CHAPTER 5. Introduction to Risk, Return, and the Historical Record INVESTMENTS BODIE, KANE, MARCUS

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2 5-2 Supply Interest Rate Determinants Households Demand Businesses Government s Net Supply and/or Demand Federal Reserve Actions

3 Real and Nominal Rates of Interest Nominal interest rate: Growth rate of your money Real interest rate: Growth rate of your purchasing power (how many Big Macs can I buy with my money?)* *The Big Mac Index is a different thing Let r n = nominal rate, r r = real rate and i = inflation rate. Then: r r r n i More precisely: 1 + r r = 1 + r n 1 + i solve r r = r n i 1 + i INVESTMENTS BODIE, KANE, MARCUS 5-3

4 5-4 Fig 5.1: Real Rate of Interest Equilibrium Determined by supply, demand, government actions, expected rate of inflation

5 5-5 Equilibrium Nominal Rate of Interest As the inflation rate increases, investors will demand higher nominal rates of return If E(i) denotes current expectations of inflation, then we get the Fisher Equation: Nominal rate = real rate + expected inflation R r E() i

6 5-6 Taxes and the Real Rate of Interest Tax liabilities are based on nominal income Given a tax rate (t) and nominal interest rate (R), the real after-tax rate of return is: R1 t i r i1 t i r1 t i t As intuition suggests, the after-tax, real rate of return falls as the inflation rate rises.

7 5-7 Rates of Return for Different Holding Periods Zero Coupon Bond Par = $100 T = maturity P = price r f (T) = total risk free return P r f T r f 100 T 1 P

8 5-8 Time Does Matter Use Annualized Rates of Return

9 5-9 Effective Annual Rate (EAR) Time matters use EAR to annualize EAR definition: percentage increase in funds invested over a 1-year horizon f T EAR T 1 r 1 1 EAR 1 r f T 1 T

10 5-10 Equation 5.8 APR Annual Percentage Rate (APR): annualizing using simple interest 1 APR T 1 APR T EAR T 1 EAR 1 T Q. You invest $1 for 30 years. Do you prefer [A] 5% APR, or [B] 5% EAR?

11 Investment End Value End Value with APR=5.0% End Value with EAR=5.0% (years) INVESTMENTS BODIE, KANE, MARCUS 5-11

12 1-12 Table 5.1 APR vs. EAR Hold the EAR fixed at 5.8% and solve for APR for each holding period Hold the APR fixed at 5.8% and solve for EAR for each holding period

13 5-13 Continuous Compounding Frequency of compounding matters At the limit to (compounding time) 0: 1 EAR e r cc Q. You invest $1 for 30 years. Which interest rate do you prefer? A. 5% EAR B. 5% R cc

14 Investment End Value End Value with APR=5.0% End Value with EAR=5.0% End Value with Rcc=5.0% (years) INVESTMENTS BODIE, KANE, MARCUS 5-14

15 How to derive R cc Let r=rate and x=compounding time T N x N T / 1 r x1 r x r x N End Value 1 Make x very small. Then use A=e ln(a) lim 1 x0 compounding N times N r x lim e ln 1rx N S x0 x Substitute N=T/x lim x0 T e ln 1rx x 1 T r 1rx 1 lime e x0 Looks like 0/0. Use de l Hôpital rt Q.E.D. lime x0 d dx T ln INVESTMENTS BODIE, KANE, MARCUS d dx 1rx x Checks: r=0 End Value=1 T=0 End Value=1

16 5-16 Table 5.2 Statistics for T-Bill Rates, Inflation Rates and Real Rates,

17 5-17 Bills and Inflation, Moderate inflation can offset most of the nominal gains on low-risk investments. One dollar invested in T-bills from grew to $20.25, but with a real value of only $1.55. Negative correlation between real rate and inflation rate means the nominal rate doesn t fully compensate investors for increased in inflation

18 5-18 Fig 5.3: Interest Rates and Inflation

19 5-19 Risk and Risk Premiums Rates of Return: Single Period HPR P P 1 P 0 0 D HPR = Holding Period Return P 0 = Beginning price P 1 = Ending price D 1 = Dividend during period one 1

20 5-20 Rates of Return: Single Period Example Ending Price = 110 Beginning Price = 100 Dividend = 4 HPR = ( ) / (100) = 14%

21 Expected Return and Standard INVESTMENTS BODIE, KANE, MARCUS 5-21 Deviation Expected (or mean) returns E( r) p( s) r( s) s s = state p(s)= probability of a state r(s) = return if a state occurs Q. What is the expected value of rolling a die? A. 1 B. Sqrt(6) C. Pi D. 3.5 E. 6

22 Scenario Returns: Example State Prob. of state r for that state Excellent Good Poor Crash E(r) = (0.25)(0.31) + (0.45)(0.14) + (0.25)( ) + (0.05)(-0.52) = = 9.76% (think of a probability-weighted avg) NOTE: use decimals instead of percentages to be safe INVESTMENTS BODIE, KANE, MARCUS 5-22

23 5-23 Variance and Standard Deviation Variance (VAR): 2 2 s p( s) r( s) E( r) Standard Deviation (STD): STD 2

24 5-24 Scenario VARiance and STD Example VARiance calculation: σ 2 = 0.25( ) ( ) ( ) ( ) 2 = = Example STD calculation:

25 5-25 Time Series Analysis of Past Rates of Return The Arithmetic Average of historical rate of return as an estimator of the expected rate of return E( r) n s1 p( s) r s 1 n n s1 r s Q. What assumptions are we implicitly making?

