Post on 19-Jan-2023
Financial Economics- CPP
Godfrey Ndlovu
AERC
January 1, 2021
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 1 / 45
Section Roadmap
Recap
Two-fund separation
Zero-covariance portfolio
Introducing risk-less asset
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 2 / 45
Application- minimum variance portfolio from previoussession
Suppose you have three stocks, i.e. X ,Y and Z whose expected return are0.10,0.12 and 0.18; respectively and standard deviation of returns are 0.04;0.10 and 0.14; respectively. You are also told that the covariance betweenthe returns of X and Y 0.002 , between the returns of X and Z is 0.004,and between the returns of Y and Z is 0.007.Determine the minimum variance portfolio.Solution in excel or other software
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 3 / 45
Portfolio Separation- Mutual (or two) Fund SeparationTheorem
According to Merton(1972)
Fund Separation Theorem
Given n assets satisfying the conditions above, there are two portfolios(’mutual funds’) constructed from these n assets, such that all risk-averseindividuals, who choose their portfolios so as to maximize utility functions(dependent only on the mean and variance of their portfolios), will beindifferent in choosing between portfolios from among the original n assetsor from these two funds.
If we identify any two efficient portfolios we should be able to identifyany portfolio that an investor would settle for; since their linearcombination will also be efficient.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 4 / 45
Fund Separation Theorem
Given any two exchange-traded(or mutual)funds in a market, eachinvested in a different portfolio, a mean-variance investor couldreplicate her optimal portfolio by dividing her wealth between the twofunds(only)- this may mean a short position on one
Let R̄1p, and R̄2p, be expected returns of two different frontierportfolios
Let x be the wealth proportion in portfolio 1, therefore (1− x) inportfolio 2
Let the third portfolio be such that:
R̄3p = xR̄1p + (1− x)R̄2p (1)
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 5 / 45
Fund Separation Theorem
We know that:ω1 = g + hR̄1p
ω2 = g + hR̄2p
Where ωi is an n × 1 vector of optimal portfolio weight
We can create a new portfolio with n × 1 vector of portfolio weightsgiven by
xω1 + (1− x)ω2 = x(g + hR̄1p) + (1− x)(g + hR̄2p)
= g + h(xR̄1p + (1− x)R̄2p)
= g + hR̄3p
= ω3
(2)
Therefore any given efficient portfolio can be replicated by twoefficient frontier portfolios
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 6 / 45
Zero Covariance Portfolios
Except for global minimum variance portfolio, all portfolios have acorresponding orthogonal pair- i.e there is another portfolio withwhich its returns have zero covariance
Covariance between portfolio 1 and portfolio 2
ω1′Vω2 = (g + hR̄1p)′V (g + hR̄2p)
=1
c+
c
bc − a2
(R̄1p −
a
c
)(R̄2p −
a
c
)(3)
Setting the above to zero, and solving for R̄2, i.e expected return ofportfolio with zero covariance with portfolio ω1
(recall Rmv = ac )
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 7 / 45
We have
R̄2p =a
c− bc − a2
c2(R̄1p − ac )
= Rmv −bc − a2
c2(R̄1p − Rmv )
(4)
The above portfolio is efficient if (R̄1p − Rmv ) > 0
By Equation (4) R̄2p < Rmv : ω2 is inefficient
Let R̄p = R̄0 +∂R̄p
∂σp
∣∣σp=σ1p
σp
Where∂R̄p
∂σp
∣∣σp=σ1p
is the slope of the hyperbola at (σ1p, R̄1p), R̄0
intercept of tangent at σp = 0
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 8 / 45
Using σp =
√1c +
c(R̄p− ac
)2
bc−a2 and∂R̄p
∂σp= bc−a2
c(R̄p− ac
