Fi8000Fi8000Valuation ofValuation of

Financial AssetsFinancial AssetsSpring Semester 2010Spring Semester 2010

Dr. Isabel TkatchDr. Isabel TkatchAssistant Professor of FinanceAssistant Professor of Finance

Risk, Return and Portfolio TheoryRisk, Return and Portfolio Theory☺ Risk and risk aversionRisk and risk aversion

☺ Utility theory and the intuition for risk aversionUtility theory and the intuition for risk aversion☺ Mean-Variance (M-V or Mean-Variance (M-V or μμ--σσ) criterion) criterion☺ The mathematics of portfolio theoryThe mathematics of portfolio theory

☺ Capital allocation and the optimal portfolioCapital allocation and the optimal portfolio☺ One risky asset and one risk-free assetOne risky asset and one risk-free asset☺ Two risky assetsTwo risky assets☺ n risky assetsn risky assets☺ n risky assets and one risk-free assetn risky assets and one risk-free asset

☺ Equilibrium in capital marketsEquilibrium in capital markets☺ The Capital Asset Pricing Model (CAPM)The Capital Asset Pricing Model (CAPM)☺ Market EfficiencyMarket Efficiency

Reward and Risk: AssumptionsReward and Risk: Assumptions

☺ Investors Investors prefer more moneyprefer more money (reward) to less: (reward) to less: all else equal, investors prefer a higher reward all else equal, investors prefer a higher reward to a lower one.to a lower one.

☺ Investors are risk averse: all else equal, Investors are risk averse: all else equal, investors investors dislike riskdislike risk. .

☺ There is a There is a tradeoff between reward and risktradeoff between reward and risk: : Investors will take risks only if they are Investors will take risks only if they are compensated by a higher reward.compensated by a higher reward.

Reward and RiskReward and Risk

Risk

Reward ☺

☺

Quantifying Rewards and RisksQuantifying Rewards and Risks☺ Reward – a measure of wealthReward – a measure of wealth

☺ The expected (average) returnThe expected (average) return

☺ RiskRisk☺ Measures of dispersion - varianceMeasures of dispersion - variance☺ Other measuresOther measures

☺ Utility – a measure of welfareUtility – a measure of welfare☺ Represents preferencesRepresents preferences☺ Accounts for both reward and riskAccounts for both reward and risk

Quantifying Rewards and RisksQuantifying Rewards and Risks

The mathematics of portfolio theory (1-3)The mathematics of portfolio theory (1-3)

Comparing Investments:Comparing Investments:an examplean example

Which investment will you prefer and why?Which investment will you prefer and why?

☺ A or B?A or B?☺ B or C?B or C?☺ C or D?C or D?☺ C or E?C or E?☺ D or E?D or E?☺ B or E, C or F (C or E, revised)? B or E, C or F (C or E, revised)? ☺ E or F?E or F?

Comparing Investments:Comparing Investments:the criteriathe criteria

☺ A vs. BA vs. B – If the return is certain look for the higher return – If the return is certain look for the higher return (reward)(reward)

☺ B vs. CB vs. C – A certain dollar is always better than a lottery – A certain dollar is always better than a lottery with an expected return of one dollarwith an expected return of one dollar

☺ C vs. DC vs. D – If the expected return (reward) is the same – If the expected return (reward) is the same look for the lower variance of the return (risk)look for the lower variance of the return (risk)

☺ C vs. EC vs. E – If the variance of the return (risk) is the same – If the variance of the return (risk) is the same look for the higher expected return (reward)look for the higher expected return (reward)

☺ D vs. ED vs. E – Chose the investment with the lower variance – Chose the investment with the lower variance of return (risk) and higher expected return (reward)of return (risk) and higher expected return (reward)

☺ B vs. EB vs. E or or C vs. FC vs. F (or (or C vs. EC vs. E) – stochastic dominance) – stochastic dominance☺ E vs. FE vs. F – maximum expected utility – maximum expected utility

Comparing InvestmentsComparing Investments

Maximum returnMaximum returnIf the return is risk-free (certain), all investors If the return is risk-free (certain), all investors

prefer the higher returnprefer the higher return

Risk aversionRisk aversionInvestors prefer a certain dollar to a lottery Investors prefer a certain dollar to a lottery

with an expected return of one dollarwith an expected return of one dollar

Comparing InvestmentsComparing InvestmentsMaximum expected returnMaximum expected return

If two risky assets have the If two risky assets have the same variancesame variance of the returns, risk-averse investors prefer of the returns, risk-averse investors prefer

the one with the higher expected returnthe one with the higher expected return

Minimum variance of the returnMinimum variance of the returnIf two risky assets have the If two risky assets have the same expected same expected

returnreturn, risk-averse investors prefer the , risk-averse investors prefer the one with the lower variance of returnone with the lower variance of return

The Mean-Variance CriterionThe Mean-Variance Criterion

Let A and B be two (risky) assets. All risk-Let A and B be two (risky) assets. All risk-averse investors prefer asset A to B ifaverse investors prefer asset A to B if

{ { μμA A ≥ ≥ μμBB and and σσAA < < σσBB } }

or ifor if{ { μμA A > > μμBB and and σσAA ≤ ≤ σσBB } }

Note that we can apply this rule only if we assume that the Note that we can apply this rule only if we assume that the distribution of returns is normal.distribution of returns is normal.

The Mean-Variance CriterionThe Mean-Variance Criterion(M-V or (M-V or μμ--σσ criterion) criterion)

STD(R) = σR

E(R) = μR ☺

☺

Other CriteriaOther CriteriaThe basic intuition is that The basic intuition is that we care about “bad” we care about “bad” surprises rather than all surprisessurprises rather than all surprises. In fact . In fact dispersion (variance) may be desirable if it dispersion (variance) may be desirable if it means that we may encounter a “good” means that we may encounter a “good” surprisesurprise..

When we assume that When we assume that returns are normally returns are normally distributeddistributed the expected-utility and the the expected-utility and the stochastic-dominance criteria result in the same stochastic-dominance criteria result in the same ranking of investments as the mean-variance ranking of investments as the mean-variance criterion.criterion.

The Normal Distribution of ReturnsThe Normal Distribution of Returns

μ μ +σ μ +2σμ - σμ - 2σ

68%

95%

R

Pr(R)

The Normal Distribution of ReturnsThe Normal Distribution of Returns

μR: Reward0 R=Return

Pr(Return)

σR: Risk

The Normal DistributionThe Normal DistributionHigher Reward (Expected Return)Higher Reward (Expected Return)

μA R=Return

Pr(Return)

μB <

The Normal DistributionThe Normal DistributionLower Risk (Standard Deviation)Lower Risk (Standard Deviation)

R=Return

Pr(Return)

A

BσA < σB

μA= μB

Practice problemsPractice problems

BKM Ch. 6: 7th edition: 1,13,14, 34;BKM Ch. 6: 7th edition: 1,13,14, 34; 8th edition : 4,13,14, CFA-8.8th edition : 4,13,14, CFA-8.

Mathematics of Portfolio Theory:Mathematics of Portfolio Theory:Read and practice parts 1-5.Read and practice parts 1-5.