7. Derivatives Part1 Pdf

Derivatives – Part 1
(LOs 27.x – 31.x)

Forwards (LO 27.x)
1

Hedging Strategies (LO 28.x)
2

3 Interest Rate Futures (LO 29.x)

4 Commodity Futures (LO 30.x)

5 Swaps (LO 31.x)

Forwards/Futures
rT
F0 E(ST ) F0 S0 e

F0
F0 FT-1 ST
ST=FT
ST-1
S0

Time (T)

Forwards/Futures
(Commodity with High
F0 = E(ST) “Convenience Yield” or
High-dividend Financial Asset)
S0
ST-1

F0
F0 FT-1 ST
ST=FT

Time (T)

Cost-of-carry model
LO 27.1: State & explain cost-of-carry model for
forward prices with & without interim cash flows
Risk-free Storage
Rate (r) Cost (U, u)

Income/ Convenience
Dividend (y) Forward
(q) (F0)
Spot
(S0)
Time (T)

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?

rT
F0 ( S0 I )e F0
( 0.05)(6/12) (.05)(1)
($50 [($1)e ])e
$51.538

Derivatives
LO 27.2: Compute the forward price given both the
price of the underlying and the appropriate carrying
costs of the underlying.

Commodity

( r u q y )T
F0 S0 e

Financial asset (e.g., stock index)
( r q )T
F0 S0 e

Cost-of-Carry Model
Cost of carry = interest to finance asset (r)
+ storage cost (u) - income earned (q)

Commodity

( r u q y )T ( r y )T
F0 S0 e F0 ( S0 U I )e
constant rates as % Present values
U = Present value, storage costs
u = storage costs
I = Present value, income
q = income (dividend)
y = convenience yield

Cost-of-Carry Model


( r q )T rT
F0 S0 e F0 ( S0 I )e
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


( r u)T (6%/12 1.5%)(4)
F0 S0 e (230)e
(.02)(4)
230e 249.16

Derivatives
LO 27.3: Calculate the value of a forward contract.

rT
f ( F0 K )e

Value of a forward contract
A 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 contract
A 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?

rT (5%)(0.25)
F0 S0 e 10e $10.126
rT (5%)(0.25)
f ( F0 K )e (10.126 8)e $2.153

Derivatives
LO 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 Contract usually closed
settlement usually occurs out prior to maturity

Derivatives
LO 27.5: Distinguish between a long futures position
and a short futures position.

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.

Derivatives
LO 27.6: Describe the characteristics of a futures
contract and explain how futures positions are settled.

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).

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)

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.

Derivatives
LO 27.7: Describe the marking-to-market procedure, the
initial margin, and the maintenance margin.

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.

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.

Derivatives
LO 27.8: Compute the variation margin.

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.

Derivatives
LO 27.9: Explain the role of the clearinghouse.

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.

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.

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.

Derivatives
LO 28.1: Differentiate between a short hedge and a
long hedge, and identify situations where each is
appropriate.
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.

Derivatives
LO 28.2: Define and calculate the basis.

Basis = Spot Price Hedged Asset –

Futures Price Futures Contract = S0 – F0

Basis =

Futures Price Futures Contract –
Spot Price Hedged Asset = F0 – S0

Hull says first is correct but second is
common for financial assets (either is okay)

Basis risk
No hedge
Spot = -$0.50
SPOT $2.50
$2.00

Short 1.67 F
Basis $0.30
$0.10
Spot = -$0.50
Forward $2.20
Future = $0.30 (1.67)
$1.90

Net = 0
T0 T1

Time (T)

Weakening of the basis = Futures price
increases more than spot

Basis risk
No hedge
Spot = -$0.50
SPOT $2.50
$2.00

Short 1.67 F
Basis $0.30
$0.30
Spot = -$0.50
Forward $2.20
Future = $0.50 (1.67)
$1.90

Net = +33.5
$1.70
T0
T1 Time (T)

Basis unchanged.
But unexpected strengthening= Hedger improved!

