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Derivatives – Part 1(LOs 27.x – 31.x)
Forwards (LO 27.x)1
4 Commodity Futures (LO 30.x)
Interest Rate Futures (LO 29.x)
2 Hedging Strategies (LO 28.x)
3
Swaps (LO 31.x)5
Forwards/Futures
Time (T)
0 0 0( ) rTTF E S F S e
S0
F0
ST-1
FT-1 F0ST
ST=F
T
Forwards/Futures
Time (T)
(Commodity with High
“Convenience Yield” or
High-dividend Financial Asset)
F0 = E(S
T)
S0
F0
ST-1
FT-1 F0ST
ST=F
T
Cost-of-carry modelLO 27.1: State & explain cost-of-carry model for forward prices with & without interim cash flows
Time (T)
Income/
Dividend
(q)
Storage
Cost (U, u)
Risk-free
Rate (r)
Spot
(S0)
Forward
(F0)
Convenience
(y)
Cost-of-carry: Question
A stock’s price today is $50. The stock will pay a $1 (2%) dividend in six months. The risk-free rate is 5% for all maturities.
What the price of a (long) forward contract (F0) to purchase the stock in one year?
Cost-of-carry: Question
A stock’s price today is $50. The stock will pay a $1 (2%) dividend in six months. The risk-free rate is 5% for all maturities.
What the price of a (long) forward contract (F0) to purchase the stock in one year?
0 0 0
( 0.05)(6/12) (.05)(1)
( )
($50 [($1) ])
$51.538
rTF S I e F
e e
DerivativesLO 27.2: Compute the forward price given both the price of the underlying and the appropriate carrying costs of the underlying.
( )0 0
r u q y TF S e
( )0 0
r q TF S e
Commodity
Financial asset (e.g., stock index)
Cost-of-Carry ModelCost of carry = interest to finance asset (r)+ storage cost (u) - income earned (q)
( )0 0( ) r y TF S U I e
( )0 0
r u q y TF S e
Present values
Commodity
constant rates as %
u = storage costs
q = income (dividend)
y = convenience yield
U = Present value, storage costs
I = Present value, income
Cost-of-Carry ModelCost of carry = interest to finance asset (r)+ storage cost (u) - income earned (q)
0 0( ) rTF S I e( )0 0
r q TF S e
Present values
Financial asset (e.g., stock index)
constant rates as %
q = income (dividend) I = Present value, income
Cost-of-Carry: Question
The spot price of corn today is 230 cents per bushel. The storage cost is 1.5% per month. The risk-free interest rate is 6% per annum.
What is the forward price in four (4) months?
Cost-of-Carry: Question
The spot price of corn today is 230 cents per bushel. The storage cost is 1.5% per month. The risk-free interest rate is 6% per annum.
What is the forward price in four (4) months?
( ) (6%/12 1.5%)(4)0 0
(.02)(4)
(230)
230 249.16
r u TF S e e
e
Derivatives
LO 27.3: Calculate the value of a forward contract.
0( ) rTf F K e
Value of a forward contractA long forward contract on a non dividend-paying stock has three months left to maturity.
The stock price today is $10 and the delivery price is $8. Also, the risk-free rate is 5%.
What is the value of the forward contract?
Value of a forward contractA long forward contract on a non dividend-paying stock has three months left to maturity.
The stock price today is $10 and the delivery price is $8. Also, the risk-free rate is 5%.
What is the value of the forward contract?
(5%)(0.25)0 0 10 $10.126rTF S e e
(5%)(0.25)0( ) (10.126 8) $2.153rTf F K e e
DerivativesLO 27.4: Describe the differences between forward and futures contracts.
Forward vs. Futures Contracts
Forward Futures
Trade over-the-counter Trade on an exchange
Not standardized Standardized contracts
One specified delivery date Range of delivery dates
Settled at contract’s end Settled daily
Delivery or final cash
settlement usually occurs
Contract usually closed
out prior to maturity
Derivatives
A long-futures position agrees to buy in the future
A short-futures position agrees to sell in the future.
Price mechanism maintains a balance between buyers and
sellers.(market equilibrium)
Most futures contracts do not lead to delivery, because
most trades ―close out‖ their positions before delivery.
Closing out a position means entering into the opposite
type of trade from the original.
LO 27.5: Distinguish between a long futures position and a short futures position.
Derivatives
An (underlying) asset
A Treasury bond futures contract is on underlying
U.S. Treasury with maturity of at least 15 years and not
callable within 15 years (15 years ≤ T bond).
A Treasury note futures contract is on the underlying
U.S. Treasury with maturity of at least 6.5 years but not
greater than 10 years (6.5 ≤ T note ≤ 10 years).
When the asset is a commodity (e.g., cotton, orange
juice), the exchange specifies a grade (quality).
LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled.
Derivatives
Contract size varies by type of futures contract
Treasury bond futures: contract size is a face value of $100,000
S&P 500 futures contract is index $250 (multiplier of 250X)
NASDAQ futures contract is index $100 (multiplier of 100X)
Recently, ―mini contracts‖ have been introduced:
S&P 500 ―mini‖ = $50 x S&P Index
NASDAQ ―mini‖ = $20 x NASDQ
(each contract is one-fifth the price, to attract smaller investors)
LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled.
Derivatives
Delivery Arrangements
The exchange specifies delivery location.
Delivery Months
The exchange must specify the delivery month; this can
be the entire month or a sub-period of the month.
LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled.
Derivatives
Margin account: Broker requires deposit.
Initial margin: Must be deposited when contract is
initiated.
Mark-to-market: At the end of each trading day, margin
account is adjusted to reflect gains or losses.
LO 27.7: Describe the marking-to-market procedure, the
initial margin, and the maintenance margin.
Derivatives
Maintenance margin: Investor can withdraw funds in the margin account in excess of the initial margin. A maintenance margin guarantees that the balance in the margin account never gets negative (the maintenance margin is lower than the initial margin).
Margin call: When the balance in the margin account falls below the maintenance margin, broker executes a margin call. The next day, the investor needs to ―top up‖ the margin account back to the initial margin level.
Variation margin: Extra funds deposited by the investor after receiving a margin call.
LO 27.7: Describe the marking-to-market procedure, the
initial margin, and the maintenance margin.
Derivatives
There is only a variation margin if and when there is a
margin call.
Variation margin = initial margin – margin
account balance
The maintenance margin is a trigger level—once
triggered, the investor must ―top up‖ to the initial
margin, which is greater than the maintenance level.
LO 27.8: Compute the variation margin.
Derivatives
The exchange clearinghouse is a division of the exchange
(e.g., the CME Clearing House is a division of the
Chicago Mercantile Exchange) or an independent
company. The clearinghouse serves as a
guarantor, ensuring that the obligations of all trades are
met.
LO 27.9: Explain the role of the clearinghouse.
Derivatives
Market order: Execute the trade immediately at the
best price available.
Limit order: This order specifies a price (e.g., buy at $30
or less)—but with no guarantee of execution.
Stop order: (aka., stop-loss order) An order to execute
a buy/sell when a specified price is reached.
LO 27.9: Explain the role of the clearinghouse.