)σp we can solve for R̄0 by evaluating (4) at (σ1p, R̄1p)
R̄0 = R̄1p −∂R̄p
∂σp
∣∣∣∣σp=σ1p
σ1p
= R̄1p −bc − a2
c(R̄1p − ac )σ1pσ1p
= R̄1p −bc − a2
c(R̄1p − ac )
[1
c+
c(R̄1p − ac )2
bc − a2
]=
a
c− bc − a2
c2(R̄1p − ac )
= R̄2p
(5)
Recall σp =
√1c +
c(R̄p− ac
)2
bc−a2
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 9 / 45
The Tangency
Therefore intercept of tangent to ω1 is expected return of itszero-covariance counterpart, ω2
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 10 / 45
Efficient Frontier and Risk-less Asset
LetRf be return from risk-less asset
Recall ω is n × 1 vector portfolio proportion on risky asset
However the constraint ω′e = 1 is not applicable since 1− ω′e isproportion invested in risk-free asset
Return on the portfolio
R̄p = (1− ω′e)Rf + ω′R̄
= Rf + ω′(R̄ − Rf e)(6)
Portfolio variance is still ω′Vω
The individual’s optimization problem becomes
minω
1
2ω′Vω + λ
(R̄p −
[Rf + ω′(R̄ − Rf e)
])(7)
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 11 / 45
Efficient Frontier and Riskless Asset
From the previous calculation
ω? = λV−1(R̄ − Rf e) (8)
Where λ =R̄p−Rf
(R̄−Rf e)′V−1(R̄−Rf e)=
R̄p−Rf
b−2aRf +cR2f
Therefore λ is non-negative when R̄p ≥ Rf , since V−1 is positivedefinite
The variance expressed in terms of ω?
σ2p = ω?
′Vω? =
(R̄p − Rf )2
b − 2aRf + cR2f
σp =(R̄p − Rf )√
b − 2aRf + cR2f
R̄p = Rf ± (b − 2aRf + cR2f )
12σp
(9)
Therefore the frontier is linear in (σp, R̄p) space
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 12 / 45
Efficient Frontier and Risk-less Asset
When we include a risk-less asset, the frontier becomes two straightlines with positive and negative slope, as shown below
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 13 / 45
Investor portfolio choice
All investors choose to hold risky assets in the same relativeproportions given by the tangency portfolio ω
A, and differ only in the
proportion of wealth allocated to this portfolio versus the risk-freeasset.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 14 / 45
Investor portfolio choice
At (0,Rf ), e ′ω? = 0, i.e. all is invested in Rf , between (0,Rf and(σA, R̄A)- investor 1’s indifference curve ,
At (σ1, R̄p1)⇒ 0 < e ′ω? < 1 - lending
At (σA, R̄A), e ′ω? = 1, - i.e. everything is allocated to risky asset,none in Rf
At (σ2, R̄p2), e ′ω? > 1, - i.e.negative allocation to risk-free(borrow)
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 15 / 45
The Capital Asset Pricing Model (CAPM)
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 16 / 45
The Capital Asset Pricing Model (CAPM)
CAPM- Jack Treynor (1961), William Sharpe (1964), John Litner (1965),Jan Mossin(1966)
From portfolio theory, assuming mean-variance optimizing investors.
For n risky assets, and a risk-free rate, optimal portfolio optimal portfolioweights, (ω?)
We have shown that:
ω? = λV−1(R̄ − Rf e)
where λ =R̄p − Rf
b − 2aRf + cR2f
a = R̄ ′V−1e = e′V−1R̄
b = R̄ ′V−1R̄
c = e′V−1e
(10)
1− e′ω? is proportion in risky asset; λ is a scalar, linear in R̄p;ω? is alsolinear in R̄p; R̄p tangent of indifference curve with efficient frontier
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 17 / 45
Recap from Mean-variance Analysis
We have shown that
σp =R̄p − Rf(
b − 2aRf + cR2f
) 12
As shown below
Source: Pennacchi
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 18 / 45
Characteristics of Tangency Portfolio
Efficient frontier given by Rf and ωm implies investors optimallychoose a combination of riskless asset and efficient frontier portfolioof risky assets with weights ωm
We know that:
e ′ω? = 1 and R̄p = R̄ ′ω?
