Introduc corporate finance ch10

The Capital Asset Pricing Model (CAPM) Chapter 10 Individual Securities  Τηε χηαραχτεριστιχσ οφ ινδιϖιδυαλ σεχυριτιεσ τηατ αρε οφ ιντερεστ αρε τηε:    Expected Return Variance and Standard Deviation Covariance and Correlation Expected Return, Variance, and Covariance Rate of Return Scenario Probability Stock fund Bond fund Recession 33.3% -7% 17% Normal 33.3% 12% 7% Boom 33.3% 28% -3% Χονσιδερ τηε φολλοωινγ τωο ρισκψ ασσετ ωορλδ Τηερε ισ α 1/3 χηανχε οφ εαχη στατε οφ τηε εχονοµψ ανδ τηε ονλψ ασσετσ αρε α στοχκ φυνδ ανδ α βονδ φυνδ Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% E (rS ) = × (−7%) + × (12%) + × (28%) 3 E (rS ) = 11 % Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% E (rB ) = × (17%) + × (7%) + × (−3%) 3 E (rB ) = 7% Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% (−7% − 11 %) = 3.24% Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% (12% − 11 %) = 01% 10.2 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% 2.05% = (3.24% + 0.01% + 2.89%) Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% 14.3% = 0.0205 The Return and Risk for Portfolios Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% Note that stocks have a higher expected return than bonds and higher risk Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks Relationship of Risk to Reward  Τηε φυνδαµενταλ χονχλυσιον ισ τηατ τηε ρατιο οφ τηε ρισκ πρεµιυµ το βετα ισ τηε σαµε φορ εϖερψ ασσετ  In other words, the reward-to-risk ratio is constant and equal to: E ( Ri ) − RF Re ward / Risk = βi Market Equilibrium  Ιν εθυιλιβριυµ, αλλ ασσετσ ανδ πορτφολιοσ µυστ ηαϖε τηε σαµε ρεωαρδ−το−ρισκ ρατιο ανδ τηεψ αλλ µυστ εθυαλ τηε ρεωαρδ−το− ρισκ ρατιο φορ τηε µαρκετ E ( RA ) − R f βA = E ( RM ) − R f βM Relationship between Risk and Expected Return (CAPM)  Εξπεχτεδ Ρετυρν ον τηε Μαρκετ: R M = RF + Market Risk Premium • Expected return on an individual security: R i = RF + β i × ( R M − RF ) Market Risk Premium This applies to individual securities held within well-diversified portfolios Expected Return on an Individual Security  Τηισ φορµυλα ισ χαλλεδ τηε Χαπιταλ Ασσετ Πριχινγ Μοδελ (ΧΑΠΜ) Expected return on a security R i = RF + β i × ( R M − RF ) = Risk+ free rate Beta of the security × Market risk premium • Assume βi = 0, then the expected return is RF • Assume βi = 1, then R i = R M Expected return Relationship Between Risk & Expected Return R i = RF + β i × ( R M − RF ) RM RF 1.0  β Τηε σλοπε οφ τηε σεχυριτψ µαρκετ λινε ισ εθυαλ το τηε µαρκετ ρισκ πρεµιυµ; ι.