Finance Lecture6

MBAM 614 Finance Capital Markets II • MBAM614 Class 6 - 1 Summary of Last Class 1. Percentage Return = Dt+1 + (Pt+1 - Pt) Pt 2. 3. 4. 5. Percentage Return = Dividend Yield + Capital Gains Yield (1 + R) = (1 + r) * (1 + h) THERE IS A REWARD FOR BEARING RISK The historic average risk premium for bearing average risk, the average Excess Return, is 9.2% • MBAM614 Class 6 - 2 1 Agenda 1. 2. 3. 4. 5. Measuring Variability Variance and Standard Deviation Random Variables and the Normal Distribution Efficient Markets Expected Returns and Variance Revisited • MBAM614 Class 6 - 3 U.S. Stock Returns 60 Average Return = 13.0% 40 20 R etu rn (% ) 0 29 32 35 62 65 68 71 59 38 41 53 56 74 77 44 47 50 80 83 89 92 26 19 19 19 19 -20 -40 -60 -What should we expect as a return? -How likely is it that we will be wrong? -If we’re wrong, how wrong are we likely to be? How much RISK is there? Function of average variation Year • MBAM614 19 86 Class 6 - 4 95 2 Mean Return Average return might be a good guess at expected future return T 1 R = R T t=1 t Σ = [R1 + R2 + R3 + … + Rt-1 + Rt] / T • MBAM614 Class 6 - 5 Average Variation A statistical measure of average variability is variance To find the variance of T returns: – First, find the mean – Second, find all the deviations from the mean and square them – Finally, divide by T-1 (sort of an “average”) A related measure is the Standard Deviation (or S.D.) which is just the square root of the variance S.D. is often preferred since it is expressed in % rather than %2 as variance is! • MBAM614 Class 6 - 6 3 Variance and Standard Deviation Variance Var(R) = σ2 T 1 [ R - R ]2 = T-1 t=1 t Σ Standard Deviation σ = √ Var(R) • MBAM614 Class 6 - 7 Example: Recent Stock Returns Actual Year Return 1994 0.0131 1995 0.3743 1996 0.2307 1997 0.3336 Total 0.9517 Avg. Return 0.2379 0.2379 0.2379 0.2379 Deviation -0.2248 0.1364 -0.0072 0.0957 0.0001 Squared Deviation 0.05054 0.01860 0.00005 0.00916 0.07835 due to rounding - Var(R) = σ2 = 0.07835/(4-1) = 0.0261 σ = √ Var(R) = √ 0.0261 = 0.1616 • MBAM614 Class 6 - 8 4 Historical R and Var(R) Historical Results: 1926-1997 Avg. Return 13.0% 6.1% 5.6% 17.7% 3.8% SD 20.3% 8.7% 9.2% 33.9% 3.2% S&P 500 Common Stocks Corporate Bonds Long U.S. Bonds Small Stocks T-Bills Highest Average Return Highest Volatility or Risk The Greater the Potential Reward, the Greater the Risk Based on historical risk premiums, “average risk” investments should return about 9.2% above the T-Bill rate • MBAM614 Class 6 - 9 Using R and Var(R) Think of stock price changes as random (in the sense that we can’t predict them due to random events) Returns (or ∆P/P) are observations of a random variable The probability that the value an observation on a random variable will be in a certain range is a function of the variable’s probability distribution function • MBAM614 Class 6 - 10 5 Normal Distribution 0.04 occurs frequently in nature math. easy to deal with completely described by mean and variance often a good approximation ±1 σ 66% of the time 0.03 Probability A random drawing will be within a certain range of the mean: 0.02 0.01 ±2 σ 95% of the time 0 0 1 2 3 0. 55 0. 65 Class 6 - 12 -4 -3 -2 Standard Deviations from Mean • MBAM614 Class 6 - 11 Frequency Distribution of Stock Returns 1926 - 1997 10 9 8 7 6 5 4 3 2 1 0 -0 .3 5 -0 .2 5 -0 .0 5 -0 .1 5 0. 05 0. 35 0. 15 0. 25 0. 