Fi8000 Valuation of Financial Assets Spring Semester 2010 Dr. Isabel Tkatch Assistant Professor of Finance

Risk, Return and Portfolio Theory Risk and risk aversion Utility theory and the intuition for risk aversion Mean-Variance (M-V or μ-σ) criterion The mathematics of portfolio theory Capital allocation and the optimal portfolio One risky asset and one risk-free asset Two risky assets n risky assets n risky assets and one risk-free asset Equilibrium in capital markets The Capital Asset Pricing Model (CAPM) Market Efficiency

Reward and Risk: Assumptions Investors prefer more money (reward) to less: all else equal, investors prefer a higher reward to a lower one. Investors are risk averse: all else equal, investors dislike risk. There is a tradeoff between reward and risk: Investors will take risks only if they are compensated by a higher reward.

Reward and Risk Reward ☺ ☺ Risk

Quantifying Rewards and Risks Reward – a measure of wealth The expected (average) return Risk Measures of dispersion - variance Other measures Utility – a measure of welfare Represents preferences Accounts for both reward and risk

Quantifying Rewards and Risks The mathematics of portfolio theory (1-3)

Comparing Investments: an example Which investment will you prefer and why? A or B? B or C? C or D? C or E? D or E? B or E, C or F (C or E, revised)? E or F?

Comparing Investments: the criteria A vs. B – If the return is certain look for the higher return (reward) B vs. C – A certain dollar is always better than a lottery with an expected return of one dollar C vs. D – If the expected return (reward) is the same look for the lower variance of the return (risk) C vs. E – If the variance of the return (risk) is the same look for the higher expected return (reward) D vs. E – Chose the investment with the lower variance of return (risk) and higher expected return (reward) B vs. E or C vs. F (or C vs. E) – stochastic dominance E vs. F – maximum expected utility

Comparing Investments Maximum return If the return is risk-free (certain), all investors prefer the higher return Risk aversion Investors prefer a certain dollar to a lottery with an expected return of one dollar

Comparing Investments Maximum expected return If two risky assets have the same variance of the returns, risk-averse investors prefer the one with the higher expected return Minimum variance of the return If two risky assets have the same expected return, risk-averse investors prefer the one with the lower variance of return

The Mean-Variance Criterion Let A and B be two (risky) assets. All risk-averse investors prefer asset A to B if { μA ≥ μB and σA < σB } or if { μA > μB and σA ≤ σB } Note that we can apply this rule only if we assume that the distribution of returns is normal.

The Mean-Variance Criterion (M-V or μ-σ criterion) E(R) = μR ☺ ☺ STD(R) = σR

Other Criteria The basic intuition is that we care about “bad” surprises rather than all surprises. In fact dispersion (variance) may be desirable if it means that we may encounter a “good” surprise. When we assume that returns are normally distributed the expected-utility and the stochastic-dominance criteria result in the same ranking of investments as the mean-variance criterion.

The Normal Distribution of Returns Pr(R) 68% 95% μ - 2σ μ - σ μ μ +σ μ +2σ R

The Normal Distribution of Returns Pr(Return) σR: Risk μR: Reward R=Return

The Normal Distribution Higher Reward (Expected Return) Pr(Return) μB μA < R=Return

The Normal Distribution Lower Risk (Standard Deviation) Pr(Return) A σA < σB B μA= μB R=Return

Practice problems BKM Ch. 6: 7th edition: 1,13,14, 34; 8th edition : 4,13,14, CFA-8. Mathematics of Portfolio Theory: Read and practice parts 1-5.