Asset Management Lecture Two

I will more or less follow the structure of the textbook “Investments” with a few exceptions. I will more or less follow the structure of the textbook “Investments” with a few exceptions. These parts of the textbook are omitted: These parts of the textbook are omitted: Part IV (fixed income) Part IV (fixed income) Part V (security analysis) Part V (security analysis) Part VI (options and other derivatives) Part VI (options and other derivatives)

Outline for today Risk aversion and utility Risk aversion and utility Estimating risk aversion Estimating risk aversion Markowitz portfolio selection model Markowitz portfolio selection model How to find the efficient frontier and the optimal risky portfolio with Excel How to find the efficient frontier and the optimal risky portfolio with Excel

Risk Aversion and utility values Risk aversion: a risk-averse investor will reject a fair gamble. Risk aversion: a risk-averse investor will reject a fair gamble. Utility value Utility value Risk-neutral investors Risk-neutral investors A=0 A=0 Risk lover Risk lover A<0 A<0

Risk Aversion and utility values σ E(r) U=0.5 U=1 A=4 A=2

Portfolio L Portfolio M Portfolio H Risk Aversion (A) E(r)=0.07 σ=0.05 E(r)=0.09 σ=0.10 E(r)=0.13 σ=

Risk Aversion and utility values Portfolio L Portfolio M Portfolio H Risk Aversion (A) E(r)=0.07 σ=0.05 E(r)=0.09 σ=0.10 E(r)=0.13 σ= Certainty equivalent rate

Estimating A Consider an insurance policy with a cost of v: Consider an insurance policy with a cost of v: Expected return Expected return Variance Variance Utility Utility -v=U -v=U ProbabilityOutcome p 1-p0

risk premium v p=0.001p=0.01 A v as a multiple of p

Two-Security Portfolios with Various Correlations 100% Stock A return  100% Stock B  = 0.2  = 1.0  = -1.0 Relationship depends on correlation coefficient Relationship depends on correlation coefficient -1.0 <  < +1.0 If  = +1.0, no risk reduction is possible If  = +1.0, no risk reduction is possible If  = –1.0, complete risk reduction is possible If  = –1.0, complete risk reduction is possible

Markowitz portfolio selection model return PP minimum variance portfolio efficient frontier Individual Assets

Markowitz portfolio selection model

rfrf return  Market portfolio Capital market line Investors allocate their money across the risk- free asset and the market portfolio Investors borrow at the risk-free rate and invest in the market portfolio Separation property: the portfolio manager offers the same risky portfolio to all investors Indifference curve

Sharpe ratio Sharpe ratio Excess return / SD of excess return Excess return / SD of excess return Reward to volatility Reward to volatility The tangency portfolio has the highest Sharpe ratio The tangency portfolio has the highest Sharpe ratio Markowitz portfolio selection model

rfrf return  Capital market line Indifference curve

How to find the efficient frontier and the optimal portfolio? How to find the efficient frontier and the optimal portfolio? Find E(r) for each asset Find E(r) for each asset Find SD for each asset Find SD for each asset Find covariance between each pair of assets Find covariance between each pair of assets As a starting point, assume a weight for each asset As a starting point, assume a weight for each asset Use Excel Solver as an optimizer Use Excel Solver as an optimizer Markowitz portfolio selection model

Individual Homework Construct a portfolio of assets with 5 financial assets Construct a portfolio of assets with 5 financial assets Explain briefly why you choose these assets for your portfolio. Explain briefly why you choose these assets for your portfolio. Use recent 36 monthly data to calculate E(r), var(r), and cov. Use recent 36 monthly data to calculate E(r), var(r), and cov. Report for your minimum variance portfolio and the tangency portfolio: Report for your minimum variance portfolio and the tangency portfolio: the weights of assets the weights of assets expected return, SD and the Sharpe ratio expected return, SD and the Sharpe ratio Repeat the exercise with no-short-sale constraint. Repeat the exercise with no-short-sale constraint. Due on Feb 13. Sent your excel file to Sérgio Gaspar Due on Feb 13. Sent your excel file to Sérgio Gaspar