds.
• And vice versa.
• How much should you invest in each bond?

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Example contd solution

• w1 invested in 1 year bonds and (1 w1) in 3
  year bonds.
• w1 D1 (1 w1 )D2
• Total amount to be invested
• Amount in 1-year bonds
• Number of 1 year bonds
• Amount in 3 year bonds
• Number of 3 year bonds

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Example contd Does the immunization strategy
work?
r after 1 year r after 1 year r after 1 year
9 10 11
Value at t 2 from reinvesting 1 year bond proceeds 438,197 442,181
Value at t 2 of 3 year bonds Value from reinvesting coupons received at t 1 Coupons received at t 2 Selling price at t 2 42,997 39,088 479,716 43,388 39,088 475,395
Total 999,998 1,000,052
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Immunization contd

• As can be seen from the table above, the
  immunization strategy appears to perform fairly
  well
• However, there are a number of assumptions needed
  for this to work. Some possible problems include
• the strategy assumes that there is no default
  risk or call risk for the bonds in the portfolio
• The strategy assumes that the term structure is
  flat and any shifts in it are parallel
• duration will change over time (even if r does
  not), so the manager may have to rebalance the
  portfolio (note that there is a tradeoff of
  accuracy from frequent rebalancing vs.
  transactions costs)
• More complicated strategies exist to handle these
  types of problems, but immunization using
  duration is still a very widely used tool in
  practice

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• Assigned questions 26.1-5, 7-9, 12-14, 17