Risk Management Dr. Keith M. Howe Summer 2008. Definition Risk and uncertainty Risk aversion Risk management The process of formulating the benefit-cost

Risk Management

Dr. Keith M. Howe

Summer 2008

Definition

Risk and uncertainty

Risk aversion

Risk management

The process of formulating the benefit-cost trade-offs of risk reduction and deciding on the course of action to take (including the decision

to take no action at all).

Two more definitions

• 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.

R i s k y U p s i d e I n c . p r i c e

1 1 0

G a i n

2 0 5 0

- $ 3 , 0 0 0

+ $ 6 , 0 0 0

Panel A. Gain from buying shares of RiskyUpside Inc. at $50 per share.

R i s k y U p s i d e I n c . p r i c e

1 1 0

G a i n

5 0

0

- $ 1 , 0 0 0

2 0

$ 5 , 0 0 0

Panel B. Gain from buying a call option onshares of Risky Upside Inc. with exercise priceof $50 for a premium of $10 per share.

E x c h a n g e r a t e

I n c o m e t o f i r m

$ 1 0 0 m i l l i o n

U n h e d g e d i n c o m e

$ 1$ 0 . 9 0

$ 9 0 m i l l i o n

Panel A. Income to Garman if it does not hedge.

Figure 1.3. Hedging with forward contract. Garman’sincome is in dollars and the exchange rate is the dollar priceof one euro.


G a i n f r o m c o n t r a c tt o f i r m

F o r w a r dg a i n

F o r w a r dl o s s

F o r w a r d r a t e $ 1$ 0 . 9

$ 1 0 m i l l i o n

Panel B. Forward contract payoff.

Panel C. Hedged firm income.


I n c o m e t o f i r m

F o r w a r dg a i n

F o r w a r dl o s s$ 1 0 0 m i l l i o n

U n h e d g e d i n c o m e

H e d g e d i n c o m e

F o r w a r d r a t e $ 1

Panel D. Comparison of income with put contract and income with forward contract.

E x c h a n g e r a t eE x c h a n g e r a t e

I n c o m e t o f i r mI n c o m e t o f i r m

G a i n w i t hG a i n w i t ho p t i o no p t i o n

L o s s w i t h o p t i o nL o s s w i t h o p t i o n$ 1 0 0 m i l l i o n

E x e r c i s e p r i c e o f $ 1

U n h e d g e di n c o m e

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.

Role of risk management

Risk management can add value to the firm by:

• Decreasing taxes

• Decreasing transaction costs (including

bankruptcy costs)

• Avoiding investment decision errors

Bankruptcy costs and costs of financial distress

• 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.

Cash flow to thefirm

Cash flow to shareholders

Unhedged cash flow

Expected cashflow $350M$250M $450M

$250M

$450M

Expected cashflow $350M

Figure 3.1. Cash flow to shareholders and operating cash flow.

Figure 3.2. Creating the unhedged firm out of the hedged firm.

Forwardgain

Forward loss

Unhedged cash flow

Hedged firm cash

flow

$350M (gold sold at forward)

$350M(hedged)

Cashflow toshareholders

Figure 3.3. Cash flow to claimholders and bankruptcy costs.

Cash flow to thefirm

Cash flow to claimholders

Unhedged cash flow

Expected cashflow $350M$250M $450M

$230M

$450M

Expected cashflow hedged$350MUnhedged$340M

Bankruptcycost

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

Analysis of decreasing transaction cost by hedging

Taxes and risk management

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

ExampleThe 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 withEqual probability. The forward price is $350.

Optimal capital structure and risk management

• 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 hedgedfirm

1 0 0 2 0 0 3 0 0 4 0 0

3 0 5

3 1 0

3 1 5

3 2 0

3 2 5

3 3 0

Principal amount of debt

Optimal amount ofdebt, $317.073M

Risk management process

Risk identification

Risk assessment

Selection of risk-mgt techniques

Implementation

Review

The rules of 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

Risk Management

Source: Riskmetrics Group (www.riskmetrics.com)

Types of risks firms face

Market risk - interest rate - foreign exchange - commodity price

Hazard risk - physical damage - liabilities - business interruption

Operational risk - industry sectors - geographical regions

Strategic risk - competition - reputation - investor support

Assignment of risk responsibilities CEOStrategic riskmanagement

CRO

Market riskmanagement

Hazard riskmanagement

Operational riskmanagement

Hedgeable Insurable Diversifiable

Three dimensions of risk transfer

•Hedging

•Insuring

•Diversifying

A new concept of risk management (VAR)• 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 over the measurement period, the 5 percent VAR of the portfolio is:

VAR = 1.65 X s X Portfolio value

Example of VAR

• 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.

More on VAR

• 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.

Instruments used in risk management

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

Forward Contracts

• 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.

Example

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

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.

Payoff after 6 months

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: F0=1100

S1=1000*1.07=1070

Payoff: 1070-1100= - $30.

Example: Valuing a Forward Contract on a Share of Stock

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?

Strategy 1---- the forward contract

One year from nowToday

Buy stock at a price of $100. Sell the share for cash at market

Strategy 2 ---- the portfolio strategyToday One year from now

Buy stock todaySell short $100 in face value of 1-year zero-coupon bonds

Sell the stockBuyback the zero-coupon bonds of $100

Buy a forward contract

Valuing a forward contract


Cost Today

Cash flow one year from now

Strategy 1Strategy 2

?$97-$92

S1- $100S1- $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 = $5In strategy 1, the obligation to buy the stock for $100 one year from now, should cost $5.

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

where S0 = current price of the stock

= the current market price of a default-free zero-coupon bond paying K, T years in the future

Tfr

KS

)1(0

Tfr

K

)1(


At no arbitrage:

T

frSKF )1(00

0)1(0

Tfr

KS

domestic

foreign

r

r

S

F

1

1

0

0

Currency Forward Rates

• 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

• 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

Forward Currency Rates

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.

Answer: (LIBOR is a zero-coupon rate based on an actual/360 day count.) So

Canada United States

Six-month interest Rate (unannualized):

The forward rate is

%4360

182%02.2 %10

360

182%06.5

.25.10506.1

0202.1

$

21.1$

US

Can

Currency Forward Rates

Futures Contracts: Preliminaries

• 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.

Futures Contracts: Preliminaries

• 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.

Daily Resettlement: An ExampleSuppose you want to speculate on a rise in the $/¥ exchange

rate (specifically you think that the dollar will appreciate).

Currently $1 = ¥140.

Currency per U.S. $ equivalent U.S. $

Wed Tue Wed TueJapan (yen) 0.007142857 0.007194245 140 1391-month forward 0.006993007 0.007042254 143 1423-months forward 0.006666667 0.006711409 150 1496-months forward 0.00625 0.006289308 160 159

The 3-month forward price is $1=¥150.

Daily Resettlement: An Example• 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:

¥150

$10¥12,500,00.04 $3,333.33

Daily Resettlement: An ExampleIf 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:

¥149

$10¥12,500,0062.892,83$

You have lost $559.28 overnight.

But ¥12,500,000 is now worth $83,892.62:

¥150

$10¥12,500,00 $83,333.33

Daily Resettlement: An Example• 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.

¥149

$10¥12,500,00.04 $3,355.70

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.

Selected Futures ContractsContract Contract Size Exchange

AgriculturalCorn 5,000 bushels Chicago BOT

Wheat 5,000 bushels Chicago & KCCocoa 10 metric tons CSCE

OJ 15,000 lbs. CTNMetals & Petroleum

Copper 25,000 lbs. CMX Gold 100 troy oz. CMX

Unleaded gasoline 42,000 gal. NYMFinancial

British Pound £62,500 IMMJapanese Yen ¥12.5 million IMM

Eurodollar $1 million LIFFE

Futures Markets

• 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

The Chicago Mercantile Exchange

• Expiry cycle: March, June, September, December.

• Delivery date 3rd 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.

CME After Hours

• 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.