Derivatives
Stop-limit: Requires two specified prices, a stop and a
limit price. Once the stop-limit price is reached, it
becomes a limit order at the limit price.
Market-if-touched: Becomes a market order once
specified price is achieved.
Discretionary (aka., market-not-held order): A
market order, but the broker is given the discretion to
delay the order in an attempt to get a better price.
LO 27.9: Explain the role of the clearinghouse.
Derivatives
A short forward (or futures) hedge is an agreement to sell in the future and is appropriate when the hedger already owns the asset.
Classic example is farmer who wants to lock in a sales price: protects against a price decline.
A long forward (or futures) hedge is an agreement to buy in the future and is appropriate when the hedger does not currently own the asset but expects to purchase in the future.
Example is an airline which depends on jet fuel and enters into a forward or futures contract (a long hedge) in order to protect itself from exposure to high oil prices.
LO 28.1: Differentiate between a short hedge and a long hedge, and identify situations where each is appropriate.
Derivatives
Basis = Spot Price Hedged Asset –
Futures Price Futures Contract = S0 – F0
Basis =
Futures Price Futures Contract –
Spot Price Hedged Asset = F0 – S0
LO 28.2: Define and calculate the basis.
Hull says first is correct but second is
common for financial assets (either is okay)
Basis risk
Time (T)
$2.50
$2.20
$2.00
$1.90
T0
$0.30$0.10
T1
No hedge
Spot = -$0.50
SPOT
Forward
Short 1.67 F
Spot = -$0.50
Future = $0.30 (1.67)
Net = 0
Basis
Weakening of the basis = Futures price
increases more than spot
Basis risk
$2.50
$2.20
$2.00
$1.90
T0
$0.30$0.30
T1
No hedge
Spot = -$0.50
SPOT
Forward
Short 1.67 F
Spot = -$0.50
Future = $0.50 (1.67)
Net = +33.5
Basis
$1.70
Time (T)
Basis unchanged.
But unexpected strengthening= Hedger improved!
Basis risk
$2.50
$2.20
$2.60
$1.90
T0
$0.30
T1
SPOT
Forward
Short 1.67 F
Spot = +$0.10
Future = -$0.40 (1.67)
Net = -56.8
Basis
$2.60
Time (T)
Basis declines
Unexpected weakening= Hedger worse!
$0.0
Derivatives
Spot price increases by more than the futures price
basis increases. This is a ―strengthening of the basis‖
When unexpected, strengthening is favorable for a short
hedge and unfavorable for a long hedge
Futures price increases by more than the spot price
basis declines. This is a ―weakening of the basis‖
When unexpected, weakening is favorable for a long hedge
and unfavorable for a short hedge
LO 28.3: Define the types of basis risk and explain how they arise in futures hedging.
Derivatives
But basis risk arises because often the characteristics
of the futures contract differ from the underlying
position.
Contract ≠ Commodity
Contract is standardized (e.g., WTI oil futures)
Commodities are not exactly commodities (e.g., hedger has a
position in different grade of oil)
LO 28.3: Define the types of basis risk and explain how they arise in futures hedging.
Basis risk higher with cross-hedging
Derivatives
But basis risk arises because often the characteristics
of the futures contract differ from the underlying
position.
Contract ≠ Commodity.
Contract is standardized (e.g., WTI oil futures)
Commodities are not exactly commodities (e.g., hedger has a
position in different grade of oil)
LO 28.3: Define the types of basis risk and explain how they arise in futures hedging.
Trade-offLiquidity(exchange)
Basis risk
Derivatives
The optimal hedge ratio (a.k.a., minimum variance hedge
ratio) is the ratio of futures position relative to the spot
position that minimizes the variance of the position.
Where is the correlation and is the standard
deviation, the optimal hedge ratio is given by:
LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio.
* S
F
h
Derivatives
For example, if the volatility of the spot price is 20%, the
volatility of the futures price is 10%, and their correlation
is 0.4, then:
LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio.
* S
F
h20%
* (0.4) 0.810%
h
Derivatives
For example, if the volatility of the spot price is 20%, the
volatility of the futures price is 10%, and their correlation
is 0.4, then:
LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio.
* S
F
h20%
* (0.4) 0.810%
h
** A
F
h NN
QNumber of
contracts
Derivatives
Given a portfolio beta ( ), the current value of the
portfolio (P), and the value of stocks underlying one
futures contract (A), the number of stock index futures
contracts (i.e., which minimizes the portfolio variance) is
given by:
LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.
PN
A
Derivatives
By extension, when the goal is to shift portfolio beta from
( ) to a target beta ( *), the number of contracts
required is given by:
LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.
( * )P
NA
Futures: QuestionLO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.
Assume:
Value of S&P 500 Index is 1240
Value of portfolio is $10 million
Portfolio beta ( ) is 1.5
How do we change the portfolio beta to 1.2?
PN ( * )
A
Hint: Contract = ($250 Index)
and # of futures is given by:
Futures: QuestionLO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.
Assume:
• Value of S&P 500 Index is 1240
• Value of portfolio is $1 million
• Portfolio beta ( ) is 1.5
We short about 10 contracts. (-) indicates short, (+) long…
( * )
$10,000,000(1.2 1.5) 9.7
(1240)(250)
PN
A
Derivatives
When the delivery date of the futures contract occurs
prior to the expiration date of the hedge, the hedger can
roll forward the hedge: close out a futures contract and
take the same position on a new futures contract with a
later delivery date.
Exposed to:
Basis risk (original hedge)
Basis risk (each new hedge) = ―rollover basis risk‖
LO 28.6: Identify situations when a rolling hedge is appropriate, and discuss the risks of such a strategy.
DerivativesLO 29.1: Identify and apply the three most common day count conventions.
Actual/actual U.S. Treasuries
30/360 U.S. corporate and
municipal bonds
Actual/360 U.S. Treasury bills and
other money market
instruments
Derivatives
The Treasury bond futures contract allows the party with
the short position to deliver any bond with a maturity of
more than 15 years and that is not callable within 15
years. When the chosen bond is delivered, the conversion
factor defines the price received by the party with the
short position:
LO 29.2: Explain the U.S. Treasury bond (T-bond) futures contract conversion factor.
Cash Received = Quoted futures price Conversion
factor + Accrued interest
= (QFP CF) + AI
Derivatives
The convexity adjustment assumes continuous
compounding. Given that ( ) is the standard deviation of
the change in the short-term interest rate in one year, t1is the time to maturity of the futures contract and t2 is
the time to maturity of the rate underlying the futures
contract:
LO 29.3: Calculate the Eurodollar futures contract convexity adjustment.
21 2
1Forward = Futures
2t t
Derivatives
The number of contracts required to hedge against an
uncertain change in the yield, given by y, is given by:
FC = contract price for the interest rate futures contract.
DF = duration of asset underlying futures contract at maturity.
P = forward value of the portfolio being hedged at the maturity
of the hedge (typically assumed to be today’s portfolio value).
DP = duration of portfolio at maturity of the hedge
LO 29.4: Formulate a duration-based hedging strategy using interest rate futures.
* P
C F
PDN
F D
DerivativesLO 29.4: Formulate a duration-based hedging strategy using interest rate futures.