45 Frequency -7.4% • MBAM614 13.0% Return -1 33.3% 4 6 Guessing the Future Given probabilities of future events, we can find measures of central tendency and dispersion (similar to mean and variance) – Say there are precisely S outcomes for an investment (usually called “states of the world” by economists) – Also, if “state” i occurs, the return will be Ri – The probability of state i occurring is pi • MBAM614 Class 6 - 13 Expected Return and Variance The Expected Return, E(R), is defined as E(R) = Σ (pi * Ri) i=1 S The Return Variance, Var(R), is defined as Var(R) = σ2 = Σ (pi * [ Ri - E(R) ]2) i=1 Class 6 - 14 S • MBAM614 7 Example: Economic Growth You are considering some stock in Sears but are unsure of the expected return. Since Sears’ performance depends on consumer spending which in turn depends on economic growth, you have made the following projections: State of Economy GNP +1% GNP +2% GNP +3% Probability of state .25 .50 .25 1.00 Return if state occ. Product -0.0125 0.0750 0.0875 E(R) = 0.1500 × × × -0.05 0.15 0.35 Expected Risk Premium = Expected Return - Risk-free rate E(risk premium) = E(R) - Rf • MBAM614 Class 6 - 15 Example: Sears’ Return Variance State of Economy GNP +1% GNP +2% GNP +3% Probability of state .25 .50 .25 1.00 Return if state occ. Dev -0.20 0.00 0.20 0.00 Sq. Dev 0.04 0.00 0.04 Product 0.01 0.00 0.01 σ2 = 0.02 × × × -0.05 0.15 0.35 S.D. = σ = 0.141 or 14.1% • MBAM614 Class 6 - 16 8 Returns on Risky Assets Since E(risk premium) = E(R) - Rf E(R) = Rf + E(risk premium) Risk Free: E(Rf) = Rf + E(risk-free risk premium) = Rf E(risk-free risk premium) = Rf - Rf = 0 Market: E(RM) = Rf + E(Market risk premium) E(Market risk premium) = E(RM) - Rf What is the expected return on other Risky Assets? • MBAM614 Class 6 - 17 Returns on Risky Assets Plot E(R) vs Risk E(R) E(RM) A Measure of RISK Rf E(R) = Rf + β [E(RM) - Rf] Called the Capital Asset Pricing Model (CAPM) Risk • MBAM614 Class 6 - 18 9 Key Points 1. The greater the potential reward, the greater the risk 2. The mean and variance of past returns may be reasonable estimates for the expected value and variance of future returns 3. If returns are random and normally distributed then returns will be within one S.D. of the mean roughly 66% of the time, within 2 S.D.s roughly 95% of the time, and within 3 S.D.s more than 99% 4. E(R) = Σ S (pi * Ri) 5. Var(R) = i=1 Σ S (pi * [ Ri - E(R) ]2) i=1 • MBAM614 Class 6 - 19 MBAM 614 Finance Portfolios & Diversification • MBAM614 Class 6 - 20 10 Agenda 1. 2. 3. 4. 5. Portfolio Weights Portfolio Expected Return & Variance Covariance and Correlation A Closer Look at Risk Diversification • MBAM614 Class 6 - 21 Portfolio Weights A Portfolio is a collection of securities (stocks, bonds,…) May be described by the percentage of the portfolio’s total value invested in each security These percentages, xi , are the Portfolio Weights Eg. You have invested $6,000 in AT&T (T) and $4,000 in Cisco Systems (CSCO). Find the portfolio weights. Total Portfolio Value = $6,000 + $4,000 = $10,000 xT = weight of AT&T = $6,000/$10,000 = 0.60 or 60% xCSCO = weight of CSCO = $4,000/$10,000 = 0.40 or 40% • MBAM614 Class 6 - 22 11 Example: Portfolio Expected Returns If you expect to earn 10% on you AT&T investment and 12% on your Cisco investment over the next year, what is you portfolio’s expected return? E($ Return on T) = E(RT) * $6,000 = 10% * $6,000 = $600 E($ Return on CSCO) = E(RCSCO) * $4,000 = 12% * $4,000 = $480 E($ Return on Portfolio) = $600 + $480 = $1,080 E(Rp) = E($ Return on Portfolio)/$10,000 = $1,080/$10,000 = 10.8% = [E(RT) * $6,000 + E(RCSCO) * $4,000]/$10,000 = [E(RT) * $6,000/$10,000] + [E(RCSCO) * $4,000/$10,000] = [xT * E(RT)] + [xCSCO * E(RCSCO)] = [0.6 * 0.1] + [0.4 * 0.12] = 0.108 or 10.8% • MBAM614 Class 6 - 23 Portfolio Expected Returns The Expected Return to a Portfolio is the sum of the product of the individual securities’ expected returns and their portfolio weights N E(Rp) = Σ [xi * E(Ri)] i=1 Often referred to as a Weighted Average • MBAM614 Class 6 - 24 12 Example: Portfolio Variance Consider a portfolio with an equal amount invested in Sears (S), CitiGroup (C), and Home Depot (HD). If your projections for possible economic outcomes and these stocks’ returns are as below, what is the portfolio expected return? What is the portfolio’s variance? State of Economy GNP +1% GNP +2% + GNP +3% + Expected Ret. Probability of state .25 .50 .25 Ret to