FIN 351: lecture 6 Introduction to Risk and Return Where does the discount rate come from?

Today’s learning objective Introduction to risk How to measure investment performance Rates of Return 73 Years of Capital Market History Measuring risk and risk premium Risk & Portfolio Diversification Two types of risk How to measure systematic risk CAPM

How to measure the performance of your investment Suppose you buy one share of IBM at $74 this year and sell it at the expected price of $102. IBM pays a dividend of $1.25 for your investment What profit do you expect to make for your investment? What profit do you expect to make for one dollar investment?

Solution Profit in total = =$29.25 Profit per one dollar=29.25/74=0.395 or 39.5%

Rates of Return

Nominal vs. Real Suppose that the inflation rate is1.6%

Market Indexes Dow Jones Industrial Average (The Dow) Value of a portfolio holding one share in each of 30 large industrial firms. Standard & Poor’s Composite Index (The S&P 500) Value of a portfolio holding shares in 500 firms. Holdings are proportional to the number of shares in the issues.

The performance of $0.1 investment

Volatility of portfolios Volatility Year

Why are stock returns so high? To invest in stocks, investors require a risk premium with respect to relative risk-free security such as government securities. The expected return on a risky security is equal to the risk- free rate plus a risk premium Expected return =risk-free rate + risk premium Risk premium =expected return –risk-free rate Example 23.3% (1981 on market portfolio)=14%+9.3% 14.1% (1999 on market portfolio)=4.8%+9.3%

How to Measure Risk We can use the variance or the standard deviation of the expected rate of return to measure risk. Variance or standard deviation measure weighted average of squared deviation of each observation from the mean.

Some formula Suppose that there are N states, then the expected rate of return (mean) is The variance of the rate of return is The standard deviation

Example of risk Stock A has the following returns depending on the state of the economy next year as follows: State of economy Probability of the stateReturn rate Good Average Bad % 10% -5%

Measure risk (continue) First, calculate the mean return or the expected rate of return. Here N=3 (three states) Expected rate of return is r-bar= p1*r1+p2*r2+p3*r3=0.6* * *(-0.05) =14.5% The variance of return is p1*(r1- r-bar) 2 +p2*(r2- r-bar) 2 +p3*(r3-r-bar) 2 = The standard deviation is =5.7%

Two types of risks Unique Risk - Risk factors affecting only that firm. Also called “firm-level risk.” Market Risk - Economy-wide sources of risk that affect the overall stock market. Also called “systematic risk.”

Can we reduce risk? Yes, we can reduce risk by diversification: that is, we invest our money in different assets or form a portfolio of different assets. Can we understand intuitively why diversification can reduce risk?

Portfolio weights Let W be the total money invested in a portfolio, a set of assets. Let x i be the proportion of total wealth invested in asset i. Then x i is called portfolio weight for asset i. The sum of portfolio weights for all the assets in the portfolio is 1, that is,

Example You invest $400 of your $1000 in IBM at a price of $74 per share and the other in Dell at a price of $28. What is the portfolio weight for IBM and Dell respectively? Are you sure that you are right?

Solution x IBM =400/1000=0.4 x Dell =600/1000=0.6 x IBM +x Dell =1

Some formula for portfolios The return of a portfolio is the weighted average of returns of the stocks in the portfolio. That is, The expected return of a portfolio is the weighted average of expected returns of the stocks in the portfolio. That is,

Risk and Diversification (example) John puts his money half in stock A and half in stock B, as shown in the following. What is the mean and variance of the return of John’s portfolio?

My solution The mean of the return of a portfolio is the weighted average of the returns of the stocks in the portfolio. Thus the mean of the return of John’s portfolio is The variance of the return of the portfolio is portfolio variance

Risk and Diversification

Measuring Market Risk Market Portfolio It is a portfolio of all assets in the economy. In practice a broad stock market index, such as the S&P 500 is used to represent the market portfolio. The market return is denoted by R m Beta (β) Sensitivity of a stock’s return to the return on the market portfolio, Mathematically,

An intuitive example for Beta Turbo Charged Seafood has the following % returns on its stock, relative to the listed changes in the % return on the market portfolio. The beta of Turbo Charged Seafood can be derived from this information.

Measuring Market Risk (example, continue)

Measuring Market Risk (continue) When the market was up 1%, Turbo average % change was +0.8% When the market was down 1%, Turbo average % change was -0.8% The average change of 1.6 % (-0.8 to 0.8) divided by the 2% (-1.0 to 1.0) change in the market produces a beta of 0.8. β=1.6/2=0.8

Another example Suppose we have following information: State MarketStock AStock B bad good -8%-10% 38% -6% 24% 32% a. What is the beta for each stock? b. What is the expected return for each stock if each scenario is equally likely? c. What is the expected return for each stock if the probability for good economy is 20%?

Solution a. b. c.

Portfolio Betas Diversification reduces unique risk, but not market risk. The beta of a portfolio will be an weighted average of the betas of the securities in the portfolio. What is the beta of the market portfolio? What is the beta of the risk-free security?

Example Suppose you have a portfolio of IBM and Dell with a beta of 1.2 and 2.2, respectively. If you put 50% of your money in IBM, and the other in Dell, what is the beta of your portfolio Beta of your portfolio =0.5* *2.2=1.7

Market risk and risk premium Risk premium for bearing market risk The difference between the expected return required by investors and the risk-free asset. Example, the expected return on IBM is 10%, the risk-free rate is 5%, and the risk premium is 10% -5%=5% If a security ( an individual security or a portfolio) has market or systematic risk, risk- averse investors will require a risk premium.

CAPM (Capital Asset Pricing Model) The risk premium on each security is proportional to the market risk premium and the beta of the security. That is,

Security market line The graphic representation of CAPM in the expected return and Beta plane rfrf Security Market Line RmRm