Since ω? = λV−1(R̄ − Rf e), if we pre-multiply by e ′
⇒ e ′λV−1(R̄ − Rf e) = 1
λ =[e ′V−1R̄ − e ′V−1Rf e)
]−1
λ = [a− cRf ]−1
(11)
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 19 / 45
Characteristics of Tangency Portfolio
Let (a− cRf )−1 = m
⇒ ωm = mV−1(R̄ − Rf e) (12)
Let σim − n × 1 vector of covariance of tangency portfolio and riskasset (for i = 1, · · · , n))
σim = Vωm
= VmV−1(R̄ − Rf e)
= m(R̄ − Rf e)
(13)
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 20 / 45
Characteristics of Tangency Portfolio
We know that σ2m = ωm′Vωm
⇒ σ2m = ωm′σm
= mωm′(R̄ − Rf e)
= m[ωm′R̄ − ωm′Rf e]
= m[R̄m − Rf ]
(14)
Recall expected return of tangency portfolio = ωm′R̄
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 21 / 45
Characteristics of Tangency Portfolio
Sinceσim = m(R̄ − Rf e)⇒ σim
m= (R̄ − Rf e) (15)
But from (14) m = σ2m
R̄m−Rfsubstituting for m in equation 15 we get;
σimσ2m
(R̄m − Rf
)= (R̄ − Rf e)
R̄ − Rf e = β(R̄m − Rf )
(16)
The last part of equation 16 is due to the fact that;
β = σimσ2m
= Cov(R̄m,R̄i )
VarR̄m
Equation 16 links the equilibrium excess return from the tangencyportfolio and any risky asset.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 22 / 45
CAPM Assumptions
To obtain the above, we used the following key assumptions;
Market Equilibrium
i Aggregate demand= aggregate supply in proportions given by ωm(full marketability & divisibility of securities)
ii Rational investor behaviour (recall VNM expected utility)
iii Investors have identical beliefs about probability distribution of assetreturns
iv All risky assets are traded
v Unlimited borrowing and lending at Rf
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 23 / 45
Application
Suppose you have three stocks, i.e. X ,Y and Z whose expected return are0.10,0.12 and 0.18; respectively and standard deviation of returns are 0.04;0.10 and 0.14; respectively. You are also told that the covariance betweenthe returns of X and Y 0.002 , between the returns of X and Z is 0.004,and between the returns of Y and Z is 0.007.What is the optimal portfolio (i.e. market portfolio) if the lending andborrowing rate is 0.05.Solution in excel or other software
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 24 / 45
Implications of CAPM
Define asset i ’s return R̃i = R̄i + ε̃i and R̃m = R̄m + ε̃m where ε̃i andε̃m are the unexpected components
Substituting the above into equation 16 we get
R̃i = Rf + βi (R̃m − ε̃m − Rf ) + ε̃i
= Rf + βi (R̃m − Rf ) + ε̃i − βi ε̃m= Rf + βi (R̃m − Rf ) + εi
(17)
Where εi = ε̃i − βi ε̃mWe can show that;
⇒ Cov(R̃m, εi ) = Cov(R̃m, ε̃i )− βiCov(R̃m, ε̃m)
= Cov(R̃m, R̃i )− βiCov(R̃m, R̃m)
= βiVar(R̃m)− βiVar(R̃m) = 0
(18)
Last part comes from the fact that βi = Cov(R̃m,R̃i )
Var(R̃m)thus
Cov(R̃m, R̃i ) = βiVar(R̃m)
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 25 / 45
Implications of CAPM
Define asset i ’s return R̃i = R̄i + ε̃i and R̃m = R̄m + ε̃m where ε̃i andε̃m are the unexpected components
Substituting the above into equation 16 we get
R̃i = Rf + βi (R̃m − ε̃m − Rf ) + ε̃i
= Rf + βi (R̃m − Rf ) + ε̃i − βi ε̃m= Rf + βi (R̃m − Rf ) + εi
(17)
Where εi = ε̃i − βi ε̃m
We can show that;
⇒ Cov(R̃m, εi ) = Cov(R̃m, ε̃i )− βiCov(R̃m, ε̃m)
= Cov(R̃m, R̃i )− βiCov(R̃m, R̃m)
= βiVar(R̃m)− βiVar(R̃m) = 0
(18)
Last part comes from the fact that βi = Cov(R̃m,R̃i )
Var(R̃m)thus