Basis risk
T1
$2.60 $2.60
SPOT $2.50
$0.0
Basis $0.30
Short 1.67 F
Spot = +$0.10
Forward $2.20 $1.90
Future = -$0.40 (1.67)
Net = -56.8
T0

Time (T)

Basis declines
Unexpected weakening= Hedger worse!

Derivatives
LO 28.3: Define the types of basis risk and explain how
they arise in futures hedging.

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

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)

Basis risk higher with cross-hedging

Derivatives


of the futures contract differ from the underlying
position.
Contract ≠ Commodity.




Trade-off
Liquidity Basis risk
(exchange)

Derivatives
LO 28.4: Define, calculate, and interpret the minimum
variance hedge ratio.

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:

S
h*
F

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:

20%
S
h* h* (0.4) 0.8
10%
F

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:

20%
S
h* h* (0.4) 0.8
10%
F
h * NA
Number of
N*
contracts QF

Derivatives
LO 28.5: Calculate the number of stock index futures
contracts to buy or sell to hedge an equity portfolio or
individual stock.

 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:

P
N
A

Derivatives
LO 28.5: Calculate the number of stock index futures
contracts to buy or sell to hedge an equity portfolio or
individual stock.

 Byextension, when the goal is to shift portfolio beta from
( ) to a target beta ( *), the number of contracts
required is given by:

P
N(* )
A

Futures: Question
LO 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?

Hint: Contract = ($250 Index)
P
N (* )
and # of futures is given by:
A

Futures: Question
LO 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

P
N (* )
A
$10,000,000
(1.2 1.5) 9.7
(1240)(250)
We short about 10 contracts. (-) indicates
short, (+) long…

Derivatives
LO 28.6: Identify situations when a rolling hedge is
appropriate, and discuss the risks of such a strategy.

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‖


Derivatives
LO 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
LO 29.2: Explain the U.S. Treasury bond (T-bond)
futures contract conversion factor.

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:

Cash Received = Quoted futures price Conversion
factor + Accrued interest
= (QFP CF) + AI

Derivatives
LO 29.3: Calculate the Eurodollar futures contract
convexity adjustment.

The convexity adjustment assumes continuous

compounding. Given that ( ) is the standard deviation of
the change in the short-term interest rate in one year, t1
is the time to maturity of the futures contract and t2 is
the time to maturity of the rate underlying the futures
contract:

1 2
Forward = Futures t 1 t2
2

Derivatives
LO 29.4: Formulate a duration-based hedging strategy
using interest rate futures.
The number of contracts required to hedge against an

uncertain change in the yield, given by y, is given by:

PDP
N*
FC DF
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

Derivatives
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?


Derivatives


price of 98.



PDP ($10 million)(6)
N* 122
FC DF (98,000)(5)

Derivatives
LO 29.5: Identify the limitations of using a duration-
based hedging strategy.

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


Derivatives
LO 30.1: Explain the derivation of the basic equilibrium
formula for pricing commodity forwards and futures.

Risk-free
Rate (r)

(r )T
F0,T E0 (ST )e
Discount
Forward
rate ( )
(F0)
Spot
(S0)
Time (T)

Derivatives
LO 30.1: Explain the derivation of the basic equilibrium
formula for pricing commodity forwards and futures.

(r )T
F0,T E0 (ST )e
E0 (ST ) Spot price of S at time T, as expected at time 0
F0,T Forward price
r Risk-free rate
Discount rate for commodity S

Derivatives
LO 30.2: Define lease rates, and discuss the
importance of lease rates for determining no-
arbitrage values for commodity futures and forwards.

 Lease rate = commodity discount rate – growth rate
 Lease rate dividend yield

(r )T
F0,T S0e
Financial asset

( r q )T
F0 S0 e

Derivatives
LO 30.3: Explain how lease rates determine whether a
forward market is in contango or backwardation.

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.

