Lecture Notes(Financial Economics)

7/30/2019 Lecture Notes(Financial Economics)

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Financial Economics

Lecture Notes

Won Joong Kimy

The materials covered here are mostly from F. Mishkin "The Economics of Money,Banking, and Financial Markets." 8th ed., and J. Hull "Fundamentals of Futures andOptions Markets." 6th ed.. Students are required to read through the textbook in additionto these lecture notes. These notes are preliminary and are not to be quoted or cited.

yAssistant Professor. Department of Economics, Kangwon National University. Email:[email protected].


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Contents

I Introduction 3

1 Why Study Money, Banking, and Financial Markets? (M. 1) 3

2 Introduction to Derivatives Markets (H. 1) 9

3 An Overview of the Financial System (M. 2) 16

4 What Is Money? (M. 3) 25

II Financial Markets 27

5 Understanding Interest Rates (M. 4) 27

6 The Behavior of Interest Rates (M. 5) 34

7 The Risk and Term Structure of Interest Rates (M. 6) 43

8 The Stock Market, the Theory of Rational Expectations, and the

Ecient Market Hypothesis (M. 7) 49

9 Capital Asset Pricing and Arbitrage Pricing Theory (BKM 7.) 55

III Futures and Options Markets 63

10 Mechanics of Futures Markets (H.2) 63

11 Hedging Strategies Using Futures (H. 3) 68

12 Determination of Forward and Futures Prices (H. 5) 75

13 Swaps (H. 7) 80

14 Credit Derivatives (H. 21) 88

15 Mechanics of Options Markets (H. 8) 93

16 Trading Strategies Involving Options (H. 10) 98

17 Introduction to Binomial Trees (H. 11) 105

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18 Valuing Stock Options:The Black-Scholes Model (H. 12) 114

19 The Greek Letters (H. 15) 119

IV International Finance and Monetary Policy 124

20 The Foreign Exchange Market (M. 17) 124

21 The International Financial System (M. 18) 129

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Part I

Introduction

1 Why Study Money, Banking, and Financial Markets?

(M. 1)

Why Study Money, Banking, and Financial Markets

To examine how nancial markets such as bond, stock and foreign ex-

change markets work

To examine how nancial institutions such as banks and insurance com-

panies work

To examine the role of money in the economy

Financial Markets

Markets in which funds are transferred from people who have an excess

of available funds to people who have a shortage of funds

The Bond Market and Interest Rates

A security (nancial instrument) is a claim on the issuers future income

or assets

A bond is a debt security that promises to make payments periodically

for a specied period of time

An interest rate is the cost of borrowing or the price paid for the rental

of funds

Interest Rates on Selected Bonds (01.108.6). Bank of Korea (BOK)

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The Stock Market

Common stock represents a share of ownership in a corporation

A share of stock is a claim on the earnings and assets of the corporation

Monthly Average Stock Prices (93.108.6). BOK

The Foreign Exchange Market

The foreign exchange market is where funds are converted from one cur-

rency into another

The foreign exchange rate is the price of one currency in terms of another

currency

The foreign exchange market determines the foreign exchange rate

Monthly average exchange rate (KRW/Foreign). BOK

Money and Business Cycles

Evidence suggests that money plays an important role in generating busi-

ness cycles

Recessions (unemployment) and booms (ination) aect all of us

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Monetary Theory ties changes in the money supply to changes in aggre-

gate economic activity and the price level

Money and Ination

The aggregate price level is the average price of goods and services in an

economy

A continual rise in the price level (ination) aects all economic players

Data shows a connection between the money supply and the price level

Aggregate Price Level and Money Supply in Korea

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Money and Interest Rates

Interest rates are the price of money

Monetary and Fiscal Policy

Monetary policy is the management of the money supply and interest

rates

Conducted in Korea by the Bank of Korea (BOK)

Fiscal policy is government spending and taxation

Budget decit is the excess of expenditures over revenues for a particular

year

Budget surplus is the excess of revenues over expenditures for a particular

year

Any decit must be nanced by borrowing

How We Will Study Money, Banking, and Financial Markets

A simplied approach to the demand for assets

The concept of equilibrium

Basic supply and demand to explain behavior in nancial markets

The search for prots

An approach to nancial structure based on transaction costs and asym-

metric information

Aggregate supply and demand analysis

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Appedix to Chapter 1: Dening Aggregate Output, Income,

the Price Level, and the Ination Rate

Aggregate Output and Aggregate Income

Aggregate Output

Gross Domestic Product (GDP) = market value of all nal goods and

services produced in the domestic economy during a particular year

Aggregate Income

Total income of the factors of production (land, capital, labor) during a

particular year

Distinction Between Nominal and Real

Nominal = values measured using current prices

Real = quantities measured with constant prices

Real vs. nominal wages, real vs. nominal GDP

An example:

Prices and Quantities in 2000 and 2004

Quantities of Prices of Quantities of Prices of

pizzas pizzas calzones calzones

2000 10 $10 15 $5

2004 20 $12 30 $6

Nominal GDP

2000 : (10)($10) + (15)($5) = $175

2004 : (20)($12) + (30)($6) = $420

Real GDP (base year: 2000)

2000 : (10)($10) + (15)($5) = $175

2004 : (20)($10) + (30)($5) = $350

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Aggregate Price Level

Aggregate Price Level is a measure of average prices in the economy

One measure of the price level is the GDP deator

GDP deator =nominal GDP

real GDP

Another measure is the Consumer Price Index (CPI)

The CPI is a measure of the average change over time in the prices paid

by urban consumers for a market basket of goods and services

Growth Rates and the Ination Rate

A growth rate is the percentage change in a variable

Growth rate(%) =xt xt1

xt1 100

GDP growth rate =$9.5 trillion $9 trillion

$9 trillion 100 = 5:6%

Ination rate=113 111

111 100 = 1:8%

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2 Introduction to Derivatives Markets (H. 1)

The Nature of Derivatives

A derivative is an instrument whose value depends on the values of other

more basic underlying variables

Examples of Derivatives

Futures Contracts

Forward Contracts

Swaps

Options

Ways Derivatives are Used

To hedge risks

To speculate (take a view on the future direction of the market)

To lock in an arbitrage prot

To change the nature of a liability

To change the nature of an investment without incurring the costs of

selling one portfolio and buying another

Futures Contracts

A futures contract is an agreement to buy or sell an asset at a certain

time in the future for a certain price

By contrast in a spot contract there is an agreement to buy or sell the

asset immediately (or within a very short period of time)

Exchanges Trading Futures

KRX (Korea Exchange)

Chicago Board of Trade, Chicago Mercantile Exchange

Euronext, Eurex

BM&F (Sao Paulo, Brazil) and many more

Futures Price

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The futures prices for a particular contract is the price at which you

agree to buy or sell

It is determined by supply and demand in the same way as a spot price

Terminology

The party that has agreed to buy has a long position

The party that has agreed to sell has a short position

Example

January: an investor enters into a long futures contract on COMEX to

buy 100 oz of gold @ $600 in April April: the price of gold $615 per oz.

What is the investors prot?

Over-the Counter Markets

The over-the counter market is an important alternative to exchanges

It is a telephone and computer-linked network of dealers who do not

physically meet

Trades are usually between nancial institutions, corporate treasurers,

and fund managers

Size of OTC and Exchange Markets

Forward Contracts

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Forward contracts are similar to futures except that they trade in the

over-the-counter market

Forward contracts are popular on currencies and interest rates

Options

A call option is an option to buy a certain asset by a certain date for a

certain price (the strike price)

A put option is an option to sell a certain asset by a certain date for a

certain price (the strike price)

American vs European Options

An American option can be exercised at any time during its life

A European option can be exercised only at maturity

Options vs Futures/Forwards

A futures/forward contract gives the holder the obligation to buy or sell

at a certain price

An option gives the holder the right to buy or sell at a certain price

Three Reasons for Trading Derivatives: Hedging, Speculation, and Arbitrage

Hedge funds trade derivatives for all three reasons

When a trader has a mandate to use derivatives for hedging or arbitrage,

but then switches to speculation, large losses can result

Hedging Examples

A US company will pay 10 million for imports from Britain in 3 months

and decides to hedge using a long position in a forward contract

An investor owns 1,000 Microsoft shares currently worth $28 per share.

