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 Risk Reward ☺ ☺

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 { μ A ≥ μ B and σ A < σ B } or if { μ A > μ B and σ A ≤ σ B } 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) STD(R) = σ R E(R) = μ R ☺ ☺

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 μμ +σμ +σμ +2σ μ - σ μ - 2σ 68% 95% R Pr(R)

The Normal Distribution of Returns μ R : Reward 0 R=Return Pr(Return) σ R : Risk

The Normal Distribution Higher Reward (Expected Return) μAμA R=Return Pr(Return) μBμB <

The Normal Distribution Lower Risk (Standard Deviation) R=Return Pr(Return) A B σ A < σ B μ A = μ B

Practice problems 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: Read and practice parts 1-5.