Sears - 0.05 0.15 0.35 0.15 Ret to C Ret to HD Ret on Portfolio × × × × × × 0.00 0.10 0.20 0.10 × × × 0.20 0.10 0.00 0.10 × × × 0.050 0.117 0.183 0.117 Given the derived portfolio expected returns, we can calculate the portfolio variance • MBAM614 Class 6 - 25 Example: Portfolio Variance σ2S = 0.25(-0.05 - 0.15)2 + 0.50(0.15 - 0.15)2 + 0.25(0.35 - 0.15)2 = 0.02 σ2C = 0.25(0.00 - 0.10)2 + 0.50(0.10 - 0.10)2 + 0.25(0.20 - 0.10)2 = 0.005 σ2HD = 0.25(0.20 - 0.10)2 + 0.50(0.10 - 0.10)2 + 0.25(0.00 - 0.10)2 = 0.005 σ2Portfolio = 0.25(0.05 - 0.117)2 + 0.50(0.117 - 0.117)2 + 0.25(0.183 - 0.117)2 = 0.00221 < (0.02 + 0.005 + 0.005)/3 = 0.01 Notice Var(Rp) ≠ Σ [xi * Var(Ri)] i=1 Class 6 - 26 N • MBAM614 13 Covariance The Covariance between two asset’s returns is the expected value of the product of their deviations, state by state, from their expected returns. Probability of state .25 .50 .25 Expected Val. Ret to Dev fr Sears E(R) Dev2 - 0.05 0.15 0.35 0.15 E(RS) -0.20 0.00 0.20 0.04 0.00 0.04 0.02 σ 2S Ret to Dev fr HD E(R) Dev2 0.20 0.10 0.00 0.10 E(RHD) 0.10 0.00 -0.10 0.01 0.00 0.01 0.005 σ2HD Product -0.02 0.00 -0.02 -0.01 Cov(RS,RHD) = σS,HD = -0.01 • MBAM614 Class 6 - 27 Correlation The Correlation Coefficient, ρAB , measures the extent to which returns on two assets move together. – if both move up and down together, they are positively correlated and ρ > 0 – if one moves up when the other moves down and vice versa, they are negatively correlated and ρ < 0 – if they are completely independent, they are uncorrelated and ρ = 0 Cov(RA,RB) = σAB = σA σB ρAB or ρAB = σAB/σA σB ρS,HD = σS,HD/σS σHD = -0.01/√0.02√0.005 = -1.0 • MBAM614 Class 6 - 28 14 Perfect Positive Correlation Stock A Return Returns move up together Returns move down together Perfectly positively correlated ρ = +1 Stock B Time Perfect Positive Correlation RB When RA is plotted versus RB, all points fall exactly on a line Slope of line is >0 so ρ > 0 RA ρ = +1 • MBAM614 Class 6 - 29 Less Than Perfect Positive Correlation Returns usually move up together Returns usually move down together Stock A Stock B Not perfectly synch. Positively correlated 0 < ρ < +1 Return Time Less Than Perfect Positive Correlation The closer the points are to the line, the closer ρ is to +1 When RA is plotted versus RB, points fall around a line 0 < ρ < +1 Class 6 - 30 RA Slope of line is >0 so ρ > 0 RB • MBAM614 15 Perfect Negative Correlation Stock B Return Returns move in exactly opposite directions Perfectly Negatively correlated ρ = -1 Stock A Time Perfect Negative Correlation Slope of line is <0 so ρ < 0 RB When RA is plotted versus RB, all points fall exactly on a line ρ = -1 RA • MBAM614 Class 6 - 31 Less Than Perfect Negative Correlation Stock B Return Returns usually move in opposite directions Not perfectly synch. Stock A Negatively correlated -1 < ρ < 0 Time Less Than Perfect Negative Correlation Slope of line is <0 so ρ < 0 When RA is plotted versus RB, points fall around a line -1 < ρ < 0 RB The closer the points are to the line, the closer ρ is to -1 RA • MBAM614 Class 6 - 32 16 Zero Correlation Stock B Return Returns do not move together Returns are entirely unrelated Stock A Time ρ=0 Zero Correlation Slope of line is 4 so ρ = 0 RB When RA is plotted versus RB, points fall all over the place Any line seems to work ρ=0 RA Slope of line is 0 so ρ = 0 • MBAM614 Class 6 - 33 Example: Portfolio S.D. and Correlation For two securities, portfolio variance is actually σP2 = xA2σA2 + xB2σB2 + 2xAxBρΑΒσAσB Portfolio Weights xA = 6/13 xB = 7/13 ρΑΒ -1.0 0.0 +1.0 Standard Deviation σA = 35% σB = 30% Expected Return E(RA) = 28% E(RB) = 19% Stock A Stock B Notice whenever ρ is < +1, diversification reduces risk (σP) Whenever ρ=-1, risk can be eliminated! • MBAM614 σP 0.000% 22.845% 32.308% Portfolio standard deviation, or risk, depends on asset correlations Class 6 - 34 17 Portfolio Theory In practice, very