Assume a portfolio value of $10 million.
The fund manager will hedge with T-bond futures (each
contract is for delivery of $100,000) with a current futures
price of 98.
She thinks the duration of the portfolio at hedge maturity will
be 6.0 and the duration of futures contract with be 5.0.
How many futures contracts should be shorted?
DerivativesLO 29.4: Formulate a duration-based hedging strategy using interest rate futures.
Assume a portfolio value of $10 million.
The fund manager will hedge with T-bond futures (each
contract is for delivery of $100,000) with a current futures
price of 98.
She thinks the duration of the portfolio at hedge maturity will
be 6.0 and the duration of futures contract with be 5.0.
How many futures contracts should be shorted?
($10 million)(6)* 122
(98,000)(5)P
C F
PDN
F D
Derivatives
Portfolio immunization or duration matching is when a
bank or fund matches the average duration of assets with
the average duration of liabilities.
Duration matching protects or ―immunizes‖ against
small, parallel shifts in the yield (interest rate) curve. The
limitation is that it does not protect against nonparallel
shifts. The two most common nonparallel shifts are:
A twist in the slope of the yield curve, or
A change in curvature
LO 29.5: Identify the limitations of using a duration-based hedging strategy.
Derivatives
LO 30.1: Explain the derivation of the basic equilibrium
formula for pricing commodity forwards and futures.
Time (T)
Discount
rate ( )
Risk-free
Rate (r)
Spot
(S0)
Forward
(F0)
( )0, 0( ) r TT TF E S e
Derivatives
LO 30.1: Explain the derivation of the basic equilibrium
formula for pricing commodity forwards and futures.
( )0, 0( ) r TT TF E S e
0
0,T
( ) Spot price of S at time T, as expected at time 0F Forward pricer Risk-free rate
Discount rate for commodity S
TE S
Derivatives
Lease rate = commodity discount rate – growth rate
Lease rate dividend yield
LO 30.2: Define lease rates, and discuss the importance of lease rates for determining no-arbitrage values for commodity futures and forwards.
( )0, 0
r TTF S e
( )0 0
r q TF S e
Financial asset
Derivatives
Contango refers to an upward-sloping forward curve
which must be the case if the lease rate is less than
the risk-free rate. Backwardation refers to a downward-
sloping forward curve which must be the case if the
lease rate is greater than the risk-free rate.
LO 30.3: Explain how lease rates determine whether a forward market is in contango or backwardation.
DerivativesLO 30.3: Explain how lease rates determine whether a forward market is in contango or backwardation.
Time (T)
Research says normal backwardation is “normal:” speculators
want compensation (risk premium) for buying the futures contract
Spot
(S0)E(ST)
Forward
(F0)
Forward
(F0)
DerivativesLO 30.4: Explain how storage costs impact commodity forward prices, and calculate the forward price of a commodity with storage costs.
Time (T)
Risk-free
Rate (r)
Spot
(S0)
Forward
(F0)
Storage Cost ( )
negative dividend
Convenience (y)
dividend
Lease rate ( )
dividend
DerivativesLO 30.4: Explain how storage costs impact commodity forward prices, and calculate the forward price of a commodity with storage costs.
Time (T)
Lease
rate ( )
Risk-free
Rate (r)
Spot
(S0)
Forward
(F0)
( )0 0
r c TF S e
Storage
Cost ( )
Convenience
(y)
DerivativesLO 30.5: Explain how a convenience yield impacts commodity forward prices, and determine the no-arbitrage bounds for the forward price of a commodity when the commodity has a convenience yield.
( ) ( )0 0 0
r c T r TS e F S e
Risk-free
Rate
Storage
Cost
Risk-free
Rate
Storage
Cost
Convenience
Yield
Commodity Futures
1460148015001520154015601580160016201640
S&P 500 Index
Rational forward curve rises by cost
of capital (risk free + premium) less
dividends
Commodity FuturesLO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures.
500
550
600
650
700
750
800
850
900
Jul-07 Nov-08 Mar-10 Aug-11 Dec-12
Gold futures
Durable, (relatively) cheap to store.
Forward curve is “uninteresting”
Commodity FuturesLO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures.
200
250
300
350
400
450
Sep
-07
Dec-0
7
Mar-
08
Jun
-08
Sep
-08
Dec-0
8
Mar-
09
Jun
-09
Sep
-09
Dec-0
9
Mar-
10
Jun
-10
Sep
-10
Dec-1
0
Corn
Commodity FuturesLO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures.
5
6
7
8
9
10
Au
g-0
7
Nov-0
7
Feb
-08
May-0
8
Au
g-0
8
Nov-0
8
Feb
-09
May-0
9
Au
g-0
9
Nov-0
9
Feb
-10
May-1
0
Au
g-1
0
Nov-1
0
Natural Gas
Costly to transport. Costly to store (storage costs). Highly seasonal
Commodity FuturesLO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures.
68
69
70
71
72
73
74
75
76
Sep
-07
Nov-0
7
Jan
-08
Mar-
08
May-0
8
Jul-
08
Sep
-08
Nov-0
8
Jan
-09
Mar-
09
May-0
9
Jul-
09
Sep
-09
Nov-0
9
Jan
-10
Mar-
10
May-1
0
Jul-
10
Sep
-10
Nov-1
0
Crude oil
Compared to natural gas, easier to store and transport. Global market. Long-run forward price less (<) volatile than short-run forward.
Commodity Futures
If we can take a long position on one commodity that is an input (e.g., oil) into another commodity that is an output (e.g., gas or heating oil), then we can take a short position in the output commodity and the difference is the commodity spread.
Assume oil is $2 per gallon, gasoline is $2.10 per gallon and heating oil is $2.50 per gallon.
If we take a long position in 2 gallons of gasoline and one gallon of heating oil, plus a short position in three gallons of oil, the commodity spread =
(2 long gasoline $2.10) + (1 long heating oil $2.50) – (3 oil $2) = +$0.70
LO 30.7: Describe and calculate a commodity spread.
Commodity Futures
The basis is the difference between the price of the futures contract and the spot price of the underlying asset.
Basis risk is the risk (to the hedger) created by the uncertainty in the basis.
The futures contract often does not track exactly with the underlying commodity; i.e., the correlation is imperfect. Factors that can give rise to basis risk include:
Mismatch between grade of underlying and contract
Storage costs
Transportation costs
LO 30.8: Define basis risk, and explain how basis risk can occur when hedging commodity price exposure.
LO 30.9: Differentiate between a
strip hedge and a stack hedge.
10 10 10 10 10 10 10 10 10 10 10 10
Jan Feb Mar
<120 <110 <100Jan Feb Mar
Commodity Futures
Oil producer to deliver
10K barrels per month
Strip hedge: contract for
each obligation
Stack hedge: Single maturity,
―stack and roll‖
Commodity Futures
A strip hedge is when we hedge a stream of obligations by offsetting each individual obligation with a futures contract that matches the maturity and quantity of the obligation. For example, if a producer must deliver X number of commodities per month, then the strip hedge entails entering into a futures contract for X commodities, to be delivered in one month; plus a futures contract for X commodities to be delivered in two months. The strip hedger matches a series of futures to the obligations.
A stack hedge is front-loaded: the hedger enters into a large future with a single maturity. In this case, our hedger would take a long position in a near-term futures contract for 12X commodities (i.e., a year’s worth). The stack hedge may have lower transaction costs but it entails speculation (implicit or deliberate) on the forward curve: if the forward curve gets steeper, the stack hedger may lose. On the other hand, if the forward curve flattens, then the stack hedger gains because he/she has locked in the commodity at a relatively lower price.
LO 30.9: Differentiate between a strip hedge and a stack hedge.
Swaps
A swap is an agreement to exchange future cash
flows
• “Plain vanilla” swap: company pays fixed rate on
notional principal and receives floating rate (pay
fixed receive floating)
• Interest rate swap: principal not exchanged
(i.e., that’s why it is called notional)
• Currency swap: principal is (typically) exchanged
at beginning (inception) and end (maturity)
LO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap.
SwapsLO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap.
Add Your
Text here
Pay
Fixed
Receive
LIBOR
Add Your
Text here
Add Your
Text here
Add Your
Text here
Receive Fixed
Receive
LIBORReceive
LIBOR
Pay
FixedPay
Fixed
Pay LIBOR“Plain-vanilla”
Counterparty
Swaps
Notional principal: $100 million (notional principal is not exchanged)
Swap agreement: Pay fixed rate of 5% and receive LIBOR
Term: 3 years with payments every six months
LO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap.
End of
Period
(6 months)
LIBOR at the
Start of
Period
Pay Fixed
Cash Flow
Receive
Floating
Cash Flow
Net Cash
Flow
1 5.0% -2.5 +2.5 0.0
2 (Year 1) 5.2% -2.5 +2.6 +0.1
3 5.4% -2.5 +2.7 +0.2
4 (Year 2) 5.0% -2.5 +2.5 0.0
5 4.8% -2.5 +2.4 -0.1
6 (Year 3) 4.6% -2.5 +2.3 -0.2
Swaps
Intel borrowing fixed-rate @ 5.2%
MSFT borrowing floating-rate @ LIBOR + 10 bps
LO 31.2: Explain how an interest rate swap can be combined with an existing asset or liability to transform the interest rate risk.
SwapsLO 31.3: Explain the advantages and disadvantages of the comparative advantage argument often used for the existence of the swap market.
Fixed Floating
BetterCreditCorp 4% LIBOR + 1%
WorseCreditCorp 6% LIBOR + 2%
Swaps
LIBOR/swap zero given: six-month = 3%, 1 year = 3.5%, 1.5 year = 4%.
The 2 year swap rate is 5% which implies that a $100 face value bond with
a 5% coupon will sell exactly at par (why? Because the 5% coupons are
discounted at 5%)
We can solve for the two year zero rate (R) because it is the unknown
LO 31.4: Explain how the discount rates in a swap are computed.
Period Cash flowLIBOR/swap
zero rates
PresentValue of
Cash Flow0.5 $2.5 3.0% $2.461.0 $2.5 3.5% $2.411.5 $2.5 4.0% $2.352.0 $102.50 X? 102.5e2R
Total PV $100.00
SwapsLO 31.4: Explain how the discount rates in a swap are computed.
Period Cash flow
LIBOR/swap zero rates
PresentValue of
Cash Flow0.5 $2.5 3.0% $2.461.0 $2.5 3.5% $2.411.5 $2.5 4.0% $2.352.0 $102.50 X? 102.5e2R
Total PV $100.00
( .5)(3%) ( 1)(3.5%) ( 1.5)(4%) 2
2
( 2 )
2.5 2.5 2.5 102.5 100
2.46 2.41 2.35 102.5 100
0.90506
4.99%
R
R
R
e e e e
e
e
R
Swaps
If two companies enter into an interest rate swap arrangement, then one of the companies has a swap position that is equivalent to a long position in floating-rate bond and a short position in a fixed-rate bond.
VSWAP = BFL - BFIX
The counterparty to the same swap has the equivalent of a long position in a fixed-rate bond and a short position in a floating-rate bond:
VSWAP Counterparty = BFIX -BFL
LO 31.5: Explain how a swap can be interpreted as two simultaneous bond positions or as a sequence of forward rate agreements (FRAs).
Swaps
LO 31.6: Calculate the value of an interest rate swap.
Add Your
Text here
Receive
½ of 7%
Pay
½ LIBOR
Add Your
Text here
Add Your
Text here
Time
0.25Time
0.75Time
1.25
Receive
½ of 7%
Pay
½ LIBOR
Receive
½ of 7%
Pay
½ LIBOR
Assumptions
Notional 100
Receive Fixed 7.0%
LIBOR Rates
3 Months (0.25) 5.0%
6 Months (0.5) 5.5%
9 Months (0.75) 6.0%
12 Months (1.0) 6.5%
SwapsLO 31.6: Calculate the value of an
interest rate swap.
Assumptions
Notional 100
Receive Fixed 7.0%
LIBOR Rates
3 Months (0.25) 5.0%
6 Months (0.5) 5.5%
9 Months (0.75) 6.0%
12 Months (1.0) 6.5%
Fixed Floating
LIBOR Disc. Cash Flows Cash Flows
Time Rates Factor FV PV FV PV
0.25 5.0% 0.988 $3.5 $3.46 $102.75 $101.47
0.75 6.0% 0.956 $3.5 $3.35
1.25 6.5% 0.922 $103.5 $95.42
Total $102.23 $101.47
Value (swap) = $102.23 - $101.47 = $0.75
SwapsLO 31.7: Explain the mechanics and calculate the value of a currency swap.
Assumptions
Principal, Dollars ($MM) 10
Principal, Yen (MM) Y 1,000
FX rate 120
US rate 5.0%
Japanese rate 2.0%
SWAP:
PAY dollars @ 5%
RECEIVE yen @ 9%
SwapsLO 31.7: Explain the mechanics and
calculate the value of a currency swap.
Assumptions
Principal, Dollars ($MM) 10
Principal, Yen (MM) Y 1,000
FX rate 120
US rate 5.0%
Japanese rate 2.0%
SWAP:
PAY dollars @ 5%
RECEIVE yen @ 9%
Dollars (MM) Yen (MM)
Time FV PV FV PV
1 0.5 $0.48 90 Y 88
2 0.5 $0.45 90 Y 86
3 0.5 $0.43 90 Y 85
3 10 $8.61 1000 Y 942
$9.97 Y 1,201
Yen bond Y 1,201
Yen bond in US dollars $10.01
Dollar bond $9.97
Swap, yen bond - dollar bond $0.04
Swaps
Because a swap involves offsetting choir position, there is
no credit risk when the swap has negative value. Credit
risk only exists when the swap has positive value.
Further, because principal is not exchanged at the end of
the life of an interest rate swap, the potential default
losses are much less than those on an equivalent loan. On
the other hand, in a currency swap, the risk is greater
because currencies are exchanged at the end of the swap.
LO 31.8: Explain the role of credit risk inherent in an
existing swap position.
