The primary financial goal is shareholder wealth maximization, which translates to maximizing stock price. Maximize stock value by: ◦ Forecasting and planning ◦ Investment and financing decisions ◦ Coordination and control ◦ Transactions in the financial markets ◦ Managing risk 1
It is the reward for undertaking the investmentThe Two Components of Return1. Yield: The income component of a security’s return2. Capital gain (loss): The change in price on a security over some period of timePutting The Two Components Together Total return = Yield + Price changeWhere: the yield components can be 0 or + the price change components can be 0, +, or - 2
“Risk comes from not knowing what you are doing” Warren Buffet The chance that the actual outcome from an investment will differ from the expected outcome. Future returns from an investment are unpredictable Risk = Probability of occurrence * Impact on objects 3
Market Risk Business Risk Liquidity Risk Exchange Rate Risk or Currency Risk Country Risk Execution Risk Interest Rate Risk Re-Investment Risk Inflation Risk 4
Total Return (TR)Percentage measure relating all cash flows on asecurity for a given time period to its purchase priceTR= Any cash payment received + Price change over the period Price at which the asset is purchasedHow to Calculate Total ReturnTR= CFt +(PE - PB) = CFt +PC PB PB 6
Example: 100 shares of data shield are purchased at $30 per share and sold one year later at $35 per share. A dividend of $2 per share is paid. Stock TR = 2+(35-30)/30 = 2+(5)/30 = 0.2333 or 23.33% 7
Example: Assume the purchase of a 10% coupon Treasury bond at a price of $960, held 1 year, and sold for $1020. The TR is Bond TR = 100+(1020-960)/960 = 100+60/960 = 0.1667 or 16.67% 8
Year S&P 500 TRs (%)1990 -3.141991 30.001992 7.431993 9.941994 1.291995 37.111996 22.681997 33.101998 28.341999 20.88 Source: Jones, Charles P, Investment; P. 148; 10th ed. ; National Book Foundation 2010 9
Arithmetic Mean = X = S x/ n = [-3.14+30+…+20.88]/10 = 187.63/10 = 18.76Geometric Mean=G=[(1+TR1)(1+TR2)…(1+TRn)]1/n - 1=[(.9687)(1.30001)(1.07432)(1.09942)(1.01286)(1.37113) (1.22683)(1.33101)(1.28338)(1.2088)]1/10 - 1 =1.18-1 =0.18 or 18% 10
Cumulative Wealth Index- Cumulative wealth over time given an initial wealthand a series of returns on some assets CWIn = WI0 (1+TR1)(1+TR2)…(1+TRn)Where CWIn = the cumulative wealth index as of the end of period n WI0 = the beginning index value, typically $1 TR1,n = the periodic TRs in decimal formCWI90-99 =1.00(.9687)(1.30001)(1.07432)(1.09942)(1.01286) (1.37113)(1.22683)(1.33101)(1.28338)(1.2088) = 5.2342 11
The future is uncertain. Investors do not know with certainty whether the economy will be growing rapidly or be in recession. Investors do not know what rate of return their investments will yield. Therefore, they base their decisions on their expectations concerning the future. The expected rate of return on a stock represents the mean of a probability distribution of possible future returns on the stock. 12
Given a probability distribution of returns, the expected return can be calculated using the following equation: N E[R] = S (piRi) i=1 Where: ◦ E[R] = the expected return on the stock ◦ N = the number of states ◦ pi = the probability of state i ◦ Ri = the return on the stock in state i. 13
Risk reflects the chance that the actual return on an investment may be different than the expected return. One way to measure risk is to calculate the variance and standard deviation of the distribution of returns. Var(R) = s2 = S pi(Ri – E[R])2 i=1 Where: ◦ N = the number of states ◦ pi = the probability of state i ◦ Ri = the return on the stock in state i ◦ E[R] = the expected return on the stock 14
The standard deviation is calculated as the positive square root of the variance:SD(R) = s = s2 = (s2)1/2 = (s2)0.5 15
The ratio of the standard deviation of adistribution to the mean of that distribution. It is a measure of RELATIVE risk. CV =s/E(R)
Probability Distribution:State Probability Return On Return On Stock X Stock Y 1 20% 5% 50% 2 30% 10% 30% 3 30% 15% 10% 4 20% 20% -10% 17
State ofEconomy Prob. Return X Pi X (in %) Pi*Ri (Ri – E[R]) (Ri – E[R])2 pi(Ri – E[R])2 1 0.25.00 1.00(7.50) 56.25 11.25 2 0.310.00 3.00(2.50) 6.25 1.875 3 0.315.00 4.502.50 6.25 1.875 4 0.220.00 4.007.50 56.25 11.25 SUM 12.5 26.25 E[R] 12.50 Variance 26.25 SD 5.12 CV 0.41 18
