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 Demand Businesses Government s Net Supply and/or Demand Federal Reserve Actions
5-3 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?) Let R = nominal rate, r = real rate and i = inflation rate. Then: r R i 1 Solve: r r 1 1 R i R i 1 i
5-4 Equilibrium Real Rate of Interest Determined by: Supply Demand Government actions Expected rate of inflation
5-5 Figure 5.1 - Real Rate of Interest Equilibrium
5-6 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
5-7 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 after tax inflation-adjusted As intuition suggests, the after-tax, real rate of return falls as the inflation rate rises.
Rates of Return for Different Holding Periods 5-8 Zero Coupon Bond Par = $100 T = maturity P = price r f (T) = total risk free return P 100 1 r f T r f 100 T 1 P
Example 5.2 Time Does Matter: Use Annualized Rates of Return 5-9
5-10 Equation 5.7 EAR Time matters use EAR to annualize Effective Annual Rate 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
5-11 Equation 5.8 APR Annual Percentage Rate (APR): annualizing using simple interest 1 APR T 1 APR T EAR T 1 EAR 1 T
Investment End Value 5.00 5-12 4.50 4.00 End Value with APR=5.0% End Value with EAR=5.0% 3.50 3.00 2.50 2.00 1.50 1.00 0 5 10 15 20 25 30 (years)
5-13 Table 5.1 APR vs. EAR
5-14 Continuous Compounding Frequency of compounding matters At the limit to (compounding time) 0: 1 EAR e r cc
Investment End Value 5.00 5-15 4.50 4.00 End Value with APR=5.0% End Value with EAR=5.0% End Value with Rcc=5.0% 3.50 3.00 2.50 2.00 1.50 1.00 0 5 10 15 20 25 30 (years)
Let r=rate and x=compounding time End Value Make x very small. Then use A=e ln(a) How to derive R cc x0 lim 1 T N * x N T / 1 r * x1 r * x 1 r * x N compounding N times N r * x lim e ln 1r* x N S x0 x Substitute N=T/x lim x0 T e ln 1r* x x 1 T r 1r* x 1 lime e x0 Looks like 0/0. Use de l Hôpital rt Q.E.D. lime x0 d dx T ln Checks: r=0 End Value=1 T=0 End Value=1 d dx 1r* x x
Table 5.2 Statistics for T-Bill Rates, Inflation Rates and Real Rates, 1926-2009 5-17
5-18 Bills and Inflation, 1926-2009 Moderate inflation can offset most of the nominal gains on low-risk investments. One dollar invested in T-bills from1926 2009 grew to $20.52, but with a real value of only $1.69. Negative correlation between real rate and inflation rate means the nominal rate responds less than 1:1 to changes in expected inflation.
Figure 5.3 Interest Rates and Inflation, 1926-2009 5-19
5-20 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
5-21 Rates of Return: Single Period Example Ending Price = 110 Beginning Price = 100 Dividend = 4 HPR = (110-100 + 4 )/ (100) = 14%
Expected Return and Standard Deviation 5-22 Expected (or mean) returns E( r) p( s) r( s) s p(s) = probability of a state r(s) = return if a state occurs s = state
5-23 Scenario Returns: Example State Prob. of State r in State Excellent 0.25 0.3100 Good 0.45 0.1400 Poor 0.25-0.0675 Crash 0.05-0.5200 E(r) = (0.25)(0.31) + (0.45)(0.14) + (0.25)(-0.0675) + (0.05)(-0.52) = 0.0976 = 9.76% (think of a probability-weighted avg) NOTE: use decimals instead of percentages to be safe
5-24 Variance and Standard Deviation Variance (VAR): 2 2 s p( s) r( s) E( r) Standard Deviation (STD): STD 2
5-25 Scenario VAR and STD Example VAR calculation: σ 2 = 0.25(0.31-0.0976) 2 + 0.45(0.14-0.0976) 2 + 0.25(-0.0675-0.0976) 2 + 0.05(-0.52-0.0976) 2 = = 0.038 Example STD calculation: 0.038 0.1949
Time Series Analysis of Past Rates of Return 5-26 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
5-27 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
Geometric Variance and Standard Deviation Formulas Recall the definition of variance 5-28 2 2 s p( s) r( s) E( r) Estimated Variance = expected value of squared deviations (from the mean) ˆ 2 1 n n s1 2 rs r
Geometric Variance and Standard Deviation Formulas Using the estimated r avg instead of the real E(r) introduces a bias: 5-29 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*: ˆ 1 n 1 n j1 2 rs r * More at http://en.wikipedia.org/wiki/unbiased_estimation_of_standard_deviation
The Reward-to-Volatility (Sharpe) Ratio 5-30 Sharpe Ratio for Portfolios: Risk Premium SD of Excess Returns
5-31 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
5-32 Figure 5.4 The Normal Distribution
5-33 Normality and Risk Measures What if excess returns are not normally distributed? Standard deviation is no longer a complete measure of risk Sharpe ratio is not a complete measure of portfolio performance Need to consider skew and kurtosis
5-34 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
Figure 5.5A Normal and Skewed Distributions 5-35
Figure 5.5B Normal and Fat-Tailed Distributions (mean = 0.1, SD =0.2) 5-36
5-37 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 5 th percentile when returns are sorted from high to low. Also referred to as 95%-ile (depends on perspective)
2.5 Normal Distribution and VaR 5-38 2 Percentile 1.5 1 0.5 VaR 0-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0
5-39 Expected Shortfall (ES) a.k.a. Conditional Tail Expectation (CTE) More conservative measure of downside risk than VaR: VaR takes the highest return from the worst cases Real life distributions are asymmetric and have fat tails ES takes an average return of the worst cases
2.5 Normal Distribution, VaR, and Expected Shortfall 5-40 2 The area is the percentile 1.5 1 0.5 Expected Shortfall VaR 0-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0
Lower Partial Standard Deviation (LPSD) and the Sortino Ratio 5-41 Issues with real life returns: Need to look at negative returns separately to account for asymmetry and fat tails Need to consider excess returns: deviations of returns from the risk-free rate. LPSD: similar to usual standard deviation, but uses only negative deviations from r f Sortino Ratio replaces Sharpe Ratio
5-42 A game with a coin Let s play a game: flip a (non-fair) 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?
5-43 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?
5-44 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?
5-45 A Portfolio of 2 stocks Portfolio = 0.5 * A + 0.5 * 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?
5-46 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?
1926-2009 5-47 S (A) (B) (C) (D) (E) 30% (A) 50% (B) 20% (D)
5-48 Historic Returns on Risky Portfolios Returns appear approximately normally distributed Returns are lower over the most recent half of the period (1986-2009) SD for small stocks became smaller; SD for long-term bonds got bigger Better diversified portfolios have higher Sharpe Ratios Negative skew
Figure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000 5-49
Figure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World, 1900-2000 5-50
Figure 5.9 Probability of Investment Outcomes After 25 Years with a Lognormal Distribution 5-51
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 1/ 2 The Terminal Value will then be: 2 T 2 1 EAR e g1/ 2 e Tg T / 2 2 T 1/ 2 2 5-52
Figure 5.10 Annually Compounded, 25-Year HPRs 5-53
Figure 5.11 Annually Compounded, 25-Year HPRs 5-54
5-55 Figure 5.12 Wealth Indices of Selected Outcomes of Large Stock Portfolios and the Average T-bill Portfolio