1
Derivatives – Part 1(LOs 27.x – 31.x)
Forwards (LO 27.x)1
4 Commodity Futures (LO 30.x)
Interest Rate Futures (LO 29.x)
2 Hedging Strategies (LO 28.x)
3
Swaps (LO 31.x)5
Forwards/Futures
Time (T)
0 0 0( ) rTTF E S F S e
S0
F0
ST-1
FT-1 F0ST
ST=F
T
Forwards/Futures
Time (T)
(Commodity with High
“Convenience Yield” or
High-dividend Financial Asset)
F0 = E(S
T)
S0
F0
ST-1
FT-1 F0ST
ST=F
T
Cost-of-carry modelLO 27.1: State & explain cost-of-carry model for forward prices with & without interim cash flows
Time (T)
Income/
Dividend
(q)
Storage
Cost (U, u)
Risk-free
Rate (r)
Spot
(S0)
Forward
(F0)
Convenience
(y)
2
Cost-of-carry: Question
A stock’s price today is $50. The stock will pay a $1 (2%) dividend in six months. The risk-free rate is 5% for all maturities.
What the price of a (long) forward contract (F0) to purchase the stock in one year?
Cost-of-carry: Question
A stock’s price today is $50. The stock will pay a $1 (2%) dividend in six months. The risk-free rate is 5% for all maturities.
What the price of a (long) forward contract (F0) to purchase the stock in one year?
0 0 0
( 0.05)(6/12) (.05)(1)
( )
($50 [($1) ])
$51.538
rTF S I e F
e e
DerivativesLO 27.2: Compute the forward price given both the price of the underlying and the appropriate carrying costs of the underlying.
( )0 0
r u q y TF S e
( )0 0
r q TF S e
Commodity
Financial asset (e.g., stock index)
Cost-of-Carry ModelCost of carry = interest to finance asset (r)+ storage cost (u) - income earned (q)
( )0 0( ) r y TF S U I e
( )0 0
r u q y TF S e
Present values
Commodity
constant rates as %
u = storage costs
q = income (dividend)
y = convenience yield
U = Present value, storage costs
I = Present value, income
3
Cost-of-Carry ModelCost of carry = interest to finance asset (r)+ storage cost (u) - income earned (q)
0 0( ) rTF S I e( )0 0
r q TF S e
Present values
Financial asset (e.g., stock index)
constant rates as %
q = income (dividend) I = Present value, income
Cost-of-Carry: Question
The spot price of corn today is 230 cents per bushel. The storage cost is 1.5% per month. The risk-free interest rate is 6% per annum.
What is the forward price in four (4) months?
Cost-of-Carry: Question
The spot price of corn today is 230 cents per bushel. The storage cost is 1.5% per month. The risk-free interest rate is 6% per annum.
What is the forward price in four (4) months?
( ) (6%/12 1.5%)(4)0 0
(.02)(4)
(230)
230 249.16
r u TF S e e
e
Derivatives
LO 27.3: Calculate the value of a forward contract.
0( ) rTf F K e
4
Value of a forward contractA long forward contract on a non dividend-paying stock has three months left to maturity.
The stock price today is $10 and the delivery price is $8. Also, the risk-free rate is 5%.
What is the value of the forward contract?
Value of a forward contractA long forward contract on a non dividend-paying stock has three months left to maturity.
The stock price today is $10 and the delivery price is $8. Also, the risk-free rate is 5%.
What is the value of the forward contract?
(5%)(0.25)0 0 10 $10.126rTF S e e
(5%)(0.25)0( ) (10.126 8) $2.153rTf F K e e
DerivativesLO 27.4: Describe the differences between forward and futures contracts.
Forward vs. Futures Contracts
Forward Futures
Trade over-the-counter Trade on an exchange
Not standardized Standardized contracts
One specified delivery date Range of delivery dates
Settled at contract’s end Settled daily
Delivery or final cash
settlement usually occurs
Contract usually closed
out prior to maturity
Derivatives
A long-futures position agrees to buy in the future
A short-futures position agrees to sell in the future.
Price mechanism maintains a balance between buyers and
sellers.(market equilibrium)
Most futures contracts do not lead to delivery, because
most trades ―close out‖ their positions before delivery.
Closing out a position means entering into the opposite
type of trade from the original.
LO 27.5: Distinguish between a long futures position and a short futures position.
5
Derivatives
An (underlying) asset
A Treasury bond futures contract is on underlying
U.S. Treasury with maturity of at least 15 years and not
callable within 15 years (15 years ≤ T bond).
A Treasury note futures contract is on the underlying
U.S. Treasury with maturity of at least 6.5 years but not
greater than 10 years (6.5 ≤ T note ≤ 10 years).
When the asset is a commodity (e.g., cotton, orange
juice), the exchange specifies a grade (quality).
LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled.
Derivatives
Contract size varies by type of futures contract
Treasury bond futures: contract size is a face value of $100,000
S&P 500 futures contract is index $250 (multiplier of 250X)
NASDAQ futures contract is index $100 (multiplier of 100X)
Recently, ―mini contracts‖ have been introduced:
S&P 500 ―mini‖ = $50 x S&P Index
NASDAQ ―mini‖ = $20 x NASDQ
(each contract is one-fifth the price, to attract smaller investors)
LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled.
Derivatives
Delivery Arrangements
The exchange specifies delivery location.
Delivery Months
The exchange must specify the delivery month; this can
be the entire month or a sub-period of the month.
LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled.
Derivatives
Margin account: Broker requires deposit.
Initial margin: Must be deposited when contract is
initiated.
Mark-to-market: At the end of each trading day, margin
account is adjusted to reflect gains or losses.
LO 27.7: Describe the marking-to-market procedure, the
initial margin, and the maintenance margin.
6
Derivatives
Maintenance margin: Investor can withdraw funds in the margin account in excess of the initial margin. A maintenance margin guarantees that the balance in the margin account never gets negative (the maintenance margin is lower than the initial margin).
Margin call: When the balance in the margin account falls below the maintenance margin, broker executes a margin call. The next day, the investor needs to ―top up‖ the margin account back to the initial margin level.
Variation margin: Extra funds deposited by the investor after receiving a margin call.
LO 27.7: Describe the marking-to-market procedure, the
initial margin, and the maintenance margin.
Derivatives
There is only a variation margin if and when there is a
margin call.
Variation margin = initial margin – margin
account balance
The maintenance margin is a trigger level—once
triggered, the investor must ―top up‖ to the initial
margin, which is greater than the maintenance level.
LO 27.8: Compute the variation margin.
Derivatives
The exchange clearinghouse is a division of the exchange
(e.g., the CME Clearing House is a division of the
Chicago Mercantile Exchange) or an independent
company. The clearinghouse serves as a
guarantor, ensuring that the obligations of all trades are
met.
LO 27.9: Explain the role of the clearinghouse.
Derivatives
Market order: Execute the trade immediately at the
best price available.
Limit order: This order specifies a price (e.g., buy at $30
or less)—but with no guarantee of execution.
Stop order: (aka., stop-loss order) An order to execute
a buy/sell when a specified price is reached.
LO 27.9: Explain the role of the clearinghouse.
7
Derivatives
Stop-limit: Requires two specified prices, a stop and a
limit price. Once the stop-limit price is reached, it
becomes a limit order at the limit price.
Market-if-touched: Becomes a market order once
specified price is achieved.
Discretionary (aka., market-not-held order): A
market order, but the broker is given the discretion to
delay the order in an attempt to get a better price.
LO 27.9: Explain the role of the clearinghouse.
Derivatives
A short forward (or futures) hedge is an agreement to sell in the future and is appropriate when the hedger already owns the asset.
Classic example is farmer who wants to lock in a sales price: protects against a price decline.
A long forward (or futures) hedge is an agreement to buy in the future and is appropriate when the hedger does not currently own the asset but expects to purchase in the future.
Example is an airline which depends on jet fuel and enters into a forward or futures contract (a long hedge) in order to protect itself from exposure to high oil prices.
LO 28.1: Differentiate between a short hedge and a long hedge, and identify situations where each is appropriate.
Derivatives
Basis = Spot Price Hedged Asset –
Futures Price Futures Contract = S0 – F0
Basis =
Futures Price Futures Contract –
Spot Price Hedged Asset = F0 – S0
LO 28.2: Define and calculate the basis.
Hull says first is correct but second is
common for financial assets (either is okay)
Basis risk
Time (T)
$2.50
$2.20
$2.00
$1.90
T0
$0.30$0.10
T1
No hedge
Spot = -$0.50
SPOT
Forward
Short 1.67 F
Spot = -$0.50
Future = $0.30 (1.67)
Net = 0
Basis
Weakening of the basis = Futures price
increases more than spot
8
Basis risk
$2.50
$2.20
$2.00
$1.90
T0
$0.30$0.30
T1
No hedge
Spot = -$0.50
SPOT
Forward
Short 1.67 F
Spot = -$0.50
Future = $0.50 (1.67)
Net = +33.5
Basis
$1.70
Time (T)
Basis unchanged.
But unexpected strengthening= Hedger improved!
Basis risk
$2.50
$2.20
$2.60
$1.90
T0
$0.30
T1
SPOT
Forward
Short 1.67 F
Spot = +$0.10
Future = -$0.40 (1.67)
Net = -56.8
Basis
$2.60
Time (T)
Basis declines
Unexpected weakening= Hedger worse!
$0.0
Derivatives
Spot price increases by more than the futures price
basis increases. This is a ―strengthening of the basis‖
When unexpected, strengthening is favorable for a short
hedge and unfavorable for a long hedge
Futures price increases by more than the spot price
basis declines. This is a ―weakening of the basis‖
When unexpected, weakening is favorable for a long hedge
and unfavorable for a short hedge
LO 28.3: Define the types of basis risk and explain how they arise in futures hedging.
Derivatives
But basis risk arises because often the characteristics
of the futures contract differ from the underlying
position.
Contract ≠ Commodity
Contract is standardized (e.g., WTI oil futures)
Commodities are not exactly commodities (e.g., hedger has a
position in different grade of oil)
LO 28.3: Define the types of basis risk and explain how they arise in futures hedging.
Basis risk higher with cross-hedging
9
Derivatives
But basis risk arises because often the characteristics
of the futures contract differ from the underlying
position.
Contract ≠ Commodity.
Contract is standardized (e.g., WTI oil futures)
Commodities are not exactly commodities (e.g., hedger has a
position in different grade of oil)
LO 28.3: Define the types of basis risk and explain how they arise in futures hedging.
Trade-offLiquidity(exchange)
Basis risk
Derivatives
The optimal hedge ratio (a.k.a., minimum variance hedge
ratio) is the ratio of futures position relative to the spot
position that minimizes the variance of the position.
Where is the correlation and is the standard
deviation, the optimal hedge ratio is given by:
LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio.
* S
F
h
Derivatives
For example, if the volatility of the spot price is 20%, the
volatility of the futures price is 10%, and their correlation
is 0.4, then:
LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio.
* S
F
h20%
* (0.4) 0.810%
h
Derivatives
For example, if the volatility of the spot price is 20%, the
volatility of the futures price is 10%, and their correlation
is 0.4, then:
LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio.
* S
F
h20%
* (0.4) 0.810%
h
** A
F
h NN
QNumber of
contracts
10
Derivatives
Given a portfolio beta ( ), the current value of the
portfolio (P), and the value of stocks underlying one
futures contract (A), the number of stock index futures
contracts (i.e., which minimizes the portfolio variance) is
given by:
LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.
PN
A
Derivatives
By extension, when the goal is to shift portfolio beta from
( ) to a target beta ( *), the number of contracts
required is given by:
LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.
( * )P
NA
Futures: QuestionLO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.
Assume:
Value of S&P 500 Index is 1240
Value of portfolio is $10 million
Portfolio beta ( ) is 1.5
How do we change the portfolio beta to 1.2?
PN ( * )
A
Hint: Contract = ($250 Index)
and # of futures is given by:
Futures: QuestionLO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.
Assume:
• Value of S&P 500 Index is 1240
• Value of portfolio is $1 million
• Portfolio beta ( ) is 1.5
We short about 10 contracts. (-) indicates short, (+) long…
( * )
$10,000,000(1.2 1.5) 9.7
(1240)(250)
PN
A
11
Derivatives
When the delivery date of the futures contract occurs
prior to the expiration date of the hedge, the hedger can
roll forward the hedge: close out a futures contract and
take the same position on a new futures contract with a
later delivery date.
Exposed to:
Basis risk (original hedge)
Basis risk (each new hedge) = ―rollover basis risk‖
LO 28.6: Identify situations when a rolling hedge is appropriate, and discuss the risks of such a strategy.
DerivativesLO 29.1: Identify and apply the three most common day count conventions.
Actual/actual U.S. Treasuries
30/360 U.S. corporate and
municipal bonds
Actual/360 U.S. Treasury bills and
other money market
instruments
Derivatives
The Treasury bond futures contract allows the party with
the short position to deliver any bond with a maturity of
more than 15 years and that is not callable within 15
years. When the chosen bond is delivered, the conversion
factor defines the price received by the party with the
short position:
LO 29.2: Explain the U.S. Treasury bond (T-bond) futures contract conversion factor.
Cash Received = Quoted futures price Conversion
factor + Accrued interest
= (QFP CF) + AI
Derivatives
The convexity adjustment assumes continuous
compounding. Given that ( ) is the standard deviation of
the change in the short-term interest rate in one year, t1is the time to maturity of the futures contract and t2 is
the time to maturity of the rate underlying the futures
contract:
LO 29.3: Calculate the Eurodollar futures contract convexity adjustment.
21 2
1Forward = Futures
2t t
12
Derivatives
The number of contracts required to hedge against an
uncertain change in the yield, given by y, is given by:
FC = contract price for the interest rate futures contract.
DF = duration of asset underlying futures contract at maturity.
P = forward value of the portfolio being hedged at the maturity
of the hedge (typically assumed to be today’s portfolio value).
DP = duration of portfolio at maturity of the hedge
LO 29.4: Formulate a duration-based hedging strategy using interest rate futures.
* P
C F
PDN
F D
DerivativesLO 29.4: Formulate a duration-based hedging strategy using interest rate futures.
Assume a portfolio value of $10 million.
The fund manager will hedge with T-bond futures (each
contract is for delivery of $100,000) with a current futures
price of 98.
She thinks the duration of the portfolio at hedge maturity will
be 6.0 and the duration of futures contract with be 5.0.
How many futures contracts should be shorted?
DerivativesLO 29.4: Formulate a duration-based hedging strategy using interest rate futures.
Assume a portfolio value of $10 million.
The fund manager will hedge with T-bond futures (each
contract is for delivery of $100,000) with a current futures
price of 98.
She thinks the duration of the portfolio at hedge maturity will
be 6.0 and the duration of futures contract with be 5.0.
How many futures contracts should be shorted?
($10 million)(6)* 122
(98,000)(5)P
C F
PDN
F D
Derivatives
Portfolio immunization or duration matching is when a
bank or fund matches the average duration of assets with
the average duration of liabilities.
Duration matching protects or ―immunizes‖ against
small, parallel shifts in the yield (interest rate) curve. The
limitation is that it does not protect against nonparallel
shifts. The two most common nonparallel shifts are:
A twist in the slope of the yield curve, or
A change in curvature
LO 29.5: Identify the limitations of using a duration-based hedging strategy.
13
Derivatives
LO 30.1: Explain the derivation of the basic equilibrium
formula for pricing commodity forwards and futures.
Time (T)
Discount
rate ( )
Risk-free
Rate (r)
Spot
(S0)
Forward
(F0)
( )0, 0( ) r TT TF E S e
Derivatives
LO 30.1: Explain the derivation of the basic equilibrium
formula for pricing commodity forwards and futures.
( )0, 0( ) r TT TF E S e
0
0,T
( ) Spot price of S at time T, as expected at time 0F Forward pricer Risk-free rate
Discount rate for commodity S
TE S
Derivatives
Lease rate = commodity discount rate – growth rate
Lease rate dividend yield
LO 30.2: Define lease rates, and discuss the importance of lease rates for determining no-arbitrage values for commodity futures and forwards.
( )0, 0
r TTF S e
( )0 0
r q TF S e
Financial asset
Derivatives
Contango refers to an upward-sloping forward curve
which must be the case if the lease rate is less than
the risk-free rate. Backwardation refers to a downward-
sloping forward curve which must be the case if the
lease rate is greater than the risk-free rate.
LO 30.3: Explain how lease rates determine whether a forward market is in contango or backwardation.
14
DerivativesLO 30.3: Explain how lease rates determine whether a forward market is in contango or backwardation.
Time (T)
Research says normal backwardation is “normal:” speculators
want compensation (risk premium) for buying the futures contract
Spot
(S0)E(ST)
Forward
(F0)
Forward
(F0)
DerivativesLO 30.4: Explain how storage costs impact commodity forward prices, and calculate the forward price of a commodity with storage costs.
Time (T)
Risk-free
Rate (r)
Spot
(S0)
Forward
(F0)
Storage Cost ( )
negative dividend
Convenience (y)
dividend
Lease rate ( )
dividend
DerivativesLO 30.4: Explain how storage costs impact commodity forward prices, and calculate the forward price of a commodity with storage costs.
Time (T)
Lease
rate ( )
Risk-free
Rate (r)
Spot
(S0)
Forward
(F0)
( )0 0
r c TF S e
Storage
Cost ( )
Convenience
(y)
DerivativesLO 30.5: Explain how a convenience yield impacts commodity forward prices, and determine the no-arbitrage bounds for the forward price of a commodity when the commodity has a convenience yield.
( ) ( )0 0 0
r c T r TS e F S e
Risk-free
Rate
Storage
Cost
Risk-free
Rate
Storage
Cost
Convenience
Yield
15
Commodity Futures
1460148015001520154015601580160016201640
S&P 500 Index
Rational forward curve rises by cost
of capital (risk free + premium) less
dividends
Commodity FuturesLO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures.
500
550
600
650
700
750
800
850
900
Jul-07 Nov-08 Mar-10 Aug-11 Dec-12
Gold futures
Durable, (relatively) cheap to store.
Forward curve is “uninteresting”
Commodity FuturesLO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures.
200
250
300
350
400
450
Sep
-07
Dec-0
7
Mar-
08
Jun
-08
Sep
-08
Dec-0
8
Mar-
09
Jun
-09
Sep
-09
Dec-0
9
Mar-
10
Jun
-10
Sep
-10
Dec-1
0
Corn
Commodity FuturesLO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures.
5
6
7
8
9
10
Au
g-0
7
Nov-0
7
Feb
-08
May-0
8
Au
g-0
8
Nov-0
8
Feb
-09
May-0
9
Au
g-0
9
Nov-0
9
Feb
-10
May-1
0
Au
g-1
0
Nov-1
0Natural Gas
Costly to transport. Costly to store (storage costs). Highly seasonal
16
Commodity FuturesLO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures.
68
69
70
71
72
73
74
75
76
Sep
-07
No
v-0
7
Jan
-08
Mar-
08
May-0
8
Jul-
08
Sep
-08
No
v-0
8
Jan
-09
Mar-
09
May-0
9
Jul-
09
Sep
-09
No
v-0
9
Jan
-10
Mar-
10
May-1
0
Jul-
10
Sep
-10
No
v-1
0
Crude oil
Compared to natural gas, easier to store and transport. Global market. Long-run forward price less (<) volatile than short-run forward.
Commodity Futures
If we can take a long position on one commodity that is an input (e.g., oil) into another commodity that is an output (e.g., gas or heating oil), then we can take a short position in the output commodity and the difference is the commodity spread.
Assume oil is $2 per gallon, gasoline is $2.10 per gallon and heating oil is $2.50 per gallon.
If we take a long position in 2 gallons of gasoline and one gallon of heating oil, plus a short position in three gallons of oil, the commodity spread =
(2 long gasoline $2.10) + (1 long heating oil $2.50) – (3 oil $2) = +$0.70
LO 30.7: Describe and calculate a commodity spread.
Commodity Futures
The basis is the difference between the price of the futures contract and the spot price of the underlying asset.
Basis risk is the risk (to the hedger) created by the uncertainty in the basis.
The futures contract often does not track exactly with the underlying commodity; i.e., the correlation is imperfect. Factors that can give rise to basis risk include:
Mismatch between grade of underlying and contract
Storage costs
Transportation costs
LO 30.8: Define basis risk, and explain how basis risk can occur when hedging commodity price exposure.
LO 30.9: Differentiate between a
strip hedge and a stack hedge.
10 10 10 10 10 10 10 10 10 10 10 10
Jan Feb Mar
<120 <110 <100Jan Feb Mar
Commodity Futures
Oil producer to deliver
10K barrels per month
Strip hedge: contract for
each obligation
Stack hedge: Single maturity,
―stack and roll‖
17
Commodity Futures
A strip hedge is when we hedge a stream of obligations by offsetting each individual obligation with a futures contract that matches the maturity and quantity of the obligation. For example, if a producer must deliver X number of commodities per month, then the strip hedge entails entering into a futures contract for X commodities, to be delivered in one month; plus a futures contract for X commodities to be delivered in two months. The strip hedger matches a series of futures to the obligations.
A stack hedge is front-loaded: the hedger enters into a large future with a single maturity. In this case, our hedger would take a long position in a near-term futures contract for 12X commodities (i.e., a year’s worth). The stack hedge may have lower transaction costs but it entails speculation (implicit or deliberate) on the forward curve: if the forward curve gets steeper, the stack hedger may lose. On the other hand, if the forward curve flattens, then the stack hedger gains because he/she has locked in the commodity at a relatively lower price.
LO 30.9: Differentiate between a strip hedge and a stack hedge.
Swaps
A swap is an agreement to exchange future cash
flows
• “Plain vanilla” swap: company pays fixed rate on
notional principal and receives floating rate (pay
fixed receive floating)
• Interest rate swap: principal not exchanged
(i.e., that’s why it is called notional)
• Currency swap: principal is (typically) exchanged
at beginning (inception) and end (maturity)
LO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap.
SwapsLO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap.
Add Your
Text here
Pay
Fixed
Receive
LIBOR
Add Your
Text here
Add Your
Text here
Add Your
Text here
Receive Fixed
Receive
LIBORReceive
LIBOR
Pay
FixedPay
Fixed
Pay LIBOR“Plain-vanilla”
Counterparty
Swaps
Notional principal: $100 million (notional principal is not exchanged)
Swap agreement: Pay fixed rate of 5% and receive LIBOR
Term: 3 years with payments every six months
LO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap.
End of
Period
(6 months)
LIBOR at the
Start of
Period
Pay Fixed
Cash Flow
Receive
Floating
Cash Flow
Net Cash
Flow
1 5.0% -2.5 +2.5 0.0
2 (Year 1) 5.2% -2.5 +2.6 +0.1
3 5.4% -2.5 +2.7 +0.2
4 (Year 2) 5.0% -2.5 +2.5 0.0
5 4.8% -2.5 +2.4 -0.1
6 (Year 3) 4.6% -2.5 +2.3 -0.2
18
Swaps
Intel borrowing fixed-rate @ 5.2%
MSFT borrowing floating-rate @ LIBOR + 10 bps
LO 31.2: Explain how an interest rate swap can be combined with an existing asset or liability to transform the interest rate risk.
SwapsLO 31.3: Explain the advantages and disadvantages of the comparative advantage argument often used for the existence of the swap market.
Fixed Floating
BetterCreditCorp 4% LIBOR + 1%
WorseCreditCorp 6% LIBOR + 2%
Swaps
LIBOR/swap zero given: six-month = 3%, 1 year = 3.5%, 1.5 year = 4%.
The 2 year swap rate is 5% which implies that a $100 face value bond with
a 5% coupon will sell exactly at par (why? Because the 5% coupons are
discounted at 5%)
We can solve for the two year zero rate (R) because it is the unknown
LO 31.4: Explain how the discount rates in a swap are computed.
Period Cash flowLIBOR/swap
zero rates
PresentValue of
Cash Flow0.5 $2.5 3.0% $2.461.0 $2.5 3.5% $2.411.5 $2.5 4.0% $2.352.0 $102.50 X? 102.5e2R
Total PV $100.00
SwapsLO 31.4: Explain how the discount rates in a swap are computed.
Period Cash flow
LIBOR/swap zero rates
PresentValue of
Cash Flow0.5 $2.5 3.0% $2.461.0 $2.5 3.5% $2.411.5 $2.5 4.0% $2.352.0 $102.50 X? 102.5e2R
Total PV $100.00
( .5)(3%) ( 1)(3.5%) ( 1.5)(4%) 2
2
( 2 )
2.5 2.5 2.5 102.5 100
2.46 2.41 2.35 102.5 100
0.90506
4.99%
R
R
R
e e e e
e
e
R
19
Swaps
If two companies enter into an interest rate swap arrangement, then one of the companies has a swap position that is equivalent to a long position in floating-rate bond and a short position in a fixed-rate bond.
VSWAP = BFL - BFIX
The counterparty to the same swap has the equivalent of a long position in a fixed-rate bond and a short position in a floating-rate bond:
VSWAP Counterparty = BFIX -BFL
LO 31.5: Explain how a swap can be interpreted as two simultaneous bond positions or as a sequence of forward rate agreements (FRAs).
Swaps
LO 31.6: Calculate the value of an interest rate swap.
Add Your
Text here
Receive
½ of 7%
Pay
½ LIBOR
Add Your
Text here
Add Your
Text here
Time
0.25Time
0.75Time
1.25
Receive
½ of 7%
Pay
½ LIBOR
Receive
½ of 7%
Pay
½ LIBOR
Assumptions
Notional 100
Receive Fixed 7.0%
LIBOR Rates
3 Months (0.25) 5.0%
6 Months (0.5) 5.5%
9 Months (0.75) 6.0%
12 Months (1.0) 6.5%
SwapsLO 31.6: Calculate the value of an
interest rate swap.
Assumptions
Notional 100
Receive Fixed 7.0%
LIBOR Rates
3 Months (0.25) 5.0%
6 Months (0.5) 5.5%
9 Months (0.75) 6.0%
12 Months (1.0) 6.5%
Fixed Floating
LIBOR Disc. Cash Flows Cash Flows
Time Rates Factor FV PV FV PV
0.25 5.0% 0.988 $3.5 $3.46 $102.75 $101.47
0.75 6.0% 0.956 $3.5 $3.35
1.25 6.5% 0.922 $103.5 $95.42
Total $102.23 $101.47
Value (swap) = $102.23 - $101.47 = $0.75
SwapsLO 31.7: Explain the mechanics and calculate the value of a currency swap.
Assumptions
Principal, Dollars ($MM) 10
Principal, Yen (MM) Y 1,000
FX rate 120
US rate 5.0%
Japanese rate 2.0%
SWAP:
PAY dollars @ 5%
RECEIVE yen @ 9%
20
SwapsLO 31.7: Explain the mechanics and
calculate the value of a currency swap.
Assumptions
Principal, Dollars ($MM) 10
Principal, Yen (MM) Y 1,000
FX rate 120
US rate 5.0%
Japanese rate 2.0%
SWAP:
PAY dollars @ 5%
RECEIVE yen @ 9%
Dollars (MM) Yen (MM)
Time FV PV FV PV
1 0.5 $0.48 90 Y 88
2 0.5 $0.45 90 Y 86
3 0.5 $0.43 90 Y 85
3 10 $8.61 1000 Y 942
$9.97 Y 1,201
Yen bond Y 1,201
Yen bond in US dollars $10.01
Dollar bond $9.97
Swap, yen bond - dollar bond $0.04
Swaps
Because a swap involves offsetting choir position, there is
no credit risk when the swap has negative value. Credit
risk only exists when the swap has positive value.
Further, because principal is not exchanged at the end of
the life of an interest rate swap, the potential default
losses are much less than those on an equivalent loan. On
the other hand, in a currency swap, the risk is greater
because currencies are exchanged at the end of the swap.
LO 31.8: Explain the role of credit risk inherent in an
existing swap position.