slides367 investment mangemnt must print

8/2/2019 Slides367 Investment Mangemnt Must Print

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Econ-367: Logistics

Professor: Jonathan Wright

Office Hour: Friday 10-12

Email: [email protected]

TAs Email Office Hour

Xiaoyu Fu [email protected] Fridays 9-10

Jiedi Mu [email protected] Mondays 4-5
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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Econ-367: Logistics

Requirements

About 6 Homeworks 20%

1 Midterm 30%

Final 50%

Exams

Midterm November 18 in class

Final December 12, 9-12.


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Econ-367: Logistics

Slides for projection

http://www.econ.jhu.edu/courses/367/index.html

The book

Investments by Bodie, Kane and Marcus
http://www.econ.jhu.edu/courses/367/index.htmlhttp://www.econ.jhu.edu/courses/367/index.html


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Minor in Financial Economics

Four required courses:

Elements of Macroeconomics

Elements of Microeconomics

Corporate Finance

Investments and Portfolio Management

Two electives courses

Plus the prerequisites for these courses

Check CFE website for detailed rules on the minor


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Major Classes of Financial Assets or

Securities

Money market

Bond market

Equity Securities Indexes

Derivative markets


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The Money Market

Treasury bills

Certificates of Deposit

Interbank Loans Eurodollars Federal Funds

Commercial Paper

Repurchase Agreements (RPs)


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Bank Discount Rate (T-Bills)

rBD

= bank discount rate

P = market price of the T-bill

n = number of days to maturity

10,000 360*10,000

BD

Pr

n

Example: 90-day Tbill, P=9,875

10,000 9,875 360* 5%

10,000 90BD

r


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The Spread between 3-month CD and

Treasury Bill Rates


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Repo Agreement

Selling an asset with an explicit agreement torepurchase the asset after a set period of time

1. Bank A sells a treasury security to Bank B at P0

2. Bank A agrees to buy the treasury back at a higher price Pf> P03. Bank B earns a rate of return implied by the difference in prices

iRA =Pf P0

P0

360

daysx


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Major Components of the Money Market


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The Bond Market

Treasury Notes and Bonds Inflation-Protected Treasury Bonds

Federal Agency Debt

International Bonds Municipal Bonds

Corporate Bonds

Mortgages and Mortgage-Backed Securities


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Treasury Notes and Bonds

Maturities

Notesmaturities up to 10 years

Bondsmaturities in excess of 10 years

30-year bond Par Value - $1,000

Quotespercentage of par


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Figure 2.4 Lisiting of Treasury Issues


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Government Sponsored Enterprises Debt

Major issuers

Federal Home Loan Bank

Federal National Mortgage Association

Government National Mortgage Association

Federal Home Loan Mortgage Corporation


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Municipal Bonds

Issued by state and local governments

Types

General obligation bondsRevenue bonds

Industrial revenue bonds

Maturitiesrange up to 30 years


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Municipal Bond Yields

Interest income on municipal bonds is not subject tofederal and sometimes not to state and local tax

To compare yields on taxable securities a TaxableEquivalent Yield is constructed


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Equivalent Taxable Yields Corresponding to

Various Tax-Exempt Yields


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Ratio of Yields on Muni to Corporate Bonds


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Corporate Bonds

Issued by private firms

Semi-annual interest payments

Subject to larger default risk than government securities Options in corporate bonds

Callable

Convertible


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Developed in the 1970s to help liquidity of financialinstitutions

Proportional ownership of a pool or a specifiedobligation secured by a pool

Mortgages and Mortgage-Backed Securities


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Mortgage-Backed Securities outstanding


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Major Components of the Bond Market


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Equity Securities

Common stock

Residual claim

Limited liability


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Stocks Traded on the NYSE


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There are several broadly based indexes computed andpublished daily

Stock Market Indexes


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Dow Jones Industrial Average

Includes 30 large blue-chip corporations Computed since 1896

Price-weighted average


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Broadly based index of 500 firms

Market-value-weighted index

Index funds

Exchange Traded Funds (ETFs)

Standard & Poors Indexes


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Other U.S. Market-Value Indexes

NASDAQ Composite

NYSE Composite

Wilshire 5000


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Investing in an index

Investor may buy stocks

Or an index tracking mutual fund

Or an Exchange-traded fund (ETF) designed to

match the index Spiders (S&P Depository Receipt) tracks S&P 500

Cube track Nasdaq 100

ETFs consist of a share in a basket of securitiescorresponding to the relevant index

There is a redemption procedure


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Derivatives Markets

Options Basic Positions

Call (Buy)

Put (Sell)

Terms

Exercise Price

Expiration Date

Assets

Futures Basic Positions

Long (Buy)

Short (Sell)

Terms

Delivery Date

Assets


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Trading Data on GE Options


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Listing of Selected Futures Contracts


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Real and Nominal Rates of Interest

Nominal interest rateGrowth rate of your money

Real interest rate

Growth rate of your purchasing power IfR is the nominal rate and rthe real rate and i is the

inflation rate:

r R i

Determination of the Equilibrium Real Rate


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Determination of the Equilibrium Real Rate

of Interest

Q ti C ti


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Quoting Conventions

APR

Suppose that interest of 1 percent is paid everymonth

The APR with monthly compounding will be 12

percent

APR = Annual Percentage Rate(periods in year) X (rate for period)

Q ti g C ti


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Quoting Conventions

EAR

Suppose again that interest of 1 percent is paidevery month

The EAR will be how much interest is earned on

$1 after 1 year.

EAR = Effective Annual Rate( 1+ rate for period)Periods per yr - 1

In this case EAR = (1.01)12 - 1 = 12.68%

After n years $1 turns becomes $(1 )n

EAR

Q ti g C ti


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Quoting Conventions

Relationship between APR and EAR

For annual compounding they are the same thing

For compounding n times a year

1/

(1 ) 1

*{[1 ] 1}

n

n

APREAR

n

APR n EAR

Q ti g C ti


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Quoting Conventions

Continuous Compounding

Suppose that I compound every second

Let r be the APR with continuous compounding

After 1 year, the value of $1 is

After n years, the value of $1 is

lim (1 ) 1 exp( ) 1nn

rEAR r

n

$exp( ) $exp( )nr rn

$exp( )r

Q ti g C ti


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Quoting Conventions

Continuous Compounding

Suppose that an asset is worth V(0) today andV(n) in years

Inverting this yields

( ) (0) *exp( )V n V rn

log( ( )) log( (0))V n V

rn


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Quoting Conventions: Example

Say we want to get an EAR of 5.8%.

What APR do we need at different compoundingfrequencies?

Frequency Periods per year Rate

Annual 1 5.80

Quarterly 4 5.68Monthly 12 5.65

Daily 365 5.64

1/

*{[1 ] 1}n

APR n EAR


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Start with $1 and after 1 year have $1.058

APR with continuous compounding is

Continuous Compounding: Example

log(1.058) log(1)0.0564

1


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A stock is worth $10 in year 1, $20 in year 2 and $10 inyear 3.

Q1. What is the return from year 1 to year 2?

ln(20)-ln(10)=0.69 Q2. What is the return from year 2 to year 3? ln(10)-ln(20)=-0.69

Q3. What is the return from year 1 to year 3?

{ln(10)-ln(10)}/2=0

Continuous compounded returns have the useful featurethat they add up

Continuous compounding and adding up


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Suppose that we have a stream of cash flows and let r be the

constant EAR

Present value:

NPV function in Excel

Present value

Time (years) Amount

1 C(1)

2 C(2)

:

T C(T)

2

(1) (2) ( )...

1 (1 ) (1 )TC C C T

PVr r r


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Let the EAR be 5% and consider the following cash flow:

Present value:

Present value example

Time (years) Amount

1 $75

2 $50

3 $100

2 3

75 50 100203.16

1.05 1.05 1.05PV


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Consider the following two projects

What are their present values at 1% and 10% interest rates?

Another Present value example

t=0 t=1 t=2 t=3

Project A -200 50 50 120

Project B 100 50 50 -220

1% 10%

PV of A 15 -23PV of B -15 21


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Suppose that we have a stream of cash flows for which an

investor is willing to pay P today. The value of r that solves theequation

is called the internal rate of return.

Internal rate of return

2

(1) (2) ( )...1 (1 ) (1 )T

C C C T

P r r r

History of T-bill Rates Inflation and Real Rates


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History of T-bill Rates, Inflation and Real Rates

for Generations, 1926-2005

1926-2005 1966-2005 1981-2005

T Bills 3.75 5.98 5.73

CPI 3.13 4.70 3.36

Real Rate 0.72 1.25 2.28


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Interest Rates and Inflation, 1926-2005


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Risk and Risk Premiums

P

DPPHPR

0

101

HPR= Holding Period Return

P0 = Beginning price

P1 = Ending price

D1 = Dividend during period one

Rates of Return: Single Period


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Ending Price = 48

Beginning Price = 40

Dividend = 2

HPR = (48 - 40 + 2 )/ (40) = 25%

Rates of Return: Single Period Example

Annual Holding Period Risk Premiums


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Annual Holding Period Risk Premiums

and Real Returns

Real Returns Risk Premium

Large Stock 9.82 9.2

Small Stock 15.59 14.97

LT Government 2.36 1.74

T Bills 0.62 --


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Expected returns

p(s) = probability of a state

r(s) = return if a state occurs

s = state

Expected Return and Standard Deviation

( ) ( ) ( )s

E r p s r s


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State Prob. of State r in State1 .1 -.05

2 .2 .05

3 .4 .15

4 .2 .255 .1 .35

E(r) = (.1)(-.05) + (.2)(.05) + (.1)(.35)E(r) = .15

Scenario Returns: Example


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Standard deviation = [variance]1/2

Variance:

Variance or Dispersion of Returns

Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2+ .1(.35-.15)2]

Var= .012

S.D.= [ .012] 1/2 = .1095

Using Our Example:

22( ) ( ) ( )

s

p s r s E r


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Covariance and correlation

Let r(1) and r(2) be two random variables

The covariance between them is

Need the joint probabilities of r(1) and r(2) to work this

out

([ (1) ( (1))][ (2) ( (2))])E r E r r E r


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Covariance Example

Let r(1) and r(2) be two returns with the followingjoint distribution

r(1)=0 r(1)=0.1 r(1)=0.3

r(2)=0 0.05 0.3 0.15r(2)=0.2 0.35 0.1 0.05


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Covariance Example

First need the means

E(r(1))=(0.4*0)+(0.4*0.1)+(0.2*0.3)=0.1

E(r(2))=(0.5*0)+(0.5*0.2)=0.1

r(1)=0 r(1)=0.1 r(1)=0.3

r(2)=0 0.05 0.3 0.15 0.5

r(2)=0.2 0.35 0.1 0.05 0.5

0.4 0.4 0.2

Covariance Example


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Covariance Example

Possible outcomes for[r(1)-E(r(1))]*[r(2)-E(r(2))]

Weight each of them by their probability

Add up

Covariance is -0.005


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Covariance and correlation

Covariance is measured in units that are hard tointerpret

Correlation avoids this problem. The correlation

between r(1) and r(2) is

It must be between -1 and +1

( (1), (2))

( (1)) ( (2))

Cov r r

Var r Var r

Th N l Di ib i


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The Normal Distribution

A Normal and Skewed Distributions


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A Normal and Skewed Distributions

(mean = 6% SD = 17%)


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Properties of stock returns

Standard deviation of S&P is about 0.17

Negative skewness Fat tails

Th R d t V l tilit (Sh ) R ti


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The Reward-to-Volatility (Sharpe) Ratio

Sharpe Ratio for Portfolios =Risk PremiumSD of Excess Return

Risk Premium: Average Excess Return

For stocks, Sharpe Ratio is about 0.4

Decision Making Under Uncertainty:


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g y

Utility and Risk Aversion

A utility function measures happiness as a function ofconsumption/wealth

Investors care about expected utility, not expected

wealth per se.

Generally think that utility function is concave

Risk Aversion

A C Utilit F ti


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A Concave Utility Function

A Risk Averse Person will always refuse a


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yfair gamble

Ri k i t d i k i


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Risk averse investors and risk premia

Have diminishing marginal utility of wealth

Reject investment portfolios that are fair games or worse

Consider only risk-free or speculative prospects with positive

risk premiums

Expected return has to be bigger than risk-free rate

This is the risk premium

C t i t E i l t V l


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Certainty Equivalent Value

The certainty equivalent value is the sum of money forwhich an individual would be indifferent between

receiving that sum and taking the gamble.

The certainty equivalent value of a gamble is less than theexpected value of a gamble for someone who is risk

averse.

Certainty Equivalence Example


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y q p

Expected Value 40

Certainty Equivalent 30

Risk Premium 10

Ri k N t lit


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Risk Neutrality

Risk Seeking


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Risk Seeking

Expected Utility


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Expected Utility

Basic premise of standard decision making underuncertainty is that agents maximize expected utility.

Say the utility function is W

Wealth is 1 or 9 each w.p.

Expected utility is 0.5*1+0.5*3=2

Coefficient of Risk Aversion


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Coefficient of Risk Aversion

Coefficient of Risk Aversion is

Consider the utility function

This has a coefficient of risk aversion of A

Constant Absolute Risk Aversion (CARA)

''( )

'( )

u W

u W

exp( )

( )

AWu W

A


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What is the certainty equivalent of

- $100,000 wp 0.5

- $50,000 wp 0.5


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What is the certainty equivalent of

- $100,000 wp 0.5

- $50,000 wp 0.5

Risk Aversion Certainty Equivalent

1 70,711

2 66,667

5 58,56610 53,991

30 51,209

Utility Function


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Utility Function

Where

U= utilityE ( r ) = expected return on the asset or portfolio

A = coefficient of risk aversion

s2

= variance of returns

21( ) 2U E r A

s

Three possible portfolios


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Three possible portfolios

L M HE(r) 0.07 0.09 0.13

(r) 0.05 0.10 0.20

Utility of three possible portfolios


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Utility of three possible portfolios

L M HE(r) 0.07 0.09 0.13

(r) 0.05 0.10 0.20

A=2 0.0675 0.08 0.09

A=5 0.6375 0.065 0.03

A=8 0.06 0.05 -0.03


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The Trade-off Between Risk and Returns ofa Potential Investment Portfolio, P

The Indifference Curve


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The Indifference Curve

Indifference Curves with A 2 and A 4


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Indifference Curves with A = 2 and A = 4


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Basic Results from Statistics

Suppose X1 and X2 are random variables

IfX1, X2, Xnare random variables

1 1 2 2 1 1 2 2

1 1 2 2

2 21 1 2 2 1 2 1 2

( ) ( ) ( )

( )

( ) ( ) 2 ( , )

E k X k X k E X k E X

Var k X k X

k Var X k Var X k k Cov X X

1 1

( ) ( )