Derivatives
LO 30.3: Explain how lease rates determine whether a
forward market is in contango or backwardation.
Forward
(F0)

Spot E(ST)
(S0)

Forward
(F0)

Time (T)

Research says normal backwardation is “normal:” speculators
want compensation (risk premium) for buying the futures contract

Derivatives
LO 30.4: Explain how storage costs impact commodity
forward prices, and calculate the forward price of a
commodity with storage costs.
Risk-free Storage Cost ( )
Rate (r) negative dividend

Lease rate ( ) Convenience (y)
Forward
dividend
dividend
(F0)
Spot
(S0)
Time (T)

Derivatives
forward prices, and calculate the forward price of a
Storage
Risk-free
Cost ( )
Rate (r)

(r c )T
F0 S0 e
Convenience
Lease
Forward
(y)
rate ( )
(F0)
Spot
(S0)
Time (T)

Derivatives
LO 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.

Risk-free Storage Risk-free Storage
Rate Cost Rate Cost

(r c )T (r )T
S0 e F0 S0 e
Convenience
Yield

Commodity Futures

S&P 500 Index
1640
1620
1600
1580
1560
1540
1520 Rational forward curve rises by cost
1500 of capital (risk free + premium) less
1480
dividends
1460

Commodity Futures
LO 30.6: Discuss the factors that impact the pricing of
gold, corn, natural gas, and oil futures.

Gold futures
900
850
800
750
700
650
Durable, (relatively) cheap to store.
600
550 Forward curve is “uninteresting”
500
Jul-07 Nov-08 Mar-10 Aug-11 Dec-12

Commodity Futures

Corn
450
400
350
300
250
200 Jun-09
Jun-08

Jun-10
Mar-09
Mar-08

Mar-10
Sep-07

Sep-08

Sep-09

Sep-10
Dec-07

Dec-08

Dec-09

Dec-10

Commodity Futures

Natural Gas
10
9
8
7
Costly to transport. Costly to store
6
(storage costs). Highly seasonal
5
May-08

May-09

May-10
Aug-07

Aug-08

Aug-09

Aug-10
Nov-07
Feb-08

Nov-08
Feb-09

Nov-09
Feb-10

Nov-10

Commodity Futures
Crude oil
76
75
74
73
72
71
70 Compared to natural gas, easier to store and
69 transport. Global market. Long-run forward price
less (<) volatile than short-run forward.
68

Jan-10
Jan-08
Mar-08

Jan-09
Mar-09

Mar-10
Sep-07

Nov-08
Nov-07

Sep-08

Sep-09
Nov-09

Sep-10
Nov-10
May-08

May-09

May-10
Jul-08

Jul-09

Jul-10

Commodity Futures
LO 30.7: Describe and calculate a commodity spread.

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

Commodity Futures
LO 30.8: Define basis risk, and explain how basis risk
can occur when hedging commodity price exposure.

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


Commodity Futures
LO 30.9: Differentiate between a
strip hedge and a stack hedge.

Oil producer to deliver

10K barrels per month
Strip hedge: contract for

each obligation
Stack hedge: Single maturity,

―stack and roll‖
<120 <110 <100
Jan Feb Mar

10 10 10 10 10 10 10 10 10 10 10 10
Jan Feb Mar

Commodity Futures
LO 30.9: Differentiate between a strip hedge and a
stack hedge.
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.

Swaps
LO 31.1: Illustrate the mechanics and compute the
cash flows of a plain vanilla interest rate swap.

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)

Swaps

Receive Fixed

“Plain-vanilla” Add Your
Pay LIBOR
Text here
Counterparty
Add Your Add Your
Add Your
Receive Receive Receive
Text here Text here
Text here
LIBOR LIBOR LIBOR

Pay Pay Pay

Fixed Fixed Fixed

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


End of LIBOR at the Receive
Period Start of Pay Fixed Floating Net Cash
(6 months) Period Cash Flow Cash Flow 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
LO 31.2: Explain how an interest rate swap can be
combined with an existing asset or liability to
transform the interest rate risk.

