Risk and return (1) Class 9 Financial Management, 15.414MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Today Risk and return • Statistics review • Introduction to stock price behavior Reading • Brealey and Myers, Chapter 7, p. 153 – 165MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Road map Part 1. Valuation Part 2. Risk and return Part 3. Financing and payout decisions 3MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Cost of capital DCF analysis NPV = CF0 + CF1 + CF2 CF3 CF4 CF5+ + + + ...r) (1 r) (1 2 r) (1 3 r) (1 4 r) (1 5+ + + + + r = opportunity cost of capital The discount rate equals the rate of return that investors demand on investments with comparable risk. Questions How can we measure risk? How can we estimate the rate of return investors require for projects with this risk level? 4MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Examples In November 1990, AT&T was considering an offer for NCR, the 5th largest U.S. computer maker. How can AT&T measure the risks of investing in NCR? What discount rate should AT&T use to evaluate the investment? From 1946 – 2000, small firms returned 17.8% and large firms returned 12.8% annually. From 1963 – 2000, stocks with high M/B ratios returned 13.8% and those with low M/B ratios returned 19.6%. What explains the differences? Are small firms with low M/B ratios riskier, or do the patterns indicate exploitable mispricing opportunities? How should the evidence affect firms’ investment and financing choices? 5MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Background The stock market Primary market New securities sold directly to investors (via underwriters) Initial public offerings (IPOs) Seasoned equity offerings (SEOs) Secondary market Existing shares traded among investors Market makers ready to buy and sell (bid vs. ask price) Market vs. limit orders NYSE and Amex: Floor trading w/ specialists NASDAQ: Electronic market Combined: 7,022 firms, $11.6 trillion market cap (Dec 2002) 6MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Background Terminology Realized return Prt = Dt + Pt − Pt 1-= tt P PPP D−+ 1-t1-t 1-t (DY + cap gain) t 1 -Expected return = best forecast at beginning of period E[rt] = E[D ] + E[P − P1-t ] P t t 1-t Risk premium, or expected excess return Risk premium = E[rt] – rf 7MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Statistics review Random variable (x) Population parameters mean = µ = E[x] variance = σ2 = E[(x – µ)2], standard deviation = σ skewness = E[(x – µ)3] / σ3 Sample of N observations 1sample mean = x =∑ixiN sample variance = s2 = N1- 1∑i(xi − x)2 , standard deviation = s sample skewness 8MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Example 0.45 0.36 0.27 0.18 0.09 0.00 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 µ = 0, σ = 1, skew = 0 µ = 0, σ = 2, skew = 0 [Probability density function: shows probability that x falls in an given range] 9MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Example 0.24 0.18 0.12 0.06 0.00 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 µ = 0, σ = 2, skew = 1.3 µ = 0, σ = 2, skew = 0 10MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Statistics review Other statistics Median 50th percentile: prob (x < median) = 0.50 Value-at-Risk (VaR) How bad can things get over the next day (or week)? 1st or 5th percentile: prob (x < VaR) = 0.01 or 0.05 ‘We are 99% certain that we won’t lose more than $Y in the next 24 hours’ 11MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Example 0.24 0.18 0.12 0.06 0.00 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 µ = 0, σ = 2, skew = -1.3 mean median 5th pctile 12MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Statistics review Normal random variables Bell-shaped, symmetric distribution x ~ N(µ, σ2) x is normally distributed with mean µ and variance σ2 ‘Standard normal’ mean 0 and variance 1 [or N(0, 1)] Confidence intervals 68% of observations fall within +/–1 std. deviation from mean 95% of observations fall within +/–2 std. deviations from mean 99% of observations fall within +/–2.6 std. deviations from mean 13MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Example 0.50 0.40 0.30 0.20 0.10 0.00 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 N(0, 1) 14MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Statistics review Estimating the mean Given a sample x1, x2, …, xN Don’t know µ, σ2 ⇒ estimate µ by sample average x estimate σ2 by sample variance s2 How precise is x ? std dev (x) ≈ s / N 95% confidence interval for µ x – 2sN x x + 2s N 15MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Application From 1946 – 2001, the average return on the U.S. stock market was 0.63% monthly above the Tbill rate, and the standard deviation of monthly returns was 4.25%. Using these data, how precisely can we estimate the risk premium? Sample: x = 0.63%, s = 4.25%, N = 672 months Std dev ( x ) = 4.25 / 672 = 0.164% 95% confidence interval Lower bound = 0.63 – 2 × 0.164 = 0.30% Upper bound = 0.63 + 2 × 0.164 = 0.96% Annual (× 12): 3.6% < µ < 11.5% 16MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Statistics review Two random variables How do x and y covary? Do they typically move in the same direction or opposite each other? Covariance = σx,y = E[(x – µx)(y – µy)] If σx,y > 0, then x and y tend to move in the same direction If σx,y < 0, then x and y tend to move in opposite directions Correlation = ρx,y = covariance = σy , x stdev ⋅ stdev σ σ x yx y -1 ≤ ρx,y ≤ 1 17MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 18Correlation -4-3-2-101234-4 -3 -2 -1 0 1 2 3 4-4-3-2-10123-4 -3 -2 -1 0 1 2 3 4 -4-3-2-101234-4 -3 -2 -1 0 1 2 3 4-4-3-2-10123-4 -3 -2 -1 0 1 2 3 4 ρ = 0 ρ = .8 ρ = .5 ρ = –.5MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Properties of stock prices Time-series behavior How risky are stocks? How risky is the overall stock market? Can we predict stock returns? How does volatility change over time? 19MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 9 Stocks, bonds, bills, and inflation Basic statistics, 1946 – 2001 Monthly, % Series Avg Stdev Skew Min Max Inflation 0.32 0.36 0.82 -0.84 1.85 Tbill (1 yr) 0.38 0.24 0.98 0.03 1.34 Tnote (10 yr) 0.46 2.63 0.61 -7.73 13.31 VW stock index 1.01 4.23 -0.47 -22.49 16.56 EW stock index 1.18 5.30 -0.17 -27.09 29.92 Motorola 1.66 10.02 0.01 -33.49 41.67 20MIT SLOAN