( ) ( , )

j j j j

n n

j j i j i j i j

E k X k E X

Var k X k k Cov X X


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Matrix algebra version

Let X be a vector and be its mean and be thevariance-covariance matrix. Then

( ' ) '

( ' ) '

E k X k

Var k x k k

Mean, Variance and Covariance of


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Suppose there are assets with returns r(1), r(2), ..r(n)

Suppose I form a portfolio with weights w(1), w(2),...w(n)

The properties of the portfolio are

Portfolios

1

1

1 1

( ) ( ) ( )

( ( )) ( ( ))

( ( )) ( ) ( ) ( ( ), ( ))

ni

ni i

n ni j

r p w i r i

E r p w E r i

Var r p w i w j Cov r i r j


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Suppose that 1 portfolio has weights v(1), v(2)...v(n) and

another has weights w(1), w(2),...w(n)

Covariance of two Portfolios

1 2 1 1( , ) ( ) ( ) ( ( ), ( ))n n

P P i jCov r r v i w j Cov r i r j

Mean, Variance and Covariance ofP f li i h M i Al b


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Matrix algebra makes the formulas simpler

Let the returns be in an nx1 vector r

Let the weights be an nx1 vector w

Let be the variance-covariance matrix of r

Then the properties of the portfolios are

Portfolios with Matrix Algebra

( ) '

( ( )) ' ( )

( ( )) '

r p w r

E r p w E r

Var r p w w

Portfolios of One Risky Asset and a Risk-F A


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Its possible to split investment funds between safe andrisky assets.

Risk free asset: proxy; T-bills

Risky asset: stock (or a portfolio)

Free Asset

Properties of combination of risk-free andi k


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rc = combined portfolio putting weight y in

the risky asset and 1-y on the risk-free asset

E (rc)=yE(rp)+(1-y) rf

sc =|y|sp

risky asset

E l


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rf= 7%,sf=0

Erp = 15%, sp=22%

y=0.75E(rc)=0.75*0.15+0.25*0.07=0.13

sc=0.75*0.22=0.165

y=0

E(rc)=0.07

sc=0

Example

E l


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rf= 7%,sf=0

Erp = 15%, sp=22%

y=1.5E(rc)=1.5*0.15-0.5*0.07=0.19

sc=1.5*0.22=0.33

Example

C it l All ti Li


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Capital Allocation Line

Capital Allocation Line with DifferentialB i g d L di g R t


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Borrowing and Lending Rates

Optimal Portfolio Using Indifference Curves


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Optimal Portfolio Using Indifference Curves

Utility as a Function of Allocation to theRi k A t


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Risky Asset, y

Risk Tolerance and Asset Allocation


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Risk Tolerance and Asset Allocation

The investor must choose one optimal portfolio,y*,

from the set of feasible choices2 22

*

2

max ( ) max ( ) (1 )2 2

( )

pcc p f

p f

p

AyAE r yE r y r

E r ry

A

ss

s

Example


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rf= 7%,sf=0

Erp = 15%, sp=22%

A=4 y*

=(0.15-0.07)/(4*0.222

)=0.41

Example


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An Optimal Portfolio of Risky Assets

Before we considered choice between one riskyasset and a risk-free asset

But there are in fact many risky assets.

Suppose we are now picking an optimal portfolio ofrisky assets (no risk-free asset)


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Two assets (debt and equity) in a portfolio

rP=yrD+(1-y)rE

2 2

( ) ( ) (1 ) ( )

( ) ( ) (1 ) ( ) 2 (1 ) ( , )

p D E

p D E D E

E r yE r y E r

Var r y Var r y Var r y y Cov r r


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D,E = Correlation coefficient of

returns

Cov(rD,rE) = DE D E

D = Standard deviation of

returns for Security D

E = Standard deviation ofreturns for Security E

Covariance


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Range of values for correlation coefficient

+ 1.0 > > -1.0

If = 1.0, the securities would be perfectly

positively correlated

If = - 1.0, the securities would beperfectly negatively correlated

Correlation Coefficients: Possible Values


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Two assets (debt and equity) in a portfolio

Suppose that

Now let . Then

2 2 2 2 2

2

(1 ) 2 (1 )

{ (1 ) }

P D E D E

D E

y y y y

y yE

E D

y 2 0P

Correlation Effects


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The relationship depends on the correlation coefficient -1.0 < < +1.0

The smaller the correlation, the greater the risk

reduction potential

If = +1.0, no risk reduction is possible

Example


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p

T-Bill yield:5 percent

Debt Equity

Expected Return 0.08 0.13

St Dev 0.12 0.20

Correlation 0.3

Covariance 0.0072

Portfolio Expected Return as a Function ofInvestment Proportions


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Investment Proportions

Portfolio Standard Deviation as a Functionof Investment Proportions


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of Investment Proportions

Portfolio Expected Return as a Function ofStandard Deviation


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Standard Deviation

Formula for Minimum Variance Portfolio


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2

2 2

( , )

2 ( , )

1

where and are the excess returns over

the riskfree rate

E D E

DE D D E

E D

D E

Cov R R

w Cov R R

w w

R R

The Opportunity Set of the Debt and


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E

D

ExpectedReturn (%)

Standard Deviation (%)

Equity Funds

The Opportunity Set of the Debt

d E it F d


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and Equity Funds

But we dont just have the two risky assets.

Theres also a risk-free asset

Combination of the risk-free asset and any point

on the opportunity set of the debt and equity

funds is the capital allocation line

The Opportunity Set of the Debt and Equity

F d d T F ibl CAL


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Funds and Two Feasible CALs

The Opportunity Set of the Debt and EquityFunds with the Optimal CAL and the


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Funds with the Optimal CAL and the

Optimal Risky Portfolio

The Sharpe Ratio


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Tangency portfolio maximizes the slope of the CAL forany possible portfolio,p

The objective function is the slope:

( )P f

P

P

E r r

S

( )p f

p

p

E r rS

Formula for Optimal Risky Portfolio


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2

2 2

( ) ( ) ( , )

( ) ( ) [ ( ) ( )] ( , )

1

where and are the excess returns over

the riskfree rate

D E E D E

DD E E D D E D E

E D

D E

E R E R Cov R Rw

E R E R E R E R Cov R R

w w

R R

Example


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T-Bill yield: 5 percent

Optimal Portfolio

0.03*0.04 0.08*0.0072 0.40.03*0.04 0.08*0.0144 (0.03 0.08)*0.0072

0.6

D

E

w

w

Debt Equity

Expected Return 0.08 0.13St Dev 0.12 0.20

Correlation 0.3

Covariance 0.0072

Example (continued)


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2 2 2

( ) (0.4*0.08) (0.6*0.13) 11%

( ) (0.4 *0.144) (0.6 * 0.4) (2 *0.4* 0.6*0.72) 0.020164

( ) 14.2%

CAL of optimal portfolio has slope of

11 50.42

14.2

E r

r

r

Determination of the Optimal OverallPortfolio


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Portfolio

Optimal Complete Portfolio


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Optimal Complete Portfolio

Consider investor with utility function

Position in portfolio will be

For example, if A=4,

2( )

2

AU E r

2

( )p f

p

E r ry

A

2

0.11 0.05

0.74394*0.142y

Optimal Complete Portfolio in Example


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Optimal Complete Portfolio in Example

Three-Security Portfolio


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2p = w1

21

2

+ w22

12

+ 2w1w2 Cov(r1,r2)

+ w32 32

Cov(r1,r3)+ 2w1w3

Cov(r2,r3)+ 2w2w3

1 1 2 2 3 3( ) ( ) ( ) ( )pE r w E r w E r w E r

Markowitz Portfolio Selection Model


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Security SelectionFirst step is to determine the risk-return

opportunities available

All risky portfolios that lie on the minimum-variance

frontier from the global minimum-variance portfolioand upward provide the best risk-return combinations

Minimum-Variance Frontier of Risky Assets


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Minimum Variance Frontier of Risky Assets

Markowitz Portfolio Selection ModelContinued


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Continued

We now search for the CAL corresponding to thetangency portfolio

The Efficient Frontier of Risky Assets withthe Optimal CAL


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t e Opt a C

Optimal Portfolio General Formula


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p

Let w be the vector or weights,

be the vector ofexpected returns and be the variance-covariance

matrix of excess returns

1

1

1

1

Minimum Variance:'

Maximum Sharpe Ratio: '

iw

i

w

Capital Allocation and the SeparationProperty


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p y

Separation property: The property that portfolio choice

problem is separated into two independent parts:

(1) determination of the optimal risky portfolio, and

(2) the personal choice of the best mix of the risky portfolio and

risk-free asset.

