Chapter 6 Risk and Rates of Return

2 Chapter 6 Objectives Inflation and rates of return How to measure risk (variance, standard deviation, beta) How to reduce risk (diversification) How to price risk (security market line, CAPM)

3 Historical Risk and Return Annual From 1926 to 1999 Avg. ReturnStd Dev. Small Stocks17.6%33.6% Large Co. Stocks13.3%20.1% L-T Corp Bonds 5.9%8.7% L-T Govt. Bonds5.5%9.3% T-Bills3.8%3.2% Inflation3.2%4.6%

4 Why are these rates different? 90-day Treasury Bill1.7% 90-day Commercial Paper1.8% 2-year US Treasury Note1.9% 10-year US Treasury Note3.8% 10-year AAA Corporate Bond5.0% 10-year BBB Corporate Bond6.1%

Inflation, Rates of Return, and the Fisher Effect Interest Rates

Interest Rates Conceptually: Nominal risk-free Interest Rate k rf = Real risk-free Interest Rate k* + Inflation- risk premium IRP

Conceptually: Nominal risk-free Interest Rate k rf = Real risk-free Interest Rate k* + Inflation- risk premium IRP Mathematically: (1 + k rf ) = (1 + k*) (1 + IRP) This is known as the “Fisher Effect” Interest Rates

8 Proof of Fisher Effect Equation You have a $100 to buy items costing a $1 each. You can buy 100 items now. Instead of spending the $100 now, you decide to invest the money at 7% (nominal risk-free rate) for a year giving you $107 at the end of the year so you can hopefully buy more than 100 items at the end of the year.

9 Proof of Fisher Effect Equation At the end of the year, the items now cost $1.04 each (4% inflation). You can buy $107/$1.04 = of these items at the end of the year. This represents a 2.88% increase in your real purchasing power (real interest rate). We used (1+krf) = (1+k*)(1+IRP) (1.07) = (1+k*)(1.04): 1+k* = 1.07/1.04 = ; k* =.0288 = 2.88%

Term Structure of Interest Rates The pattern of rates of return for debt securities that differ only in the length of time to maturity. yield to maturity time to maturity (years)

Term Structure of Interest Rates yield to maturity time to maturity (years) The yield curve may be downward sloping or “inverted” if rates are expected to fall.

12 Recent US Treasury Yield Curves

For a Treasury security, what is the required rate of return? Since Treasuries are essentially free of default risk, the rate of return on a Treasury security is considered the “risk-free” rate of return. Required rate of return = Risk-free return

For a corporate stock or bond, what is the required rate of return? How large of a risk premium should we require to buy a corporate security? Required rate of return = += += += + Risk-free returnRiskpremium

Returns Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc. Required Return - the return that an investor requires on an asset given its risk and market interest rates.

16 Holding Period (Actual) Returns The realized return over a period of time (HPR). HPR=(Ending Price - Beginning Price + Distributions Received)/Beginning Price Example: What is your HPR if you buy a stock for $20, receive $1 in dividends, and then sell it for $25. HPR = ($25-$20+$1)/$20 = 0.3 = 30%

17 Calculation of Expected Returns Expected Rate of Return (Expected Value) given a probability distribution of possible returns(k i ): E(k) or k _ n E(k)=k =  k i P(k i ) i=1 Realized or Average Return on Historical Data: - n k = 1/n  k i i=1

18 Expected Return and Standard Deviation Example MAD E(r) =.25(80%) +.60(30%) +.15(-30%) = 33.5% CON E(r) =.25(5%) +.60(10%) +.15(15%) = 9.5%

19 Definition of Risk Risk is an uncertain outcome or chance of an adverse outcome. Concerned with the riskiness of cash flows from financial assets. Namely, the chance that actual cash flows will be different from forecasted cash flows. Standard Deviation can measure this type of risk. For a stock, we can examine the standard deviation of the stock’s returns.

Standard Deviation = (k i - k) 2 P(k i )  n i=1 

21 Expected Return and Standard Deviation Example MAD E(r) =.25(80%) +.60(30%) +.15(-30%) = 33.5% CON E(r) =.25(5%) +.60(10%) +.15(15%) = 9.5%

MAD, Inc. ( 80% %) 2 (.25) = (30% %) 2 (.6) = 7.35 (-30% %) 2 (.15) = Variance = % Stand. dev. = = 34.0% MAD, Inc. ( 80% %) 2 (.25) = (30% %) 2 (.6) = 7.35 (-30% %) 2 (.15) = Variance = % Stand. dev. = = 34.0% = (k i - k) 2 P(k i )  n i=1 

23 Expected Return and Standard Deviation Example MAD E(r) =.25(80%) +.60(30%) +.15(-30%) = 33.5% CON E(r) =.25(5%) +.60(10%) +.15(15%) = 9.5%

Contrary Co. (5% - 9.5%) 2 (.25) = 5.06 (10% - 9.5%) 2 (.6) = 0.15 (15% - 9.5%) 2 (.15) = 4.54 Variance = 9.75% Stand. dev. = 9.75 = 3.1% = (k i - k) 2 P(k i )  n i=1 

Which stock would you prefer? How would you decide?