State ofEconomy Prob. Return Pi Y (in %) Pi*Ri (Ri – E[R]) (Ri – E[R])2 pi(Ri – E[R])2 1 0.250.00 10.0037.50 1,406.25 281.25 2 0.330.00 9.0017.50 306.25 91.88 3 0.310.00 3.00(2.50) 6.25 1.88 4 0.2(10.00) -2.00(22.50) 506.25 101.25 SUM 20.00 476.25 E[R] 20.00 Variance 476.25 SD 21.82 CV 1.09 19
The variance and standard deviation for stock X is calculated as follows: E[R]X = .2(5%) + .3(10%) + .3(15%) + .2(20%) = 12.5%s2X = .2(.05 -.125)2 + .3(.1 -.125)2 + .3(.15 -.125)2 + .2(.2 -.125)2 = .002625sX = (.002625)0.5 = .0512 = 5.12%CV = 5.12/12.5 = 0.41 20
E[R]Y = .2(50%) + .3(30%) + .3(10%) + .2(-10%) =20 %s2y = .2(.50 -.20)2 + .3(.30 -.20)2 + .3(.10 -.20)2 + .2(-.10 - .20)2 = .042sy = (.042)0.5 = .2049 = 20.49%CV = 20.49 / 20 = 1.09 Although Stock Y offers a higher expected return than Stock X, it is also riskier since its variance and standard deviation are greater than Stock Xs. 21
Certainty Equivalent (CE) is the amount of cashsomeone would require with certainty at a point in time to make the individual indifferent between that certain amount and an amount expected to be received with risk at the same point in time.
Certainty equivalent > Expected value Risk PreferenceCertainty equivalent = Expected value Risk IndifferenceCertainty equivalent < Expected value Risk Aversion Most individuals are Risk Averse.
Risk Attitude Example You have the choice between (1) a guaranteed dollar reward or (2) a coin-flip gamble of $100,000 (50% chance) or $0 (50% chance). The expected value of the gamble is $50,000.• Mary requires a guaranteed $25,000, or more, to call off the gamble.• Raleigh is just as happy to take $50,000 or take the risky gamble.• Shannon requires at least $52,000 to call off the gamble.
What are the Risk Attitude tendencies of each?Mary shows “risk aversion” because her“certainty equivalent” < the expected value ofthe gamble.Raleigh exhibits “risk indifference” because her“certainty equivalent” equals the expected valueof the gamble.Shannon reveals a “risk preference” because her“certainty equivalent” > the expected value ofthe gamble.
Weak Form ME: ◦ fully reflect all currently available security market data about price and volume. Semi Strong ME: ◦ fully reflect all publically available information Strong Form ME; ◦ fully reflect all information from both public and private 8- 26
The Expected Return on a Portfolio is the weighted average of the expected returns on the stocks which comprise the portfolio. This can be expressed as follows: N E[Rp] = S wiE[Ri] i=1 Where: ◦ E[Rp] = the expected return on the portfolio ◦ N = the number of stocks in the portfolio ◦ wi = the proportion of the portfolio invested in stock i ◦ E[Ri] = the expected return on stock i 27
The variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together. ◦ Covariance is a measure that combines the variance of a stock’s returns with the tendency of those returns to move up or down at the same time other stocks move up or down. ◦ Correlation coefficient, is often used to measure the degree of co-movement between two variables. The correlation coefficient simply standardizes the covariance. ◦ Its range is from –1.0 (perfect negative correlation), through 0 (no correlation), to +1.0 (perfect positive correlation). 28
The Covariance between the returns on two stocks can be calculated as follows: NCov(RX,RY) = sX,Y = S pi(RXi - E[RX])(RYi - E[RY]) i=1 Where: ◦ sX,Y = the covariance between the returns on stocks X and Y ◦ N = the number of states ◦ pi = the probability of state i ◦ RXi = the return on stock X in state i ◦ E[RX] = the expected return on stock X ◦ RYi = the return on stock Y in state i ◦ E[RY] = the expected return on stock Y 29
The Correlation Coefficient between the returns on two stocks can be calculated as follows: sX,YCorr(RX,RY) = rX,Y = sX sy Where: ◦ rX,Y =the correlation coefficient between the returns on stocks X and Y ◦ sX,Y =the covariance between the returns on stocks X and Y, ◦ sX =the standard deviation on stock X, and ◦ sy =the standard deviation on stock Y 30
The covariance between stock X and stock Y is as follows:sX,Y = .2(.05-.125)(.5-.2) + .3(.1-.125)(.3-.2) + .3(.15-.125)(.1-.2) +.2(.2-.125)(-.1-.2) = -.0105 The correlation coefficient between stock X and stock Y is as follows: -.0105rX,Y = (.0512)(.2049) = -1.00 31
Most investors do not hold stocks in isolation. Instead, they choose to hold a portfolio of several stocks. When this is the case, a portion of an individual stocks risk can be eliminated, i.e., diversified away. From our previous calculations, we know that: ◦ the expected return on Stock X is 12.5% ◦ the expected return on Stock Y is 20% ◦ the variance on Stock X is .00263 ◦ the variance on Stock Y is .04200 ◦ the standard deviation on Stock X is 5.12% ◦ the standard deviation on Stock Y is 20.49% 32
Using either the correlation coefficient or the covariance, the Variance on a Two-Asset Portfolio can be calculated as follows:s2p = (wA)2s2x + (wy)2s2y + 2wxwy rX,Y sxsy ORs2p = (wA)2s2x + (wy)2s2y + 2wxwy sx,y The Standard Deviation of the Portfolio equals the positive square root of the variance. 33
Expected Return, Variance and standard deviation of a Two Asset portfolio:Investment Proportions: 75% stock X and 25% stock Y: E[Rp] = 0.75 (0.125) +0.25(0.20) =0.14375 or 14.375%s2p =(.75)2(.0512)2+(.25)2(.2049)2+2(.75)(.25)(-1)(.0512)(.2049) = .00026sp = .00016 = .0128 = 1.28% 34
Summary of the Portfolio Return andRisk Calculation X Y Portfolio Weights 0.75 0.25 E [R] 12.50 20.00 14.375 Variance 26.25 476.25 2.60 SD 5.12 21.82 1.613 CV 0.4099 1.0912 0.1122 Corr (x,y) -1.00
Diversification and the Correlation Coefficient Combination SECURITY X SECURITY Y X and YINVESTMENT RETURN TIME TIME TIME
RF asset has zero SD and zero correlation of returns with risky Portfolio SD of Portfolio = (Wa) (SDa) 8- 38
While diversification of portfolio, there are two kinds of risk we will deal with:- Unsystematic Risk- Systematic Risk- So,Total Risk = Unsystematic Risk + Systematic Risk
An unsystematic risk, also called diversifiable, unique, firm specific risk, is one that is particular to a single asset or, at most, a small group.For example, if the asset under consideration is stock in a single company,- The positive NPV projects( successful new products, cost saving) will tend to increase the value of stock.- Unanticipated lawsuits, industrial accidents, strikes etc will decrease the FCF’s and thereby decrease the share values.
Unsystematic risk is essentially eliminated by diversification, so a portfolio with many assets has almost no unsystematic risk.
Systematic risk, also called as undiversifiable, unavoidable, market risk, is due to factors that affect overall market, such as,- Changes in nation’s economy- Tax reforms- Change is world energy situation These are the risks that affect securities overall( whether in a portfolio or single) and, consequently, cannot be diversified away. An investor who holds a well-diversified portfolio will be exposed to this type of risk.
Total Risk = Systematic Risk + Unsystematic RiskSTD DEV OF PORTFOLIO RETURN Unsystematic risk Total Risk Systematic risk NUMBER OF SECURITIES IN THE PORTFOLIO
The systematic risk principle states that the reward for bearing risk depends only on the systematic risk of an investment.- The underlying rationale for this principle is straight forward: because unsystematic risk can be eliminated at virtually no cost(by diversifying), there’s no reward for bearing it.- No matter how much total risk an asset has, only the systematic risk is relevant determining the expected return and risk premium on that asset.
Beta is an index of systematic risk. Beta (β) of a stock or portfolio is a number describing the relation of its returns with those of the market as a whole. The sensitivity of an asset’s return on the market index. A measure of the volatility, or systematic risk, of a security or a portfolio in comparison to the market as a whole. Beta is a standardized measure of the covariance of the asset’s return with the market return. 45
Beta = Covariance of Asset’s return with market return / variance of market return = Cov im/ s2m The beta of a portfolio is simply a weighted average of the individual stock betas in the portfolio.
bp = Weighted average = 0.5(bX) + 0.5(bY) = 0.5(1.29) + 0.5(-0.86) = 0.22 47
The typical analysis involves either monthly or weekly returns on the stocks and on the market index for 3-5 years. Many analysts use the S&P 500 to find the market return. Analysts typically use four or five years’ of monthly returns to establish the regression line like Merill Lynch. Some analysts use 52 weeks of weekly returns like Value Line. Go to http://finance.yahoo.comEnter the ticker symbol for a “Stock Quote”, such as IBM or Dell, then click GO. 48
•Obtaining Betas • Can use historical data if past best represents the expectations of the future.•Adjusted Beta • There appears to be a tendency for the measured betas of individual securities to revert eventually toward the beta of the market portfolio. • This might be due to the economic factors affecting the operations and financing of the firm.