Implication of the separation property: the optimal risky

portfolio P is the same for all clients of a fund manager.

The Power of Diversification


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We have

Suppose that all securities in the portfolio have thesame weight, have variance and covariance

We can then express portfolio variance as:

21 1 ( , )

n np i j i j i jw w Cov r r

2 21 1p

nCov

n n

2 Cov

Risk Reduction of Equally Weighted

Portfolios


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PortfoliosNumber of Stocks =0 =0.4

1 0.5000 0.5000

2 0.3536 0.4183

10 0.1581 0.3391

100 0.0500 0.3186

Infinite 0.0000 0.3162

Portfolio Risk as a Function of the Number of

Stocks in the Portfolio


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Stocks in the Portfolio

Market risk =2lim

n PCov

An ExampleS th t k


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Suppose there are many stocks

All have mean 15%

Standard deviation 60%

Correlation 0.5

1. What is the expected return and standard deviation of

a portfolio of 25 stocks?

2. How many stocks are needed to get the standarddeviation down to 43%

An Example


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3. What is the systematic (market) risk in this

universe?

4. If T bills are available and yield 10% what is theslope of the CAL?

Where next?


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Analysis using optimal portfolio choicerequires a lot of parameters to be estimated

Hard to use in practice

So there are some models that are better athandling many assets

Single Index Model

CAPMAPT

Excess Returns on HP and S&P 500 April2001 March 2006


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Single-Index Model


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i = index of a securities particular return tothe market

( ) ( ) ( )i i i M i

R t R t e t

The alphas and betas of stocks

Al h i f hi h k


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Alphas is a measure of which stocks are

under and overpriced

Beta is a measure of market sensitivity

Which stocks have high beta?Which stocks have low beta?

Single Index Model Decomposition


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( ) ( ) ( )i i i m iR t R t e t

is stock market mispricing

is component of return due to systematic

risk is component of return due to idiosyncratic

risk (or firm-specific risk)

i

( )i MR t

( )ie t

How to estimate betas

C h f h k


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Compute the excess returns of the stock

Compute the excess returns of the marketportfolio (proxied by S&P 500 Index)

Regress excess returns of the stock on the

excess returns of the S&P 500 index Slope coefficient is beta

Also gives alpha and error estimates

Security Characteristic Line Plots excess returns of a single asset against excess


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Plots excess returns of a single asset against excess

market returns for different time periods

Single-Index Model Continued


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Risk:

Total risk = Systematic risk+Idiosyncratic risk:2 2 2 2

( )i i M i

e2 2 2 2 ( )i i M ie

Portfolios With the Single-Index ModelF tf li ( ) ( ) ( )R t R t t


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For a portfolio

Alpha and beta of a portfolio:

Variance of a portfolio:

When ngets large, becomes negligible2ep

1 1

andn n

p i i p i i

i i

x x

2 2 2 2

p p m ep

( ) ( ) ( )P P P M PR t R t e t

The Variance of an Equally WeightedPortfolio with Risk Coefficient

pin the


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p

Single-Factor Economy

Single-Index Model Input List


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Risk premium on the S&P 500 portfolio

Estimate of the SD of the S&P 500 portfolio

n sets of estimates of

Alpha values

Beta coefficient

Stock residual variances

Fewer inputs than with a general covariance matrix

Optimal Risky Portfolio of the Single-IndexModel


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Expected return, SD, and Sharpe ratio:

Maximize the Sharpe ratio for optimal portfolio

1 1

1 1

1

2 21 1 1

2 2 2 2 2 22

1 1

( ) ( ) ( )

( ) ( )

( )

n n

P P M P i i M i i

i i

n n

P P M P M i i i i

i i

P

P

P

E R E R w E R w

e w w e

E RS

Efficient Frontier with full covariance


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matrix and the single index model

Capital Asset Pricing Model (CAPM)


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It is the equilibrium model that underlies all modernfinancial theory

Markowitz, Sharpe, Lintner and Mossin are

researchers credited with its development

Assumptions


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Investors are price takers Single-period investment horizon

Same expectations

Perfect marketsNo transactions costs or taxes

Short selling and borrowing/lending at risk-free

rate are allowed

Investors have preferences on the mean and

variance of returns

Resulting Equilibrium Conditions


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Expected return is a function of

Beta

Market return

Risk-free return

The CAPM Pricing Equation


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( ) [ ( ) ]

where

( , ) / ( )

i f i m f

i i f m f m f

E r r E r r

Cov r r r r Var r r

This is also known as the security market

line (SML), or the Expected Return-Betarelationship

CAPM Implications


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Ifi=1,E(r

i)=E(r

m)

Ifi>1,E(ri)>E(rm)

Ifi


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Suppose you have the following info:

Risk free rate: 3.5%Expected market return: 8.5%

Beta of IBM is 0.75

Q. What is the expected return on IBM stock?

A. 0.035+0.75*[0.085-0.035]=7.25%

The Security Market Line


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Expected Return-Beta Relationship


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CAPM holds for any portfolio because:

This also holds for the market portfolio:

P

( ) ( ) and

( ) [ ( ) ]

P k k

k

k k

k

P f P m f

E r w E r

w

E r r E r r

( ) ( )M f M M f

E r r E r r 1

M

Different Agents


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CAPM does not assume that all agents are the

same

They can have different risk aversion

This will affect only the mix of the market and

riskfree asset that they hold.

Market Risk Premium


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The risk premium on the market portfolio will be

proportional to its risk and the degree of risk

aversion of the average investor:

2

2( )

where is the variance of the market portolio and

is the average degree of risk aversion across investors

M f M

M

E r r A

A

CAPM Intuition: Systematic andUnsystematic Risk


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Unsystematic risk can be diversified and is

irrelevant

Systematic risk cannot be diversified and is

relevant

Measured by beta

Beta determines the level of expected return

The risk-to-reward ratio should be the same

across assets

CAPM versus the Index model


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The index model

The CAPM index model beta coefficient turns out to

be the same beta as that of the CAPM expectedreturn-beta relationship

Indeed, if you take the index model, take expectationsand let you get the CAPM

( ) ( ) ( )i i i m iR t R t e t

( ( )) ( ) [ ( ( )) ( )]i f i m f E R t R t E R t R t

( )(1 )i f i

R t

CAPM versus the Index model


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CAPM is an equilibrium model

Derived assuming investors maximize utility

Index model is not

The CAPM implies no intercept

Index model says nothing about expected market

return

The CAPM and Reality


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Testing the CAPM

Proxies must be used for the market portfolio

And for the risk-free rate

Estimates of Individual Mutual Fund Alphas,1972-1991


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Does Beta Explain the Cross-section ofReturns?


