CORPORATE FINANCE:AN INTRODUCTORY COURSEDISCUSSION NOTESMODULE #111RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM)I. Summary of Key Points:- The definition of risk is a major issue in finance. Risk is a "slippery" concept, i.e., it is not easy to define nor does it have a natural “unit.”- We assume that investors like expected return, E(r) (the more the better!), and dislike risk (theless the better!). In short, investors are risk averse--they must be compensated for bearing risk. Therefore, the relationship between expected (ex ante) return and risk must be upward sloping. After the fact, however, realized (ex post) returns can have upward, downward, or no relationship with risk. I will elaborate on this important distinction between expected andrealized returns as a function of risk. However, over longer holding periods, return data, which we will discuss shortly, illustrate that risk has been rewarded historically with higher realized returns. We expect this for a long history.- Don’t ever think of E(r) without thinking about the risk associated with that E(r). Risk and return go together, like hand and glove! [Practice the following exercise in the shower everymorning. Toss your bar of soap back-and-forth between hands and simultaneously say "risk versus return, risk versus return," where one hand with soap represents E(r) and the other hand with soap represents risk!] Okay maybe that’s a stupid thing to do but do whatever it takes to link these two words in your mind! High E(r) equates to high risk! If anyone tells you otherwise, beware! They are either a fool or a crook! Capital markets provide no (few?) free lunches, i.e., high returns with low risk!II. E(r) and Volatility of Individual Securities:The equation for the expected return on a security for time period t isE(rt) = (E(Pt) - Pt-1 + E(Divt))/Pt-1.Note again that the "E" represents expectations, or an expected but not realized value.Assume that at t = 0, today, a stock costs $60, or P0. This stock does not pay dividends. You expect one of the following three states of nature to occur in the next year:1 This lecture module is designed to complement Chapters 11 and 12 in B&D.1Statei Probability of StateiP1i rtiRecession 1/3 $ 50-16.7%Normal Times 1/3 $ 75 25.0%Boom Times 1/3 $100 66.7%1.0The expected price of the security at t = 1 isE(P1) = (1/3)*$50 + (1/3)*$75 + (1/3)*$100 = $75.The expected return on the security in time period 1is E(r1) = (1/3)(-16.7%) + (1/3)(25.0%) + (1/3)(66.7%) = 25.0%, or 3E(r1) = Σ Probi*r1i, where i=1 - E(r1) is the expected return on the security in period 1,- i indexes the three states that may occur, - Probi = the probability of state i occurring, and - r1i is the return on the security if state i occurs. Note that the probabilities of the states must sum to 1.0.Alternatively, we could calculate the expected return in one-period, E(r1), as aboveE(r1) = (E(P1) - P0)/P0 = ($75 - $60)/$60 = 25.0% (note you still have to find E(P1)).How might we measure the risk of the return over the coming year for this security? Candidate measures include:- Range of outcomes- Variance of outcomes- Standard Deviation of outcomes- Other measures?The range of outcomes is -16.7% to 66.7%. However, this measure lacks precision and lacks a relationship to the expected outcome. Further, probabilities of outcomes are not considered.The variance of outcomes, σ2, is calculated as N σ2 = Σ Probi(r1i – E(r1))2, where i=1 - i indexes the state i, 2- N represents the number of possible states, - Probi represents the probability of state i, - r1i is the return in state i, and - E(r1) represents the expected return over all states.σ2 = (1/3)(-0.167 - 0.250)2 + (1/3)(0.250 - 0.250)2 + (1/3)(0.667 - 0.250)2σ2 = 0.1159, or 11.59%2 in our numerical example.The standard deviation of the outcomes, σ, is calculated asσ = (σ2)1/2 or (0.1159)1/2 = 0.3405, or 34.05%.Review the relationship of σ to the distribution of r1i's as per the material in an introductory statistics course, i.e., (+/-) 1 σ incorporates about 68% of the distribution centered on E(r), (+/-) 2σ incorporated about 95% of the distribution, (+/-) 3σ includes 99+% of the distribution, assuming the distribution is normally distributed.The other possible measures of risk will be developed below.A source of confusion is when to use Probi versus 1/N versus 1/(N-1) as the weight in the variance calculation, where N equals sample size. If all outcomes are equally likely, Probi = 1/N. Therefore, if all of the outcomes (here future states) are not equally likely, then 1/N is not appropriate to use as the weights in calculating the variance of a distribution--use Probi.If you'll recall from your introductory statistics course, if you are analyzing actual realized data and the sample size is small, you should use 1/(N-1), when you have N observations, to correct for the small sample size bias problem. If the sample is large (N > 30), however, the sample size bias problem is negligible, using 1/N is a very close approximation. III. Risk and Diversification--The Intuition:Is the dispersion of the returns, actual or expected, measured by variance or standard deviation, the correct definition of risk for any single security? The answer is "yes" and "no." The answer is "yes" if you are constrained to hold only one security. However, if you have the opportunity to diversify your assets, the answer is "no," variance is not a good measure of risk fora single asset. In our development of risk measures, we assume that the great majority of investors have the opportunity to diversify. Therefore, we must examine the impact diversificationhas on determining the appropriate measure of risk for an individual security.What do we mean by diversification? Let's use an intuitive but extreme example.We are examining an investment in two companies, The Umbrella Company, UC, and the Sun Tan Oil Company, STO. Details regarding these investments are as follows:States of Nature Probability3Umbrella Company Sun TanOil CompanySunny Year 0.50.06 0.18Rainy Year 0.5 0.18 0.06 NE(r) = Σ Probi *ri, where i=1 - E(r) is the expected return on security j in a given time period,- N is the number of states of nature that can occur in that time period, here two,- Probi is the probability of state i occuring in that time period, here two, and- ri is the return in state i