26 5-26 Geometric Average Return TV n ( 1 r )(1 r2 )...(1 r 1 n ) TV = Terminal Value of the Investment Solve for a rate g that, if compounded n times, gives you the same TV TV n 1 g g 1/ n 1 TV g = geometric average rate of return

27 5-27 Estimating Variance and Standard Deviation Estimated Variance = expected value of squared deviations (from the mean) 2 2 s p( s) r( s) E( r) Recall the definition of variance ˆ 2 1 n n s1 2 rs r

28 Geometric Variance and Standard Deviation Formulas Using the estimated r avg instead of the real E(r) introduces a bias: we already used the n observations to estimate r avg we really have only (n-1) independent observations correct by multiplying by n/(n-1) When eliminating the bias, Variance and Standard Deviation become*: n 1 ˆ n 1 2 rs r * More at INVESTMENTS BODIE, KANE, MARCUS 5-28 j1

29 5-29 The Reward-to-Volatility (Sharpe) Ratio Excess Return The difference in any particular period between the actual rate of return on a risky asset and the actual risk-free rate Risk Premium The difference between the expected HPR on a risky asset and the risk-free rate Sharpe Ratio SD of Risk Premium Excess Returns

30 5-30 The Normal Distribution Investment management math is easier when returns are normal Standard deviation is a good measure of risk when returns are symmetric If security returns are symmetric, portfolio returns will be, too Assuming Normality, future scenarios can be estimated using just mean and standard deviation

31 5-31 Figure 5.4 The Normal Distribution

32 5-32 Normality and Risk Measures What if excess returns are not normally distributed? Standard deviation is no longer a complete measure of risk Sharpe ratio would not be a complete measure of portfolio performance Need to consider higher moments, like skew and kurtosis

33 5-33 Skew and Kurtosis skew this is zero for symmetric distributions average R 3 ˆ R 3 kurtosis R R average ˆ 4 this equals 3 for a Normal distribution 4 3

34 5-34 Fig.5.5A Normal and Skewed Distributions Mean = 6% SD = 17%

35 5-35 Fig 5.5B Normal & Fat-Tailed Distributions Mean = 0.1 SD = 0.2

36 5-36 Value at Risk (VaR) A measure of loss most frequently associated with extreme negative returns VaR is the quantile of a distribution below which lies q% of the possible values of that distribution The 5% VaR, commonly estimated in practice, is the return at the 5th percentile when returns are sorted from high to low. Also referred to as 95%-ile (depends on perspective)

37 2.5 Normal Distribution and VaR 2 The area is the percentile VaR INVESTMENTS BODIE, KANE, MARCUS 5-37

38 5-38 Expected Shortfall (ES) a.k.a. Conditional Tail Expectation (CTE) More conservative measure of downside risk than VaR: VaR = highest return from the worst cases Real life distributions are asymmetric and have fat tails ES = average return of the worst cases

39 2.5 Normal Distribution, VaR, and Expected Shortfall 2 The area is the percentile Expected Shortfall VaR INVESTMENTS BODIE, KANE, MARCUS

40 5-40 A game with a coin Let s play a game: flip one coin, and receive $1 if heads Assume Pr[Heads]= p (for example p=50%) Q. What is the game s expected outcome? Q. What is the Variance? Q. What is the St.Dev?

41 5-41 A game with two coins Let s play a game: flip 2 fair coins, and receive $1 for each head Q. What is the portfolio expected return? Q. What is the portfolio Variance? Q. What is the portfolio St.Dev?

42 5-42 A lot more coins Let s play a game: flip 30 fair coins, and receive $1 for each head. Q. What is the portfolio expected return? Q. What is the portfolio Variance? Q. What is the portfolio St.Dev?

43 5-43 A Portfolio of 2 stocks Portfolio = 0.5 * A * B A: r A = 0.08 StDev A = 0.1 B: r B = 0.10 StDev B = 0.1 Q. What is the portfolio Expected Return? Q. What is the portfolio Variance? Q. What is the portfolio Standard Deviation?

44 5-44 A Portfolio of 3 stocks Portfolio = w A A + w B B + w C C Q. What is the portfolio expected return? Q. What is the portfolio Variance? Q. What is the portfolio Standard Deviation? Q. What is if you have N stocks?

45 5-45 (A) (B) (C) Q. Which portfolio has best Sharpe? (D) (E) 30% (A) 50% (B) 20% (D)

46 Historic Returns on Risky Portfolios Normal distribution is generally a good approximation of portfolio returns VaR indicates no greater tail risk than is characteristic of the equivalent normal The ES does not exceed 0.41 of the monthly SD, presenting no evidence against the normality However Negative skew is present in some of the portfolios some of the time, and positive kurtosis is present in all portfolios all the time INVESTMENTS BODIE, KANE, MARCUS 5-46

47 5-47 Figure 5.7 Nominal and Real Equity Returns Around the World,

48 Figure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World, INVESTMENTS BODIE, KANE, MARCUS 5-48

49 Figure 5.9 Probability of Investment Outcomes After 25 Years with a Lognormal Distribution INVESTMENTS BODIE, KANE, MARCUS 5-49

50 5-50 Terminal Value with Continuous Compounding When the continuously compounded rate of return on an asset is normally distributed, the effective rate of return will be lognormally distributed. Remember: E so Geom. Avg EArithm.Avg m g 0.5 The Terminal Value will then be: EAR T = e g+0.5 σ2 T = e Tg+0.5Tσ 2 2

CHAPTER 5. Introduction to Risk, Return, and the Historical Record INVESTMENTS BODIE, KANE, MARCUS. McGraw-Hill/Irwin

CHAPTER 5. Introduction to Risk, Return, and the Historical Record INVESTMENTS BODIE, KANE, MARCUS. McGraw-Hill/Irwin CHAPTER 5 Introduction to Risk, Return, and the Historical Record McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved. 5-2 Interest Rate Determinants Supply Households