ε., τηε ρεωαρδ φορ βεαρινγ αν αϖεραγ ε αµουντ οφ σψστεµατιχ ρισκ Expected return Relationship Between Risk & Expected Return 13.5% β i = 1.5 RF = 3% R M = 10% 3% 1.5 β R i = 3% + 1.5 × (10% − 3%) = 13.5% Total versus Systematic Risk  Χονσιδερ τηε φολλοωινγ ινφορµατιον:      Standard Deviation Security C 20% Security K 30% Beta 1.25 0.95 Ωηιχη σεχυριτψ ηασ µορε τοταλ ρισκ? Ωηιχη σεχυριτψ ηασ µορε σψστεµατιχ ρισκ? Ωηιχη σεχυριτψ σηουλδ ηαϖε τηε ηιγ ηερ εξπεχτεδ ρετυρν? Summary and Conclusions   Τηισ χηαπτερ σετσ φορτη τηε πρινχιπλεσ οφ µοδερν πορτφολιο τηεορψ Τηε εξπεχτεδ ρετυρν ανδ ϖαριανχε ον α πορτφολιο οφ τωο σεχυριτιεσ Α ανδ Β αρε γ ιϖεν βψ E (rP ) = wA E (rA ) + wB E (rB ) σ P2 = (wAσ A )2 + (wB σ B )2 + 2(wB σ B )(wAσ A )ρ AB • By varying wA, one can trace out the efficient set of portfolios We graphed the efficient set for the two-asset case as a curve, pointing out that the degree of curvature reflects the diversification effect: the lower the correlation between the two securities, the greater the diversification • The same general shape holds in a world of many assets Summary and Conclusions Τηε εφφιχιεντ σετ οφ ρισκψ ασσετσ χαν βε χοµβινεδ ωιτη ρισκλεσσ βορροωινγ ανδ λενδινγ Ιν τηισ χασε, α ρατιοναλ ινϖεστορ ωιλλ αλωαψσ χηοοσε το ηολδ τηε πορτφολιο οφ ρισκψ σεχυριτιεσ ρεπρεσεντεδ βψ τηε µαρκετ πορτφολιο • Then with borrowing or lending, the investor selects a point along the CML return  CM L efficient frontier M rf σP Summary and Conclusions  Τηε χοντριβυτιον οφ α σεχυριτψ το τηε ρισκ οφ α ωελλ− διϖερσιφιεδ πορτφολιο ισ προπορτιοναλ το τηε χοϖαριανχε οφ τηε σεχυριτψ∋σ ρετυρν ωιτη τηε µαρκετ ’σ ρετυρν Τηισ χοντριβυτιον ισ χαλλεδ τηε βετα βi = Cov ( Ri , RM ) σ ( RM ) • The CAPM states that the expected return on a security is positively related to the security’s beta: R i = RF + β i × ( R M − RF ) Expected (Ex-ante) Return, Variance and Covariance  Εξπεχτεδ Ρετυρν: Ε(Ρ) = Σ (πσ ξ Ρσ)  ςαριανχε: σ2 = Σ {πσ ξ [Ρσ − Ε(Ρ)]2}  Στανδαρδ ∆εϖιατιον =  Χοϖαριανχε: − Ε(ΡΒ)]}  Χορρελατιον Χοεφφιχιεντ: σ σΑΒ = Σ {πσ ξ [Ρσ,Α − Ε(ΡΑ)] ξ [Ρσ,Β ρΑΒ = σΑΒ / (σΑ σΒ) Risk and Return Example Στατε ΗΜ Προβ Τ−Βιλλσ ΞΨΖ Μαρκετ Πορτ Ρεχεσσιον 0 % (2 %) Βελοω Αϖγ Αϖεραγ ε Αβοϖε Αϖγ Βοοµ 0.20 0.50 0.20 0.05 8.0 8.0 8.0 8.0 (2.0) 20.0 35.0 50.0 Ε(Ρ)= σ = ΙΒΜ % 0 % 0.0 (10.0) (20.0) (1 0 ) 1.0 7.0 15.0 45.0 30.0 43.0 (1 %) Expected Return and Risk of IBM Ε(ΡΙΒΜ)= 0 ∗(−2 )+0 ∗(−2 ) +0.50∗(20)+0.20∗(35)+0.05∗(50) = 18% σΙΒΜ2 = 0.05∗(−22−18) 2+0.20∗(−2−18) +0.50∗(20− 18) 2+0.20∗(35−18) +0.05∗(5018) = σΙΒΜ = % Covariance and Correlation ΧΟς ΙΒΜ&ΞΨΖ = 0 ∗(−2 −1 )(1 −1 )+ ∗(−2 −1 )(−1 −1 )+0 ∗(2 −1 )(7 − 12.5)+ ∗(3 −1 )(4 −1 )+0 ∗(5 −1 )(3 − 12.5) = 194 Χορρελατιον = /(1 )(1 )= 5 Risk and Return for Portfolios (2 assets)  Εξπεχτεδ Ρετυρν οφ α Πορτφολιο: Ε(Ρπ) = ΞΑ Ε(Ρ) Α + ΞΒ Ε(Ρ) Β  ςαριανχε οφ α Πορτφολιο: σπ2 = ΞΑ 2σΑ + ΞΒ2σΒ2 + ΞΑ ΞΒ σΑΒ