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Early tests found strong evidence for an expected

return-beta relationship

Black, Jensen and Scholes (1972)

More recent results have been less positive.

Tests of the CAPM: Fama MacBethProcedure


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Tests of the expected return beta relationship:

First Pass Regression

Estimate betas (and maybe variance of residuals) Second Pass: Using estimates from the first pass to

determine if model is supported by the data

0 1

0 1 2 ( )

i i i

i i it i

R

R Var e

( ) ( )i f i M f E r r E r r

Classic Test of the CAPM(Black, Jensen and Scholes, 1972)


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Tests of the CAPM: Fama MacBethProcedure Results


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Coefficient Estimate Standard Error

0 0.127 0.006

1 0.042 0.006

2 0.310 0.026

Single Factor Test Results


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Return %

Beta

Predicted

Actual

Measurement Error in Beta


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If beta is measured with error in the first stage,

second stage estimate of1will be biaseddownwards

Excessively flat estimated security market line

could result from measurement error in beta

Does Beta Explain Everything?


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Does it explain anything?

Fama and French (1992) results

Cross sectional regression

Book to market and size matter

Beta explains almost nothing!

0 1 2 3( / ) log( )

i i i ir B M ME

Average Return as a Function of Book-To-

Market Ratio, 19262006


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Average Annual Return for 10 Size-BasedPortfolios, 1926 2006


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Liquidity and the CAPM


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Liquidity

Illiquidity Premium

Research supports a premium for illiquidity.

Amihud and Mendelson

Acharya and Pedersen

The Relationship Between Illiquidity andAverage Returns


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The CAPM and Reality


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There are problems with the CAPM

Other factors are important in explaining returns

Firm size effect

Ratio of book value to market valueLiquidity

Still the benchmark asset pricing model

The CAPM without a risk-free asset


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Suppose there is no riskfree asset (e.g. due to

inflation uncertainty)

wherez is the return on any zero-beta portfolio

This is called the zero-beta CAPM (Black (1972))

Q. How do we find a zero-beta portfolio?

( ) ( ) ( ( ) ( ))i z i m zE R E R E R E R

Finding a zero-beta portfolio


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Suppose that there are two assets and the market portfolio

has a weight w in the first and 1-w in the second

Let v and 1-v be portfolio weights for zero beta portfolio

Same idea works with many assets in market portfolio

1 2 1 2

1 2 1 2

1 2 2

1 2 1 2 1 2

2 1 2

1

( (1 ) , (1 ) ) 0

( ) (1 )(1 ) ( ) [(1 ) (1 ) ] ( , ) 0

( ) (1 ) ( ) (1 ) ( )

( , ) ( , ) 2 ( , ) 0

( 1) ( ) ( , )( ) (1

Cov wr w r vr v r

wvVar r w v Var r w v v w Cov r r

wvVar r w Var r v w Var r

vCov r r wCov r r wvCov r r

w Var r wCov r r v

wVar r w

2 1 2) ( ) (1 2 ) ( , )Var r w Cov r r

Factor Models


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Returns on a security come from two sources

Common factor

Firm specific events

Possible common factors

Market excess return

Gross Domestic Product Growth

Interest Rates

Single Factor Model Equation


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ri= Return for security i

= Factor sensitivity or factor loading or factor beta

Estimate by regression

F= Surprise in macro-economic factor (E(F)=0)

(F could be positive, negative or zero)

ei= Firm specific events

i

( )i i i i

r E r F e

Arbitrage Pricing Theory (APT)


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Arbitrage - arises if an investor can construct a zero

investment portfolio with a sure profit

Since no investment is required, an investor can

create large positions to secure large levels of profit

In efficient markets, profitable arbitrage opportunitieswill quickly disappear

APT & Well-Diversified Portfolios


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rP = E (rP) +PF + eP

F= some factor

For a well-diversified portfolio: eP approaches zero

Returns as a Function of the SystematicFactor

Well Diversified Portfolios Individual Stocks


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Pricing


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Implication of the factor structure for a well-diversified portfolio:

If not, theres an arbitrage (show next slide)

Expected returns proportional to factor beta

Intuition: only systematic risk should command

higher expected returns

( )i f i

E r r

APT Pricing Derivation

Suppose there is are 2 well-diversified portfolios( )r E r f


179/453

Suppose I invest w in 1 and 1-w in 2

Let . Then

----or else there is an arbitrage opportunity

1 1 1

2 2 2

( )

( )

r E r f

r E r f

1 2 1 2( ) (1 ) ( ) [ (1 ) ]pr wE r w E r w w f

2 2 1/ ( )w

2 11 2

2 1 2 1

( ) ( )p fr E r E r r

2 11 2

2 1 2 1

2 11

2 22 2

2 1

11 1

2 1 2

1 2

2 1

1

1

2

2 1

( )

( ) ( )(

( )

) (

( ) )

)

(

p f

p f

r E r E r r

r

E r

E r E r r

r

E r E r

E

APT Pricing Derivation

S 22 1 2 1 2( )[ ( ) ( )] ( ) ( )fE r rE r E r E r E r


180/453

So

And

Therefore

And so

22 1 2 1 22

2 1 2 1 2

( )[ ( ) ( )] ( ) ( )( )

f

f

E r E r E r E rE r r

11 1 2 1 21

2 1 1 1 2

( )[ ( ) ( )] ( ) ( )( )

f

f

E r rE r E r E r E rE r r

1 2

1 2

( ) ( )f fE r r E r r

1 1

2 2

( )

( )f

f

E r r

E r r

APT and CAPM


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Suppose that

Then and for any well-

diversified portfolio

where

This is the CAPM, except that it applies only to well

diversified portfolios

( )m m

F r E r

( ) [ ( ) ]i f i m f E r r E r r

( ) [ ( )]i i i m m i

r E r r E r e

( , ) / ( )i i f m f f

Cov r r r r Var r

APT li ll di ifi d f li d

APT and CAPM Compared


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APT applies to well diversified portfolios and not

necessarily to individual stocks

Possible for some individual stocks to be mispriced

- not lie on the SML

APT doesnt need investors to have mean-varianceutility (which CAPM does)

APT doesnt necessarily require us to measure

market portfolio APT can have more than one factor.

Factor model


183/453

Variance of returns is

( )i i i ir E r F e

2

( ) ( )i iVar F Var e

Factor model: Example


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The risk-free rate is 4 percent and X and Y are two well-

diversified portfolios:

Q. What is the expected return on Y?

A. 10=4+ and so =6. The expected return on Y is

4+ *0.5=4+3=7

Portfolio Beta Expected Return

X 1 10

Y 1/2 ?

Factor model: An arbitrage Suppose instead that the expected return on Y were 9%


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Suppose instead that the expected return on Y were 9%

Q. What can an investor do?

1. Invest $100 in Y. Payoff: $100*[1+0.09+0.5F]= 109+50F

2. Invest -$50 in the riskfree asset:

Payoff: -50*1.04=-52 3. Invest -$50 in X.

Payoff: -50*[1+1.10+F]= -55-50F

Total payoff: $2: This is an arbitrage.

Portfolio Beta Expected Return

X 1 10

Y 1/2 9

Multifactor Models

U th f t


186/453

Use more than one factor

Examples include gross domestic product,

expected inflation, interest rates etc.

Estimate a beta or factor loading for each factor

using multiple regression.

Multifactor Model Equation

( )E F F


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ri = Return for security i

= Factor sensitivity for first factor (e.g. GDP)

= Factor sensitivity for second factor (e.g. interest rate)

ei = Firm specific events

1i

2i

1 1 2 2( )i i i i ir E r F F e

Pricing with Multiple Factors

1 1 2 2( )i f i iE r r 1 1 2 2( )i f i iE r r


188/453

= Factor sensitivity for GDP

GDP

= Risk premium for GDP

= Factor sensitivity for Interest Rate

IR = Risk premium for Interest Rate

,i GDP

1 1 2 2( )i f i i 1 1 2 2( )i f i iE r r

, ,e.g. ( )i f i GDP GDP i IR IRE r r

,i IR

There are 3 factors

APT Example


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There are 3 factors

The betas are 1.1, 0.5 and 2

The lambdas are 10%, 10% and 1%

The riskfree rate is 5%

Q: What is the expected return on stock i?

A: 0.05+(1.1*0.1)+(0.5*0.1)+(2*0.01)=23%

What Factors?

Factors that are important to performance of the


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Factors that are important to performance of the

general economy (Chen, Roll and Ross)

Fama-French Three Factor Model

Example of the Multifactor Approach

Work of Chen Roll and Ross Factors are


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Work of Chen, Roll, and Ross. Factors are Change in Industrial Production

Change in Expected Inflation

Change in Unexpected Inflation

Excess returns of corporate bonds over Treasuries

Excess returns of long-term Treasuries over T Bills

Fama French Three Factor Model

The factors chosen are variables that on past evidence


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The factors chosen are variables that on past evidence

seem to predict average returns well and may capture therisk premiums

where:

SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excessof the return on a portfolio of large stocks

HML = High Minus Low, i.e., the return of a portfolio of stocks with a high book

to-market ratio in excess of the return on a portfolio of stocks with a low book-to-

market ratio

it i iM Mt iSMB t iHML t it r R SMB HML e

it i iM Mt iSMB t iHML t it r R SMB HML e

Tests of the Multifactor Model

Chen, Roll and Ross 1986 Study


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, y

FactorsGrowth rate in industrial production

Changes in expected inflation

Unexpected inflationChanges in corporate spreads

Change in term spread

Study Structure & Results

Method: Two stage regression


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Method: Two -stage regression

Stage 1: Estimate the betas

Stage 2: Run a cross-sectional regression

Findings

Significant factors: industrial production, riskpremium on bonds and unanticipated inflation

Market index returns were not statistically

significant in the multifactor model

Chen, Roll and Ross Second Stage CoefficientEstimates


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Variable Coefficient T-stat

Value Weighted Return -2.403 -0.63

IP 11.756 3.05Change in Expected Inflation -0.123 -1.60

Unanticipated Inflation -0.795 -2.38

Change in corporate spread 8.274 2.97

Change in term spread -5.905 -1.88

Constant 10.713 2.76

Kendall (1953) found that log stock prices were

Stock Prices as a Random Walk


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Kendall (1953) found that log stock prices were

approximately a random walk

Erratic market behavior, or

A well functioning market (where prices reflect allavailable information) ?

1t t tp p

Efficient Market Hypothesis (EMH) EMH says stock prices already reflect all available


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EMH says stock prices already reflect all available

information

A forecast about favorable future performance leads to

favorable current performance, as market participants

rush to trade on new information.

Efficient Market Hypothesis (EMH) New information is unpredictable; if it could be


198/453

New information is unpredictable; if it could be

predicted, then the prediction would be part of todaysinformation.

Stock prices that change in response to new

(unpredictable) information also must move

unpredictably.

Competition ensures that prices reflect all information.

Stock price changes follow a random walk

Versions of the EMH


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Weak (Past trading data) Semi-strong (All public info)

Strong (All info)

Even if the market is efficient a role exists for

Market Efficiency & PortfolioManagement


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Even if the market is efficient a role exists for

portfolio management:

Appropriate risk level

Tax considerations

Cumulative Abnormal Returns BeforeTakeover Attempts: Target Companies


201/453

Stock Price Reaction to CNBC Reports


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Cumulative Abnormal Returns inResponse to Earnings Announcements


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Strong-Form Tests: Inside Information

The ability of insiders to trade profitability in their


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The ability of insiders to trade profitability in their

own stock has been documented in studies byJaffe, Seyhun, Givoly, and Palmon

SEC requires all insiders to register their trading

activity

Momentum

Weak or Semistrong Tests: Anomalies


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Momentum

Long-term reversals

Size and book-to-market effects

January effect

Post-Earnings Announcement Price Drift

Persistence in mutual fund performance

Relations between dividend yields & future returns

Magnitude Issue

Are Markets Efficient?


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Magnitude Issue

Difficulty in detecting a small gain in performance

Selection Bias Issue

Results of tests of market efficiency are only

reported if there is evidence against efficiency Lucky Event Issue

Some method will appear to work well ex post by

chance alone

Dividend yield and stock returns


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Dividend yields since 1900


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Some evidence of persistent positive and negative

Mutual Fund Performance


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v p p v a ga v

performance (hot hands)

Mutual fund fees

Mutual funds in ranking & post ranking quarter


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Non-traditional mutual fund with 3 key

Hedge Funds


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y

characteristicsAttempts to earn returns in rising or falling markets

Subject to lighter regulation than mutual funds

Not open to the general publicCharges higher fees (2 and 20 is typical)

Interpreting the Evidence

Risk Premiums or market inefficiencies


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Fama and French argue that these effects can beexplained are due to risk premia

Lakonishok, Shleifer, and Vishney argue thatthese effects are evidence of markets not fully

incorporating available information

Equity Premium Puzzle

The average equity premium is about 7%


213/453

g q y p

The volatility (standard deviation) is about 16%

Why?

Investors very risk averse

Irrational fear of stocksThe history in the U.S. in the last 100 years was

a fluke

Survivor bias

Cross-Country Equity Returns


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Behavioral Finance

Investors Do Not Always Process Information


215/453

y

Correctly

Investors Often Make Inconsistent or

Systematically Suboptimal Decisions

Asset Price Bubbles

Information Processing Critique

Forecasting Errors


216/453

g

Too much weight on recent experience

Overconfidence

Conservatism

Initially too slow to update beliefs in response to news

Sample Size Neglect and Representativeness

Behavioral Biases

Loss Aversion


217/453

Framing

Mental Accounting

Ambiguity Aversion

Loss aversion People are more motivated by avoiding a loss than


218/453

p y g

acquiring a similar gain. Once I own something, not having it becomes more

painful, because it is a loss.

If I dont yet own it, then acquiring it is less important,

because it is a gain.

Loss aversion Imagine that you have just been given $1,000 and have been


219/453

g y j g

asked to choose between two options. Option A: Guaranteed additional $500.

Option B: heads you get additional $1000, tails gain nothing?

Now imagine that you have been given $2000 and have to

choose between two options.

Option A: Guaranteed losing $500.

Option B: heads you lose $1000, tails lose nothing?

Loss Aversion

Coval and Shumway (2005) investigate morning and


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afternoon trades of 426 traders of CBOT Assume significantly more risk in the afternoon trading

following morning losses than gains

Prospect Theory


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Ambiguity Aversion

You are to draw a ball at random from a bag


222/453

containing 100 balls. You win a prize if the balldrawn is red. One bag contains 50/50 blue/red

balls. The distribution of balls in the other bag

is unknown. Which bag would you prefer todraw the ball from?