few assets have ρ = +1 or ρ = -1 Diversification reduces risk but does not eliminate it The Opportunity Set is made up of every portfolio that can be formed by investing in some combination of every possible asset that can be purchased Narrowing this down to only those portfolios with the lowest risk for a given expected return leaves the Feasible Set The Minimum Variance Portfolio will have the lowest risk The Efficient Frontier or Efficient Set excludes all portfolios that fall below the minimum variance portfolio • MBAM614 Class 6 - 35 Possible Portfolios Efficient Frontier Portfolio Expected Return, E(RP) A Opportunity Set C B A dominates B because A is less risky with same expected return A dominates C because A has greater expected return but same risk Risk, σP Minimum Variance Portfolio Feasible Set • MBAM614 Class 6 - 36 18 Expected and Unexpected Returns Actual or Realized returns have two components – the Expected Return, E(R), the normal profit from an investment – the Unexpected Return, U, the risky aspect of investing only becomes known after the investment period due to “surprises” or unexpected events can be negative (bad events) or positive (good events) R = E(R) + U • MBAM614 Class 6 - 37 Announcements and Surprises An Announcement is the release of information not previously available in its entirety Announcements have two parts: – Expected News, used to formulate E(R) and set market prices – Unexpected News or Surprises, affects U and is news that was not previously reflected in prices Expected news is said to have been “Discounted” since it has already been factored into market prices • MBAM614 Class 6 - 38 19 Example: Diamond Fields Resources When DFR first made the Voisey Bay discovery, their share price jumped from around $0.30 to $3.00. When the preliminary test results were announced, share prices jumped again to around $30.00. - Although no test results had been announced, the market had certain expectations which were discounted into the value of DFR and the price rose to $3.00 per share. The value of the expected news was $2.70. - When the actual announcement was made, there was a BIG surprise. The value of the unexpected news was around $27.00 • MBAM614 Class 6 - 39 Risk Assets are risky because of surprises which are random and come in two forms: Systematic Risk or Market Risk, m - surprises that affect most or all assets (sometimes to varying degrees) Eg. Changes in GNP, interest rates, or inflation affect all firms Unsystematic Risk or Unique Risk, 0 - surprises that affect only a small number of assets Eg. A strike, a plant accident, a takeover, a CEO’s resignation • MBAM614 Class 6 - 40 20 Components of Return Total Return = Expected Return + Unexpected Return R = E(R) + U R = E(R) + Systematic Portion + Unsystematic Portion R = E(R) + m + 0 • MBAM614 Class 6 - 41 Portfolio Realized Returns Suppose we have a portfolio with N (a very large number) stocks. The realized return, RP, on the portfolio is the weighted average of the individual stock realized returns, Ri. Non-random expected returns with a non-0 avg. Portfolio Returns have an expected return component Portfolio Returns have a market or systematic risk component • MBAM614 R1 = E(R1) + m1 + 01 R2 = E(R2) + m2 + 02 R3 = E(R3) + m3 + 03 . . . RN = E(RN) + mN + 0N Random but related surprises with a non-0 avg. Random but unrelated surprises. These will average out to 0. Portfolio Returns have no unique or unsystematic risk component Class 6 - 42 21 Key Points 1. E(Rp) = Σ [xi * E(Ri)] i=1 N 2. A portfolio’s variance is, in general, less than the weighted average of its security variances and depends on the correlation between portfolio securities. 3. Total Return = Expected Return + Unexpected Return 4. R = E(R) + Systematic Portion + Unsystematic Portion 5. Diversification reduces or eliminates the Unsystematic Portion • MBAM614 Class 6 - 43 22