A two-month put with a strike price of $27.50 costs $1. The investor

decides to hedge by buying 10 contracts

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Value of Microsoft Shares with and without Hedging

20,000

25,000

30,000

35,000

40,000

20 25 30 35 40

Stock Price ($)

Value of

Holding ($)

No Hedging

Hedging

Speculation Example

An investor with $2,000 to invest feels that Amazon.coms stock price

will increase over the next 2 months. The current stock price is $20 and

the price of a 2-month call option with a strike of $22.50 is $1

What are the alternative strategies?

Purchase 100 shares of the stock

Options like futures requires only a small amount of cash to be deposited

by the speculator in what is termed a margin account

The futures and options market allows speculator to obtain leverage

Arbitrage Example

Arbitrage involves locking in a riskless prot by simultaneously entering

into transactions in two or more markets

A stock price is quoted as 100 in London and $182 in New York The current exchange rate is 1.8500

What is the arbitrage opportunity with 100 shares of the stocks (assum-

ing zero transaction cost)?

Buys 100 shares in New York and sells the shares in London Converts the sale proceeds from pound to dollars This leads to a prot of

[$185 $182] 100 = $300

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Gold: An Arbitrage Opportunity?

Suppose that:

The spot price of gold is US$600 The quoted 1-year futures price of gold is US$650 The 1-year US$ interest rate is 5% per annum No income or storage costs for gold

Is there an arbitrage opportunity?

The Futures Price of Gold

If the spot price of gold is S and the futures price is for a contractdeliverable in T years is F, then

F = S(1 + r)T

where r is the 1-year (domestic currency) risk-free rate of interest.

In our examples, S = 600, T = 1, and r = 0:05 so that

F = 600(1 + 0:05) = 630

Oil: An Arbitrage Opportunity?

Suppose that:

The spot price of oil is US$70 The quoted 1-year futures price of oil is US$80 The 1-year US$ interest rate is 5% per annum

The storage costs of oil are 2% per annum


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3 An Overview of the Financial System (M. 2)

Function of Financial Markets

Perform the essential function of channeling funds from economic players

that have saved surplus funds to those that have a shortage of funds

Promotes economic eciency by producing an ecient allocation of cap-

ital, which increases production

Directly improve the well-being of consumers by allowing them to time

purchases better

Structure of Financial Markets

Debt and Equity Markets

Debt: bond, mortgage In terms of maturity: short-term debt (less than a year), long-

term debt (ten years or longer)

Equity: residual claim Primary and Secondary Markets

Investment Banks underwrite securities in primary markets Brokers and dealers work in secondary markets

Brokers: match buyers with sellers of securities Dealers: link buyers and sellers by buying and selling securities

at stated prices

Exchanges and Over-the-Counter (OTC) Markets

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Money and Capital Markets

Money markets deal in short-term debt instruments

Capital markets deal in longer-term debt and equity instruments

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Internationalization of Financial Markets

Foreign Bondssold in a foreign country and denominated in that coun-

trys currency Eurobondbond denominated in a currency other than that of the coun-

try in which it is sold

Eurocurrenciesforeign currencies deposited in banks outside the home

country

EurodollarsU.S. dollars deposited in foreign banks outside the U.S.or in foreign branches of U.S. banks

World Stock Markets

Function of Financial Intermediaries: Indirect Finance

Lower transaction costs

Economies of scale Liquidity services

Reduce Risk

Risk Sharing (Asset Transformation) Diversication

Asymmetric Information

Adverse Selection (before the transaction) more likely to select riskyborrower

Moral Hazard (after the transaction) less likely borrower will repayloan

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Regulation of the Financial System

To increase the information available to investors:

Reduce adverse selection and moral hazard problems Reduce insider trading

To ensure the soundness of nancial intermediaries:

Restrictions on entry Disclosure Restrictions on Assets and Activities Deposit Insurance Limits on Competition Restrictions on Interest Rates

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4 What Is Money? (M. 3)

Meaning of Money

Money (money supply) anything that is generally accepted in payment

for goods or services or in the repayment of debts; a stock concept

Wealth the total collection of pieces of property that serve to store

value

Income ow of earnings per unit of time

Functions of Money

Medium of Exchange promotes economic eciency by minimizing thetime spent in exchanging goods and services

Must be easily standardized Must be widely accepted Must be divisible Must be easy to carry Must not deteriorate quickly

Unit of Account used to measure value in the economy Store of Value used to save purchasing power; most liquid of all assets

but loses value during ination

Evolution of the Payments System

Commodity Money

Money made up of precious metals or another valuable commodity

Fiat Money

Currency decreed by government as legal tender (meaning that legallyit must be accepted as payment for debts) but not convertible into

coins or precious metal

Checks

Electronic Payment

E-Money

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How Reliable are the Money Data?

Revisions are issued because:

Small depository institutions report infrequently Adjustments must be made for seasonal variation

We probably should not pay much attention to short-run movements in

the money supply numbers, but should be concerned only with longer-

run movements

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Part II

Financial Markets

5 Understanding Interest Rates (M. 4)

Present Value

A dollar paid to you one year from now is less valuable than a dollar

paid to you today

Discounting the Future

Let i = 0:1.

In one year $100(1 + 0:1) = $110. In two years $110(1 + 0:1) = $121or 100 (1 + 0:1)2

In n years, the present value of$100 is equal to $100(1 + i)n. Likewise,the future value of $100 in n years is equal to $100

(1+i)n(< $100) today.

Simple Present Value

PV = todays (present) value

CF = future cash ow (payment) (in n years)

i = interest rate

P V =CF

(1 + i)n

In 1626, Manhattan was sold by the Indians to the Dutch at $24 dollars

Example 1 If we assume that interest rate is 10% and has not been changed

over time, then $24 is worth (in 2008):

$24 (1:10)20041626 = $24 (1:10)382 ' $155; 674; 318; 134; 231; 000!!

Four Types of Credit Market Instruments

Simple Loan

Fixed Payment Loan

Coupon Bond

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Discount Bond

Yield to Maturity

The interest rate that equates the present value of cash ow payments

received from a debt instrument with its value today

Simple Loan Yield to Maturity

PV = amount borrowed = $100

CF = cash ow in one year = $110

n = number of years = 1

$100 =$110

(1 + i)1) (1 + i) = 110

100= 1:1 ) i = 0:1 = 10%

For simple loans, the simple interest rate equals the yield to maturity

Fixed Payment Loan Yield to Maturity

The same cash ow every period throughout the life of the loan

LV = loan value

FP = xed yearly payment (assuming FP is paid from the next year)

n= years to maturity

LV =F P

1 + i+

F P

(1 + i)2+

F P

(1 + i)3+ + F P

(1 + i)n

Coupon Bond Yield to Maturity

Using the same strategy used for the xed-payment loan P = price of coupon bond C = yearly coupon payment (assuming C is paid from the next year) F = face value of the bond n = years to maturity n= number of years until maturity

P =C

1 + i

+C

(1 + i)

2 +C

(1 + i)

3 +

+

C+ F

(1 + i)

n

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When the coupon bond is priced at its face value, the yield to maturity

equals the coupon rate

The price of a coupon bond and the yield to maturity are negatively

related

The yield to maturity is greater than the coupon rate when the bond

price is below its face value

Consol or Perpetuity

A bond with no maturity date that does not repay principal but pays

xed coupon payments forever

Remark 2 (Math Review) Let

Sn = a + ar1 + ar2 + ar3 + + arn1| {z }

total number of summation = n

(1)

rSn = 0 + ar1 + ar2 + ar3 + + arn1 + arn: (2)

Subtract (2) from (1) ((2) (1))to get

(1 r) Sn = a (1 rn) ) Sn = a (1 rn)

(1 r) (3)

If jrj < 1, then limn!1 rn = 0, and we have

limn!1

Sn =a

(1 r)

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The price of console is calculated as

Pc =C

(1 + ic)+

C

(1 + ic)2 + +

C

(1 + ic)1

=

az }| {C

1 + ic0BB@1 11 + ic| {z }

r

1CCA

=C

1+ic1+ic11+ic

=C

ic

where Pc is the price of console, C is the yearly interest payment, ic is

the yield to maturity.