OpenOpen High Low Settle Change High Low Interest

July 179 180 178¼ 178½ -1½ 312 177 2,837Sept 186 186½ 184 186 -¾ 280 184 104,900Dec 196 197 194 196½ -¼ 291¼ 194 175,187

Sept 117-05 117-21 116-27 117-05 +5 131-06 111-15 647,560Dec 116-19 117-05 116-12 116-21 +5 128-28 111-06 13,857

Sept 11200 11285 11145 11241 -17 11324 7875 18,530Dec 11287 11385 11255 11349 -17 11430 7987 1,599

Lifetime

Corn (CBT) 5,000 bu.; cents per bu.

TREASURY BONDS (CBT) - $1,000,000; pts. 32nds of 100%

DJ INDUSTRIAL AVERAGE (CBOT) - $10 times average

Expiry month

Opening price

Highest price that day

Lowest price that day

Closing price Daily Change

Highest and lowest prices over the lifetime of the contract.

Number of open contracts

Wall Street Journal Futures Price Quotes

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.

Hedging

• 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

Hedging and Speculating Example

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

Hedging: How many contacts?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:

Now you can quit worrying about the price of corn and get back to worrying about the weather.

contracts 10contractper bushels 000,5

bushels 000,50

Interest Rate Futures Contracts

Pricing of Treasury BondsConsider a Treasury bond that pays a semiannual coupon of $C

for the next T years:• The yield to maturity is r

TT rr

C

r

FPV

)1(

11

)1(

Value of the T-bond under a flat term structure= PV of face value + PV of coupon payments

C…

0 1 2 3 2T

C FC C

Pricing of Treasury BondsIf the term structure of interest rates is not flat, then

we need to discount the payments at different rates depending upon maturity

TTr

FC

r

C

r

C

r

CPV

)1()1()1()1( 23

32

21

= PV of face value + PV of coupon payments

C…

0 1 2 3 2T

C FC C

Pricing of Forward Contracts

An N-period forward contract on that T-Bond C

…0 N N+1 N+2 N+3 N+2T

C FC CforwardP

Can be valued as the present value of the forward price.

NN

TTNNNN

r

rFC

rC

rC

rC

PV)1(

)1()1()1()1( 23

32

21

NN

forward

r

PPV

)1(

Futures Contracts

• The pricing equation given above will be a good approximation.

• The only real difference is the daily resettlement.

Hedging in Interest Rate Futures

• 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.

Duration Hedging

• 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

Duration Hedging

Duration Formula

N

tt

t

N

tt

t

T

r

Cr

tC

D

PV

TCPVCPVCPVD

1

1

21

)1(

)1(

)(2)(1)(

Calculating DurationCalculate 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 PVYears Cash flow factor value / Bond price

0.5 $40.00 0.96154 $38.46 0.01921 $40.00 0.92456 $36.98 0.0370

1.5 $40.00 0.88900 $35.56 0.05332 $40.00 0.85480 $34.19 0.0684

2.5 $40.00 0.82193 $32.88 0.08223 $1,040.00 0.79031 $821.93 2.4658

$1,000.00 2.7259 yearsBond price Bond duration

Calculating Duration

Duration is expressed in units of time; usually years.

Duration

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

Swaps Contracts: Definitions

• 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.

The Swap Bank

• 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.


• 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 floating-

rate notes at LIBOR + ½ percent.

• Firm B would prefer to borrow at a fixed rate.


The borrowing opportunities of the two firms are: COMPANY B BANK A

Fixed rate 11.75% 10%

Floating rate LIBOR + .5% LIBOR


Bank

A

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



Swap

Bank

LIBOR – 1/8%

10 3/8%

COMPANY B BANK A




Here’s what’s in it for Bank A: They can borrow externally at 10% fixed and have a net borrowing position of

-10 3/8 + 10 + (LIBOR – 1/8) =

LIBOR – ½ % which is ½ % better than they can borrow floating without a swap.

10%

½% of $10,000,000 = $50,000. That’s quite a cost savings per year for 5 years.

Swap

Bank

LIBOR – 1/8%

10 3/8%

Bank

A


Company

B

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

Bank10 ½%

LIBOR – ¼%

COMPANY B BANK A



COMPANY B BANK A




They can borrow externally at

LIBOR + ½ % and have a net

borrowing position of

10½ + (LIBOR + ½ ) - (LIBOR - ¼ ) = 11.25% which is ½% better than they can borrow floating.