 Intel
borrowing fixed-rate @ 5.2%
 MSFT borrowing floating-rate @ LIBOR + 10 bps

Swaps
LO 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
LO 31.4: Explain how the discount rates in a swap are
computed.
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


Present
LIBOR/swap Value of
Period Cash flow zero rates Cash Flow
0.5 $2.5 3.0% $2.46
1.0 $2.5 3.5% $2.41
1.5 $2.5 4.0% $2.35
102.5e2R
2.0 $102.50 X?
Total PV $100.00

Swaps
LO 31.4: Explain how the discount rates in a swap are
computed.
Present
LIBOR/swap Value of
zero rates
Period Cash flow Cash Flow
0.5 $2.5 3.0% $2.46
1.0 $2.5 3.5% $2.41
1.5 $2.5 4.0% $2.35
102.5e2R
2.0 $102.50 X?
Total PV $100.00

2.5e( .5)(3%) 2.5e( 1)(3.5%) 2.5e( 1.5)(4%) 2R
102.5e 100
2.46 2.41 2.35 102.5e 2 R 100
e( 2 R) 0.90506
R 4.99%

Swaps
LO 31.5: Explain how a swap can be interpreted as two
simultaneous bond positions or as a sequence of
forward rate agreements (FRAs).
 Iftwo 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

Swaps
LO 31.6: Calculate the value of an interest rate swap.

Assumptions
Notional 100
Receive Fixed 7.0%

Time Time Time
LIBOR Rates
0.25 0.75 1.25
3 Months (0.25) 5.0%
6 Months (0.5) 5.5%
Receive Receive Receive
9 Months (0.75) 6.0%
12 Months (1.0) 6.5%
½ of 7% ½ of 7% ½ of 7%

AddPay AddPay Pay
Your Add Your
Your
Text here Text here
Text here
½ LIBOR ½ LIBOR ½ LIBOR

Assumptions

Swaps Notional 100
Receive Fixed 7.0%
LO 31.6: Calculate the value of an
interest rate swap. 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

Swaps
LO 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%

Assumptions
Principal, Dollars ($MM) 10

Swaps Principal, Yen (MM) Y 1,000
FX rate 120
LO 31.7: Explain the mechanics and US rate 5.0%
calculate the value of a currency swap. 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
LO 31.8: Explain the role of credit risk inherent in an
existing swap position.

 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.

Forwards/Futures
Derivatives – Part 1
(LOs 27.x – 31.x)
S0 erT
F0 E(ST ) F0
Forwards (LO 27.x)
1

F0
F0 FT-1 ST
Hedging Strategies (LO 28.x)
2

ST=FT
3 Interest Rate Futures (LO 29.x)
ST-1
S0
4 Commodity Futures (LO 30.x)

5 Swaps (LO 31.x) Time (T)

Forwards/Futures Cost-of-carry model
(Commodity with High LO 27.1: State & explain cost-of-carry model for
F0 = E(ST) forward prices with & without interim cash flows
“Convenience Yield” or
High-dividend Financial Asset) Risk-free Storage
Rate (r) Cost (U, u)
S0
ST-1

F0
F0 FT-1 ST
Income/ Convenience
Dividend (y)
ST=FT Forward
(q) (F0)
Spot
(S0)
Time (T)

Time (T)

1

Cost-of-carry: Question Cost-of-carry: Question
A stock’s price today is $50. The stock will pay A stock’s price today is $50. The stock will pay
a $1 (2%) dividend in six months. The risk- a $1 (2%) dividend in six months. The risk-
free rate is 5% for all maturities. free rate is 5% for all maturities.

What the price of a (long) forward contract What the price of a (long) forward contract
(F0) to purchase the stock in one year? (F0) to purchase the stock in one year?

I )erT
F0 ( S0 F0
($50 [($1)e( 0.05)(6/12)
])e(.05)(1)
$51.538

Derivatives Cost-of-Carry Model
LO 27.2: Compute the forward price given both the
price of the underlying and the appropriate carrying
costs of the underlying.
Commodity

( r u q y )T
I )e(r y )T
Commodity
F0 S0 e F0 ( S0 U
( r u q y )T
F0 S0 e constant rates as % Present values
U = Present value, storage costs
u = storage costs
I = Present value, income
q = income (dividend)
Financial asset (e.g., stock index) y = convenience yield

S 0 e( r q )T
F0

2

Cost-of-Carry Model Cost-of-Carry: Question
Cost of carry = interest to finance asset (r) The spot price of corn today is 230 cents per
+ storage cost (u) - income earned (q) bushel. The storage cost is 1.5% per month.

( r q )T
I )e rT What is the forward price in four (4) months?
F0 S0 e F0 ( S0
q = income (dividend) I = Present value, income

Cost-of-Carry: Question Derivatives
The spot price of corn today is 230 cents per LO 27.3: Calculate the value of a forward contract.
The risk-free interest rate is 6% per annum. rT
f ( F0 K )e

S 0 e( r u)T
(230)e(6%/12 1.5%)(4)
F0
230e(.02)(4) 249.16

3

Value of a forward contract Value of a forward contract
A long forward contract on a non dividend-paying stock has A long forward contract on a non dividend-paying stock has
three months left to maturity. three months left to maturity.

The stock price today is $10 and the delivery price is $8. The stock price today is $10 and the delivery price is $8.
Also, the risk-free rate is 5%. Also, the risk-free rate is 5%.

What is the value of the forward contract? What is the value of the forward contract?

S0 e rT 10e(5%)(0.25)
F0 $10.126
rT
(10.126 8)e(5%)(0.25)
f ( F0 K )e $2.153

Derivatives Derivatives
LO 27.4: Describe the differences between forward LO 27.5: Distinguish between a long futures position
and futures contracts. and a short futures position.

A long-futures position agrees to buy in the future

Forward vs. Futures Contracts A short-futures position agrees to sell in the future.

Forward Futures
Price mechanism maintains a balance between buyers and

Trade over-the-counter Trade on an exchange
sellers.(market equilibrium)
Not standardized Standardized contracts
Most futures contracts do not lead to delivery, because

One specified delivery date Range of delivery dates
most trades ―close out‖ their positions before delivery.
Settled at contract’s end Settled daily
Closing out a position means entering into the opposite
Delivery or final cash Contract usually closed type of trade from the original.
settlement usually occurs out prior to maturity

4

LO 27.6: Describe the characteristics of a futures LO 27.6: Describe the characteristics of a futures
contract and explain how futures positions are settled. contract and explain how futures positions are settled.

Contract size varies by type of futures contract
An (underlying) asset
 Treasury bond futures: contract size is a face value of
 A Treasury bond futures contract is on underlying
$100,000
U.S. Treasury with maturity of at least 15 years and not
 S&P 500 futures contract is index $250 (multiplier of 250X)
callable within 15 years (15 years ≤ T bond).
 NASDAQ futures contract is index $100 (multiplier of
 A Treasury note futures contract is on the underlying 100X)
U.S. Treasury with maturity of at least 6.5 years but not
greater than 10 years (6.5 ≤ T note ≤ 10 years). Recently, ―mini contracts‖ have been introduced:

 When the asset is a commodity (e.g., cotton, orange S&P 500 ―mini‖ = $50 x S&P Index

NASDAQ ―mini‖ = $20 x NASDQ
juice), the exchange specifies a grade (quality). 
(each contract is one-fifth the price, to attract smaller investors)

LO 27.6: Describe the characteristics of a futures LO 27.7: Describe the marking-to-market procedure, the
contract and explain how futures positions are settled. initial margin, and the maintenance margin.

Delivery Arrangements Margin account: Broker requires deposit.

 The exchange specifies delivery location. Initial margin: Must be deposited when contract is

initiated.
Delivery Months
Mark-to-market: At the end of each trading day, margin
 The exchange must specify the delivery month; this can 
account is adjusted to reflect gains or losses.
be the entire month or a sub-period of the month.

5

LO 27.8: Compute the variation margin.