Discount Bond - Yield to Maturity

For any one year discount bond

P =F

(1 + i)1! (1 + i) = F

P! i = F P

P

where F is the face value of the discount bond, P is the current price of

the discount bond

The yield to maturity equals the increase in price over the year divided

by the initial price. As with a coupon bond, the yield to the maturity is

negatively related to the current bond price

Yield on a Discount Basis

Less accurate but less dicult to claculate

idb =

F

P

P 360

days to maturityidb = yield on a discount basis

F= face value of the Treasury bill (discount bond)

P = purchase price of the discount bond

Uses the percentage gain on the face value

Puts the yield on a annual basis using 360 instead of 365 days

Always understates the yields to maturity (relative to compounding

method)

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The understatement becomes more severe the longer the maturity

Following the Financial News: Bond Prices and Interest Rates

Colons in bid-and-asked quotes represent 32nds; 101:01 means 101 1/32

Net changes in quotes in hundredths, quoted on terms of a rate of dis-

count

Rate of Return

The payment to the owner plus the change in value expressed as a fraction

of the purchase price

Example 3 (One period case) Let

Pt =C

(1 + RR)+

Pt+1(1 + RR)

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Multiply both sides by (1+RR)Pt

to get

(1 + RR) =C

Pt+

Pt+1Pt

! RR = CPt

+Pt+1 Pt

PtRR = return from holding bond from t to t + 1

Pt (Pt+1) = price of bond at time t (t + 1)

C= coupon paymentC

Pt= current yield (= ic)

Pt+1 PtPt

= rate of capital gain

Rate of Return and Interest Rates (yield to maturity)

The return equals the yield to maturity only if the holding period equals

the time to maturity

A rise in interest rates is associated with a fall in bond prices, resulting

in a capital loss if time to maturity is longer than the holding period

The more distant a bonds maturity, the greater the size of the percentage

price change associated with an interest-rate change

The more distant a bonds maturity, the lower the rate of return the

occurs as a result of an increase in the interest rate

Even if a bond has a substantial initial interest rate, its return can be

negative if interest rates rise

Interest-Rate Risk

Prices and returns for long-term bonds are more volatile than those for

shorter-term bonds

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There is no interest-rate risk for any bond whose time to maturity matches

the holding period

Real and Nominal Interest Rates Nominal interest rate makes no allowance for ination

Real interest rate is adjusted for changes in price level so it more accu-

rately reects the cost of borrowing

Ex ante real interest rate is adjusted for expected changes in the price

level

Ex post real interest rate is adjusted for actual changes in the price level

Fisher Equation

When the real interest rate is low, there are greater incentives to borrow

and fewer incentives to lend

The real interest rate is a better indicator of the incentives to borrow

and lend

i = r + e

i = nominal interest rate

r = real interest rate

e = expected ination rate

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Fisher-Eect

The tendency for nominal interest rates to be high when ination is high

and low when ination is low

6 The Behavior of Interest Rates (M. 5)

Determining the Quantity Demanded of an Asset

Wealth the total resources owned by the individual, including all assets

Expected Return the return expected over the next period on one asset

relative to alternative assets

Risk the degree of uncertainty associated with the return on one assetrelative to alternative assets

Liquidity the ease and speed with which an asset can be turned into

cash relative to alternative assets

Theory of Asset Demand

Holding all other factors constant (ceteris paribus):

The quantity demanded of an asset is positively related to wealth The quantity demanded of an asset is positively related to its ex-

pected return relative to alternative assets

The quantity demanded of an asset is negatively related to the riskof its returns relative to alternative assets

The quantity demanded of an asset is positively related to its liquid-ity relative to alternative assets

Supply and Demand for Bonds At lower prices (higher interest rates), ceteris paribus, the quantity de-

manded of bonds is higher an inverse relationship

At lower prices (higher interest rates), ceteris paribus, the quantity sup-

plied of bonds is lower a positive relationship

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Market Equilibrium

Occurs when the amount that people are willing to buy (demand) equals

the amount that people are willing to sell (supply) at a given price

Shifts in the Demand for Bonds

Wealth in an expansion with growing wealth, the demand curve forbonds shifts to the right

Expected Returns higher expected interest rates in the future lower

the expected return for long-term bonds, shifting the demand curve to

the left

Expected Ination an increase in the expected rate of inations lowers

the expected return for bonds, causing the demand curve to shift to the

left

Risk an increase in the riskiness of bonds causes the demand curve to

shift to the left

Liquidity increased liquidity of bonds results in the demand curve shift-

ing right

Shifts in the Supply of Bonds

Expected protability of investment opportunities in an expansion, the

supply curve shifts to the right

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Expected ination an increase in expected ination shifts the supply

curve for bonds to the right

Government budget increased budget decits shift the supply curve to

the right

The Fisher Eect: the tendency for nominal interest rates to be high whenination is high and low when ination is low

When expected ination rises, the expected return on bonds relative to

real assets falls

As a result, the demand for bonds falls The real cost of borrowing declines

The supply curve shifts to the right

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Changes in the Interest Rate Due to a Business Cycle Expansion

Depending on whether the supply curve shifts more than the demand

curve, or vice versa, the new equilibrium interest rate can either rise orfall

The Liquidity Preference Framework

Keynesian model that determines the equilibrium interest rate in terms

of the supply and the demand for money

There are two main categories of assets that people use to store their

wealth: money and bonds

Total wealth of the economy

Bs + Ms = Bd + Md

!Bs

Bd = Md

Ms

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If the market for money is in equilibrium

Ms = Md

, then the bond

market is also in equilibrium

Bs = Bd

Shifts in the Demand for Money

Income Eect a higher level of income causes the demand for money at

each interest rate to increase and the demand curve to shift to the right

Price-Level Eect a rise in the price level causes the demand for moneyat each interest rate to increase and the demand curve to shift to the

right

Shifts in the Supply of Money

Assume that the supply of money is controlled by the central bank

An increase in the money supply engineered by the Federal Reserve will

shift the supply curve for money to the right

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Everything Else Remaining Equal?

Liquidity preference framework leads to the conclusion that an increasein the money supply will lower interest rates the liquidity eect

Income eect of an increase in the money supply nds interest rates rising

Because increasing the money supply is an expansionary inuenceon the economy, it should raise national income and wealth

Then interest rates will rise due to a shift upward in money demand Price-Level eect predicts an increase in the money supply leads to a rise

in interest rates in response to the rise in the price level

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A rise in price level force people to have more money causing themoney demand curve to shift upward. It will cause the interest rate

to rise

Expected-Ination eect shows an increase in interest rates because an

increase in the money supply may lead people to expect a higher price

level in the future

An increase in the money supply may lead people to expect a higherprice level in the futureand hence the expected ination rate will

be higher

Then this increase in ination will lead to a higher level of interestrates

M

DM

0

SM

0i

1i

1

SM

(A) Liquidity

Effect

i

M

0

DM

0

SM

0i

1i

1

SM

(B) Income Effect,

Price-level Effect

i

1

DM

(1)

(2)

Price-Level Eect and Expected-Ination Eect

A one time increase in the money supply will cause prices to rise to a

permanently higher level by the end of the year. The interest rate will

rise via the increased prices

Price-level eect remains even after prices have stopped rising.

A rising price level will raise interest rates because people will expect

ination to be higher over the course of the year. When the price level

stops rising, expectations of ination will return to zero

Expected-ination eect persists only as long as the price level continues

to rise

Does a Higher Rate of Growth of the Money Supply Lower Interest rates?

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Liquidity eect indicates that a higher rate of money growth will cause

a decline in interest rates

In contrast, the income, price-level, and expected-ination eects indi-

cate that interest rates will rise when money growth is higher

Which of these eects are largest, and how quickly do the take eects?

Generally, the liquidity eect from the greater money growth takeseect immediately because the rising money supply leads to an im-

mediate decline in the equilibrium interest rate

The income and price-level eects take time to work The expected-ination eect can be slow or fast, depending on whether

people adjust their expectations of ination slowly or quickly whenthe money growth rate is increased

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7 The Risk and Term Structure of Interest Rates (M. 6)

Risk Structure of Interest Rates

Default risk occurs when the issuer of the bond is unable or unwilling

to make interest payments or pay o the face value

U.S. T-bonds are considered default free Risk premium the spread between the interest rates on bonds with

default risk and the interest rates on T-bonds

Liquidity the ease with which an asset can be converted into cash

Income tax considerations

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Term Structure of Interest Rates

Bonds with identical risk, liquidity, and tax characteristics may have

dierent interest rates because the time remaining to maturity is dierent

Yield curve a plot of the yield on bonds with diering terms to maturity

but the same risk, liquidity and tax considerations

Upward-sloping: long-term rates are above short-term rates

Flat" short- and long-term rates are the same Inverted: long-term rates are below short-term rates

Facts that Theory of the Term Structure of Interest Rates Must Explain

Interest rates on bonds of dierent maturities move together over time

When short-term interest rates are low, yield curves are more likely to

have an upward slope; when short-term rates are high, yield curves are

more likely to slope downward and be inverted

Yield curves almost always slope upward

Three Theories to Explain the Three Facts

Expectations theory explains the rst two facts but not the third

Segmented markets theory explains fact three but not the rst two

Liquidity premium theory combines the two theories to explain all three

facts

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Expectations Theory

The interest rate on a long-term bond will equal an average of the short-

term interest rates that people expect to occur over the life of the long-term bond

Buyers of bonds do not prefer bonds of one maturity over another; they

will not hold any quantity of a bond if its expected return is less than

that of another bond with a dierent maturity

Bonds like these are said to be perfect substitutes

Expectations Theory Example

Let the current rate on one-year bond be 6%

You expect the interest rate on a one-year bond to be 8% next year

Then the expected return for buying two one-year bonds averages (6%

+ 8%)/2 = 7%

The interest rate on a two-year bond must be 7% for you to be willing

to purchase it

Expectations Theory In General

Let it is todays interest rate on a one-period bond, i2t is todays interest

on the two-period bond, iet+1 is interest rate on a one-period bond for

next period

Expected return over the two periods from investing $1 in the two-period

bond and holding it for the two periods is

(1 + i2t) (1 + i2t) 1 = 1 + 2i2t + (i2t)2 1 = 2i2t + (i2t)2

Since (i2t)2 is small, the expected return for holding the two-period bonds

for two periods is 2i2t

If two one-period bonds are bought with $1 investment, the expected

return is

(1 + it)

1 + iet+1 1 = 1 + it + iet+1 + (it) iet+1 1

= it + iet+1 + (it)

iet+1

Since (it)

iet+1

is small, simplifying we get it + i

et+1

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Both bonds will be held only if the expected returns are equal

2i2t = it + iet+1 ! i2t =

it + iet+1

2

The two-period rate must equal the average of the two one-period rates

For bonds with longer maturities

int =it + i

et+1 + i

et+2 + + iet+(n1)

n

The n-period interest rate equal the average of the one-period interest

expected to occur over the n-period life of the bond

Expectations Theory

Explains why the term structure of interest rates changes at dierent

times

Explains why interest rates on bonds with dierent maturities move to-

gether over time (fact 1)

Explains why yield curves tend to slope up when short-term rates are

low and slope down when short-term rates are high (fact 2)

Cannot explain why yield curves usually slope upward (fact 3)

Segmented Markets Theory

Bonds of dierent maturities are not substitutes at all

The interest rate for each bond with a dierent maturity is determined

by the demand for and supply of that bond

Investors have preferences for bonds of one maturity over another

If investors have short desired holding periods and generally prefer bonds

with shorter maturities that have less interest-rate risk, then this explains

why yield curves usually slope upward (fact 3)

Liquidity Premium & Preferred Habitat Theories

The interest rate on a long-term bond will equal an average of short-term

interest rates expected to occur over the life of the long-term bond plus

a liquidity premium that responds to supply and demand conditions for

that bond

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Bonds of dierent maturities are substitutes but not perfect substitutes

Liquidity Premium Theory

int =it + iet+1 + iet+2 + + iet+(n1)

n+ lnt

where lnt is the liquidity premium for the n-period bond at time t. lnt is

always positive and rise with term to maturity

Preferred Habitat Theory

Investors have a preference for bonds of one maturity over another

They will be willing to buy bonds of dierent maturities only if they earna somewhat higher expected return

Investors are likely to prefer short-term bonds over longer-term bonds

Liquidity Premium and Preferred Habitat Theories, Explanation of the Facts

Interest rates on dierent maturity bonds move together over time; ex-plained by the rst term in the equation

Yield curves tend to slope upward when short-term rates are low and to

be inverted when short-term rates are high; explained by the liquidity

premium term in the rst case and by a low expected average in the

second case

Yield curves typically slope upward; explained by a larger liquidity pre-

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mium as the term to maturity lengthens

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Proof. Write the genearalized stock valuation equation as

P0 =D1

(1 + ke)1 +

D2

(1 + ke)2 + +

Dn1(1 + ke)

n1

=D0 (1 + g)

1

(1 + ke)1 +

D0 (1 + g)2

(1 + ke)2 + +

D0 (1 + g)n1

(1 + ke)n1

= D0

"1 + g

1 + ke

+

1 + g

1 + ke

2+ +

1 + g

1 + ke

n1#(1)

Mitiply both sides of (1) by

1+g1+ke

to get

1 + g1 + keP0 = D0 "0 + 1 + g1 + ke2

+ 1 + g1 + ke3

+ + 1 + g1 + ken1

+ 1 + g1 + ken

#(2)

Subtract (2) from (1) to get

P0

1 + g

1 + ke

P0 = D0

1 + g

1 + ke

1 + g

1 + ke

n

P0

ke g1 + ke

= D0

1 + g

1 + ke

1 + g

1 + ke

n

If n !1, 1+g1+ken ! 0 (because ke > g), and we haveP0

ke g1 + ke

= D0

1 + g

1 + ke

! P0 = D0 (1 + g)

ke g =D1

ke g

How the Market Sets Prices

The price is set by the buyer willing to pay the highest price

The market price will be set by the buyer who can take best advantage

of the asset

Superior information about an asset can increase its value by reducing

its risk

Adaptive vs. Rational Expectation

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Rational expectation implies

P= P +

E(P) = Pe = P + E() = P:

On the other hand, adaptive expectation implies

Pt =1Xi=1

iPti + t; 0 < < 1

E(Pt) = Pet =

1Xi=1

iPti:

Theory of Rational Expectations

Expectations will be identical to optimal forecasts using all available

information

Even though a rational expectation equals the optimal forecast using

all available information, a prediction based on it may not always be

perfectly accurate

It takes too much eort to make the expectation the best guess

possible

Best guess will not be accurate because predictor is unaware of somerelevant information

Formal Statement of the Theory of Rational Expectations

Xe = Xof

where Xe is the expectation of the variable that is being forecast, Xof is the

optimal forecast using all available information

Implications

If there is a change in the way a variable moves, the way in which ex-

pectations of the variable are formed will change as well

The forecast errors of expectations will, on average, be zero and cannot

be predicted ahead of time

Ecient Markets Application of Rational Expectations

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Recall that the rate of return from hodling a security equals the sum of

the capital gain on security, plus any cash payment divided by the initial

purchase price of the security

R =Pt+1 Pt + C

Pt

where Pt (Pt+1) is the price of the security at time t (t + 1), the beginning

(end) of the holding period, C is the cas payment (coupon or dividend)

made during the holding period

At the beginning of the holding period, we know Pt and C. Pt+1 is

unknown and we must form an expectation of it. The expected return

then isRe =

Pet+1 Pt + CPt

Expectations of future prices are equal to optimal forecasts using all

currently available information so

Pet+1 = Poft+1 ) Re = Rof

Supply and demand analysis states Re will be equal the equilibrium

return R soRof = R

Ecient Markets

Current prices in a nancial market will be set so that the optimal fore-

cast of a securitys return using all available information equals the se-

curitys equilibrium return

In an ecient market, a securitys price fully reects all available infor-

mation

Rationale

Rof> R ) Pt ") Rof #; Rof < R ) Pt #) Rof "until

Rof= R

In an ecient market, all unexploited prot opportunities will be eliminated

Evidence in Favor of Market Eciency

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Having performed well in the past does not indicate that an investment

advisor or a mutual fund will perform well in the future

If information is already publicly available, a positive announcement does

not, on average, cause stock prices to rise

Stock prices follow a random walk

Future changes in stock prices should, for all practical purposes, beunpredictatble. Formally,

pt =pt1 + ut or pt pt1 = pt = utut i:i:d N(0; 1)

The change in stock price is randomly determined.

Technical analysis cannot successfully predict changes in stock prices

Evidence Against Market Eciency

Small-rm eect

Small rms earned abnormally high returns over long periods oftime, even when the greater risk for these rms have been taken into

account January Eect

Abnormal price rise from December to January that is predictableand hence inconsistent with random-walk behavior

Market Overreaction

Excessive Volatility

Mean Reversion

New information is not always immediately incorporated into stock prices

Application Investing in the Stock Market

Recommendations from investment advisors cannot help us outperform

the market

A hot tip is probably information already contained in the price of the

stock

Stock prices respond to announcements only when the information is new

and unexpected

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A buy and hold strategy is the most sensible strategy for the small

investor

Behavioral Finance The lack of short selling (causing over-priced stocks) may be explained

by loss aversion

The large trading volume may be explained by investor overcondence

Stock market bubbles may be explained by overcondence and social

contagion

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9 Capital Asset Pricing and Arbitrage Pricing Theory (BKM

7.)

Capital Asset Pricing Model (CAPM) Equilibrium model that underlies all modern nancial theory

Derived using principles of diversication with simplied assumptions

Markowitz, Sharpe, Lintner and Mossin are researchers credited with its

development

Assumptions

Individual investors are price takers

Single-period investment horizon

Investments are limited to traded nancial assets

No taxes, and transaction costs (frictionless market)

Information is costless and available to all investors

Investors are rational mean-variance optimizers

Homogeneous expectations

Resulting Equilibrium Conditions

All investors will hold the same portfolio for risky assets market port-

folio

Market portfolio contains all securities and the proportion of each secu-

rity is its market value as a percentage of total market value

Risk premium on the market depends on the average risk aversion of all

market participants

Risk premium on an individual security is a function of its covariance

with the market

CAPMs Expected Return-Beta Relationship

E(ri) rf = i [E(rM) rf]

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where

E(ri) : expected return on stock i

rf : return from a risk-free asset

E(rM) : expected return on market portfolio

i : sensitivity of stock i on market risk premium

The Ecient Frontier and the Capital Market Line

Ecient Frontier: Graph representing a set of portfolios that maximizes

expected return at each level of portfolio risk.

Capital Allocation Line (CAL): Plot of risk-return combinations avail-able by varying portfolio allocation between a risk-free asset and a risky

portfolio.

Capital Market Line (CML): The capital allocation line using the market

index portfolio as the risky asset.

Aggressive

Portfolio

Market

Portfolio

Conservative

Portfolio

M

( )rE

Efficient

Frontier

rf

( )rME

( )

Capital Market

Line CML

Expected Return and Risk on Individual Securities

If all investors use identical mean-variance analysis (assumption 5), apply it to the sameuniverse of securities (assumption 3), with an identical time horizon (assumption 2), use the samesecurity analysis (assumption 6), and experience identical tax consequences (assumption 4), theyall must arrive at the same determination of the optimal risky portfolio. That is, they all deriveidentical ecient frontiers and nd the same tangency portfolio for the capital allocation line(CAL) from T-bills (the risk-free rate, with zero standard deviation) to that frontier. Because

each investor uses the market portfolio for the optimal risky portfolio, the CAL in this case iscalled the capital market line, or CML

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The risk premium on individual securities is a function of the individual

securitys contribution to the risk of the market portfolio

Individual securitys risk premium is a function of the covariance of re-

turns with the assets that make up the market portfolio

The Security Market Line and Positive Alpha Stock

Security Market Line (SML): Graphical representation of the expected

returnbeta relationship of the CAPM

Alpha (): The abnormal rate of return on a security in excess of what

would be predicted by an equilibrium model such as the CAPM.

E(r) rf= [E(rM) rf]E(r) = rf + [E(rM) rf]| {z }

Slope

Stock 4

Stock 3

Stock 2

MarketPortfolio

Stock 1

f

100

Security MarketLine (SML)

Stock 1

rf

( )rME

Stock 2

Stock 3

Stock 4

Capital MarketLine (CML)

( )rE

= 0 rf

= 12

= 1

MarketPortfolio

( )rE

( )rME

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SML Relationships

=cov (ri; rM)

MSlope SM L = E(rM) rf

= market risk premium

SM L = rf + [E(rM rf)]

An example:

Suppose the return on the market is expected to be 14%, a stock has

a beta of 1.2, and the T-bill rate is 6%. The SML would predict an

expected return on the stock of

E(r) = rf + [E(rM) rf] = 0:06 + [0:14 0:06] 1:2 = 0:156 (15:6%)

If one believes the stock will provide instead a return of 17%, its implied

alpha would be 1.4%.

Estimating the Index Model

The CAPM has two limitations: It relies on the theoretical market port-

folio, which includes all assets, and it deals with expected as opposed to

actual returns. To implement the CAPM, we cast it in the form of an

index model and use realized, not expected, returns

Using historical data on T-bills, S&P 500 and individual securities Regress risk premiums for individual stocks against the risk premi-

ums for the S&P 500

Slope is the beta for the individual stock

ri rf| {z }excess return on i

= i + i (rM rf)| {z }excess return on marekt portpolio

+ et

where where ri is the holding-period return (HPR) on asset i, and

i and i are the intercept and slope of the line that relates asset

is realized excess return to the realized excess return of the index.

The ei measures rm-specic eects during the holding period; it is

the deviation of security is realized HPR from the regression line,

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that is, the deviation from the forecast that accounts for the indexs

HPR.

Security Characteristic Line (SCL) A plot of a securitys expected excess return over the risk-free rate as a

function of the excess return on the market.

Multifactor Models Limitations for CAPM

Market Portfolio is not directly observable

Research shows that other factors aect returns

Fama French Research

Returns are related to factors other than market returns

Size

Book value relative to market value

Three factor model better describes returns

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Regression Statistics for the Single-index and FF Three-factor Model

Arbitrage

Arises if an investor can construct a zero beta investment portfolio with

a return greater than the risk-free rate, or

Arises when an investor can construct a zero-investment portfoliothat will yield a sure prot

If two portfolios are mispriced, the investor could buy the low-priced

portfolio and sell the high-priced portfolio

In ecient markets, protable arbitrage opportunities will quickly dis-appear

Arbitrage Pricing Theory (Stephen Ross (1976))

A theory of risk-return relationships derived from no-arbitrage consider-

ations in large capital markets

By showing that mispriced portfolios would give rise to arbitrage op-

portunities, the APT arrives at an expected returnbeta relationship for

portfolios identical to that of the CAPM

Security Characteristic Lines

Figure below illustrates the dierence between a single security with a

beta of 1.0 and a well-diversied portfolio with the same beta. For the

portfolio (Panel A), all the returns plot exactly on the security character-

istic line. There is no dispersion around the line, as in Panel B, because

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the eects of rm-specic events are eliminated by diversication.

Mathematical Illustration of APT

In its simple form, just like the CAPM, the APT posits a single-factor se-

curity market. Thus, the excess rate of return on each security, Ri (= ri rf),can be represented by

ri rf = i + i [rM rf] + e

A well-diversied portfolio has zero rm-specic risk, we can write its

returns as

rp rf = p + p [rM rf]

The only value for alpha that rules out arbitrage opportunities is zero,

i.e.,

rp rf = p [rM rf]

Hence, we arrive at the same expected returnbeta relationship as the

CAPM without any assumption about either investor preferences or ac-

cess to the all-inclusive (and elusive) market portfolio.

APT and CAPM Compared

APT applies to well diversied portfolios and not necessarily to individ-

ual stocks

With APT it is possible for some individual stocks to be mispriced - not

lie on the SML

APT is more general in that it gets to an expected return and beta

relationship without the assumption of the market portfolio

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APT can be extended to multifactor models

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maintenance margin is US$1,500/contract (US$3,000 in total)

Daily Cumulative Margin

Futures Gain Gain Account MarginPrice (Loss) (Loss) Balance Call

Day (US$) (US$) (US$) (US$) (US$)

400.00 4,000

5-Jun 397.00 (600) (600) 3,400 0. . . . . .. . . . . .. . . . . .

13-Jun 393.30 (420) (1,340) 2,660 1,340. . . . . .. . . . .. . . . . .

19-Jun 387.00 (1,140) (2,600) 2,740 1,260. . . . . .. . . . . .. . . . . .

26-Jun 392.30 260 (1,540) 5,060 0

+

= 4,000

3,000

+

= 4,000

E(ST)),

the situation is known as contango

Nowadays, it is also called contango when Ft;T > St Oil market typically shows a contango

Questions

When a new trade is completed what are the possible eects on the open

interest?

Can the volume of trading in a day be greater than the open interest?

Regulation of Futures

Regulation is designed to protect the public interest

Regulators try to prevent questionable trading practices by either indi-

viduals on the oor of the exchange or outside groups

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Accounting & Tax

It is logical to recognize hedging prots (losses) at the same time as the

losses (prots) on the item being hedged It is logical to recognize prots and losses from speculation on a mark to

market basis

Roughly speaking, this is what the accounting and tax treatment of

futures in the U.S.and many other countries attempts to achieve

Forward Contracts

A forward contract is an OTC agreement to buy or sell an asset at a

certain time in the future for a certain price

There is no daily settlement (unless a collateralization agreement requires

it). At the end of the life of the contract one party buys the asset for

the agreed price from the other party

Prot from a Long Forward or Futures Position

Profit

Price of Underlying

at Maturity

Prot from a Short Forward or Futures PositionProfit

Price of Underlying

at Maturity

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Forward Contracts vs Futures Contracts

Foreign Exchange Quotes

Futures exchange rates are quoted as the number of USD per unit of theforeign currency

Forward exchange rates are quoted in the same way as spot exchange

rates. This means that GBP, EUR, AUD, and NZD are USD per unit of

foreign currency. Other currencies (e.g., CAD and JPY) are quoted as

units of the foreign currency per USD.

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11 Hedging Strategies Using Futures (H. 3)

Long & Short Hedges

A long futures hedge is appropriate when you know you will purchase an

asset in the future and want to lock in the price

A short futures hedge is appropriate when you know you will sell an asset

in the future & want to lock in the price

Arguments in Favor of Hedging

Companies should focus on the main business they are in and take steps

to minimize risks arising from interest rates, exchange rates, and other

market variables

Arguments against Hedging

Shareholders are usually well diversied and can make their own hedging

decisions

It may increase risk to hedge when competitors do not

Explaining a situation where there is a loss on the hedge and a gain on

the underlying can be dicult

Convergence of Futures to Spot (Hedge initiated at time t1 and closed out attime t2)

Basis Risk

Basis is the dierence between spot & futures

Basis risk arises because of the uncertainty about the basis when the

hedge is closed out

Long Hedge

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In the future, you must buy some products at the market price

Suppose that F1 : Initial Futures Price, F2 : Final Futures Price, S2 :

Final Asset Price

You hedge the future purchase of an asset by entering into a long futures

contract

Cost of Asset = S2 (F2 F1) = F1 + Basis

An example: It is January 15. A copper fabricator knows it will require

100,000 pounds of copper on May 15 to meet a certain contract. The

spot price of copper is 340 cents per pound and the May futures price

is 320 cents per pound. The fabricator can hedge with the following

transactions:

January 15: Take a long position in four May futures on copper (onecontract contains 25,000 pounds of copper)

May 15: Close out the position Suppose that the price of copper on May 15 proves to be 325 cents

per pound. Because May is the delivery month for the futures con-

tract, this should be very close to the futures price. The fabricator

therefore gains approximately

100; 000 ($3:25 3:20) = $5; 000

on the futures contracts. It pays 100; 000$3:25 = $325; 000 for thecopper, making the total cost approximately $325; 000 $5; 000 =$320; 000: (or 320 cents per pound)

Short Hedge

In the future, you must sell your product at the market price

Suppose that F1 : Initial Futures Price, F2 : Final Futures Price, S2 :

Final Asset Price

You hedge the future sale of an asset by entering into a short futures

contract

Price Realized = S2 + (F1 F2) = F1 + Basis

An example: It is May 15. An oil producer has negotiated a contract to

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sell 1 million barrels of crude oil. The price in the sales contract is the

spot price on August 15.

Quotes:

Spot price of crude oil: $60 per barrel August oil futures price: $59 per barrel

The oil producer can hedge with the following transactions: May 15: Short 1,000 August futures contracts on crude oil (1

contract = 1,000 barrel)

August 15: Close out futures position Suppose that the spot price on August 15 proves to be $55 per barrel.

The company realize $55 million for the oil under its sales contract.Because August is the delivery month for the futures contract, the

future price on August 15 should be very close to the spot price of

$55 on that date. The company therefore gains approximately

$59-$55=$4 per barrel, or $4 million in total from the shortfutures position

The total amount realized from both the futures position andthe sales contract is therefore approximately $59 per barrel, or

$59 million in total

Choice of Contract

Choose a delivery month that is as close as possible to, but later than,

the end of the life of the hedge

When there is no futures contract on the asset being hedged, choose

the contract whose futures price is most highly correlated with the asset

price. There are then 2 components to basis

Dene S2 as the price of the asset underlying the futures contractat time t2

As before, S2 is the price of the asset being hedged at time t2 By hedging, a company ensures that the price that will be paid (or

received) for the asset is

S2 + F1 F2

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To hedge the risk in a portfolio the number of contracts that should be

shorted is

P

F

where P is the value of the portfolio, is its beta, and F is the current

value of one futures (=futures price times contract size)

Reasons for Hedging an Equity Portfolio

Desire to be out of the market for a short period of time. (Hedging may

be cheaper than selling the portfolio and buying it back.)

Desire to hedge systematic risk (Appropriate when you feel that you have

picked stocks that will outpeform the market.)

Example

Futures price of S&P 500 is 1,000, Size of portfolio is $5 million, Beta of

portfolio is 1.5, One contract is on $250 times the index

What position in futures contracts on the S&P 500 is necessary to hedge

the portfolio?

N = SF

F=$250 1; 000 = 250; 000N = 1:5 5; 000; 000

250; 000= 30 (short)

Changing Beta

What position is necessary to reduce the beta of the portfolio to 0.75?

N = 0:75 5; 000; 000

250; 000 = 15 (short)

Therefore, contract should be reduced by 15.

What position is necessary to increase the beta of the portfolio to 2.0?

N = 2:0 5; 000; 000250; 000

= 40 (short)

Therefore, contract should be increased by 10.

Rolling The Hedge Forward

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We can use a series of futures contracts to increase the life of a hedge

Each time we switch from 1 futures contract to another we incur a type

of basis risk

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12 Determination of Forward and Futures Prices (H. 5)

Consumption vs Investment Assets

Investment assets are assets held by signicant numbers of people purely

for investment purposes (Examples: gold, silver)

Consumption assets are assets held primarily for consumption (Example:

oil)

Short Selling

Short selling involves selling securities you do not own

Your broker borrows the securities from another client and sells them inthe market in the usual way

At some stage you must buy the securities back so they can be replaced

in the account of the client

You must pay dividends and other benets the owner of the securities

receives

Notation

S0 : Spot price today

F0 : Futures or forward price today

T : Time until delivery date

r : Risk-free interest rate for maturity T

Gold: An Arbitrage Opportunity?

Suppose that: The spot price of gold is US$600 The quoted 1-year futures price of gold is US$650 The 1-year US$ interest rate is 5% per annum No income or storage costs for gold


The Futures Price of Gold

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Forward vs Futures Prices

Forward and futures prices are usually assumed to be the same. When

interest rates are uncertain they are, in theory, slightly dierent: A strong positive correlation between interest rates and the asset price

implies the futures price is slightly higher than the forward price

A strong negative correlation implies the reverse

Stock Index

Can be viewed as an investment asset paying a dividend yield

The futures price and spot price relationship is therefore

F0 = S0e(rq)T

where q is the dividend yield on the portfolio represented by the index

during life of contract

For the formula to be true it is important that the index represent an

investment asset

In other words, changes in the index must correspond to changes in the

value of a tradable portfolio

The Nikkei index viewed as a dollar number does not represent an in-

vestment asset

Index Arbitrage

When F0 > S0e(rq)T an arbitrageur buys the stocks underlying the index

and sells futures

When F0 < S0e(rq)T an arbitrageur buys futures and shorts or sells the

stocks underlying the index

Index arbitrage involves simultaneous trades in futures and many dier-

ent stocks

Very often a computer is used to generate the trades

Occasionally (e.g., on Black Monday) simultaneous trades are not possi-

ble and the theoretical no-arbitrage relationship between F0 and S0 does

not hold

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Futures and Forwards on Currencies

A foreign currency is analogous to a security providing a dividend yield

The continuous dividend yield is the foreign risk-free interest rate

It follows that if rf is the foreign risk-free interest rate

F0 = S0e(rrf)T

Why the Relation Must Be True

1000 units of

foreign currency

at time zero

units of foreign

currency at time T

Trfe1000

dollars at time T

TrfeF01000

1000S0 dollars

at time zero

dollars at time T

rTeS01000

1000 units of

foreign currency

at time zero

units of foreign

currency at time T

Trfe1000

dollars at time T

TrfeF01000

1000S0 dollars

at time zero

dollars at time T

rTeS01000

Futures on Consumption Assets

F0 S0e(r+u)T

where u is the storage cost per unit time as a percent of the asset value.

Alternatively,

F0 (S0 + U)erT

where U is the present value of the storage costs.

The Cost of Carry

The cost of carry, c, is the storage cost plus the interest costs less the

income earned

For an investment asset F0 = S0ecT

For a consumption asset F0 S0ecT

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The convenience yield on the consumption asset, y, is dened so that

F0 = S0e(cy)T

Futures Prices & Expected Future Spot Prices

Suppose k is the expected return required by investors on an asset

We can invest F0erT now to get ST back at maturity of the futures

contract

This shows that

F0 = E(ST)e(rk)T

If the asset has

no systematic risk, then k = r and F0 is an unbiased estimate of ST positive systematic risk, then k > r and F0 < E(ST) negative systematic risk, then k < r and F0 > E(ST)

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13 Swaps (H. 7)

Nature of Swaps

A swap is an agreement to exchange cash ows at specied future times

according to certain specied rules

An Example of a Plain Vanilla Interest Rate Swap

An agreement by Microsoft to receive 6-month LIBOR & pay a xed rate

of 5% per annum every 6 months for 3 years on a notional principal of

$100 million, and in return Intel agrees to pay Microsoft the six month

LIBOR rate on the same principal

Cash Flows to Microsoft

---------Millions of Dollars---------

LIBOR FLOATING FIXED Net

Date Rate Cash Flow Cash Flow Cash Flow

Mar.5, 2007 4.2%

Sept. 5, 2007 4.8% +2.10 2.50 0.40

Mar.5, 2008 5.3% +2.40 2.50 0.10

Sept. 5, 2008 5.5% +2.65 2.50 +0.15

Mar.5, 2009 5.6% +2.75 2.50 +0.25

Sept. 5, 2009 5.9% +2.80 2.50 +0.30

Mar.5, 2010 6.4% +2.95 2.50 +0.45

Typical Uses of an Interest Rate Swap

Converting a liability from

xed rate to oating rate oating rate to xed rate

Converting an investment from

xed rate to oating rate oating rate to xed rate

Intel and Microsoft (MS) Transform a Liability

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When Financial Institution is Involved

Intel F.I. MS

LIBOR LIBOR

4.7%

5.015%4.985%

LIBOR-0.2%

Quotes By a Swap Market Maker

The average of the bid and oer xed rate is known as the swap rate

The Comparative Advantage Argument

AAACorp wants to borrow oating

BBBCorp wants to borrow xed

Fixed Floating

AAACorp 4.00% 6-month LIBOR + 0.30%

BBBCorp 5.20% 6-month LIBOR + 1.00%

BBB pays 1.2% more than AAA in xed-rate markets and only 0.7%

more than AAA in oating-rate markets

BBB appears to have a comparative advantage in oating-rate mar-kets, whereas AAA appears to have a comparative advantage in

xed-rate markets

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This is because the lender can enter into a swap where income from the

LIBOR loans is exchanged for the 5-year swap rate

Valuation of an Interest Rate Swap Interest rate swaps can be valued as the dierence between the value of

a xed-rate bond and the value of a oating-rate bond

Alternatively, they can be valued as a portfolio of forward rate agree-

ments (FRAs)

Valuation in Terms of Bonds

The xed rate bond is valued in the usual way

The oating rate bond is valued by noting that it is worth par immedi-

ately after the next payment date

Example

Receive 8% per annum and pay oating semiannually on a principalof $100 million.

1.25 years to go and next oating payment is $5.1 million

The LIBOR rates with continuous compounding for 3-month, 9-month, and 15-month maturities are 10%, 10.5%, and 11%

The 6-month LIBOR rate at the last payment was 10.2% (with semi-annual compounding)

Time Fixed Floating Disc PV fixed PV f loating

Bond Bond Factor Bond Bond

0.25 4 105.1 0.9753 3.901 102.5045

0.75 4 0.9243 3.697

1.25 104 0.8715 90.64

98.238 102.505

Swap value (long position in a xed-rate bond and a short positionin a oating-rate bond)

Vswap = Bfix Boat= 98:238 102:505 = 4:267

Valuation in Terms of FRAs

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Credit Risk

A swap is worth zero to a company initially

At a future time its value is liable to be either positive or negative

The company has credit risk exposure only when its value is positive

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14 Credit Derivatives (H. 21)

Credit Default Swaps (CDS)

Buyer of the instrument acquires protection from the seller against adefault by a particular company or country (the reference entity)

Example: Buyer pays a premium of 90 bps per year for $100 millionof 5-year protection against company X

Premium is known as the credit default spread. It is paid for life of

contract or until default

If there is a default, the buyer has the right to sell bonds with a face value

of $100 million issued by company X for $100 million (Several bonds maybe deliverable)

CDS Structure

Default

Protection

Buyer, A

Default

Protection

Seller, B

90 bps per year

Payoffif there is a default byreference entity=100(1-

R)

Recovery rate, R, is the ratio of the value of the bond issued by reference

entity immediately after default to the face value of the bond

Attractions of the CDS Market

Allows credit risks to be traded in the same way as market risks

Can be used to transfer credit risks to a third party

Can be used to diversify credit risks

CDS Spreads and Bond Yields

Portfolio consisting of a 5-year par yield corporate bond that provides a

yield of 6% and a long position in a 5-year CDS costing 100 basis points

per year is (approximately) a long position in a riskless instrument paying

5% per year

This shows that CDS spreads should be approximately the same as bond

yield spreads

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Valuation

Suppose that conditional on no earlier default a reference entity has a

(risk-neutral) probability of default of 2% in each of the next 5 years Assume that the risk-free (LIBOR) rate is 5% per annum with continuous

compounding

Assume payments are made annually in arrears, that defaults always

happen half way through a year, and that the expected recovery rate is

40%

Suppose that the breakeven CDS rate is s per dollar of notional principal

Unconditional Default and Survival Probabilities

Calculation of PV of Payments (Principal=$1)

Time (yrs) Survival

Prob

Expected

Paymt

Discount

Factor

PV of Exp

Pmt

1 0.9800 0.9800s 0.9512 0.9322s

2 0.9604 0.9604s 0.9048 0.8690s

3 0.9412 0.9412s 0.8607 0.8101s

4 0.9224 0.9224s 0.8187 0.7552s

5 0.9039 0.9039s 0.7788 0.7040s

Total 4.0704s

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Payos from Options: What is the Option Position in Each Case?

