ASSET PRICINGASSET PRICING

FACULTY F MATHEMATICSFACULTY F MATHEMATICS

BELGRADE BELGRADE 20102010

IntroductionIntroduction

• Investors are concerned withInvestors are concerned with– RiskRisk– ReturnsReturns

• What determines the required What determines the required compensation for risk?compensation for risk?

• It will depend onIt will depend on– The risk faced by investorsThe risk faced by investors– The tradeoff between risk and return they The tradeoff between risk and return they

faceface

return = R = change in asset value + income

initial value

Measuring ReturnMeasuring ReturnMeasuring ReturnMeasuring Return

• R is typically annualizedR is typically annualized• R is typically annualizedR is typically annualized

example 1example 1example 1example 1

• Bond: 1 month holding periodBond: 1 month holding period

• buy for $9488, sell for $9528buy for $9488, sell for $9528

• 1 month R:1 month R:

• Bond: 1 month holding periodBond: 1 month holding period

• buy for $9488, sell for $9528buy for $9488, sell for $9528

• 1 month R:1 month R:

9528 - 9488

9488=0 .0042 = 0.42%

• annualized R:annualized R:• annualized R:annualized R:

(1.0042)12 - 1 = 0.052 = 5.2%

example 2example 2example 2example 2

• 100 shares IBM, 9 months 100 shares IBM, 9 months

• buy for $62, sell for $101.50buy for $62, sell for $101.50

• $.80 dividends$.80 dividends

• 9 month R:9 month R:

• 100 shares IBM, 9 months 100 shares IBM, 9 months

• buy for $62, sell for $101.50buy for $62, sell for $101.50

• $.80 dividends$.80 dividends

• 9 month R:9 month R:

101.50 - 62 + .80

62=0 .65 =65%

• annualized R:annualized R:• annualized R:annualized R:

(1.65)12/9 - 1 =0 .95 = 95%

example 3example 3example 3example 3

R Prob(R)

10% 0.2 5% 0.4-5% 0.4

E(R) = (0.2)10% + (0.4)5% + (0.4)(-5%)

= 2%

example 4example 4example 4example 4

= (0.2)(10%-2%)2

= 0.0039

+ (0.4)(5%-2%)2

+ (0.4)(-5%-2%)2

= 6.24%

8%9%10%11%12%13%14%15%16%17%

0% 10% 20% 30% 40%

Std. Dev.

Ex

p.

Re

t.

r=1r=0r=-1

(10%,16%)

(16%,30%)

Two risky assetsTwo risky assets

DiversificationDiversification

# assets

systematicrisk

unsystematic risk

totalrisk

Feasible portfoliosFeasible portfolios

Expected

return (Ei)

Std dev (i)

Efficient

frontier

Feasible Set

Dominated and Efficient Dominated and Efficient PortfoliosPortfolios

Expected

return (Ei)

Std dev (i)

A

B

C

Efficient frontierEfficient frontier

• 1. Find all asset expected returns 1. Find all asset expected returns and standard deviations.and standard deviations.

• 2. Pick one expected return and 2. Pick one expected return and minimize portfolio risk.minimize portfolio risk.

• 3. Pick another expected return 3. Pick another expected return and minimize portfolio risk.and minimize portfolio risk.

• 4. Use these two portfolios to map 4. Use these two portfolios to map out the efficient frontier.out the efficient frontier.

CAPMCAPM

• CAPM Characteristics:CAPM Characteristics:– bi = sismrim/sm2bi = sismrim/sm2

• Asset Pricing Equation:Asset Pricing Equation:– E(ri) = rf + bi[E(rm)-rf]E(ri) = rf + bi[E(rm)-rf]

• CAPM is a model of what expected CAPM is a model of what expected returns returns should beshould be if if everyoneeveryone solves the solves the samesame passive portfolio passive portfolio problemproblem

Expected

return (Ei)

Std dev (i)

D

Utility maximizing

risky-asset portfolio

Utility MaximizationUtility Maximization

Tobin Separation Tobin Separation TheoremTheorem• When the riskfree asset is introduced,When the riskfree asset is introduced,

• All investors prefer a combination ofAll investors prefer a combination of

• 1) The riskfree asset and1) The riskfree asset and

• 2) The market portfolio2) The market portfolio

• Such combinations dominate all other Such combinations dominate all other assets and portfolios (in a sense of assets and portfolios (in a sense of risk-return)risk-return)

All risky assets

and portfoliosExpected

return (Ei)

Std dev (i)

Riskless

asset Minimum

Variance

Portfolio

Market

Portfolio

Efficient

frontier

CMLCML

%25.7)E(r

75.0035.0085.0035.0)E(r

r)E(rr)E(r

IBM

IBM

ifmfi

CAPMCAPM

• Suppose you have the following Suppose you have the following information:information:rf = 3.5% E(rm)=8.5% rf = 3.5% E(rm)=8.5% bIBM=0.75bIBM=0.75

• What should E(rIBM) be?What should E(rIBM) be?

• Answer:Answer: