10_Oheads

10_Oheads - Lecture Notes 10 The Capital Asset Pricing...

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1 Lecture Notes 10 The Capital Asset Pricing Model Expected return, variance and standard deviation • The expected return on a security may be based: i. on the historical returns earned in the past, or ii. a fundamental analysis implying deviations from past returns. • We now want to contrast these two approaches. 1. Historical data • As we saw in the previous chapter, historical data on prices and dividends can be used to calculate: i. the percentage returns over T years R 1 , R 2 , R 3 , …, R T ; ii. the sample average percentage return R (1) iii. sample variance ( σ 2 ), and standard deviation ( σ ), of returns . (2) R R 1 R 2 R T +++ T -------------------------------------------- 1 T --- R t t 1 = T = σ 2 1 T 1 ------------ R t R () 2 t 1 = T =
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2 2. Fundamental analysis • In a fundamental analysis, we try to explain the key economic forces leading to stock price movements. • We build up a model of the underlying determinants of returns and use the model to forecast returns. • For example, as we noted at the beginning of the course, the business cycle is an important determinant of stock returns. • In particular, recessions correspond to the major periods of loss- es and booms the major periods of above-average gains. • We also examine economic factors affecting particular indus- tries (demand or cost shocks, new technologies or products), and then firms within each industry sector. • The fundamental approach is needed when the economic envi- ronment changes. Historical returns may then be of little use for forecasting future returns.
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3 A simple model based on fundamental analysis • We might postulate a number of possible “states”, or situations: i. four “business cycle states” ( depression,” “recession,” “nor- mal” and “boom”); ii. three “industry demand” states; iii. two “firm demand share states”; and iv. three “firm cost states”. In total, there would be 4 × 3 × 2 × 3=72 possible “states of the world.” • Let π s be the joint probability of state s = 1,…,72 and R s the like- ly return in state s . The expected return on the stock would be (3) The expected variance ( σ 2 ) (and standard deviation σ ) would be (4) Numerical example . Suppose we have a model that predicts the ER π s R s s 1 = 72 = E σ 2 π s R s () 2 s 1 = 72 =
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4 following returns on two stocks A and B in three states: • Expected returns: μ A = (0.25)(0.20) + (0.50)(0.10) + (0.25)(0.00) = 0.l0 μ B = (0.25)(0.05) + (0.50)(0.10) + (0.25)(0.15) = 0.l0 Variances: Standard deviations: σ A = 0.005 = 0.07071, σ B = 0.00125 = 0.03536. Covariance and correlation • Covariance measures how two random variables are related. Table 1: Data for a fundamental analysis Outcome s Probability π s R A R B Boom 0.25 20% 5% Normal 0.50 10% 10% Recession 0.25 0% 15% σ A 2 0.25 () 0.20 0.1 2 0.5 0.1 0.1 2 0.25 0.0 0.1 2 ++ 0.005 = = σ B 2 0.25 0.05 0.1 2 0.5 0.1 0.1 2 0.25 0.15 0.1 2 0.00125 = =
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5 • From two historical samples of T returns on stocks A and B : i. calculate the sample mean returns (5) ii. calculate the sample covariance using (6) If when R B > R B , R A > R A , and vice versa, returns on A and B will have a positive covariance .
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10_Oheads - Lecture Notes 10 The Capital Asset Pricing...

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