Maintenance margin: Investor can withdraw funds in the There is only a variation margin if and when there is a
 
margin account in excess of the initial margin. A maintenance margin call.
margin guarantees that the balance in the margin account
Variation margin = initial margin – margin
never gets negative (the maintenance margin is lower than the
account balance
initial margin).
Margin call: When the balance in the margin account falls The maintenance margin is a trigger level—once
 
below the maintenance margin, broker executes a margin call. triggered, the investor must ―top up‖ to the initial
The next day, the investor needs to ―top up‖ the margin
margin, which is greater than the maintenance level.
account back to the initial margin level.
Variation margin: Extra funds deposited by the investor

after receiving a margin call.

LO 27.9: Explain the role of the clearinghouse. LO 27.9: Explain the role of the clearinghouse.

The exchange clearinghouse is a division of the exchange Market order: Execute the trade immediately at the
 
(e.g., the CME Clearing House is a division of the best price available.
Chicago Mercantile Exchange) or an independent Limit order: This order specifies a price (e.g., buy at $30

company. The clearinghouse serves as a or less)—but with no guarantee of execution.
guarantor, ensuring that the obligations of all trades are Stop order: (aka., stop-loss order) An order to execute

met. a buy/sell when a specified price is reached.

6

LO 28.1: Differentiate between a short hedge and a
long hedge, and identify situations where each is
appropriate.
Stop-limit: Requires two specified prices, a stop and a

A short forward (or futures) hedge is an agreement to
limit price. Once the stop-limit price is reached, it sell in the future and is appropriate when the hedger already
becomes a limit order at the limit price. owns the asset.
Classic example is farmer who wants to lock in a sales price:
Market-if-touched: Becomes a market order once 

protects against a price decline.
specified price is achieved.
A long forward (or futures) hedge is an agreement to buy
Discretionary (aka., market-not-held order): A
 in the future and is appropriate when the hedger does not
market order, but the broker is given the discretion to currently own the asset but expects to purchase in the future.
delay the order in an attempt to get a better price. 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.

Basis risk
Derivatives
LO 28.2: Define and calculate the basis.
No hedge
Spot = -$0.50
Basis = Spot Price Hedged Asset –

SPOT $2.50
Futures Price Futures Contract = S0 – F0 $2.00

Short 1.67 F
Basis $0.30
$0.10
Basis = Spot = -$0.50

Forward $2.20
Futures Price Futures Contract – Future = $0.30 (1.67)
$1.90
Spot Price Hedged Asset = F0 – S0 Net = 0
T0 T1

Time (T)
Hull says first is correct but second is
common for financial assets (either is okay) Weakening of the basis = Futures price
increases more than spot

7

Basis risk Basis risk

No hedge T1
Spot = -$0.50
$2.60 $2.60
SPOT SPOT
$2.50 $2.50
$2.00 $0.0

Short 1.67 F
Basis Basis
$0.30 $0.30
Short 1.67 F
$0.30
Spot = -$0.50 Spot = +$0.10
Forward $2.20 Forward $2.20
Future = $0.50 (1.67)
$1.90 $1.90
Future = -$0.40 (1.67)
Net = +33.5
$1.70
T0 T0 Net = -56.8
T1 Time (T) Time (T)

Basis unchanged. Basis declines
But unexpected strengthening= Hedger improved! Unexpected weakening= Hedger worse!

LO 28.3: Define the types of basis risk and explain how LO 28.3: Define the types of basis risk and explain how
they arise in futures hedging. they arise in futures hedging.

Spot price increases by more than the futures price  

basis increases. This is a ―strengthening of the basis‖ of the futures contract differ from the underlying
position.
When unexpected, strengthening is favorable for a short

hedge and unfavorable for a long hedge Contract ≠ Commodity

Futures price increases by more than the spot price  Contract is standardized (e.g., WTI oil futures)
 
basis declines. This is a ―weakening of the basis‖ Commodities are not exactly commodities (e.g., hedger has a

When unexpected, weakening is favorable for a long hedge

and unfavorable for a short hedge

Basis risk higher with cross-hedging

8

LO 28.3: Define the types of basis risk and explain how LO 28.4: Define, calculate, and interpret the minimum
they arise in futures hedging. variance hedge ratio.