LIBOR + ½%

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 years.

Swap

Bank

Company

B

10 ½%

LIBOR – ¼%

An Example of an Interest Rate SwapThe swap bank makes money too. ¼% of $10 million

= $25,000 per year for 5 years.

LIBOR – 1/8 – [LIBOR – ¼ ]= 1/8

10 ½ - 10 3/8 = 1/8

¼

Swap

Bank

Company

B

10 ½%

LIBOR – ¼%LIBOR – 1/8%

10 3/8%

Bank

A

COMPANY B BANK A




Swap

Bank

Company

B

10 ½%

LIBOR – ¼%LIBOR – 1/8%

10 3/8%

Bank

A

B saves ½%A saves ½%

The swap bank makes ¼%

COMPANY B BANK A



An Example of a Currency Swap

• 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 mirror-image financing need they may both benefit from a swap.

• If the spot exchange rate is S0($/£) = $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%

$9.4%


$ £


Company B 10.0% 12.0%

Firm

B

$8% £12%

Swap

Bank

Firm

A

£11%

$8%

£12%


$8% £12%

$ £


Company B 10.0% 12.0%

Firm

B

Swap

Bank

Firm

A

£11%

$8% $9.4%

£12%

A’s net position is to borrow at £11%

A saves £.6%


$8% £12%

$ £


Company B 10.0% 12.0%

Firm

B

Swap

Bank

Firm

A

£11%

$8% $9.4%

£12%

B’s net position is to borrow at $9.4%

B saves $.6%


$8% £12%

$ £


Company B 10.0% 12.0%

Firm

B

The swap bank makes money too:

At S0($/£) = $1.60/£, that is a gain of $124,000 per

year for 5 years. The swap bank faces exchange rate risk, but maybe they can lay it off (in another swap).

1.4% of $16 million financed with 1% of £10 million per year

for 5 years.

Swap

Bank

Firm

A

£11%

$8% $9.4%

£12%

Variations of Basic Swaps• 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.

Pricing a 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.

Options

• 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.

Options Contracts: Preliminaries

• 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.

Options Contracts: Preliminaries• 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

• 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 ST - E.

• If the call is out-of-the-money, it is worthless.

CaT = CeT = Max[ST - E, 0]

• Where

ST is the value of the stock at expiry (time T)

E is the exercise price.

CaT is the value of an American call at expiry

CeT is the value of a European call at expiry

Call Option Payoffs

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Buy a call

Exercise price = $50

Call Option Payoffs

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Write a call


Call Option Profits

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

rofi

ts (

$)

Write a call

Buy a call

Exercise price = $50; option premium = $10

Put Options

• 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 - ST.

• If the put is out-of-the-money, it is worthless.

PaT = PeT = Max[E - ST, 0]

Put Option Payoffs

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Buy a put


Put Option Payoffs

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Op

tio

n p

ayo

ffs

($)

write a put


Stock price ($)

Put Option Profits

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

rofi

ts (

$)

Buy a put

Write a put

Exercise price = $50; option premium = $10

10

-10

Selling Options• The seller (or writer) of an

option has an obligation.• The purchaser of an option

has an option.

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

rofi

ts (

$)

Buy a put

Write a put

10

-10

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

rofi

ts (

$)

Write a call

Buy a call

Reading The Wall Street Journal

Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16

138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--



138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

This option has a strike price of $135;

a recent price for the stock is $138.25

July is the expiration month



138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

This makes a call option with this exercise price in-the-money by $3.25 = $138¼ – $135.

Puts with this exercise price are out-of-the-money.



138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

On this day, 2,365 call options with this exercise price were traded.



138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

The CALL option with a strike price of $135 is trading for $4.75.

Since the option is on 100 shares of stock, buying this option would cost $475 plus commissions.



138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

On this day, 2,431 put options with this exercise price were traded.



138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

The PUT option with a strike price of $135 is trading for $.8125.

Since the option is on 100 shares of stock, buying this option would cost $81.25 plus commissions.