A line that describes the relationship between an individual security’s returns and returns on the market portfolio. It is useful to deal with returns in excess of the risk free rate. The excess return is simply the expected return less the risk free return. There are two ways of determining the relationship b/w excess return on stock and market portfolio.- Historical data (with the assumption that relationship will continue in future)- Security Analysts.
Characteristic Line EXCESS RETURN Unsystematic Risk ON STOCK RiseBeta = Run EXCESS RETURN ON MARKET PORTFOLIO Characteristic Line
Greater the slope, greater the systematic risk. Alpha is intercept of characteristic line on vertical axis. If excess returns for market portfolio were zero, alpha would be the expected excess return for the stock. Beta is slope of characteristic Line = Rise/Runi.e. Change in Stock’s Return/ Change in Market Return The monthly returns are calculated as:(Div paid) + (Ending price – beginning price)/Beginning price
Characteristic Lines and Different Betas EXCESS RETURN Beta > 1 ON STOCK (aggressive) Beta = 1 Each characteristic line has a Beta < 1 different slope. (defensive) EXCESS RETURN ON MARKET PORTFOLIO
Aggressive Investment: A slope steeper than 1 means that the stock’s excess return varies more than proportionally with the excess return of market portfolio, it has more systematic risk than market. Defensive Investment: A slope less than 1 means that the stock’s excess returns varies less than proportionally with the excess return of the market portfolio. It has less systematic risk than market.
If b = 1.0, stock has average risk. If b > 1.0, stock is riskier than average. If b < 1.0, stock is less risky than average. Most stocks have betas in the range of 0.5 to 1.5. A positive beta means that the assets returns generally follow the markets returns, in the sense that they both tend to be above their respective averages together, or both tend to be below their respective averages together. A negative beta means that the assets returns generally move opposite the markets returns: one will tend to be above its average when the other is below its average 55
The CAP model was introduced by Jack Treynor, John Lintner, William Sharpe and Jan Mossin in the early 1960’s. According to CAP model the investor needs to be compensated in two ways, for time value of money (risk free rate) and for taking systematic risk. In a competitive market, the expected risk premium varies in direct proportion to beta. This model states the linear relationship between risk (systematic) and expected (required) return. A security’s expected return is risk free rate plus a premium based on the systematic risk of security. Rj = Rf + bj(RM – Rf)
Capital markets are efficient. Homogeneous investor expectations over a given period. Investors all think in terms of a single holding period. There are no taxes and no transactions costs. All investors are price takers, that is, investors buying and selling won’t influence stock prices. Quantities of all assets are given and fixed. Risk-free asset return is certain. Market portfolio contains only systematic risk (use S&P 500 Index or similar as a proxy). Investors can borrow or lend unlimited amounts at the risk-free rate.
The least risky investment is T-bills, since the return on them is fixed, it is unaffected by what happens to the market. (beta = 0), The riskier investment is market portfolio of common stocks (average beta = 1) Risk premium(excess return) is expected returns minus risk free return. The relationship between systematic risk and expected return in financial markets is usually called the security market line (SML).
The relationship between an individual security’s expected rate of return and it’s systematic risk as measured by beta will be linear, this relationship is called as Security Market Line.Required Return RM Risk Premium Rf Risk-free Return bM = 1.0 Systematic Risk (Beta)
Now, if everyone holds the market portfolio, and if beta measures each security’s contribution to the market portfolio risk, then it’s no surprise that the risk premium demanded by investors is proportional to beta. This is what the CAPM says!
What if a stock does not lie on SML? Stock X (Underpriced) Required Return Direction of Movement Direction of Movement Rf Stock Y (Overpriced) Systematic Risk (Beta)
Investors require some extra return for taking risk, that is why common stocks are given higher returns on average than t-bills. Investors are not concerned with those risks that they cannot diversify, hence the systematic risk the relevant risk only.
Maturity of Risk free Security :CAPM is one period model and investors are concerned about the long term capital investment returns. Faulty use of the market index CAPM/SML concepts are based on expectations, yet betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness.