CHAPTER 7 Capital Asset Pricing and Arbitrage Pricing Theory

7.1 THE CAPITAL ASSET PRICING MODEL

Capital Asset Pricing Model (CAPM) Equilibrium model that underlies all modern financial theory Derived using principles of diversification with simplified assumptions Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development

Assumptions Individual investors are price takers (perfect market) Single-period investment horizon No limit for borrowing or lending No taxes nor transaction costs Information is costless and available to all investors Investors are rational mean-variance optimizers Homogeneous expectations

Example of one factor model: Capital Asset Pricing Model (CAPM) Since every body knows that a considerable extent of risk (i.e., unique risk) can be diversified away by making a portfolio of only around 15-20 securities, so all the investors assume a diversified portfolio. Risk premium is then due only for market risk, represented by beta (rather than total risk represented by sigma). Rj-rf=f(βj) The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio Beta risk measures the volatility of security return in response to the volatility of the market return. βj= COV(Rj,Rm)/ σ2Rm In an equilibrium situation expected return of each security is composed of the risk free return and the risk premium. E(Rj)=Rf+[E(Rm)-Rf]βj

Calculation of Expected Return (CAPM) Economic state Probability Return j Market Return Depression 0.25 5 7 Normal 0.5 10 Inflation 20 15 COV(Rj,Rm)=∑(Rj-Rj)(Rm-Rm)(Pi)=15.6 σ2Rm=∑(Rm-Rm)2(Pi)=8.25 βj= COV(Rj,Rm)/ σ2Rm=15.6/8.25=1.9 E(Rj)=Rf+(Rm-Rf) βj=5%+(10.5%-5%)(1.9)=15.45%

Work Sheet of CAPM βj= COV(Rj,Rm)/ σ2Rm=15.6/8.25=1.9 Economic State Probability Return J Market Return ExptdValue j Expd. Value M Dev (Rj) [Rj-E(Rj)] Dev (Rm) [Rm-E(Rm)] x Dev(Rm) Dev(Rj) X Pi Depression 0.25 5 7 1.25 1.75 -6.25 -3.5 21.875 5.46875 3.0625 Normal 0.5 10 -1.25 -0.5 0.625 0.3125 0.125 Inflation 20 15 3.75 8.75 4.5 39.375 9.84375 5.0625  Sum   11.25 10.5 15.6 8.25 βj= COV(Rj,Rm)/ σ2Rm=15.6/8.25=1.9 E(Rj)=Rf+(Rm-Rf) βj=5%+(10.5%-5%)(1.9)=15.45%

Security Market Line (SML) Conservative Return Aggressive Note: This point does not refer to borrowing. It is investment in a security which assumes more risk and return compared to market average. It is optimum for aggressive people. Concentrate on the IC curve. Normal SML E(Rj) E(Rm) Rf=5% Systematic Risk /Beta Risk βm=1 βj=1.9

Monthly rates of return Monthly Excess Returns GM   Monthly rates of return Monthly Excess Returns Month GM S&P500 T-bills 1 -24.57 a-2.19 0.5 -25.07 -2.69 2 -17.79 2.39 0.49 -18.28 1.9 3 -1.94 -1.63 0.51 -2.45 -2.14 4 22.94 6.07 0.52 22.42 5.55 5 -7.14 -5.35 -7.66 -5.87 6 -4.42 -0.49 -4.94 -1.01 7 -20.32 -8.01 0.53 -20.85 -8.54 8 2.9 0.41 2.4 -0.09 9 5.42 3.46 0.44 4.98 3.02 10 -0.71 -9.23 0.42 -1.13 -9.65 11 -2.76 -6.42 0.38 -3.14 -6.8 12 5.71 7.68 0.33 5.38 7.35 13 3.81 0.31 3.5 0.2 14 13.09 -2.5 0.3 12.79 -2.8 15 -1.17 -1.07 -1.47 -1.37 16 -13.92 -6.41 0.29 -14.21 -6.7 17 -21.64 -8.17 0.22 -21.86 -8.39 18 -3.68 1.81 0.18 -3.86 1.63 19 20.28 7.52 0.16 20.12 7.36 20 -2.21 0.76 0.14 -2.35 0.62 21 5.23 -1.56 5.09 -1.7 22 3.6 -2.08 0.15 3.45 -2.23 23 14.1 3.67 13.95 3.52 24 6.12 -6.14 5.97 -6.29 25 -3.12 -0.91 -3.27 -1.06 26 -14 -7.25 -14.14 -7.39 27 -12.91 -7.9 -13.05 -8.04 28 2.81 2.67 0.35 29 -18.72 -11 -18.86 -11.14 30 -14.52 8.64 0.13 -14.65 8.51 31 19.4 0.1 19.3 5.61 32 -7.15 -6.03 -6.13 33 -1.44 -2.74 -1.54 -2.84 34 -7.05 -1.8 35 -0.44 0.84 -0.54 0.74 36 7.23 8.1 7.13 37 -2 0.09 -2.09 38 1.13 0.08 1.82 1.05 39 3.97 1.62 3.89 1.54 40 9.8 1.79 9.72 1.71 41 -0.41 -1.19 -1.27 42 4.25 5.5 4.17 43 0.26 0.71 0.63 44 24.82 5.08 24.74 45 -6.97 1.73 1.65 46 1.22 -3.22 1.14 47 -1.81 -1.64 -1.89 -1.72 48 0.36 -1.68 0.28 -1.76 49 -4.28 1.21 -4.37 1.12 50 2.64 1.8 0.11 2.53 1.69 51 -7.41 -3.43 -7.52 -3.54 52 -4.24 0.23 53 2.83 0.94 2.69 0.8 54 -9.25 1.4 -9.4 1.25 55 3.86 -0.08 3.68 56 3.25 0.19 3.62 3.06 57 -8.11 -2.53 -8.31 -2.73 58 -3.15 1.89 -3.37 1.67 59 -17.56 -1.91 60 -12.15 -3.65 0.24 -12.39 -3.89

