Dr. Keith M. Howe
Summer 2008

Risk and uncertainty
Risk management
Risk aversion
The process of formulating the benefit-cost tradeoffs of risk reduction and deciding on the course of action to take (including the decision to take no action at all).

•
Derivatives
• financial assets (e.g., stock option, futures, forwards, etc) whose values depend upon the value of the underlying assets.
•
Hedge
• the use of financial instruments or of other tools to reduce exposure to a risk factor.

Figure 1.2. Gains and losses from buying shares and a call option on Risky
Upside Inc.
Gain
+$6,000
Risky Upside Inc. price
-$3,000
20 50
110
Panel A. Gain from buying shares of Risky
Upside Inc. at $50 per share.

Gain
$5,000
0
-$1,000
Risky Upside Inc. price
110
20 50
Panel B. Gain from buying a call option on shares of Risky Upside Inc. with exercise price of $50 for a premium of $10 per share.

Figure 1.3. Hedging with forward contract. Garman’s income is in dollars and the exchange rate is the dollar price of one euro.
Unhedged income
Income to firm
$100 million
$90 million
Exchange rate
$0.90
$1
Panel A. Income to Garman if it does not hedge.

$10 million
Gain from contract to firm
Forward gain
Forward loss
$0.9
Forward rate $1
Panel B. Forward contract payoff.
Exchange rate

Income to firm
$100 million
Forward gain
Forward loss
Unhedged income
Hedged income
Exchange rate
Forward rate $1
Panel C. Hedged firm income.

Unhedged income
$100 million
Exercise price of $1
Panel D. Comparison of income with put contract and income with forward contract.

Risk management irrelevance proposition
•
Bottom line : hedging a risk does not increase firm value when the cost of bearing the risk is the same whether the risk is borne within the firm or outside the firm by the capital markets.
•
This proposition holds when financial markets are perfect.

Risk management irrelevance proposition
• Allows us to find out when homemade risk management is not equivalent to risk management by the firm.
• This is the case whenever risk management by a firm affects firm value in a way that investors cannot mimic.
• For risk management to increase firm value, it must be more expensive to take a risk within the firm than to pay the capital markets to take it.

Risk management can add value to the firm by:
•
Decreasing taxes
•
Decreasing transaction costs (including bankruptcy costs)
•
Avoiding investment decision errors

• Costs incurred as a result of a bankruptcy filing are called bankruptcy costs.
• The extent to which bankruptcy costs affect firm value depends on their extent and on the probability that the firm will have to file for bankruptcy.
•
The probability that a firm will be bankrupt is the probability that it will not have enough cash flow to repay the debt.

•
Direct bankruptcy costs
• Average ratio of direct bankruptcy costs to total assets: 2.8%
•
Indirect bankruptcy costs
• Many of these indirect costs start accruing as soon as a firm’s financial situation becomes unhealthy, called costs of financial distress
•
Managers of a firm in bankruptcy lose control of some decisions. They might not allowed to undertake costly new projects, for example.

Figure 3.1. Cash flow to shareholders and operating cash flow.
Cash flow to shareholders
$450M
Expected cash flow $350M
$250M
Unhedged cash flow
$250M
Expected cash flow $350M $450M
Cash flow to the firm

Figure 3.2. Creating the unhedged firm out of the hedged firm.
Unhedged cash flow shareholders
$350M
(hedged)
Forward gain
Forward
loss
Hedged firm cash flow
$350M (gold sold at forward)

Figure 3.3. Cash flow to claimholders and bankruptcy costs .
Cash flow to claimholders
$450M
Expected cash flow hedged
$350M
Unhedged
$340M
$230M
Bankruptcy cost
Unhedged cash flow
$250M
Expected cash flow $350M $450M
Cash flow to the firm

Value of firm unhedged = PV (C – Bankruptcy costs)
= PV (C) – PV (Bankruptcy costs)
= value of firm without bankruptcy costs – PV (bankruptcy costs)
Gain from risk management
= value of firm hedged – value of firm unhedged
= PV( bankruptcy costs)
Value of firm unhedged
+ gain from risk management
= value of firm hedged
= value of firm without bankruptcy costs

Tax rationale for risk management: If it moves a dollar away from a possible outcome in which the taxpayer is subject to a high tax rate and shifts it to a possible outcome where the taxpayer incurs a low tax rate, a firm or an investor reduces the present value of taxes to be paid. It applies whenever income is taxed differently at different levels.
- Carrybacks and carryforwards
- Tax shields
- Personal taxes

The firm pays taxes at the rate of 50 percent on cash flow in excess of
$300 per ounce. For simplicity, the price of fold is either $250 or $450 with
Equal probability. The forward price is $350.

•
In general, firms cannot eliminate all risk, debt is risky.
• By having more debt, firms increase their tax shield from debt but increase the present value of costs of financial distress.
•
The optimal capital structure of a firm:
•
Balances the tax benefits of debt against the costs of financial distress.
• Through risk management:
• A firm can reduce the present value of the costs of financial distress by making financial distress less likely.
• As a result, it can take on more debt.

Should the firm hedge to reduce the risk of large undiversified shareholders?
• Large undiversified shareholders can increase firm value
•
Risk and the incentives of managers
•
Large shareholders, managerial incentives, and homestake

Figure 3.6. Firm after-tax cash flow and debt issue.
After tax cash flow of hedged firm
330
325
320
315
310
305
100 200 300
Optimal amount of debt, $317.073M
400
Principal amount of debt

Risk identification
Risk assessment
Review
Selection of risk-mgt techniques
Implementation

Risk Management
• There is no return without risk
• Be transparent
• Seek experience
• Know what you don’t know
• Communicate
• Diversify
• Show discipline
• Use common sense
• Get a RiskGrade
Source: Riskmetrics Group (www.riskmetrics.com)

Market risk
- interest rate
- foreign exchange
- commodity price
Operational risk
- industry sectors
- geographical regions
Hazard risk
- physical damage
- liabilities
- business interruption
Strategic risk
- competition
- reputation
- investor support

CEO
Strategic risk management
Market risk management
Hedgeable
CRO
Hazard risk management
Insurable
Operational risk management
Diversifiable

•
•
•

•
Value-at-risk (VAR) is a category of risk measures that describe probabilistically the market risk of mostly a trading portfolio.
• It summarizes the predicted maximum loss (or worst loss) over a target horizon within a given confidence interval.
• If the portfolio return is normally distributed, has zero mean, and has volatility s over the measurement period, the 5 percent
VAR of the portfolio is:
VAR = 1.65 X s X Portfolio value

•
The US bank J.P. Morgan states in its 2000 annual report that its aggregate VAR is about
$22m.
• The bank, one of the pioneers in risk management, may say that for 95 percent of the time it does not expect to lose more than $22m on a given day.

