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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
Is there an arbitrage opportunity?
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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
Is there an arbitrage opportunity?
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
Stock price Payo from Payo from Total
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
Stock price Payo from Payo from Total
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