SUMMARY OUTPUT OF REGRESSION R Square 0.33 ANOVA   df SS MS F Significance F Regression 1 2126 2126. 28.35 1.72E-06 Residual 58 4351 75.02 Total 59 6478 Coefficients Standard Error t Stat P-value Lower 95% Upper 95.0% Intercept -1.1 1.125 -0.9 0.3 -3.3 1.1 -3.36 X Variable 1 1.323 0.248 5.3 1.72E-6 0.8 1.82 1.8

Calculation of beta Covariance (Rm,Rq)= 0.003155434 Var (Rm)= Month Closing DGEN Values Market Return (Rm) Closing Share Price Returns of Qassem (Rq) Jul-04 1289.1405 14.80 Aug-04 1513.28894 0.173874329 15.30 0.033783784 Sep-04 1633.02272 0.079121559 18.50 0.209150327 Oct-04 1710.44837 0.047412476 19.40 0.048648649 Nov-04 1877.04943 0.097401981 17.30 -0.108247423 Dec-04 1971.31331 0.050219178 22.10 0.277456647 Jan-05 1843.94786 -0.06460944 -0.162895928 Feb-05 1835.62097 -0.004515795 17.50 -0.054054054 Mar-05 1919.253 0.04556062 16.90 -0.034285714 Apr-05 1537.8699 -0.198714344 -0.124260355 May-05 1648.28467 0.071797211 14.40 -0.027027027 Jun-05 1713.17356 0.039367526 14.10 -0.020833333 Jul-05 1510.10911 -0.11853116 12.90 -0.085106383 Aug-05 1613.17397 0.068249943 14.00 0.085271318 Sep-05 1673.20767 0.037214647 15.80 0.128571429 Oct-05 1694.61758 0.012795728 14.70 -0.069620253 Nov-05 1694.4199 -0.000116652 -0.047619048 Dec-05 1677.345 -0.010077136 11.80 -0.157142857 Jan-06 1643.33863 -0.020273927 11.30 -0.042372881 Feb-06 1531.43 -0.068098338 10.10 -0.10619469 Mar-06 1491.7723 -0.025895862 11.1 0.099009901 Apr-06 1361.26953 -0.087481695 10.4 -0.063063063 May-06 1355.04 -0.004576265 10.80 0.038461538 Jun-06 1339.525 -0.011449846 10.60 -0.018518519 Jul-06 1406.81 0.050230492 10.30 -0.028301887 Aug-06 1587.076 0.128138128 12.50 0.213592233 Sep-06 1562.53 -0.015466178 -0.056 Oct-06 1541.65 -0.013362943 11.10 -0.059322034 Nov-06 1527.29 -0.009314695 8.6 -0.225225225 Dec-06 1609.51 0.053833915 10.2 0.186046512 Jan-07 1805.11998 0.121533871 11.3 0.107843137 Feb-07 1791.54 -0.007523035 12 0.061946903 Mar-07 1760.88 -0.017113768 11.5 -0.041666667 Apr-07 1743.33 -0.009966608 10.9 -0.052173913 May-07 2003.58 0.149283268 10.6 -0.027522936 Jun-07 2149.32 0.072739796 0.047169811 Jul-07 2384.18 0.10927177 11.6 0.045045045 Aug-07 2455.086 0.029740204 12.3 0.060344828 Sep-07 2548.49 0.038045103 15.9 0.292682927 Oct-07 2850.81 0.118627109 18.1 0.13836478 Nov-07 2971.11 0.042198533 21.1 0.165745856 Dec-07 3017.2133 0.015517197 20.8 -0.014218009 Jan-08 2907.166 -0.036473159 25.2 0.211538462 Feb-08 2931.38 0.008329074 29.50 0.170634921 Mar-08 3016.489 0.029033766 32.50 0.101694915 Apr-08 3072.85 0.018684305 34.80 0.070769231 May-08 3167.99 0.030961485 34.10 -0.020114943 Jun-08 3000.497 -0.052870432 38.10 0.117302053 Covariance (Rm,Rq)= 0.003155434 Var (Rm)= 0.00467312 Beta 0.675

Calculation of CAPM Return Covarience (Rm,Rq) 0.003155434 Var Rm 0.00467312 Beta 0.67523075 Average Monthly Return  0.020484084 Annualized Market Return Rm 24.5809% Risk-free Rate Rf   7% ke   18.8712%

CML vs. SML CML assumes that all investors may not hold diversified portfolio, and as such the concerned risk is total risk represented by sigma or standard deviation. SML assumes that all investors hold diversified portfolio, as such there is only systematic risk concerned, represented by beta. In case of CML, risk premium is the price for total risk. In case of SML, risk premium is the price for systematic risk. Among all the points of CML, there is only one point that belongs to the market which is optimum portfolio and the rest of the points are derived by means of borrowing or lending at the risk free rate. All the points of SML are available from the market that refers to risk-return of different stocks, and different portfolios including the market portfolio.

THE CAPM AND THE REAL WORLD (Limitations of CAPM) Empirical Estimates are different from theoretical expectation. Endogenous risk free rate is higher than the true rate, and endogenous market risk premium is lower than the true one. Research shows that other factors affect returns January Effect (April Effect) Small Firm Effect Returns are related to factors other than market returns like book value relative to market value Choice of market portfolio Thin trading problem

7.5 FACTOR MODELS AND THE ARBITRAGE PRICING THEORY

Arbitrage Pricing Theory Arbitrage arises if an investor can construct a zero investment portfolio with a sure profit. Since no investment is required, an investor can create large positions to secure large levels of profit. In efficient markets, profitable arbitrage opportunities will quickly disappear.

Equilibrium rate of return Role of Arbitrage: Simultaneous buying and selling of securities k k S1 S1 S2 9 % 8 % 7 % D D2 D1 Q Q Market A Market B

Factor Models: Announcements, Surprises, and Expected Returns The return on any security consists of two parts. First, the expected returns Second, the unexpected or risky returns A way to write the return on a stock in the coming month is:

Factor Models: Announcements, Surprises, and Expected Returns Any announcement can be broken down into two parts, the anticipated (or expected) part and the surprise (or innovation): Announcement = Expected part + Surprise. The expected part of any announcement is the part of the information the market uses to form the expectation, R, of the return on the stock. The surprise is the news that influences the unanticipated return on the stock, U.

Risk: Systematic and Unsystematic A systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree. An unsystematic risk is a risk that specifically affects a single asset or small group of assets. Unsystematic risk can be diversified away. Examples of systematic risk include uncertainty about general economic conditions, such as GNP, interest rates or inflation, foreign exchange rates, etc. On the other hand, announcements specific to a single company are examples of unsystematic risk.

Risk: Systematic and Unsystematic We can break down the total risk of holding a stock into two components: systematic risk and unsystematic risk: 2 Total risk  Nonsystematic Risk:  Systematic Risk: m n

Systematic Risk and Betas For example, suppose we have identified three systematic risks: inflation, GNP growth, and the foreign exchange spot rate S($=Tk). Our model is: F = the surprise in the factor.

Systematic Risk and Betas: Example Suppose we have made the following estimates: bI = -2.30 bGNP = 1.50 bS = 0.50 Finally, the firm was able to attract a “superstar” CEO, and this unanticipated development contributes 1% to the return.

Systematic Risk and Betas: Example We must decide what surprises took place in the systematic factors. If it were the case that the inflation rate was expected to be 3%, but in fact was 8% during the time period, then: FI = Surprise in the inflation rate = actual – expected = 8% – 3% = 5%

Systematic Risk and Betas: Example If it were the case that the rate of GNP growth was expected to be 4%, but in fact was 1%, then: FGNP = Surprise in the rate of GNP growth = actual – expected = 1% – 4% = – 3%

Systematic Risk and Betas: Example If it were the case that the dollar-Taka spot exchange rate, S($=Tk), was expected to increase by 10%, but in fact remained stable during the time period, then: FS = Surprise in the exchange rate = actual – expected = 0% – 10% = – 10%

Systematic Risk and Betas: Example Finally, if it were the case that the expected return on the stock was 8%, then:

Relationship Between b & Expected Return SML Expected return D A B C b

The Capital Asset Pricing Model and the Arbitrage Pricing Theory APT applies to well diversified portfolios and not necessarily to individual stocks. With APT it is possible for some individual stocks to be mispriced - not lie on the SML. APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio. APT can be extended to multifactor models. For example, there may be GNP growth beta, inflation beta, foreign exchange beta, P:E beta and so on.