• The main appeal of VAR was to describe risk in dollars - or whatever base currency is used - making it far more transparent and easier to grasp than previous measures.
• VAR also represents the amount of economic capital necessary to support a business, which is an essential component of
“economic value added” measures.
• VAR has become the standard benchmark” for measuring financial risk.

•
Forward contracts
•
Futures contracts
•
Hedging
•
Interest rate futures contracts
•
Duration hedging
•
Swap contracts
•
Options

• A forward contract specifies that a certain commodity will be exchanged for another at a specified time in the future at prices specified today.
• Its not an option: both parties are expected to hold up their end of the deal.
•
If you have ever ordered a textbook that was not in stock, you have entered into a forward contract.

Suppose S&P index price is $1050 in 6 months. A holder who entered a long position at a forward price of
$1020 is obligated to pay $1020 to acquire the index, and hence earns $1050 - $1020 = $30 per unit of the index. The short is likewise obligated to sell for $1020, and thus loses $30.

If the index price in 6 months = $1020, both the long and short have a 0 payoff.
If the index price > $1020, the long makes money and the short loses money.
If the index price < $1020, the long loses money and the short makes money.
S&R Index S&R Forward in 6 months long short
900 -$120 $120
950 -70 70
1000 -20 20
1020 0 0
1050 30 -30
1100 80 -80

Problem: The current S&P index is $1000. You have just purchased a 6- month forward with a price of $1100. If the index in 6 months has appreciated by 7%, what is the payoff of this position?
Solution: F
0
=1100
S
1
=1000*1.07=1070
Payoff: 1070-1100= - $30.

Consider the obligation to buy a share of Microsoft stock one year from now for $100. Assume that the stock currently sells for $97 per share and that Microsoft will pay no dividends over the coming year. One-year zero-coupon bonds that pay
$100 one year from now currently sell for $92. At what price are you willing to buy or sell this obligation?

Valuing a forward contract
Strategy 1---- the forward contract
Today One year from now
Buy a forward contract Buy stock at a price of $100.
Sell the share for cash at market
Strategy 2 ---- the portfolio strategy
Today One year from now
Buy stock today
Sell short $100 in face value of 1-year zero-coupon bonds
Sell the stock
Buyback the zero-coupon bonds of $100

Valuing a forward contract
Strategy 1
Strategy 2
Cost
Today
?
$97-$92
Cash flow one year from now
S
1
- $100
S
1
- $100
Since strategies 1 and 2 have identical cash flows in the future, they should have the same cost today to prevent arbitrage.
? = $97 - $92 = $5
In strategy 1, the obligation to buy the stock for $100 one year from now, should cost $5.

Valuing a forward contract
The no-arbitrage value of a forward contract on a share of stock (the obligation to buy a share of stock at a price of K, T years in the future), assuming the stock pays no dividends prior to T, is
S
0 where
S
0
= current price of the stock

K
( 1
 r f
)
T
( 1

K r f
)
T
= the current market price of a default-free zero-coupon bond paying K, T years in the future
At no arbitrage:
F
0

S
0

K
( 1
 r f
)
T
K

S
0

0
( 1
 r f
) T

•
Currency forward rates are a variation on forward price of stock.
•
In the absence of arbitrage, the forward currency rate F0 (for example,
Euros/dollar) is related to the current exchange rate (or spot rate) S0, by the equation
F
0
S
0

1
 r foreign
1
 r domestic
• where r = the return (unannualized) on a domestic or foreign risk-free security over the life of the forward agreement, as measured in the respective country's currency

Example: The Relation Between Forward Currency
Rates and Interest Rates
Assume that six-month LIBOR on Canadian funds is 4 percent and the US$ Eurodollar rate (six-month LIBOR on U.S. funds) is 10 percent and that both rates are default free. What is the six-month forward Can$/US$ exchange rate if the current spot rate is Can$1.25/US$? Assume that six months from now is
182 days.

Currency Forward Rates
Answer: ( LIBOR is a zero-coupon rate based on an actual/360 day count.) So
Canada United States
Six-month interest
2 .
02 %
Rate (unannualized):

182
360

4 % 5 .
06 %

182
360

10 %
The forward rate is Can $ 1 .
21
US $

1 .
0202
1 .
0506

1 .
25 .

•
A futures contract is like a forward contract:
• It specifies that a certain commodity will be exchanged for another at a specified time in the future at prices specified today.
• A futures contract is different from a forward:
• Futures are standardized contracts trading on organized exchanges with daily resettlement (“marking to market”) through a clearinghouse.

•
Standardizing Features:
•
Contract Size
•
Delivery Month
•
Daily resettlement
•
Minimizes the chance of default
•
Initial Margin
•
About 4% of contract value, cash or T-bills held in a street name at your brokerage.

Suppose you want to speculate on a rise in the $/¥ exchange rate (specifically you think that the dollar will appreciate).
Currency per
U.S. $ equivalent U.S. $
Wed Tue
Japan (yen) 0.007142857
0.007194245
1-month forward 0.006993007
0.007042254
3-months forward 0.006666667
0.006711409
6-months forward 0.00625
0.006289308
Wed
140
143
150
160
Tue
139
142
149
159
Currently $1 = ¥140.
The 3-month forward price is $1=¥150.