Let K = Strike price, ST = Price of asset at maturity

Payoff Payoff

ST STK

K

Payoff Payoff

ST

ST

K

K

Assets Underlying Exchange-Traded Options

Stocks

Foreign Currency

Stock Indices

Futures

Specication of Exchange-Traded Options

Expiration date

Strike price

European or American

Call or Put (option class)

Terminology

Moneyness :

At-the-money option In-the-money option Out-of-the-money option

Option class

Option series

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Intrinsic value

Time value

Dividends & Stock Splits Suppose you own N options with a strike price of K :

No adjustments are made to the option terms for cash dividends When there is an n-for-m (from m to n) stock split,

the strike price is reduced to mK=n the number. of options is increased to nN=m

Stock dividends are handled in a manner similar to stock splits

Consider a call option to buy 100 shares for $20/share

How should terms be adjusted:

for a 2-for-1 stock split? for a 5% stock dividend?

Market Makers

Most exchanges use market makers to facilitate options trading A market maker quotes both bid and ask prices when requested

The market maker does not know whether the individual requesting the

quotes wants to buy or sell

Margins

Margins are required when options are sold

Warrants

Warrants are options that are issued (or written) by a corporation or a

nancial institution

The number of warrants outstanding is determined by the size of the

original issue & changes only when they are exercised or when they expire

Warrants are traded in the same way as stocks

The issuer settles up with the holder when a warrant is exercised

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Problem 5 An investor buys for $3 a call with a strike price of $30 and

sells for $1 call with a strike price of $35. What is the prot from this

transaction?

K1

K2

Profit

ST

payo from a bull spread created using calls

Stock price Payo from Payo from Total

range long call option short call option payo

ST K1 0 0 0K1 < ST < K2 ST K1 0 ST K1ST

K2

ST

K1

K2

ST

K2

K1

Bull Spread Using Puts

Buy a put option with a low price (K1) and sell a put with a high strike

price (K2)

Both options have the same expiration date

K1

K2

Profit

ST

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Bear Spread Using Puts

K1

K2

Profit

ST

Bear Spread Using Calls

K1

K2

Profit

ST

Box Spread

A combination of a bull call spread and a bear put spread

If all options are European, a box spread is worth the present value of

the dierence between the strike prices

If they are American this is not necessarily so

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Payo from a box spread


range bull call spread bear put spread payoST K1 0 K2 K1 K2 K1K1 < ST < K2 ST K1 K2 ST K2 K1ST K2 K2 K1 0 K2 K1

The value of a box spread is always (K2 K1) erT (if and only if it isan European).

Buttery Spread

Involves positions in the same types of options (call or put) with three

dierent strike prices

Appropriate strategy for an investor who feels that large stock price

moves are unlikely

Buttery Spread Using Calls

Buy a call with strike price of K1, buy a call with K3, and sell 2 call

options with K2

K1 K3

Profit

STK2

Buttery Spread Using Puts

Buy a put with strike price of K1, buy a put with K3, and sell 2 put

options with K2

Calendar Spread

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The options have the same strike price and dierent expiration date

Sell a option with short-maturity and buy a option with long-maturity

Understanding the prot diagram of callendar spread (call option case) If the stock price is very low when the short-maturity option expires

The short-maturity option is worthless and the value of long-maturityoption is close to zero

Investor therefore incurs a loss that is close to the cost of settingup the spread initially

If the stock price; ST, is very high when the short-maturity option expires

The short-maturity option cost investor ST K, and the long-marurity option is worth a little more than ST K Again, the investor makes a net loss that is close to the cost of

setting up the spread initially

If ST is close to K

the short-maturity option costs the investor either a small amountor nothing at all

However, the long-maturity option is still quite valuable In this case, a signicant net prot is made

Prot diagram is drawn on the assumption the long-maturity option is

sold when the short-maturity option expires

Calendar Spread Using Calls

Prot diagrams show the prot when the short-maturity option expires

on the same day the long-maturity option is sold

TS

K

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Diagonal Spread

Similar to bull and bear spread except that diagonal spread the expiration

date of the option is dierent

Combination

Taking a position in both calls and puts on the same stock

A Straddle

Buying a call and put with the same strike price and expiration date

If the stock price close to this strike price, the straddle leads to aloss

Profit

ST

K


range call put payo

ST K 0 K ST K STK1 > K ST K 0 ST K

Strip & Strap

Strip: buying 1 call and 2 puts with the same strike price and expiration

date

Makes more money when the stock price falls signicantly Strap: buying 2 calls and 1 put with the same strike price and expiration

date

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The value of the portfolio in 3 months when the stock price becomes $22

is

22 0:25 1 = 4:50

The value of the portfolio in 3 months when the stock price becomes $18

is

18 0:25 = 4:50

The value of the portfolio in 2 months becomes 4.50 whether the stock

price rises to $22 or falls to $18

The value of the portfolio today is

4:5e0:120:25 = 4:3670

Valuing the Option

The portfolio that is

long 0.25 shares

short 1call option

is worth 4.367

The value of the shares is 5:000(= 0:25 20) The value of the option is therefore 0:633(= 5:000 4:367)

Generalization

A derivative lasts for time T and is dependent on a stock

Su

u

Sd

d

S

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Consider the portfolio that is long shares and short 1 derivative

The portfolio is riskless when Su fu = Sd fd or

=fu fd

Su Sd

Value of the portfolio at time T is

Su fu

Value of the portfolio today is

(Su fu)erT

Another expression for the portfolio value today is S f

Hence

f = S (Su fu)erT

Substituting for we obtain

f = [pfu + (1 p)fd]erT

where

p =erT du

d

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Proof.

f= S (Su fu)erT; = fu fdSu

Sd

f= S fu fdSu Sd (Su

fu fdSu Sd fu)e

rT

= (Sfu fd

Su Sd erT Su fu fd

Su Sd + fu)erT

(fu fdu d e

rT u fu fdu d fu)e

rT

= (erT u + (u d)

u d fu erT u

u d )erT

= (erT du

d

fu +u erT

u

d

)erT

= (pfu + (1 p) fd) ert

(note : 1 erT du d =

u d erT + du d =

u erTu d )

Risk-Neutral Valuation

f = [pfu + (1 p)fd]erT

The variables p and (1 p ) can be interpreted as the risk-neutral prob-abilities of up and down movements

The value of a derivative is its expected payo in a risk-neutral world

discounted at the risk-free rate

Irrelevance of Stocks Expected Return

When we are valuing an option in terms of the underlying stock the

expected return on the stock is irrelevant

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Valuing a Call Option

20

1.2823

22

18

24.2

3.2

19.8

0.0

16.2

0.0

2.0257

0.0

A

B

C

D

E

F

Value at node B

= e0:120:25(0:6523 3:2 + 0:3477 0) = 2:0257

Value at node A

= e0:120:25(0:6523 2:0257 + 0:3477 0) = 1:2823

A Put Option Example

K = 52; t = 1yr; r = 5%

50

4.1923

60

40

72

0

48

4

32

20

1.4147

9.4636

A

B

C

D

E

F

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What Happens When an Option is American

50

5.0894

60

40

72

0

48

4

32

20

1.4147

12.0

A

B

C

D

E

F

Delta

Delta () is the ratio of the change in the price of a stock option to the

change in the price of the underlying stock

It is the number of units of the stock we should hold for each option

shortened in order to create riskless hedge

The value of varies from node to node

Deta is calculated as:

=fu fd

Su SdStock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?

In the previous example, wec can calculate the value of delta of the call

as:

=1

0

22 18 = 0:25:This because when the stock price changes from $18 to $22, the option

price changes from $0 to $1

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E(ST) = S0erT

V ar(ST) = S20e

2T

e2T 1

Remark 6 The log normal distribution has the probability density function

f(x; ; ) =1

xp

2e

(ln(x))2

22

for x > 0 where and are the mean and standard deviation of the vari-

ables natural loga

Documents

Lecture Notes(Financial Economics)