CHAPTER 26: DERIVATIVES AND HEDGING
RISK
TOPICS:
• 26.1
• 26.2
• 26.3
• 26.4
• 26.5
Forward Contracts
Futures
Hedging
Interest Rate Futures Contracts
Duration Hedging
1
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
2
How risk is managed
Plant
May
Harvest
Sept.
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
3
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)
4
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.
5
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
6
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.
7
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).
8
Example: Corn Futures at CBOT
Contract Size
5,000 bushels
Deliverable Grades
No. 2 Yellow at par, No. 1 yellow at 1 1/2 cents per bushel over contract
price, No. 3 yellow at 1 1/2 cents per bushel under contract price
Price Quote
Cents/bushel
Last Trading Day
The business day prior to the 15th calendar day of the contract month.
Last Delivery Day
Second business day following the last trading day of the delivery month.
Exp
Last
Net Open High Low Close ETS
Chg
Vol
06Dec 312'4 -3'2 316'0 319'0 311'4 312'4 37537
4 Digit Price Quote: Fourth digit is 1/8 cent/bu
9
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.
10
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 ______.
11
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.)
12
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 Cash added to
Margin $
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
(1) 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) What’s his ultimate gain/loss?
13
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
14
Hedging with futures—Locking 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
15
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
16
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
17
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 − F 0 = S T − F 0
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 => the objective of hedging is to reduce risk by making the
outcome less variable
18
Interest Rate Futures Contracts
• Futures contract whose underlying security is a debt
obligation.
• We’ll consider interest rate futures
• Interest rate futures are used to lock into the forward term
structure (lock into future interest rates).
19
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
20
Pricing of Government Bonds
• Consider a Government of Canada bond that pays a semiannual coupon of $C for the next T/2 years (Note that there
is a total of T=2(T/2) payments):
0
C
C
C
1
2
3
…
CF
T
C
C
C
CF
PV 



2
3
(1  r1 ) (1  r2 ) (1  r3 )
(1  rT )T
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
21
Pricing of Interest Rate Forward Contracts
• An N-period forward contract on that Government Bond
 Pforward C
0
N
N+1
C
C
N+2 N+3
…
CF
N+T
Can be valued as the present value of the forward price:
PV 
Pforward
(1  rN ) N
C
C
C
CF




N 1
N 2
N 3
(1  rN 1 )
(1  rN  2 )
(1  rN 3 )
(1  rN T ) N T
• In the above, PV is the current value of the forward contract.
• Pforward is the forward contract price (the price you’ll pay in the
future). One implies the other.
22
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?
23
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
24
,84-165 equals 84 16.5/32
25
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% (semiannual). These bonds are therefore selling at $1,000.
40

50
t 11.05
1,000

t
1.05
 $1,000
40
If the term structure shifts up uniformly to 5.5%, the new price per bond is:
40

50
t 11.055
t

1,000
1.055
40
 $919 .77
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?
26
Example cont’d: 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.
27
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)
28
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%.
29
Interest rate risk cont’d
• 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:
Bond
Coupon (%)
A
B
C
9
10
4
Maturity
(Years)
8
7
6
30
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:
31
Duration cont’d
• Duration is defined as:
• Or:
CFt /( 1  r )t
Wt 
Bond Pr ice
CFt = cash flow at t
T
D   tWt
t 1
• 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)
32
Example
Calculate the duration for a 3 year bond, P = 1,026.25, 8% annual
coupon, r = 7%
1
2
3
80
80
1,080
PV of cash Flow
74.77
69.88
881.60
Relative value (Wt)
0.0729
0.0681
.8591
Payment or cash flow
Weighted maturity (tWt)
Duration =
33
Duration and Interest Rate Risk
A bit of algebra yields:
dP
1

D dr
P
1 r
• 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)
34
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%
35
Portfolio Duration
• The duration of a portfolio P containing M bonds is:
M
DP   wi Di
i 1
where wi is the percentage weight of bond i in P.
36
Examples
• D = 3.3, P=1,000, 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. What’s the duration for the fund’s portfolio?
37
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
38
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
39
Example cont’d: 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 portfolio’s 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?
40
Example cont’d: 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 =
41
Example cont’d: Does the immunization strategy work?
r after 1 year
9%
10%
11%
$438,197
$442,181
Value at t = 2 of 3 year bonds:
Value from reinvesting coupons received at t = 1
$42,997
$43,388
Coupons received at t = 2;
$39,088
$39,088
Selling price at t = 2;
$479,716
$475,395
Total
$999,998 $1,000,052
Value at t = 2 from reinvesting 1 year bond
proceeds:
42
Immunization cont’d
• 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
43
Assigned questions # 26.1-5, 7-9, 12-14, 17
44