• Currently $1 = ¥140 and it appears that the dollar is strengthening.
•
If you enter into a 3-month futures contract to sell
¥ at the rate of $1 = ¥150 you will make money if the yen depreciates . The contract size is ¥12,500,000
• Your initial margin is 4% of the contract value:
$3,333.33

.04

¥12,500,00 0

$1
¥150

If tomorrow, the futures rate closes at $1 = ¥149, then your position’s value drops.
Your original agreement was to sell ¥12,500,000 and receive $83,333.33:
$83,333.33

¥12,500,00 0

$1
¥150
But ¥12,500,000 is now worth $83,892.62:
$ 83 , 892 .
62

¥12,500,00 0

$1
¥149
You have lost $559.28 overnight.

•
The $559.28 comes out of your $3,333.33 margin account, leaving $2,774.05
•
This is short of the $3,355.70 required for a new position.
$3,355.70

.04

¥12,500,00 0

$1
¥149
Your broker will let you slide until you run through your maintenance margin. Then you must post additional funds or your position will be closed out.
This is usually done with a reversing trade .

Contract Contract Size Exchange
Agricultural
Corn 5,000 bushels Chicago BOT
Wheat 5,000 bushels Chicago & KC
Cocoa 10 metric tons CSCE
OJ 15,000 lbs.
CTN
Metals & Petroleum
Copper
Gold
Unleaded gasoline
Financial
25,000 lbs.
100 troy oz.
42,000 gal.
CMX
CMX
NYM
British Pound £62,500 IMM
Japanese Yen ¥12.5 million IMM
Eurodollar $1 million LIFFE

•
The Chicago Mercantile Exchange (CME) is by far the largest.
•
Others include:
•
The Philadelphia Board of Trade (PBOT)
•
The MidAmerica Commodities Exchange
• The Tokyo International Financial Futures Exchange
• The London International Financial Futures Exchange

•
Expiry cycle: March, June, September, December.
• Delivery date 3 rd Wednesday of delivery month.
•
Last trading day is the second business day preceding the delivery day.
• CME hours 7:20 a.m. to 2:00 p.m. CST.

•
Extended-hours trading on GLOBEX runs from 2:30 p.m. to 4:00 p.m dinner break and then back at it from
6:00 p.m. to 6:00 a.m. CST.
•
Singapore International Monetary Exchange
(SIMEX) offer interchangeable contracts.
• There’s other markets, but none are close to CME and
SIMEX trading volume.

Wall Street Journal Futures Price Quotes
Highest price that day Lifetime Open
Open High Low Settle Change High Low Interest
Highest and lowest prices over the lifetime of the contract.
Corn (CBT) 5,000 bu.; cents per bu.
179 180 178¼ 178½ -1½ 312 177 2,837 July
Sept 186
Dec 196
186½ 184
197 194
186
196½
-¾
-¼
280 184 104,900
291¼ 194 175,187
TREASURY BONDS (CBT) - $1,000,000; pts. 32nds of 100%
Sept 117-05 117-21 116-27 117-05 +5 131-06 111-15 647,560
Dec 116-19 117-05 116-12 116-21
Opening price Closing price
+5 128-28 111-06 13,857
Daily Change
DJ INDUSTRIAL AVERAGE (CBOT) - $10 times average
Sept 11200 11285 11145 11241 -17
Dec 11287 11385 11255 11349 -17
11324 7875 18,530
11430 7987 1,599
Lowest price that day
Number of open contracts
Expiry month

Basic Currency Futures Relationships
•
Open Interest refers to the number of contracts outstanding for a particular delivery month.
• Open interest is a good proxy for demand for a contract.
• Some refer to open interest as the depth of the market. The breadth of the market would be how many different contracts (expiry month, currency) are outstanding.

•
Two counterparties with offsetting risks can eliminate risk.
•
For example, if a wheat farmer and a flour mill enter into a forward contract, they can eliminate the risk each other faces regarding the future price of wheat.
•
Hedgers can also transfer price risk to speculators and speculators absorb price risk from hedgers.
•
Speculating: Long vs. Short

You speculate that copper will go up in price, so you go long
10 copper contracts for delivery in 3 months. A contract is
25,000 pounds in cents per pound and is at $0.70 per pound or $17,500 per contract.
If futures prices rise by 5 cents, you will gain:
Gain = 25,000 × .05 × 10 = $12,500
If prices decrease by 5 cents, your loss is:
Loss = 25,000 × -.05 × 10 = -$12,500

You are a farmer and you will harvest 50,000 bushels of corn in
3 months. You want to hedge against a price decrease. Corn is quoted in cents per bushel at 5,000 bushels per contract. It is currently at $2.30 cents for a contract 3 months out and the spot price is $2.05.
To hedge you will sell 10 corn futures contracts:
50 , 000 bushels
5 , 000 bushels per contract

10 contracts
Now you can quit worrying about the price of corn and get back to worrying about the weather.

Consider a Treasury bond that pays a semiannual coupon of $ C for the next T years:
•
The yield to maturity is r
C C C
…
C

F
0 1 2 3 2T
Value of the T-bond under a flat term structure
= PV of face value + PV of coupon payments
PV

F
( 1
 r )
T

C r

 1

1
( 1
 r )
T



If the term structure of interest rates is not flat, then we need to discount the payments at different rates depending upon maturity
C C C
…
C

F
0 1 2 3 2T
= PV of face value + PV of coupon payments
PV

C
( 1
 r
1
)

C
( 1
 r
2
)
2

C
( 1
 r
3
)
3
  
C
( 1

 r
2 T
F
)
T

An Nperiod forward contract on that T-Bond

P forward
C C C
…
C

F
0 N N+1 N+2 N+3 N+2T
Can be valued as the present value of the forward price.
PV

( 1

C r
N

1
)

PV

P forward
( 1
 r
N
)
N
C
( 1
 r
N

2
)
2

( 1
 r
N

3
)
3
( 1
 r
N
)
C
N
  
( 1

C

F r
N

2 T
)
T

•
The pricing equation given above will be a good approximation.
•
The only real difference is the daily resettlement.