Combinations of Options

• 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

Buy a put with an exercise price of $50

Buy the stock

Protective Put strategy has downside protection and upside potential

$50

$0

$50

Value at expiry

Value of stock at expiry

Protective Put Strategy Profits

Buy a put with exercise price of $50 for $10

Buy the stock at $40

$40

Protective Put strategy has

downside protection and upside potential

$40

$0

-$40

$50

Value at expiry


Covered Call Strategy

Sell a call with exercise price of $50 for $10


$40

Covered call

$40

$0

-$40

$10

-$30

$30 $50


Value at expiry

Long Straddle: Buy a Call and a Put

Buy a put with an exercise price of

$50 for $10$40

A Long Straddle only makes money if the stock price moves $20 away from $50.

$40

$0

-$20$50

Buy a call with an exercise price of $50 for $10

-$10

$30

$60$30 $70


Value at expiry

Short Straddle: Sell a Call and a Put

Sell a put with exercise price of$50 for $10

$40

A Short Straddle only loses money if the stock price moves $20 away from $50.

-$40

$0

-$30$50

Sell a call with an exercise price of $50 for $10

$10

$20

$60$30 $70


Value at expiry

Long Call Spread

Sell a call with exercise price of $55 for $5

$55

long call spread$5$0

$50

Buy a call with an exercise price of $50 for $10

-$10-$5

$60


Value at expiry

Put-Call Parity

Sell a put with an exercise price of $40

Buy the stock at $40 financed with some debt: FV = $XBuy a call option with

an exercise price of $40

$0

-$40

$40-P0

rTXe40$

$40


040$ C)40($ rTXe

-[$40-P0]0C

0P

In market equilibrium, it mast be the case that option prices are set such that:

000 SPXeC rT

Otherwise, riskless portfolios with positive payoffs exist.


Value at expiry

Valuing Options

• 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.

Option Value DeterminantsCall Put

1. Stock price + –2. Exercise price – +3. Interest rate + –4. Volatility in the stock price + +5. Expiration date + +

The value of a call option C0 must fall within

max (S0 – E, 0) < C0 < S0.

The precise position will depend on these factors.

Market Value, Time Value and Intrinsic Value for an American Call

CaT > Max[ST - E, 0]

Profit

loss E ST

Market Value

Intrinsic value

S T - E

Time value

Out-of-the-money In-the-money

S T

The value of a call option C0 must fall within max (S0 – E, 0) < C0 < S0.

An Option‑Pricing Formula

• 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 real-world option valuation.

Binomial Option Pricing Model

Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. S0= $25 today and in one year S1is either $28.75 or $21.25. The risk-free rate is 5%. What is the value of an at-the-money call option?

$25

$21.25

$28.75

S1S0


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.

$25

$21.25

$28.75

S1S0 C1

$3.75

$0

Binomial Option Pricing ModelBorrow 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.

$25

$21.25

$28.75S1S0 debt

- $21.25portfolio$7.50

$0

( - ) ==

=

C1

$3.75

$0- $21.25

Binomial Option Pricing Model The levered equity portfolio value today is

today’s value of one share less the present value of a $21.25 debt:

)1(

25.21$25$

fr

$25

$21.25

$28.75S1S0 debt


$0

( - ) ==

=

C1

$3.75

$0- $21.25


We can value the option today as half of the value of the levered equity

portfolio:

)1(

25.21$25$

2

10

frC

$25

$21.25

$28.75S1S0 debt


$0

( - ) ==

=

C1

$3.75

$0- $21.25

If the interest rate is 5%, the call is worth:The Binomial Option Pricing Model

38.2$24.2025$2

1

)05.1(

25.21$25$

2

10

C

$25

$21.25

$28.75S1S0 debt


$0

( - ) ==

=

C1

$3.75

$0- $21.25

If the interest rate is 5%, the call is worth:The Binomial Option Pricing Model

38.2$24.2025$2

1

)05.1(

25.21$25$

2

10

C

$25

$21.25

$28.75S1S0 debt


$0

( - ) ==

=

C1

$3.75

$0- $21.25

$2.38

C0


the replicating portfolio intuition.the replicating portfolio intuition.

Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities.