Limits to Arbitrage

Fundamental Risk


223/453

Things may get worse before they get better

Implementation Costs

Costly to implement strategy

Model RiskMaybe it wasnt really an arbitrage

Limits to Arbitrage

Siamese Twin Companies


224/453

Equity Carve-outs

Closed-End Funds

Pricing of Royal Dutch Relative to Shell(Deviation from Parity)


225/453

3Com and Palm

In March 2000 3Com announced that at the end of


226/453

the year it would give their shareholders 1.5 sharesin Palm for each 3Com share they owned.

Afterwards:

Price of Palm: $95.06

Price of 3 Com: $81.11

Value of Non-Palm 3 Com: -$61

Intrinsic or Fundamental Value

Equity Valuation Methods


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Book ValueFair price given required return and expected

future dividends

Trading Signal

IV > Market Price Buy

IV < Market Price Sell or Short Sell

IV = Market Price Hold or Fairly Priced

Dividend Discount Models:Expected Holding Period Return


228/453

The return on a stock investment comprises cashdividends and capital gains or losses

Assuming a one-year holding period

1 1 0

0

( ) ( )Expected HPR= ( ) E D E P PE rP

Required Return


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CAPM gave us required return:

If the stock is priced correctly Required return should equal expected return

( )f M fk r E r r

1 2 H H

Specified Holding Period


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1 2

1 20 ...(1 ) (1 ) (1 )

H H

HD D D PV k k k

PH= the expected sales price for the stock at timeH

H= the specified number of years the stock is

expected to be held

Dividend Discount Models: GeneralModel


231/453

VD

ko

t

t

t

( )11

V0 = Value of Stock

Dt

= Dividend

k = required return

No Growth Model


232/453

V Dk

o

Stocks that have earnings and dividends thatare expected to remain constant

Preferred stock (assuming divs get paid)

Do not share in the earnings or earnings growth Do not have the security of bonds

Higher yield than bonds

TARP investments in banks are in preferred stock

No Growth Model: Example


233/453

D1 = $5.00

k= .15

V0 = $5.00 /.15 = $33.33

V Dk

o

Estimating Dividend Growth Rates


234/453

g ROE b

g = growth rate in dividends

ROE= Return on Equity for the firm

b = plowback or retention percentage rate

(1- dividend payout percentage rate)

Dividend Growth for Two EarningsReinvestment Policies


235/453

Constant Growth Model2

0 0

0 02

(1 ) (1 ) 1*

1oD g D g

V D D


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g = constant perpetual growth rate

1

0 0 0

1 (1 ) 11

1 1*

gk kk

k g DD D D

k g k g k g

Constant Growth Model: Example


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1(1 )oD g DVok g k g

E1 = $5.00 b = 40% k= 15%

(1-b) = 60% D1= $3.00 g = 8%

V0 = 3.00 / (.15 - .08) = $42.86

Present Value of Growth Opportunities


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If the stock price equals its IV, and growth rate issustained, the stock should sell at:

1

0

DP

k g

Face or par value

Bond Characteristics


239/453

Coupon rate

Zero coupon bond

Compounding and payments

Accrued Interest

Different Issuers of Bonds

U.S. Treasury


240/453

Notes and Bonds Corporations

Municipalities

International Governments and Corporations

Innovative Bonds

Floaters and Inverse Floaters

Asset-Backed

Indexe-Linked BondsCatastrophe Bonds

Listing of Treasury Issues


241/453

Listing of Corporate Bonds


242/453

Secured or unsecured

Provisions of Bonds


243/453

Call provision Convertible provision

Floating rate bonds

Preferred Stock

Innovations in the Bond Market

Floaters and Inverse Floaters


244/453

Asset-Backed Bonds Income paid from a pool of underlying assets

(mortgage loans, auto loans, credit card receivables)

Catastrophe BondsA form of reinsurance

Indexed Bonds

Linked to inflation CDOs

Asset backed bond split into different tranches

T

l

Bond Pricing: Present value calculation


245/453

1 (1 )(1 )TB t

t

ParValueCPrr

PB = Price of the bondCt = interest or coupon payments

T = number of periods to maturity

r = semi-annual discount rate or the semi-annual yield to

maturity

Price: 10-yr, 8% Coupon, Face = $1,000

40 1000P


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Ct = 40 (Semi Annual)P = 1000

T = 20 periods

r = 3% (Semi Annual)

201 201.03 1.03

$1,148.77

t tP

P

Can also be calculated using the Excel PRICE function

Prices and Yields (required rates of return) have an

i l i hi

Bond Prices and Yields


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inverse relationship When yields get very high the value of the bond

will be very low

When yields approach zero, the value of the bondapproaches the sum of the cash flows

The Inverse Relationship Between BondPrices and Yields


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Bond Prices at Different Interest Rates(8% Coupon Bond)

Ti M i 4% 6% 8% 10% 12%


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Time to Maturity 4% 6% 8% 10% 12%1 year 1038.83 1029.13 1000.00 981.41 963.33

10 years 1327.03 1148.77 1000.00 875.35 770.60

20 years 1547.11 1231.15 1000.00 828.41 699.07

30 years 1695.22 1276.76 1000.00 810.71 676.77

Yield to Maturity

Interest rate that makes the present value of the bonds

l i i


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payments equal to its priceSolve the bond formula for r

1 (1 )(1 )

T

Tt

t

BParValueCP

rr

Yield to Maturity Example

1000352 0


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)1(1000

)1(

359502 0

1 rrT

t

t

10 yr Maturity Coupon Rate = 7%

Price = $950

Solve for r = semiannual rate r = 3.8635%

Yield MeasuresBond Equivalent Yield

7.72% = 3.86% x 2

Eff i A l Yi ld


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Effective Annual Yield

(1.0386)2 - 1 = 7.88%

Current Yield

Annual Interest / Market Price$70 / $950 = 7.37 %

Bond Prices: Callable and Straight Debt

Callable bonds are always worth less thanregular debt


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Prices over Time of 30-Year Maturity,6.5% Coupon Bonds


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Perpetuity (Coupons, but never redeemed)

Special Bonds

1C C


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Zero coupon bonds (no coupons)

Treasury STRIPSeffectively zero-coupon bonds

1 11( )

1(1 )

tt tt

C CP Cr rr

(1 )TParP

r

Yields and Prices to Maturities on Zero-Coupon Bonds ($1,000 Face Value)


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Maturity (years) Yield to Maturity Price

1 5% 952.38=1000/1.05

1 7.01% 934.502 6% 890.00

3 7% 816.30

4 8% 735.03

The Price of a 30-Year Zero-Coupon Bondover Time at a Yield to Maturity of 10%


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Principal and Interest Payments for TreasuryInflation Protected Security (TIPS)


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Rates on TIPS relative to nominal Treasury bonds are called

breakeven rates and contain information about inflation expectations.