•
A mortgage lender who has agreed to loan money in the future at prices set today can hedge by selling those mortgages forward.
•
It may be difficult to find a counterparty in the forward who wants the precise mix of risk, maturity, and size.
• It’s likely to be easier and cheaper to use interest rate futures contracts however.

•
As an alternative to hedging with futures or forwards, one can hedge by matching the interest rate risk of assets with the interest rate risk of liabilities.
•
Duration is the key to measuring interest rate risk.

•
Duration measures the combined effect of maturity, coupon rate, and YTM on bond’s price sensitivity
• Measure of the bond’s effective maturity
• Measure of the average life of the security
• Weighted average maturity of the bond’s cash flows

D

PV ( C
1
)

1

PV ( C
2
)

2
  
PV ( C
T
)

T
PV
D
 t
N 

1 t
N 

1
(
C
1 t

 r t
) t
( 1
C t
 r ) t

Calculate the duration of a three-year bond that pays a semi-annual coupon of $40, has a $1,000 par value when the YTM is 8% semiannually?

Discount Present Years x PV
Years Cash flow factor value / Bond price
0.5
$40.00
0.96154
$38.46
1 $40.00
0.92456
$36.98
1.5
$40.00
0.88900
$35.56
2 $40.00
0.85480
$34.19
0.0192
0.0370
0.0533
0.0684
2.5
$40.00
0.82193
$32.88
3 $1,040.00
0.79031
$821.93
0.0822
2.4658
$1,000.00
2.7259 years
Bond price Bond duration
Duration is expressed in units of time; usually years.

The key to bond portfolio management
•
Properties:
• Longer maturity, longer duration
• Duration increases at a decreasing rate
• Higher coupon, shorter duration
•
Higher yield, shorter duration
•
Zero coupon bond: duration = maturity

•
In a swap, two counterparties agree to a contractual arrangement wherein they agree to exchange cash flows at periodic intervals.
•
There are two types of interest rate swaps:
•
Single currency interest rate swap
• “Plain vanilla” fixed-for-floating swaps are often just called interest rate swaps .
• Cross-Currency interest rate swap
•
This is often called a currency swap ; fixed for fixed rate debt service in two (or more) currencies.

• A swap bank is a generic term to describe a financial institution that facilitates swaps between counterparties.
•
The swap bank can serve as either a broker or a dealer.
• As a broker, the swap bank matches counterparties but does not assume any of the risks of the swap.
•
As a dealer, the swap bank stands ready to accept either side of a currency swap, and then later lay off their risk, or match it with a counterparty.

An Example of an Interest Rate Swap
• Consider this example of a “plain vanilla” interest rate swap.
• Bank A is a AAA-rated international bank located in the U.K. and wishes to raise $10,000,000 to finance floating-rate
Eurodollar loans.
• Bank A is considering issuing 5-year fixed-rate Eurodollar bonds at 10 percent.
•
It would make more sense to for the bank to issue floating-rate notes at LIBOR to finance floating-rate Eurodollar loans.

An Example of an Interest Rate Swap
•
Firm B is a BBB-rated U.S. company. It needs
$10,000,000 to finance an investment with a five-year economic life.
• Firm B is considering issuing 5-year fixed-rate Eurodollar bonds at
11.75 percent.
•
Alternatively, firm B can raise the money by issuing 5-year floatingrate notes at LIBOR + ½ percent.
• Firm B would prefer to borrow at a fixed rate.

An Example of an Interest Rate Swap
The borrowing opportunities of the two firms are:
COMPANY B BANK A
Fixed rate 11.75%
Floating rate LIBOR + .5%
10%
LIBOR

An Example of an Interest Rate Swap
Bank
A
Swap
Bank
10 3/8%
LIBOR – 1/8%
The swap bank makes this offer to Bank A: You pay LIBOR – 1/8 % per year on $10 million for 5 years and we will pay you 10 3/8% on $10 million for 5 years
COMPANY B BANK A
Fixed rate 11.75%
Floating rate LIBOR + .5%
10%
LIBOR

An Example of an Interest Rate Swap
½% of $10,000,000 =
$50,000. That’s quite
Here’s what’s in it for Bank A: a cost savings per year
Swap
They can borrow externally at for 5 years.
Bank 10% fixed and have a net
10 3/8% borrowing position of
10%
Bank
A
LIBOR – 1/8%
-10 3/8 + 10 + (LIBOR – 1/8) =
LIBOR – ½ % which is ½ % better than they can borrow floating without a swap.
COMPANY B BANK A
Fixed rate 11.75%
Floating rate LIBOR + .5%
10%
LIBOR

An Example of an Interest Rate Swap
The swap bank makes this offer to company B: You pay us 10½% per year on $10 million for 5 years and we will pay you
LIBOR – ¼ % per year on $10 million for 5 years.
Swap
Bank
10 ½%
LIBOR – ¼%
Company
COMPANY B BANK A
Fixed rate 11.75%
Floating rate LIBOR + .5%
10%
LIBOR
B

An Example of an Interest Rate Swap
Here’s what’s in it for B:
½ % of $10,000,000 =
$50,000 that’s quite a cost savings per year for 5
10 ½% years.
They can borrow externally at
Swap
Bank
LIBOR + ½ % and have a net
LIBOR – ¼%
Company borrowing position of
10½ + (LIBOR + ½ ) - (LIBOR - ¼ ) = 11.25% which is ½% better than they can borrow floating.
B
LIBOR
+ ½%
COMPANY B BANK A
Fixed rate 11.75%
Floating rate LIBOR + .5%
10%
LIBOR

An Example of an Interest Rate Swap
The swap bank makes money too.
Swap
¼% of $10 million
= $25,000 per year for 5 years.
Bank
10 3/8% 10 ½%
Bank
A
LIBOR – 1/8% LIBOR – ¼%
Company
LIBOR – 1/8 – [LIBOR – ¼ ]= 1/8
10 ½ - 10 3/8 = 1/8
¼
COMPANY B BANK A
B
Fixed rate 11.75%
Floating rate LIBOR + .5%
10%
LIBOR

An Example of an Interest Rate Swap
The swap bank makes ¼%
Swap
Bank
10 3/8% 10 ½%
LIBOR – 1/8% LIBOR – ¼%
Bank Company
A
A saves ½%
B
B saves ½%
COMPANY B BANK A
Fixed rate 11.75%
Floating rate LIBOR + .5%
10%
LIBOR

•
Suppose a U.S. MNC wants to finance a £10,000,000 expansion of a British plant.
• They could borrow dollars in the U.S. where they are well known and exchange for dollars for pounds.
•
This will give them exchange rate risk: financing a sterling project with dollars.
• They could borrow pounds in the international bond market, but pay a premium since they are not as well known abroad.

•
If they can find a British MNC with a mirrorimage financing need they may both benefit from a swap.
•
If the spot exchange rate is S
0
($/£) = $1.60/£, the U.S. firm needs to find a British firm wanting to finance dollar borrowing in the amount of $16,000,000.

Consider two firms A and B: firm A is a U.S.–based multinational and firm B is a U.K.–based multinational.
Both firms wish to finance a project in each other’s country of the same size. Their borrowing opportunities are given in the table below.
$ £
Company A 8.0% 11.6%
Company B 10.0% 12.0%

Swap
Bank
$8%
Firm
A
$8%
£11%
$ £
Company A 8.0% 11.6%
Company B 10.0% 12.0%
$9.4%
£12%
Firm
B
£12%

A’s net position is to borrow at £11%
Swap
$8%
Bank
$8%
£11%
Firm
A
A saves £.6%
$ £
Company A 8.0% 11.6%
Company B 10.0% 12.0%
$9.4%
£12%
Firm
B
£12%

B’s net position is to borrow at $9.4%
Swap
Bank
$8%
£11%
$9.4%
£12%
$8%
Firm
A
Firm
B
£12%
$ £
Company A 8.0% 11.6%
Company B 10.0% 12.0%
B saves $.6%

The swap bank makes money too:
$8%
Swap
Bank
1.4% of $16 million financed with 1% of
£10 million per year for 5 years.
$9.4%
$8%
£11% £12%
Firm
A
At S
0
($/£) = $1.60/£, that is a gain of $124,000 per year for 5 years.
$ £
Company A 8.0% 11.6%
Firm
B
£12%
The swap bank faces exchange rate risk, but maybe
Company B 10.0% 12.0% they can lay it off
(in another swap).

•
Currency Swaps
• fixed for fixed
• fixed for floating
• floating for floating
• amortizing
• Interest Rate Swaps
• zero-for floating
• floating for floating
•
Exotica
•
For a swap to be possible, two humans must like the idea. Beyond that, creativity is the only limit.

Risks of Interest Rate and
Currency Swaps
•
Interest Rate Risk
• Interest rates might move against the swap bank after it has only gotten half of a swap on the books, or if it has an unhedged position.
• Basis Risk
• If the floating rates of the two counterparties are not pegged to the same index.
•
Exchange Rate Risk
• In the example of a currency swap given earlier, the swap bank would be worse off if the pound appreciated.

Risks of Interest Rate and
Currency Swaps
•
Credit Risk
• This is the major risk faced by a swap dealer—the risk that a counter party will default on its end of the swap.
• Mismatch Risk
• It’s hard to find a counterparty that wants to borrow the right amount of money for the right amount of time.
•
Sovereign Risk
•
The risk that a country will impose exchange rate restrictions that will interfere with performance on the swap.

•
A swap is a derivative security so it can be priced in terms of the underlying assets:
•
How to:
• Plain vanilla fixed for floating swap gets valued just like a bond.
•
Currency swap gets valued just like a nest of currency futures.

• Many corporate securities are similar to the stock options that are traded on organized exchanges.
•
Almost every issue of corporate stocks and bonds has option features.
•
In addition, capital structure and capital budgeting decisions can be viewed in terms of options.

•
An option gives the holder the right, but not the obligation , to buy or sell a given quantity of an asset on (or perhaps before) a given date, at prices agreed upon today.
•
Calls versus Puts
•
Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset at some time in the future, at prices agreed upon today. When exercising a call option, you “call in” the asset.
• Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset at some time in the future, at prices agreed upon today. When exercising a put, you “put” the asset to someone.

•
Exercising the Option
•
The act of buying or selling the underlying asset through the option contract.
• Strike Price or Exercise Price
• Refers to the fixed price in the option contract at which the holder can buy or sell the underlying asset.
• Expiry
•
The maturity date of the option is referred to as the expiration date, or the expiry.
• European versus American options
•
European options can be exercised only at expiry.
• American options can be exercised at any time up to expiry.

•
In-the-Money
• The exercise price is less than the spot price of the underlying asset.
•
At-the-Money
• The exercise price is equal to the spot price of the underlying asset.
•
Out-of-the-Money
• The exercise price is more than the spot price of the underlying asset.

•
Intrinsic Value
• The difference between the exercise price of the option and the spot price of the underlying asset.
• Speculative Value
• The difference between the option premium and the intrinsic value of the option.
Option
Premium
=
Intrinsic
Value
+
Speculative
Value

•
Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset on or before some time in the future, at prices agreed upon today.
• When exercising a call option, you “call in” the asset.

Basic Call Option Pricing
Relationships at Expiry
•
At expiry, an American call option is worth the same as a
European option with the same characteristics.
•
If the call is in-the-money, it is worth S
T
E.
•
If the call is out-of-the-money, it is worthless.
C aT
= C eT
= Max [ S
T
- E , 0]
• Where
S
T is the value of the stock at expiry (time T )
E is the exercise price.
C aT is the value of an American call at expiry
C eT is the value of a European call at expiry

60
40 Buy a call
20
0
0 10 20 30 40 50
-20
60 70 80 90
Stock price ($)
100
-40
-60
Exercise price = $50

60
40
20
0
-20
0 10 20 30 40 50
Stock price ($)
60 70 80
Write a call
-40
90 100
-60
Exercise price = $50

60
40
Buy a call
20
0
-20
0 10 20 30 40 50
Stock price ($)
60 70 80
Write a call
90 100
-40
-60
Exercise price = $50; option premium = $10

•
Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset on or before some time in the future, at prices agreed upon today.
• When exercising a put, you “put” the asset to someone.

Basic Put Option Pricing
Relationships at Expiry
•
At expiry, an American put option is worth the same as a European option with the same characteristics.
•
If the put is in-the-money, it is worth E - S
T
.
•
If the put is out-of-the-money, it is worthless.
P aT
= P eT
= Max [ E - S
T
, 0]

60
40 Buy a put
20
0
0 10 20 30 40 50
-20
60 70 80 90
Stock price ($)
100
-40
-60
Exercise price = $50

60
40
20
0
0 10 20 30 40 50
-20
60 70 80 90
Stock price ($)
100
-40
-60 write a put
Exercise price = $50

60
40
20
10
0
-10
-20
0 10 20 30 40 50
Stock price ($)
Write a put
60 70 80
Buy a put
90 100
-40
-60
Exercise price = $50; option premium = $10

•
The seller (or writer) of an option has an obligation.
•
The purchaser of an option has an option.
Buy a call
Write a put
10
-10
0 10 20 30 40 50
Buy a put
Write a call

--Call---Put--
Option/Strike Exp.
Vol.
Last Vol.
Last
IBM 130 Oct
138¼ 130 Jan
364
112
15¼
19½
107
420
5¼
9¼
138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94
138¼ 140 Jul 1826 1¾ 427
138¼ 140 Aug 2193 6½ 58
5½
2¾
7½

This option has a strike price of $135;
--Call---Put--
Option/Strike Exp.
Vol.
Last Vol.
Last
IBM 130 Oct
138¼ 130 Jan
364
112
15¼
19½
107
420
5¼
9¼
138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94
138¼ 140 Jul 1826 1¾ 427
138¼ 140 Aug 2193 6½ 58
5½
2¾
7½ a recent price for the stock is $138.25
July is the expiration month

This makes a call option with this exercise price in-themoney by $3.25 = $138¼ – $135.
--Call---Put--
Option/Strike Exp.
Vol.
Last Vol.
Last
IBM 130 Oct
138¼ 130 Jan
364
112
15¼
19½
107
420
5¼
9¼
138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94
138¼ 140 Jul 1826 1¾ 427
138¼ 140 Aug 2193 6½ 58
5½
2¾
7½
Puts with this exercise price are out-of-the-money.

--Call---Put--
Option/Strike Exp.
Vol.
Last Vol.
Last
IBM 130 Oct
138¼ 130 Jan
364
112
15¼
19½
107
420
5¼
9¼
138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94
138¼ 140 Jul 1826 1¾ 427
138¼ 140 Aug 2193 6½ 58
5½
2¾
7½
On this day, 2,365 call options with this exercise price were traded.

The CALL option with a strike price of $135 is trading for
$4.75.
--Call---Put--
Option/Strike Exp.
Vol.
Last Vol.
Last
IBM 130 Oct
138¼ 130 Jan
364
112
15¼
19½
107
420
5¼
9¼
138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94
138¼ 140 Jul 1826 1¾ 427
138¼ 140 Aug 2193 6½ 58
5½
2¾
7½
Since the option is on 100 shares of stock, buying this option would cost $475 plus commissions.

--Call---Put--
Option/Strike Exp.
Vol.
Last Vol.
Last
IBM 130 Oct
138¼ 130 Jan
364
112
15¼
19½
107
420
5¼
9¼
138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94
138¼ 140 Jul 1826 1¾ 427
138¼ 140 Aug 2193 6½ 58
5½
2¾
7½
On this day, 2,431 put options with this exercise price were traded.

The PUT option with a strike price of $135 is trading for
$.8125.
--Call---Put--
Option/Strike Exp.
Vol.
Last Vol.
Last
IBM 130 Oct
138¼ 130 Jan
364
112
15¼
19½
107
420
5¼
9¼
138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94
138¼ 140 Jul 1826 1¾ 427
138¼ 140 Aug 2193 6½ 58
5½
2¾
7½
Since the option is on 100 shares of stock, buying this option would cost $81.25 plus commissions.

•
Puts and calls can serve as the building blocks for more complex option contracts.
•
If you understand this, you can become a financial engineer, tailoring the risk-return profile to meet your client’s needs.

Protective Put Strategy: Buy a Put and Buy the Underlying Stock: Payoffs at Expiry
Value at expiry
Protective Put strategy has downside protection and upside potential
$50
Buy the stock
$0
Buy a put with an exercise price of $50
$50
Value of stock at expiry

Value at expiry
$40 Buy the stock at $40
Protective Put strategy has downside protection and upside potential
$0
Buy a put with exercise price of
$50 for $10
-$40
$40 $50
Value of stock at expiry

Value at expiry
Buy the stock at $40 $40
$10
$0
-$30
-$40
Covered call
Value of stock at expiry
$30 $40 $50
Sell a call with exercise price of
$50 for $10

Long Straddle: Buy a Call and a Put
Value at expiry
$40
$30
Buy a call with an exercise price of
$50 for $10
$0
-$10
-$20
$30 $40 $50 $60 $70
Buy a put with an exercise price of
$50 for $10
A Long Straddle only makes money if the stock price moves $20 away from $50.
Value of stock at expiry

Short Straddle: Sell a Call and a Put
Value at expiry
$20
$10
$0
-$30
-$40
A Short Straddle only loses money if the stock price moves $20 away from $50.
Sell a put with exercise price of
$50 for $10
$30 $40
Value of stock at expiry
$50 $60 $70
Sell a call with an exercise price of $50 for $10

Value at expiry
$5
$0
-$5
-$10
Buy a call with an exercise price of
$50 for $10 long call spread
Value of stock at expiry $50
$55
$60
Sell a call with exercise price of $55 for $5

In market equilibrium, it mast be the case that option prices are set such that: C
0

Xe
 rT 
P
0

S
0
Otherwise, riskless portfolios with positive payoffs exist.
Value at expiry
Buy a call option with an exercise price of $40
Buy the stock at $40 financed with some debt: FV = $ X
Buy the stock at $40
P
0

$0
C
0

-[$40P
($ 40

Xe
0
 rT
]
)
-$40
$40
$40P
0
$ 40

Xe
 rT
Sell a put with an exercise price of $40
$ 40

C
0
Value of stock at expiry

•
The last section concerned itself with the value of an option at expiry.
•
This section considers the value of an option prior to the expiration date.
•
A much more interesting question.

