# Portfolio Theory

Fall 2012

Joni Kokkonen

General outline

1. “Technological” part of asset allocation

– How can we characterize the opportunities available to the investor given the features of the broad asset markets in which they can invest? – The investment opportunity set

2. “Personal” part of asset allocation

– How should an individual investor choose the best risk-return combination from the set of feasible combinations?

3. Equilibrium

– When all investors optimize their portfolios, how are asset returns determined in equilibrium?

Agenda

• • • • • Risk, risk aversion, and utility Portfolio risk and return Diversification Allocation between one risky and a risk-free asset Optimal risky portfolios and the efficient frontier

“OCTOBER: This is one of the peculiarly dangerous months to speculate in stocks in. The other are July, January, September, April, November, May, March, June, December, August, and February.” (Mark Twain)

Key ideas of portfolio theory

• Risk of a single investment vs. the new investment as part of one’s existing portfolio? • Diversification = “Don't put all your eggs in one basket” • Finding the least risky portfolio for any level of target return • Finding the portfolio with the highest expected return for any level of target risk • Assessing the risk-return relationship of various investments • Selecting an optimal portfolio

Practical value

Asset management • Asset allocation • Portfolio optimization • Performance measurement Risk management • Scenario analysis • Value-at-Risk (VaR) Banking • Credit risk pricing

Risk aversion

• Assume we want to invest 100 000 and have two possible investment strategies: 1. A risky investment giving a profit of 50 000 with probability 0.6 and a loss of 20 000 with a probability 0.4 2. A risk-free investment giving a profit of 5000

• The expected profit from the risky strategy is 22 000 • The incremental profit of the risky strategy over the riskfree investment is 22 000 – 5000 = 17 000 • This is the risk premium in monetary terms • Is this risk premium adequate compensation for the investment’s risk?

Mean-variance criterion

• Portfolio A dominates portfolio B if

and

• In addition, one of the inequalities must be strict

Mean-variance criterion

Risk aversion and utility

• Prospects that have a zero risk premium are called fair games • Risk averse investors reject investment portfolios that are fair game or worse • Risk averse investors are willing to consider only investments that are risk-free or have a positive risk premia • To quantify risk aversion, we can assign a utility score to competing investment portfolios • More attractive portfolios receive higher utility scores

Utility scoring

• One reasonable, and quite common, function is

• A is an index of the investor’s risk aversion, and 0.005 is a scaling convention • Higher expected return leads to higher utility and increased risk to lower utility • To what extent the variance decreases utility depends on the risk aversion (A) • What is the utility provided by a risk-free investment?

Utility scoring: example

• In our previous example, the expected return on the risky investment is 22% with a standard deviation of 34% • The risk-free investment provides a return of 5% • The risk premium is thus 17% • Assume the investor has a coefficient of risk aversion equal to 3 • The utility value of the risky portfolio is

• Which investment would the investor choose?

Certainty equivalent

• The utility value can be interpreted as the certainty equivalent rate of return • Certainty equivalent: The rate a risk-free investment would have to provide with certainty to be equally attractive as the risky portfolio • Generally, the certainty equivalent C of a random variable x is defined by:

Alternative attitudes toward risk

Risk neutral: • Judge prospects only based on their expected return • The level of risk is irrelevant • Portfolio’s...

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