It depends on your tolerance for risk! Remember, there’s a tradeoff between risk and return. Return Risk

27 Coefficient of Variation A relative measure of risk. Whereas,  is an absolute measure of risk. Relates risk to expected return. CV =  /E(k) MAD’s CV = 34%/33.5% = 1.01 CON’s CV = 3.1%/9.5% = 0.33 CONtrary is the less risky of the two investments. Would choose CON if risk averse.

Portfolios Expected Portfolio Return is weighted average of the expected returns of the individual stocks = Σw j k j. However, portfolio risk (standard deviation) is NOT the weighted average of the standard deviations of the individual stocks. Combining several securities in a portfolio can actually reduce overall risk. How does this work?

Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated). rate of return time kAkA kBkB

rate of return time kpkp kAkA kBkB What has happened to the variability of returns for the portfolio?

Diversification Investing in more than one security to reduce risk. If two stocks are perfectly positively correlated, diversification has no effect on risk. If two stocks are perfectly negatively correlated, the portfolio is perfectly diversified.

32 Some risk can be diversified away and some cannot. Market risk (systematic risk) is nondiversifiable. This type of risk cannot be diversified away. Company-unique risk (unsystematic risk) is diversifiable. This type of risk can be reduced through diversification.

As you add stocks to your portfolio, company-unique risk is reduced. portfolio risk number of stocks Market risk

As you add stocks to your portfolio, company-unique risk is reduced. portfolio risk number of stocks Market risk company- unique risk

Note: The market compensates investors for accepting risk - but only for market risk. Company-unique risk can and should be diversified away. So - we need to be able to measure market risk. We use beta as a measure of market risk.

36 The Concept of Beta Beta(  ) measures how the return of an individual asset (or even a portfolio) varies with the market portfolio.  = 1.0 : same risk as the market  < 1.0 : less risky than the market  > 1.0 : more risky than the market Beta is the slope of the regression line (y = a +  x) between a stock’s return(y) and the market return(x) over time,  from simple linear regression.  i = Covariance i,m /Mkt. Var. =  im  i  m /  m 2

37 Relating Market Risk and Required Return: the CAPM Here’s the word story: a stock’s required rate of return = risk-free rate + the stock’s risk premium. The main assumption is investors hold well diversified portfolios = only concerned with market risk. A stock’s risk premium = measure of systematic risk X market risk premium.

38 CAPM Equation k rp = market risk premium = k m - k rf stock risk premium =  j (k rp ) k j = k rf +  j (k m - k rf ) = k rf +  j (k rp ) Example: What is Yahoo’s required return if its b = 1.75, the current 3-mo. T-bill rate is 1.7%, and the historical market risk premium of 9.5% is demanded? Yahoo k = 1.7% (9.5%) = 18.3%

39 Question: If Yahoo’s exp. Return = 15%, what to do?

40 Portfolio Beta and CAPM The  for a portfolio of stocks is the weighted average of the individual stock  s.  p =  w j  j Example: The risk-free rate is 6%, the market return is 16%. What is the required return for a portfolio consisting of 40% AOL with b = 1.7, 30% Exxon with b = 0.85, and 30% Fox Corp. with b = Bp =.4(1.7)+.3(0.85)+.3(1.15) = 1.28 k p = 6% (16% - 6%) = 18.8%

41 More CAPM/SML Fun! According to the CAPM and SML equation with k = 6% +  (16% - 6%) How would a change in inflation affect required returns? (Say inflation increases 2% points) How would a change in risk aversion (market risk premium) affect required returns? (Say market risk premium decreases 2% points.)

42 Changes to SML

43 Changes to SML

44 Changes to SML

45 Limitations of CAPM/SML Don’t really know what the market portfolio is, which makes it hard to estimate market expected or required return. Beta estimates can be unstable and might not reflect the future. Maturity debate over proper risk-free estimate. Most investors focus on more than systematic risk.