1.
Stock price
2.
Exercise price
3.
Interest rate
4.
Volatility in the stock price
5.
Expiration date
+
Call Put
+
–
–
+
–
+ +
+ +
The value of a call option C
0 max ( S
0 must fall within
–
E, 0) < C
0
< S
0
.
The precise position will depend on these factors.

Market Value, Time Value and Intrinsic Value for an
American Call
Profit
The value of a call option C
0 max ( S
0
–
E, 0) < C
0
< S
0
.
must fall within loss
C aT
> Max [ S
T
- E, 0]
Market Value
Time value
Intrinsic value
E
Out-of-the-money In-the-money
S
T

•
We will start with a binomial option pricing formula to build our intuition.
• Then we will graduate to the normal approximation to the binomial for some realworld option valuation.

Suppose a stock is worth $25 today and in one period will either be worth
15% more or 15% less. S
0
= $25 today and in one year S
1 is either $28.75 or
$21.25. The risk-free rate is 5%. What is the value of an at-the-money call option?
S
0
S
1
$28.75
$25
$21.25

1.
A call option on this stock with exercise price of $25 will have the following payoffs.
2.
We can replicate the payoffs of the call option. With a levered position in the stock.
S
0
S
1
C
1
$28.75
$3.75
$25
$21.25
$0

Borrow the present value of $21.25 today and buy 1 share.
The net payoff for this levered equity portfolio in one period is either $7.50 or $0.
The levered equity portfolio has twice the option’s payoff so the portfolio is worth twice the call option value.
S
0 debt ) =
$28.75
- $21.25
= portfolio
$7.50
C
1
$3.75
$25
$21.25
- $21.25
= $0 $0

The levered equity portfolio value today is today’s value of one share less the present value of a $21.25 debt:
$ 25

$ 21 .
25
( 1
 r f
)
S
0 debt ) =
$28.75
- $21.25
= portfolio
$7.50
C
1
$3.75
$25
$21.25
- $21.25
= $0 $0

We can value the option today as half of the value of the levered equity portfolio:
S
0 debt
C
0

1
2
$ 25

$ 21 .
25
( 1
 r f
)
) =
$28.75
- $21.25
= portfolio
$7.50
C
1
$3.75
$25
$21.25
- $21.25
= $0 $0

If the interest rate is 5%, the call is worth:
C
0

1
2 


$ 25

$ 21 .
25
( 1 .
05 ) 



1
2

$ 25

20 .
24


$ 2 .
38
S
0 debt ) =
$28.75
- $21.25
= portfolio
$7.50
C
1
$3.75
$25
$21.25
- $21.25
= $0 $0

If the interest rate is 5%, the call is worth:
C
0

1
2 


$ 25

$ 21 .
25
( 1 .
05 ) 



1
2

$ 25

20 .
24


$ 2 .
38
S
0
C
0 debt ) =
$28.75
- $21.25
= portfolio
$7.50
C
1
$3.75
$25 $2.38
$21.25
- $21.25
= $0 $0

The most important lesson (so far) from the binomial option pricing model is:
Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities.

The Risk-Neutral Approach to Valuation
S ( U ), V ( U ) q
S (0), V (0)
1- q
S ( D ), V ( D )
We could value V (0) as the value of the replicating portfolio.
An equivalent method is risk-neutral valuation
V ( 0 )
 q

V ( U )

( 1
 q )

V ( D )
( 1
 r f
)

The Risk-Neutral Approach to Valuation
S ( U ), V ( U ) q
S (0), V (0) q is the risk-neutral probability of an “up” move.
1- q
S (0) is the value of the underlying asset today.
S ( D ),
S ( U ) and S ( D ) are the values of the asset in the next period following an up move and a down move, respectively.
V ( D )
V ( U ) and V ( D ) are the values of the asset in the next period following an up move and a down move, respectively.

S ( U ), V ( U ) q
S (0), V (0)
V ( 0 )
 q

V ( U )

( 1
 q )

V ( D )
( 1
 r f
)
1- q
S ( D ), V ( D )
•
The key to finding q is to note that it is already impounded into an observable security price: the value of S (0):
S ( 0 )
 q

S ( U )

( 1
 q )

S ( D )
( 1
 r f
)
A minor bit of algebra yields: q

( 1
 r f
)

S
S ( U )

( 0 )
S

S
( D )
( D )

Example of the Risk-Neutral Valuation of a Call:
Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. The risk-free rate is 5%. What is the value of an at-the-money call option?
The binomial tree would look like this:
$ 28 .
75

$ 25

( 1 .
15 )
$25 , C (0) q
1- q
$28.75
, C ( D )
$ 21 .
25

$ 25

( 1

.
15 )
$21.25
, C ( D )

Example of the Risk-Neutral Valuation of a Call:
The next step would be to compute the risk neutral probabilities q

( 1
 r f
)

S ( 0 )

S
S ( U )

S ( D )
( D ) q

( 1 .
05 )

$ 25

$ 21 .
25
$ 28 .
75

$ 21 .
25

$ 5
$ 7 .
50

2 3
2/3
$28.75
, C ( D )
$25 , C (0)
1/3
$21.25
, C ( D )

Example of the Risk-Neutral Valuation of a Call:
After that, find the value of the call in the up state and down state.
C ( U )

$ 28 .
75

$ 25
$25 , C (0)
2/3
$28.75
, $3.75
C ( D )
 max[$ 25

$ 28 .
75 , 0 ]
1/3
$21.25
, $0

Example of the Risk-Neutral Valuation of a Call:
Finally, find the value of the call at time 0:
C ( 0 )
 q