The most important lesson (so far) from the binomial option pricing model is:

The Risk-Neutral Approach to Valuation

We could value V(0) as the value of the replicating portfolio. An equivalent method is risk-neutral valuation

S(0), V(0)

S(U), V(U)

S(D), V(D)

q

1- q

)1(

)()1()()0(

fr

DVqUVqV


S(0) is the value of the underlying asset today.

S(0), V(0)

S(U), V(U)

S(D), V(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.

q

1- q

V(U) and V(D) are the values of the asset in the next period following an up move and a down move, respectively.

q is the risk-neutral probability of an “up” move.


• 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), V(0)

S(U), V(U)

S(D), V(D)

q

1- q

)1(

)()1()()0(

fr

DVqUVqV

)1(

)()1()()0(

fr

DSqUSqS

A minor bit of algebra yields:)()(

)()0()1(

DSUS

DSSrq f

Example of the Risk-Neutral Valuation of a Call:

$21.25,C(D)

q

1- q

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:

$25,C(0)

$28.75,C(D)

)15.1(25$75.28$

)15.1(25$25.21$


$21.25,C(D)

2/3

1/3

The next step would be to compute the risk neutral probabilities

$25,C(0)

$28.75,C(D)

)()(

)()0()1(

DSUS

DSSrq f

3250.7$

5$

25.21$75.28$

25.21$25$)05.1(

q


$21.25, $0

2/3

1/3

After that, find the value of the call in the up state and down state.

$25,C(0)

$28.75, $3.75

25$75.28$)( UC

]0,75.28$25max[$)( DC


Finally, find the value of the call at time 0:

$21.25, $0

2/3

1/3

$25,C(0)

$28.75,$3.75

)1(

)()1()()0(

fr

DCqUCqC

)05.1(

0$)31(75.3$32)0(

C

38.2$)05.1(

50.2$)0( C

$25,$2.38

This risk-neutral result is consistent with valuing the call using a replicating portfolio.

Risk-Neutral Valuation and the Replicating Portfolio

38.2$24.2025$2

1

)05.1(

25.21$25$

2

10

C

38.2$05.1

50.2$

)05.1(

0$)31(75.3$320

C

The Black-Scholes ModelThe Black-Scholes Model is

)N()N( 210 dEedSC rT

Where

C0 = the value of a European option at time t = 0r = the risk-free interest rate.

T

Tσ

rESd

)2

()/ln(2

1

Tdd 12

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.

The Black-Scholes Model

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.

The Black-Scholes ModelLet’s try our hand at using the model. If you have a calculator handy,

follow along.

Then,

T

TσrESd

)5.()/ln( 2

1

First calculate d1 and d2

31602.05.30.052815.012 Tdd

5282.05.30.0

5).)30.0(5.05(.)150/160ln( 2

1

d

The Black-Scholes Model

N(d1) = N(0.52815) = 0.7013

N(d2) = N(0.31602) = 0.62401

5282.01 d

31602.02 d

)N()N( 210 dEedSC rT

92.20$

62401.01507013.0160$

0

5.05.0

C

eC

Stocks and Bonds as Options• 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-the-money call, they will not pay the bondholders (i.e. the shareholders will declare bankruptcy) and let the call expire.

Stocks and Bonds as Options• 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-of-

the-money put, they will not exercise the option (i.e. NOT

declare bankruptcy) and let the put expire.

Stocks and Bonds as Options

• It all comes down to put-call parity.

Value of a call on the

firm

Value of a put on the

firm

Value of a risk-free

bond

Value of the firm= + –

TreXPSC 00

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 BV MV Liabilities BV MV

Cash $200 $200 LT bonds $300 ?

Fixed Asset $400 $0 Equity $300 ?

Total $600 $200 Total $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 Probability Payoff

Win Big 10% $1,000

Lose Big 90% $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

133$50.1

100$200$

NPV

NPV

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.

Mergers and Options

• 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.

Summary and Conclusions

• 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 parity00 PSeXC Tr

Summary and Conclusions

• 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.

Documents

Risk Management Dr. Keith M. Howe Summer 2008. Definition Risk and uncertainty Risk aversion Risk management The process of formulating the benefit-cost