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

2. 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

3. 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.

4. Reward and Risk
Reward ☺ ☺
Risk

5. 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

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

7. 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?

8. 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

9. 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

10. 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

11. 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.

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

13. 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.

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

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

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

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

18. 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.