Capital Asset Pricing Model Applied covariance: Project Part 1

Review question Asset A has an expected rate of return of .15. Asset B has an expected rate of return of .25. Consider a portfolio consisting 30% asset A and 70% asset B. What is the expected rate of return on the portfolio?

Answer Expected rate of return is .3*.15+.7*.25 = .22

Review variance, covariance Variance: square the deviations and take expectation. Covariance: multiply the deviations and take expectation.

Notation Variance Covariance Portfolio weights

Portfolio variance The role of covariance. Equation 9

It all happens because

Portfolio risk and return,

Portfolio deviation Deviation squared

Portfolio variance

Portfolio variance depends on covariance of the assets. Positive covariance raises the variance of the portfolio.

Correlation coefficient

Application Asset B is the market portfolio Call it asset M. Everyone prefers to hold M, in theory Asset A is any asset. Think of adding a little A to the market portfolio.

Question does adding a little of asset A to the market portfolio increase the risk? Yes if No if

Derivation

Beta measures risk How much risk is added depends on the relation of sigma AM and sigma squared M Define beta

Beta item Download price data for your stock and the market (S&P 500). Construct rates of return. Compute variances and covariances. Compute beta for the stock. Don’t use the financial formulas, except as a check on your work

Another check on your work Regression Idea: take some points in (Dev M,Dev A) space and fit a line to them. Let b*Dev M be an estimate of Dev A. Minimize sum of squared errors.

Sum of squared errors Minimize it

Divide by T-1

The estimate of b Is the ratio of sample covariance over variance of the market. It’s beta, except for using sample statistics instead of population values.

Problem 8.1; read Ch 8.2 If the product is marketed now, its chance of success is .5 and the payoff is 20M in present value. Failure = 5M If the product is tested and improved, launch is delayed one year. The cost is 2M and the chance of success is .75. Discount at 15%. Question: Launch now or later?

The story of CAPM Investors prefer higher expected return and dislike risk. All have the same information. Two (mutual) funds are sufficient to satisfy all such investors:

The two funds: 1) The "risk-free" asset, i.e., Treasury Bills 2) The market portfolio consisting of all risky assets held in proportion to their market value.

The market portfolio Its expected return is 8.5% over the T-Bill rate It bears the market risk Its beta is unity by definition.

Capital asset pricing model T-bill rate is known. Market premium is known, approximately 8.5%. Estimate beta as in the project

Security market line It’s straight. Risk-return relation is a straight line.

Why is it a straight line? Beta is the measure of risk that matters. Given beta construct a portfolio with the same beta by a mix of T-Bills (beta = 0) and the market portfolio (beta = 1) Expected return on the portfolio is on the SML. So any asset with the same beta must also be on the SML.

Security market line Rate of return expected by the market E[RM] Rf beta 1

Examinations Samples on the web page. 1. A midterm from the past. 2. Sample questions for midterm and final. Practice the technique of answering in short essays.

Review item Return on asset A has a std dev of .05 Return on asset B has a std dev of .07 Correlation of return on asset A with return on asset B is 1. What is the covariance of the returns?

Answer: sAB = rAB*sA*sB=.0035