Holding-Period Return: Single Period

HPR = [ I + ( P0 - P1 )] /P0

h


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whereI= interest payment

P1= price in one period

P0 = purchase price

Holding-Period Return Example

CR = 8% YTM = 8% N=10 years

Semiann al Compo nding P0 $1000


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Semiannual Compounding P0 = $1000In six months the rate falls to 7%

P1 = $1068.55

HPR = [40 + ( 1068.55 - 1000)] / 1000HPR = 10.85% (semiannual)

Note that HPR and yield are two completelydifferent things

Rating companies Moodys Investor Service

d d

Default Risk and Ratings


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Standard & Poors

Fitch

Rating Categories

Investment grade BBB or higher for S&P

BAA or higher for Moodys and Fitch

Speculative grade/Junk Bonds

Coverage ratios

Leverage ratios

Factors Used to assess financial stability


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Leverage ratios

Liquidity ratios Profitability ratios

Cash flow to debt

Debt Assets Debt+Equityor =

Equity Equity Equity

Financial Ratios by S&P Ratings Class


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Default Rates

Default Rate by S&P Bond Rating

(15 Years)

60 00%


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0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

S&P Bond Rating

DefaultRate

Default Rate 0.52% 1.31% 2.32% 6.64% 19.52% 35.76% 54.38%

AAA AA A BBB BB B CCC

Source: "The Credit-Raters: How They Work and How They Might Work Better",Business Week (8 April 2002), p. 40

Yield Spreads between long-termcorporates and Treasuries


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Components of a corporate risk spread

Expected default

Recovery rate


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Recovery rate Risk premium

Treasury yields


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Information on expected future short term rates

can be implied from the yield curve

Overview of Term Structure


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can be implied from the yield curve

The yield curve is a graph that displays the

relationship between yield and maturity

Treasury Yield Curves


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Yield Curve Under Certainty

An upward sloping yield curve is evidence that

short term rates are going to be higher next year


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short-term rates are going to be higher next year

Say we knew next years interest rate

Two 2-Year Investment Programs


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The two programs must have the same yield

2(1 ) (1 ) * (1 )y r r


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So

2 1 2

1

22 1 2

(1 ) (1 ) * (1 )

1 (1 ) * (1 )

y r r

y r r

2 1 2 1if and only ify y r r

Short Rates versus Spot Rates


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(1 ) (1 ) * (1 ) * (1 )ny r r r


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1 2

1

1 2

(1 ) (1 ) (1 ) ....(1 )

1 (1 ) * (1 ) *...(1 )

n n

nn n

y r r r

y r r r

Interest Rate Uncertainty What can we say when future interest rates are not

known today?


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Suppose that todays rate is 5% and the expected

short rate for the following year isE(r2) = 6%

Expectations Hypothesis

Treat expected future rates as though they were known

Theories of Term Structure


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Treat expected future rates as though they were known

Liquidity Preference Upward bias over expectations

Investors require extra yield to compensate them for the risk

of capital loss if interest rates go up

2

2 1 2

2

(1 ) (1 )*[1 ( )] 1.05*1.06

5.5%

y r E r

y

Expectations Hypothesis

Observed long-term rate is a function of todays

short-term rate and expected future short-term


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short term rate and expected future short termrates

1 2

11 2

(1 ) (1 )*(1 ( ))*...(1 ( ))

( ( )... ( ))

n

n n

n n

y r E r E r

y n r E r E r

Long-term bonds are more risky

Liquidity Premium Theory


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Investors will demand a premium for the risk

associated with long-term bonds

The yield curve has an upward bias built into

the long-term rates because of the risk premium

(or term premium)

Risk premium may change over time

Engineering a Synthetic Forward Loan

The one-year rate next year is not known

But the one- and two-year yields today are


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But the one and two year yields today are Can

Sell a two-year zero coupon bond for

Buy one-year zero coupon bonds for

Cash flows:

Today: Nothing

In one year receive $

In two years pay $1000

22

1000

(1 )y

1

22

1

(1 )

y

y

22

1000(1 )y

21 21000*(1 ) / (1 )y y

Engineering a Synthetic Forward Loan

In this way, I can lock in a rate to borrow at some

point in the future


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point in the future In one year, I receive and in

two years I pay $1000

So the interest rate that I am locking in is

and this is called a forward rate

21 21000*(1 ) / (1 )y y

2

2

1

(1 )1

1

yf

y

1

(1 )1

(1 )

n

nn n

yf

Forward Rates from Observed Rates


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1

1

1(1 )

n n

n

f y

fn = one-year forward rate for period n

yn = yield for a security with a maturity of n

Example: Forward Rates

4 yr = 8.00% 3yr = 7.00% fn = ?


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4 yr = 8.00% 3yr = 7.00% fn = ?fn =(1.08)

4/(1.07)3 -1

fn = .1106 or 11.06%

Downward Sloping Spot Yield CurveExample

Zero-Coupon Rates Bond Maturity


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Zero Coupon Rates Bond Maturity12% 1

11.75% 2

11.25% 310.00% 4

9.25% 5

Forward Rates for Downward SlopingY C Example

1yr Forward Rates

1 [(1 1175)2 / 1 12] 1 0 115006


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1yr [(1.1175)2 / 1.12] - 1 = 0.115006

2yrs [(1.1125)3 / (1.1175)2] - 1 = 0.102567

3yrs [(1.1)4 / (1.1125)3] - 1 = 0.063336

4yrs [(1.0925)5 / (1.1)4] - 1 = 0.063008

Forward rates and future interestrates

If the path of future interest rates is known for certain,

then forward rates are equal to future interest rates


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then forward rates are equal to future interest rates

With interest rate uncertainty,

Under the expectations hypothesis, forward rates are equalto expected future short-term interest rates

Under liquidity preference theory, forward rates are greater

than expected future short-term interest rates

Typically yield curve slopes up

Slope of yield curve related to bank profitabilityB k l d l t d fi th l h t t

Yield curve slope


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p y p y Banks lend long-term and finance themselves short-term

Why yield curves slope up?

If the yield curve is to rise as one moves to longer

maturities


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A longer maturity results in the inclusion of a

new forward rate that is higher than the average

of the previously observed rates

Reason:

Higher expectations for forward rates or

Liquidity premium

Inverse relationship between price and yield

An increase in a bonds yield to maturity results in

Bond Pricing Relationships


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y ya smaller price decline than the gain associated

with a decrease in yield

Long-term bonds tend to be more price sensitivethan short-term bonds

Change in Bond Price as a Function ofChange in Yield to Maturity


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As maturity increases, price sensitivity increases

at a decreasing rate

Bond Pricing Relationships Continued


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g Price sensitivity is inversely related to a bonds

coupon rate

Price sensitivity is inversely related to the yield tomaturity at which the bond is selling

Prices of 8% Coupon Bond (CouponsPaid Semiannually)

Yield to Maturity 1 year 10 year 20 year8% 1000 00 1000 00 1000 00


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8% 1000.00 1000.00 1000.00

9% 990.64 934.96 907.99

Percent Fall in Price 0.94% 6.50% 9.20%

Prices of Zero-Coupon Bond

Yield to Maturity 1 year 10 year 20 year8% 924 56 456 39 208 29


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8% 924.56 456.39 208.29

9% 915.73 414.64 171.93

Percent Fall in Price 0.96% 9.15% 17.46%

A measure of the effective maturity of a bond

The weighted average of the times until each

payment is received, with the weights

Duration


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p y , g

proportional to the present value of the

payment

Duration is shorter than maturity for all bondsexcept zero coupon bonds

Duration is equal to maturity for zero coupon

bonds

tCF y ice( )1 Pr

Duration: Calculation


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t tw CF y ice ( )1 Pr

twtD

T

t 1

CF Ca sh Flo w fo r p erio d t t

SpreadsheetCalculating the Duration of Two Bonds


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Suppose that there is a level change in interest rates

Duration/Price Relationship

*P y

D


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modified durationwith semiannual compounding y is yield per 6 months

*1

yD

P y

*MODP D yP

/ (1 )MODD D y

Rules for DurationRule 1 The duration of a zero-coupon bond equals

its time to maturity

Rule 2 Holding maturity constant, a bonds durationis higher when the coupon rate is lower


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g pRule 3 Holding the coupon rate constant, a bonds

duration generally increases with its time tomaturity

Rule 4 Holding other factors constant, the durationof a coupon bond is higher when the bonds yiel

slides367 investment mangemnt must print

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