C ( U )

( 1
 q )

C ( D )
( 1
 r f
)
C ( 0 )

2 3

$ 3 .
75

( 1 3 )

$ 0
( 1 .
05 )
C ( 0 )

$ 2 .
50
( 1 .
05 )

$ 2 .
38
$25 , C (0)
2/3
$28.75
,$3.75
1/3
$21.25
, $0

Risk-Neutral Valuation and the Replicating Portfolio
This risk-neutral result is consistent with valuing the call using a replicating portfolio.
C
0

2 3

$ 3 .
75

( 1 3 )

$ 0
( 1 .
05 )

$ 2 .
50
1 .
05

$ 2 .
38
C
0

1
2 


$ 25

$ 21 .
25
( 1 .
05 ) 



1
2

$ 25

20 .
24


$ 2 .
38

The Black-Scholes Model is
C
0

S

N( d
1
)

Ee
 rT 
N( d
2
)
Where
C
0
= the value of a European option at time r = the risk-free interest rate.
t = 0 d
1
 ln( S / E )

( r

σ 2
) T
2 s
T d
2
 d
1
 s
T
N( d ) = Probability that a standardized, normally distributed, random variable will be less than or equal to d .
The Black-Scholes Model allows us to value options in the real world just as we have done in the 2-state world.

Find the value of a six-month call option on the Microsoft with an exercise price of $150
The current value of a share of Microsoft is $160
The interest rate available in the U.S. is r = 5%.
The option maturity is 6 months (half of a year).
The volatility of the underlying asset is 30% per annum.
Before we start, note that the intrinsic value of the option is
$10—our answer must be at least that amount.

Let’s try our hand at using the model. If you have a calculator handy, follow along.
First calculate d
1 and d
2 d
1
 ln( S / E )

( r

.
5
σ 2
) T s
T d
1
 ln( 160 / 150 )

(.
05

.
5 ( 0 .
30 )
2
).
5

0 .
5282
0 .
30 .
5
Then, d
2
 d
1
 s
T

0 .
52815

0 .
30 .
5

0 .
31602

C
0

S

N( d
1
)

Ee
 rT 
N( d
2
) d d
2
1

0 .
5282

0 .
31602
N( d
1
) = N(0.52815) = 0.7013
N( d
2
) = N(0.31602) = 0.62401
C
0
C
0

$ 160

0 .
7013

150 e

.
05

.
5 
0 .
62401

$ 20 .
92

• Levered Equity is a Call Option.
• The underlying asset comprise the assets of the firm.
• The strike price is the payoff of the bond.
• If at the maturity of their debt, the assets of the firm are greater in value than the debt, the shareholders have an in-the-money call, they will pay the bondholders and “call in” the assets of the firm.
•
If at the maturity of the debt the shareholders have an out-of-themoney call, they will not pay the bondholders ( i.e.
the shareholders will declare bankruptcy) and let the call expire.

•
Levered Equity is a Put Option.
• The underlying asset comprise the assets of the firm.
•
The strike price is the payoff of the bond.
•
If at the maturity of their debt, the assets of the firm are less in value than the debt, shareholders have an in-the-money put.
• They will put the firm to the bondholders.
•
If at the maturity of the debt the shareholders have an out-ofthe-money put, they will not exercise the option ( i.e.
NOT declare bankruptcy) and let the put expire.

• It all comes down to put-call parity.
C
0
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S
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P
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Value of a call on the firm
=
Value of the firm
+
Value of a put on the firm
–
Value of a risk-free bond
Stockholder’s position in terms of call options
Stockholder’s position in terms of put options

Capital-Structure Policy and Options
•
Recall some of the agency costs of debt: they can all be seen in terms of options.
•
For example, recall the incentive shareholders in a levered firm have to take large risks.

Balance Sheet for a Company in Distress
Assets
Cash
BV MV
$200 $200
Fixed Asset $400 $0
Total $600 $200
Liabilities BV MV
LT bonds $300 ?
Equity
Total
$300
$600
?
$200
What happens if the firm is liquidated today?
The bondholders get $200; the shareholders get nothing.

Selfish Strategy 1: Take Large Risks
(Think of a Call Option)
The Gamble
Win Big
Lose Big
Probability
10%
90%
Payoff
$1,000
$0
Cost of investment is $200 (all the firm’s cash)
Required return is 50%
Expected CF from the Gamble = $1000 × 0.10 + $0 = $100
NPV
 
$ 200

$ 100
1 .
50
NPV
 
$ 133

Selfish Stockholders Accept Negative NPV Project with Large
Risks
• Expected cash flow from the Gamble
•
To Bondholders = $300 × 0.10 + $0 = $30
•
To Stockholders = ($1000 - $300) × 0.10 + $0 = $70
•
PV of Bonds Without the Gamble = $200
•
PV of Stocks Without the Gamble = $0
•
PV of Bonds With the Gamble = $30 / 1.5 = $20
•
PV of Stocks With the Gamble = $70 / 1.5 = $47
The stocks are worth more with the high risk project because the call option that the shareholders of the levered firm hold is worth more when the volatility is increased.

•
This is an area rich with optionality, both in the structuring of the deals and in their execution.

Investment in Real Projects & Options
•
Classic NPV calculations typically ignore the flexibility that real-world firms typically have.
•
The next chapter will take up this point.

•
The most familiar options are puts and calls.
•
Put options give the holder the right to sell stock at a set price for a given amount of time.
• Call options give the holder the right to buy stock at a set price for a given amount of time.
• Put-Call parity
C
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S
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P
0

• The value of a stock option depends on six factors:
1. Current price of underlying stock.
2. Dividend yield of the underlying stock.
3. Strike price specified in the option contract.
4. Risk-free interest rate over the life of the contract.
5. Time remaining until the option contract expires.
6. Price volatility of the underlying stock.
•
Much of corporate financial theory can be presented in terms of options.
1.
Common stock in a levered firm can be viewed as a call option on the assets of the firm.
2.
Real projects often have hidden option that enhance value.