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Finance: a Quantitative Introduction Chapter 3 - Part 3 CAPM and APT Capital Asset Pricing Model A summarizing digression Arbitrage Pricing Theory Finance: A Quantitative Introduction Chapter 3 - part 3 CAPM and APT Nico van der Wijst 1 Finance: A Quantitative Introduction c Cambridge University Press Capital Asset Pricing Model A summarizing digression Arbitrage Pricing Theory 1 Capital Asset Pricing Model 2 A summarizing digression 3 Arbitrage Pricing Theory 2 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests E[rp] Ind.1 C Ind.2 M B rf A σp The Capital Market Line 3 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests Capital Asset Pricing Model CAPM Capital Market Line only valid for efficient portfolios combinations of risk free asset and market portfolio M all risk comes from market portfolio What about inefficient portfolios or individual stocks? don't lie on the CML, cannot be priced with it need a different model for that What needs to be changed in the model: the market price of risk ((E(rm) − rf )/σm); or the measure of risk σp? 4 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests CAPM is more general model, developed by Sharpe Consider a two asset portfolio: one asset is market portfolio M, weight (1 − x) other asset is individual stock i, weight x Note that this is an inefficient portfolio Analyse what happens if we vary proportion x invested in i begin in point I, 100% in i, x=1 in point M, x=0, but asset i is included in M with its market value weight to point I', x<0 to eliminate market value weight of i 5 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests E[rp] C I' M I B rf A σp Portfolios of asset i and market portfolio M 6 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests Risk-return characteristics of this 2-asset portfolio: E(rp) = xE(ri ) + (1 − x)E(rm) q 2 2 2 2 σp = [x σi + (1 − x) σm + 2x(1 − x)σi;m] Expected return and risk of a marginal change in x are: @E(r ) p = E(r ) − E(r ) @x i m @σ 1 − 1 p = x2σ2 + (1 − x)2σ2 + 2x(1 − x)σ 2 @x 2 i m i;m 2 2 2 × 2xσi − 2σm + 2xσm + 2σi;m − 4xσi;m 7 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests First term of @σ =@x is 1 ; so: p 2σp @σ 2xσ2 − 2σ2 + 2xσ2 + 2σ − 4xσ p = i m m i;m i;m @x 2σp xσ2 − σ2 + xσ2 + σ − 2xσ = i m m i;m i;m σp Isolating x gives: @σ x(σ2 + σ2 − 2σ ) + σ − σ2 p = i m i;m i;m m @x σp 8 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests At point M all funds are invested in M so that: x = 0 and σp = σm Note also that: i is already included in M with its market value weight economically x represents excess demand for i in equilibrium M excess demand is zero This simplifies marginal risk to: 2 2 @σp σi;m − σm σi;m − σm = = @x x=0 σp σm 9 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests So the slope of the risk-return trade-off at equilibrium point M is: @E(rp)=@x E(ri ) − E(rm) = 2 @σp=@x x=0 (σi;m − σm) /σm But at point M the slope of the risk-return trade-off is also the slope of the CML, so: E(ri ) − E(rm) E(rm) − rf 2 = (σi;m − σm) /σm σm Solving for E(ri ) gives: σi;m E(ri ) = rf + (E(rm) − rf ) 2 σm = rf + (E(rm) − rf )βi 10 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests E(ri ) = rf + (E(rm) − rf )βi This is the Capital Asset Pricing Model Sharpe was awarded the Nobel prize for this result Its graphical representation is known as the Security Market Line Pricing relation for entire investment universe including inefficient portfolios including individual assets clear price of risk: E(rm) − rf clear measure of risk: β 11 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests CAPM formalizes risk-return relationship: well-diversified investors value assets according to their contribution to portfolio risk if asset i increases portf. risk E(ri ) > E(rp) if asset i decreases portf. risk E(ri ) < E(rp) expected risk premium proportional to β Offers other insights as well. Look at 4 of them: 1 Systematic and unsystematic risk 2 Risk adjusted discount rates 3 Certainty equivalents 4 Performance measures 12 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests 1. Systematic & unsystematic risk The CML is pricing relation for efficient portfolios: E(rm) − rf E(rp) = rf + σp σm E(rm)−rf is the price per unit of risk σm σp is the volume of risk. The SML valid for all investments, incl. inefficient portfolios and individual stocks: E(rp) = rf + (E(rm) − rf )βp 13 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests we can write β as: covp;m σpσmρp;m σpρp;m βp = 2 = 2 = σm σm σm so that the SML becomes: σpρp;m E(rp) = rf + (E(rm) − rf ) ; σm E(rm) − rf E(rp) = rf + σpρp;m σm Compare with CML: E(rm) − rf E(rp) = rf + σp σm 14 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests The difference between CML and SML is in volume part: SML only prices the systematic risk is therefore valid for all investment objects. CML prices all risks only valid when all risk is systematic risks, i.e. for efficient portfolios otherwise, CML uses 'wrong' risk measure difference is correlation term, that is ignored in CML efficient portfolios only differ in proportion M in it so all efficient portfolios are perfectly positively correlated: ρM;(1−x)M = 1 if ρp;m = 1 ) σpρp;m = σp and CML = SML 15 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests E[rp] E[rp] Capital market Security market line line C C F M F M G B G B rf rf σp 1 βp Systematic and unsystematic risk 16 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests 2. CAPM and discount rates Recall general valuation formula for investments: t X Exp [Cash flowst ] Value = t (1 + discount ratet ) Uncertainty can be accounted for in 3 different ways: 1 Adjust discount rate to risk adjusted discount rate 2 Adjust cash flows to certainty equivalent cash flows 3 Adjust probabilities (expectations operator) from normal to risk neutral or equivalent martingale probabilities 17 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests Use of CAPM as risk adjusted discount rate is easy CAPM gives expected (=required) return on portfolio P as: E(rp) = rf + (E(rm) − rf )βp But return is also: E(Vp;T ) − Vp;0 E(rp) = Vp;0 Discount rate: links expected end-of-period value, E(Vp;T ); to value now, Vp;0 found by equating the two expressions: 18 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests E(Vp;T ) − Vp;0 = rf + (E(rm) − rf )βp Vp;0 solving for Vp;0 gives: E(Vp;T ) Vp;0 = 1 + rf + (E(rm) − rf )βp rf is the time value of money (E(rm) − rf )βp is the adjustment for risk together they form the risk adjusted discount rate 19 Finance: A Quantitative Introduction c Cambridge University Press Derivation of the CAPM Capital Asset Pricing Model Insights from the CAPM A summarizing digression Underlying assumptions Arbitrage Pricing Theory Empirical tests 3.
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