But basis risk arises because often the characteristics The optimal hedge ratio (a.k.a., minimum variance hedge
 
of the futures contract differ from the underlying ratio) is the ratio of futures position relative to the spot
position. position that minimizes the variance of the position.
Where is the correlation and is the standard
Contract ≠ Commodity.

deviation, the optimal hedge ratio is given by:


S
h*
Trade-off
Liquidity Basis risk
F
(exchange)

LO 28.4: Define, calculate, and interpret the minimum LO 28.4: Define, calculate, and interpret the minimum
variance hedge ratio. variance hedge ratio.

For example, if the volatility of the spot price is 20%, the For example, if the volatility of the spot price is 20%, the
 
volatility of the futures price is 10%, and their correlation volatility of the futures price is 10%, and their correlation
is 0.4, then: is 0.4, then:

20% 20%
S S
h* h*
h* (0.4) 0.8 h* (0.4) 0.8
10% 10%
F F
h * NA
Number of
N*
contracts QF

9

LO 28.5: Calculate the number of stock index futures LO 28.5: Calculate the number of stock index futures
contracts to buy or sell to hedge an equity portfolio or contracts to buy or sell to hedge an equity portfolio or
individual stock. individual stock.

 By extension, when the goal is to shift portfolio beta from
 Given a portfolio beta ( ), the current value of the
portfolio (P), and the value of stocks underlying one ( ) to a target beta ( *), the number of contracts
futures contract (A), the number of stock index futures required is given by:
contracts (i.e., which minimizes the portfolio variance) is
given by:
P
P N(* )
N A
A

Futures: Question Futures: Question
LO 28.5: Calculate the number of stock index futures contracts to LO 28.5: Calculate the number of stock index futures contracts to
buy or sell to hedge an equity portfolio or individual stock. buy or sell to hedge an equity portfolio or individual stock.

Assume: Assume:
• Value of S&P 500 Index is 1240
Value of S&P 500 Index is 1240

• Value of portfolio is $1 million
Value of portfolio is $10 million • Portfolio beta ( ) is 1.5


Portfolio beta ( ) is 1.5 P

N (* )
How do we change the portfolio beta to 1.2? A
$10,000,000
(1.2 1.5) 9.7
Hint: Contract = ($250 Index)
P (1240)(250)
and # of futures is given by: N (* )
A We short about 10 contracts. (-) indicates
short, (+) long…

10

LO 28.6: Identify situations when a rolling hedge is LO 29.1: Identify and apply the three most common
appropriate, and discuss the risks of such a strategy. day count conventions.

When the delivery date of the futures contract occurs

prior to the expiration date of the hedge, the hedger can Actual/actual U.S.Treasuries
roll forward the hedge: close out a futures contract and
30/360 U.S. corporate and
take the same position on a new futures contract with a
municipal bonds
later delivery date.
Actual/360 U.S.Treasury bills and
Exposed to:

other money market
Basis risk (original hedge)

instruments
Basis risk (each new hedge) = ―rollover basis risk‖


LO 29.2: Explain the U.S. Treasury bond (T-bond) LO 29.3: Calculate the Eurodollar futures contract
futures contract conversion factor. convexity adjustment.

The Treasury bond futures contract allows the party with The convexity adjustment assumes continuous
 
the short position to deliver any bond with a maturity of compounding. Given that ( ) is the standard deviation of
more than 15 years and that is not callable within 15 the change in the short-term interest rate in one year, t1
years. When the chosen bond is delivered, the conversion is the time to maturity of the futures contract and t2 is
factor defines the price received by the party with the the time to maturity of the rate underlying the futures
short position: contract:

1
Cash Received = Quoted futures price Conversion 2
Forward = Futures t 1 t2
factor + Accrued interest
2
= (QFP CF) + AI

11

LO 29.4: Formulate a duration-based hedging strategy LO 29.4: Formulate a duration-based hedging strategy
using interest rate futures. using interest rate futures.
The number of contracts required to hedge against an Assume a portfolio value of $10 million.


uncertain change in the yield, given by y, is given by: The fund manager will hedge with T-bond futures (each

PDP price of 98.
N*
FC DF 
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 LO 29.5: Identify the limitations of using a duration-
using interest rate futures. based hedging strategy.

