Risk, Return, and Discount Rates Capital Market History The Risk/Return Relation Application to Corporate Finance

How Are Risk and Expected Return Related? There are two main reasons to be concerned with this question. (1) When conducting discounted cash flow analysis, how should we adjust discount rates to allow for riskiness in the cash flow projections? (2) When saving/investing, what is the tradeoff between taking risks and our expected future wealth? In this course we will concentrate on the first of these questions. For your own concerns, do not lose sight of the second. The answers are opposite sides of the same coin.

Discounting Risky Cash Flows n How should the discount rate change in our standard NPV calculation if the cash flows are not riskless? n The question is more easily answered from the “other side.” How must the return on an asset change so you will be happy to own it if it is a risky rather than riskless asset? –Risk averse investors will say that to hold a risky asset they require a higher expected return than they would require for holding a riskless asset. E(r risky ) = r f + . –Note now that we have to start to talk about expected returns since risk has been introduced.

Review: Rates of Return Returns have two components: –Dividends (or Interest) –Capital Gains (Price Appreciation) n The percentage return (R) on an asset is defined as: If we wait until we see the outcomes (what happens) we are describing a realized return. If we wait until we see the outcomes (what happens) we are describing a realized return. If we do the computation based on forecasts (expectations) we are describing an expected return. If we do the computation based on forecasts (expectations) we are describing an expected return. We have to make our decisions based on expected returns, but past realized returns often contain useful information for forming our expectations about the future. We have to make our decisions based on expected returns, but past realized returns often contain useful information for forming our expectations about the future.

n Suppose you purchased a share of Intel for $100 at the beginning of the year. During the year Intel paid $3.50 in dividends and the share price at the end of the year was $110. What was your return (R) for the year? n Answer: R = (3.5+[ ])/100 = 3.5/ /100 = 3.5% + 10% = 13.5%. Rate of Return Example:

More Returns n Four years ago you bought some GM stock. n For the last four years the return on GM stock has been 17.5%, 22.22%, %, and 21.9% respectively. n What return have you received for the four years? –Your “Holding Period Return” has been: (1+r HP ) = (1+r 1 )(1+r 2 )(1+r 3 )(1+r 4 ) = (1.175)(1.2222)(0.8462)(1.219) = –This is how your wealth increased over the 4 years. –Your “compound annual return” for this investment was: (1+r CA ) 4 = (1+r HP ) = so (1+r CA ) = (1.4813) 1/4 = or 10.32% –This tells you that if you had held the stock for four years and in each year the return was 10.32% you’d have the same value today. –Compound annual return or geometric average return tells you the average annual rate of wealth increase over the holding period.

Still More Returns n For the last four years the return on GM stock have been 17.5%, 22.22%, %, and 21.9%. n What did the typical year’s return look like? n The average return for the four years was (17.5% % % %)/4 = 11.56% n This represents the average year, and is also a good place to start if someone asks what you expect the return will be next year. –Be careful to note that your wealth didn’t increase an average of 11.56% over the four years. –Rather, if you had to make a guess about what the return will be next year, 11.56% is a good one.

What should you expect for next year’s return? There is general agreement that expected returns should increase with risk. Expected Return Risk But, how should risk be measured? at what rate does the line slope up? is the relation linear? Lets look at some important historical evidence.

The Future Value of an Investment of $1 in 1926 $40.22 $15.64 Source: © Stocks, Bonds, Bills, and Inflation 2000 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.

Rates of Return Source: © Stocks, Bonds, Bills, and Inflation 2000 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.

Risk, More Formally n Many people think intuitively about risk as the possibility of an outcome that is worse than what one expected. –For those who hold more than one asset, is it the risk of each asset they care about, or the risk of their whole portfolio? n A useful construct for thinking rigorously about risk: –The “probability distribution.” –A list of all possible outcomes and their probabilities.

Example: Two Probability Distributions on Tomorrow's Share Price. n Which implies more risk?

n In some very simple cases, we try to specify probability distributions completely. n More often, we rely on parameters of the probability distribution to summarize the important information. These include: u The expected value, which is the center or mean of the distribution. u The variance or standard deviation, which are measures of the dispersion of possible outcomes around the mean. Risk and Probability Distributions

Summary Statistics for a Probability Distribution over Returns Summary Statistics for a Probability Distribution over Returns n The expected return is a weighted sum of the possible returns, where each return is weighted by its probability of occurring, p. The Variance of Returns Measures the dispersion of returns around the expectation. The Standard Deviation (STD or  ) is the square root of the variance. It is in the same units as the returns

Example E[R A ]= -25*0.1+5*0.2+15*0.4+30*0.2+45*0.1 = 15% E[R B ] = -40*0.1+0*0.2+16*0.4+40*0.2+66*0.1 = 17% VAR A = (-25-15) 2 *0.1+(5-15) 2 *0.2+(15-15) 2 *0.4+ (30-15) 2 *0.2+(45-15) 2 *0.1 = 315 VAR B = (-40-17) 2 *0.1+(0-17) 2 *0.2+(16-17) 2 *0.4+ (40-17) 2 *0.2+(66-17) 2 *0.1 = 729 STD A = 17.75% STD B = 27%

Calculating Sample Statistics n When we want to describe the returns on an asset (e.g. a stock) we usually don't (never) really know the actual probability distribution. But we typically have observations of returns in the past --- that is we have some observations drawn from the probability distribution. We can estimate the variance and expected return using the arithmetic mean of past returns and the sample variance. n Average = R = (R 1 + R 2 + R R T )/T Sample Variance =  2 = "Average" of [R t - R] 2.

Example: Calculate the arithmetic mean and sample standard deviation of returns on stocks A & B. R A = ( )/4 = 0.1 = 10% R B = ( )/4 = 0.2 = 20% VAR A = [( ) 2 + (0 -.1) 2 + ( ) 2 + (.2 -.1) 2 ]/3 = = 83.3% 2 ;STD A = 9.13% VAR B = [(.3 -.2) 2 + ( ) 2 + (.2 -.2) 2 + (.5 -.2) 2 ]/3 =.0866 = 866% 2 ;STD B = 29.4%

Normal Distribution? Source: © Stocks, Bonds, Bills, and Inflation 2000 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.

Historical Returns, Source: © Stocks, Bonds, Bills, and Inflation 2000 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved. – 90%+ 90%0% Average Standard Series Annual Return DeviationDistribution Large Company Stocks13.0%20.3% Small Company Stocks Long-Term Corporate Bonds Long-Term Government Bonds U.S. Treasury Bills Inflation3.24.5

The Risk-Return Tradeoff

Risk Premium n We wrote the expected return on an asset as E(r risky ) = r f + . –Refer to  as the risk premium, the return you get for bearing risk. n Defining the S&P500 portfolio to have one unit of risk, the premium per unit risk is the expected return on this portfolio less the riskless rate. –Measured this way, using the historical average, the risk premium is about 9% per year. –Many believe this to be overstated. –We can do a much more complicated analysis. –For example note the next slide.

Stock Market Volatility Source: © Stocks, Bonds, Bills, and Inflation 2000 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved. The volatility of stocks is not constant from year to year.

Summary n An old saying on Wall Street is “You can either sleep well or eat well.”