Portfolio immunization or duration matching is when a


bank or fund matches the average duration of assets with
the average duration of liabilities.
price of 98.
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
PDP ($10 million)(6) 
N* 122 A change in curvature

FC DF (98,000)(5)

12

LO 30.1: Explain the derivation of the basic equilibrium LO 30.1: Explain the derivation of the basic equilibrium
formula for pricing commodity forwards and futures. formula for pricing commodity forwards and futures.

Risk-free
Rate (r)
E0 (ST )e(r )T
F0,T
E0 (ST )e(r )T
F0,T E0 (ST ) Spot price of S at time T, as expected at time 0
F0,T Forward price
r Risk-free rate
Discount
Forward Discount rate for commodity S
rate ( )
(F0)
Spot
(S0)
Time (T)

LO 30.2: Define lease rates, and discuss the LO 30.3: Explain how lease rates determine whether a
importance of lease rates for determining no- forward market is in contango or backwardation.
arbitrage values for commodity futures and forwards.
Contango refers to an upward-sloping forward curve

 Lease rate = commodity discount rate – growth rate
which must be the case if the lease rate is less than
 Lease rate dividend yield 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.

S0e(r )T
F0,T
Financial asset

S 0 e( r q )T
F0

13

LO 30.3: Explain how lease rates determine whether a LO 30.4: Explain how storage costs impact commodity
forward market is in contango or backwardation. forward prices, and calculate the forward price of a
Forward
(F0)
Risk-free Storage Cost ( )
Rate (r) negative dividend
Spot E(ST)
(S0)

Forward
(F0)
Lease rate ( ) Convenience (y)
Forward
dividend dividend
(F0)
Spot
Time (T)
(S0)
Research says normal backwardation is “normal:” speculators
Time (T)
want compensation (risk premium) for buying the futures contract

LO 30.5: Explain how a convenience yield impacts commodity
forward prices, and determine the no-arbitrage bounds for the
forward prices, and calculate the forward price of a forward price of a commodity when the commodity has a
commodity with storage costs. convenience yield.

Storage
Risk-free
Cost ( )
Rate (r)
Risk-free Storage Risk-free Storage
Rate Cost Rate Cost
(r c )T
F0 S0 e
S 0 e( r c )T
S 0 e( r )T
F0
Convenience
Lease
Forward
(y)
rate ( )
(F0)
Convenience
Spot Yield
(S0)
Time (T)

14

Commodity Futures Commodity Futures
S&P 500 Index
1640
Gold futures
1620
1600 900
1580 850
1560
800
1540
750
1520 Rational forward curve rises by cost
700
1500 of capital (risk free + premium) less
650
1480
Durable, (relatively) cheap to store.
dividends 600
1460
550 Forward curve is “uninteresting”
500
Jul-07 Nov-08 Mar-10 Aug-11 Dec-12

Commodity Futures Commodity Futures
LO 30.6: Discuss the factors that impact the pricing of LO 30.6: Discuss the factors that impact the pricing of
gold, corn, natural gas, and oil futures. gold, corn, natural gas, and oil futures.

Corn Natural Gas
450 10
9
400
8
350
7
300 Costly to transport. Costly to store
6
250 (storage costs). Highly seasonal
5
200
Nov-08

Nov-09
Aug-07
Nov-07

Aug-08

Aug-09

Aug-10
Nov-10
May-08

May-09

May-10
Feb-08

Feb-09

Feb-10
Mar-08

Mar-09

Mar-10
Dec-07

Dec-08

Dec-09

Dec-10
Sep-07

Sep-08

Sep-09

Sep-10
Jun-10
Jun-08

Jun-09

15

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