Cov(R̃m, R̃i ) = βiVar(R̃m)
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 25 / 45
Implications of CAPM
Define asset i ’s return R̃i = R̄i + ε̃i and R̃m = R̄m + ε̃m where ε̃i andε̃m are the unexpected components
Substituting the above into equation 16 we get
R̃i = Rf + βi (R̃m − ε̃m − Rf ) + ε̃i
= Rf + βi (R̃m − Rf ) + ε̃i − βi ε̃m= Rf + βi (R̃m − Rf ) + εi
(17)
Where εi = ε̃i − βi ε̃mWe can show that;
⇒ Cov(R̃m, εi ) = Cov(R̃m, ε̃i )− βiCov(R̃m, ε̃m)
= Cov(R̃m, R̃i )− βiCov(R̃m, R̃m)
= βiVar(R̃m)− βiVar(R̃m) = 0
(18)
Last part comes from the fact that βi = Cov(R̃m,R̃i )
Var(R̃m)thus
Cov(R̃m, R̃i ) = βiVar(R̃m)
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 25 / 45
Since R̃i = Rf + βi (R̃m − Rf ) + εi
⇒ Var(R̃i ) = Var(Rf ) + Var [βi (R̃m − Rf )] + Var(εi )
σ2i = β2σ2
m + σ2εi
(19)
Equation 19, above uses variance of independent random variablesi σ2
i - total varianceii β2σ2
m- proportion of variance or market portfolioiii σ2
εi variance of εi , it is orthogonal to portfolio’s return (diversifiable) asillustrated below
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 26 / 45
Since R̃i = Rf + βi (R̃m − Rf ) + εi
⇒ Var(R̃i ) = Var(Rf ) + Var [βi (R̃m − Rf )] + Var(εi )
σ2i = β2σ2
m + σ2εi
(19)
Equation 19, above uses variance of independent random variablesi σ2
i - total varianceii β2σ2
m- proportion of variance or market portfolioiii σ2
εi variance of εi , it is orthogonal to portfolio’s return (diversifiable) asillustrated below
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 26 / 45
Implications of CAPM
Therefore, using equation 18, the unbiased estimate of R̃i can beobtained via OLS
We can also show that
R̄i − Rf =σmi
σm
(R̄m − Rf )
σm
= ρimσi(R̄m − Rf )
σm= ρimσiSe
(20)
Where Se = (R̄m−Rf )σm
is the equilibrium excess return on marketportfolio per unit of risk AKA Sharpe Ratio (in honour of William F.Sharpe, 1966). It is market price for systematic risk- Slope of thecapital market line(CML).
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 27 / 45
Implications of CAPM
Let ωmi weight of i in market portfolio, Vi is i th row of covariance
matrix V
Therefore∂σm∂ωm
i
=1
2σm
∂σ2m
∂ωmi
=1
2σm
∂ωm′Vωm
∂ωmi
=1
2σm2Viω
m =1
σm
N∑j=1
ωmj σij
(21)
Since R̄m =∑n
j=1 ωmj R̄j
Cov(R̄i , R̄m) = Cov
(R̄i ,
n∑j=1
ωmj R̄j
)=
n∑j=1
ωmj σij . (22)
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 28 / 45
Implications of CAPM
⇒ ∂σm∂ωm
i
=1
σmCov(R̄i , R̄m)
= ρimσi
(23)
ρimσi is the marginal increase in market risk(σm) from a marginalincrease in asset i- its the asset’s systematic or non-diversifiable risk
Beta represents an asset’s systematic (market or non-diversifiable) risk
The CAPM, at the point, ‘M’ on the efficient frontier gives the riskadjusted equilibrium return on asset ‘i ′
i E (Ri )− Rf = βi [E (Rm − Rf )]ii βi = Cov(Ri ,Rm)/σ2
m
How do we get β and Rf ?
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 29 / 45
Implications of CAPM
⇒ ∂σm∂ωm
i
=1
σmCov(R̄i , R̄m)
= ρimσi
(23)
ρimσi is the marginal increase in market risk(σm) from a marginalincrease in asset i- its the asset’s systematic or non-diversifiable risk
Beta represents an asset’s systematic (market or non-diversifiable) risk
The CAPM, at the point, ‘M’ on the efficient frontier gives the riskadjusted equilibrium return on asset ‘i ′
i E (Ri )− Rf = βi [E (Rm − Rf )]ii βi = Cov(Ri ,Rm)/σ2
m
How do we get β and Rf ?
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 29 / 45
CAPM and risk premium
The risk premium is βi [E (Rm − Rf )] and represents the reward fortaking risk above that of the risk-free rate
The numerator represents the systematic risk of asset
The denominator represents the total risk of the market portfolio
Beta is an index of the amount of share i ’s systematic risk relative tothe market portfolio
The beta value will tell us how much the expected return on a shareshould rise or fall relative to the market
What are the implications for β = 1;β > 1, β < 1
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 30 / 45
CAPM and risk premium
Plot the risk premium of the asset against the risk premium of themarket – slope is beta
Regress risk premium of the asset against the risk premium of themarket – beta is the regression coefficient(Ri − Rf ) = αi + βi (Rm − Rf ) + εi ;αi = 0
αi = 0 means that when the market risk premium is zero so shouldthe individual shares that make up the portfolio have zero riskpremium.
So what if αi 6= 0?I Over a long period alpha values should be zero.I If αi < 0 shares should be sold
F Negative alpha- returns are below the equilibrium predicted by theCAPM, therefore share prices will fall until yields rise to the equilibrium.
F The act of selling will drive down the price
I αi > 0 should be bought- i.e. share price of i is underpriced.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 31 / 45
CAPM and risk premium
Plot the risk premium of the asset against the risk premium of themarket – slope is beta
Regress risk premium of the asset against the risk premium of themarket – beta is the regression coefficient(Ri − Rf ) = αi + βi (Rm − Rf ) + εi ;αi = 0
αi = 0 means that when the market risk premium is zero so shouldthe individual shares that make up the portfolio have zero riskpremium.
So what if αi 6= 0?
I Over a long period alpha values should be zero.I If αi < 0 shares should be sold
F Negative alpha- returns are below the equilibrium predicted by theCAPM, therefore share prices will fall until yields rise to the equilibrium.
F The act of selling will drive down the price
I αi > 0 should be bought- i.e. share price of i is underpriced.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 31 / 45
CAPM and risk premium
Plot the risk premium of the asset against the risk premium of themarket – slope is beta
Regress risk premium of the asset against the risk premium of themarket – beta is the regression coefficient(Ri − Rf ) = αi + βi (Rm − Rf ) + εi ;αi = 0
αi = 0 means that when the market risk premium is zero so shouldthe individual shares that make up the portfolio have zero riskpremium.
So what if αi 6= 0?I Over a long period alpha values should be zero.
I If αi < 0 shares should be soldF Negative alpha- returns are below the equilibrium predicted by the
CAPM, therefore share prices will fall until yields rise to the equilibrium.F The act of selling will drive down the price
I αi > 0 should be bought- i.e. share price of i is underpriced.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 31 / 45
CAPM and risk premium
Plot the risk premium of the asset against the risk premium of themarket – slope is beta
Regress risk premium of the asset against the risk premium of themarket – beta is the regression coefficient(Ri − Rf ) = αi + βi (Rm − Rf ) + εi ;αi = 0
αi = 0 means that when the market risk premium is zero so shouldthe individual shares that make up the portfolio have zero riskpremium.
So what if αi 6= 0?I Over a long period alpha values should be zero.I If αi < 0 shares should be sold
F Negative alpha- returns are below the equilibrium predicted by theCAPM, therefore share prices will fall until yields rise to the equilibrium.
F The act of selling will drive down the price
I αi > 0 should be bought- i.e. share price of i is underpriced.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 31 / 45
CAPM and risk premium
Plot the risk premium of the asset against the risk premium of themarket – slope is beta
Regress risk premium of the asset against the risk premium of themarket – beta is the regression coefficient(Ri − Rf ) = αi + βi (Rm − Rf ) + εi ;αi = 0
αi = 0 means that when the market risk premium is zero so shouldthe individual shares that make up the portfolio have zero riskpremium.
So what if αi 6= 0?I Over a long period alpha values should be zero.I If αi < 0 shares should be sold
F Negative alpha- returns are below the equilibrium predicted by theCAPM, therefore share prices will fall until yields rise to the equilibrium.
F The act of selling will drive down the price
I αi > 0 should be bought- i.e. share price of i is underpriced.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 31 / 45
The Security Market Line
The CAPM can also be illustrated in graphical form as the securitymarket line (SML).
Similar to the CML, the SML shows the trade-off between risk andexpected return as a straight line intersecting the vertical axis (i.e.,zero-risk point) at the risk-free rate.
However,I CML measures risk by the standard deviation (i.e., total risk), while the
SML considers only the systematic component.I CML can be applied only to a fully diversified portfolio, whereas SML
can be applied to any individual asset or portfolio of assets
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 32 / 45
The Security Market Line
The CAPM can also be illustrated in graphical form as the securitymarket line (SML).
Similar to the CML, the SML shows the trade-off between risk andexpected return as a straight line intersecting the vertical axis (i.e.,zero-risk point) at the risk-free rate.
However,I CML measures risk by the standard deviation (i.e., total risk), while the
SML considers only the systematic component.I CML can be applied only to a fully diversified portfolio, whereas SML
can be applied to any individual asset or portfolio of assets
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 32 / 45
The Security Market Line
Using the SML, we can compare this required rate of return to theasset’s estimated rate of return over a specific investment horizon todetermine whether it would be an appropriate investment.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 33 / 45
The Security Market Line
Using the SML, we can compare this required rate of return to theasset’s estimated rate of return over a specific investment horizon todetermine whether it would be an appropriate investment.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 33 / 45
The Security Market Line- Identifying under/Over-valuedassets
Suppose you have the following
Company β Pt Pt+1 Dt+1
1 0.70 35 36 1.002 1.00 50 52 0.603 1.20 66 68 1.004 1.35 94 98 -5 -0.80 50 53 1.30
Rf = 5.5;Rm = 10%
Assess whether the above stocks are under/over-valued.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 34 / 45
The Security Market Line- Identifying under/Over-valuedassets
Suppose you have the following
Company β Pt Pt+1 Dt+1 Actual Ret(%) Req. Ret(%)
1 0.70 35 36 1.00 5.71 8.52 1.00 50 52 0.60 5.20 10.003 1.20 66 68 1.00 4.55 11.004 1.35 94 98 - 4.26 11.805 -0.80 50 53 1.30 8.60 -4.00
Rf = 5.5;Rm = 10%
Assess whether the above stocks are under/over-valued.
Actual Return= Pt+1+Dt+1−Pt
Pt; Required Return=Rf + βi (Rm − Rf )
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 35 / 45
Changes in the SML
Three changes can occur with respect to the initial security marketline, namely;
1 Individual investments can change positions on the SML- movementsalong the SML
2 Slope of SML can change3 SML can experience a shift
Movements along the SMLI Reflect changes in the perceived risk of a security/firm.I Due to change in firm’s risk, e.g. business risk, financial risk etc - it
will move along the SML.I NB: SML is unchanged; only the position of individual asset changes.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 37 / 45
Changes in the SML
Three changes can occur with respect to the initial security marketline, namely;
1 Individual investments can change positions on the SML- movementsalong the SML
2 Slope of SML can change3 SML can experience a shift
Movements along the SMLI Reflect changes in the perceived risk of a security/firm.I Due to change in firm’s risk, e.g. business risk, financial risk etc - it
will move along the SML.I NB: SML is unchanged; only the position of individual asset changes.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 37 / 45
Changes in Slope of SML
Reflects changes in the return required by investors per unit of risk.
SML rotates clockwise, about the risk-free rate, when there is adecrease in the risk and vice versa.
Reasons for changes in slope are not clearly understood, but we doknow that there are changes in the yield differences between assetswith different levels of risk, even though the intrinsic risk differencesare constant.
Often implies, or is an outcome of, change in risk premium.
Affects the required rate of return for all risky assets - however itsunique risk characteristics remain unchanged.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 38 / 45
Shifts in SML
Shift may occur due to changes in any of the following:1 Expected real growth;2 Capital market conditions; or3 Expected inflation rate.
Increase in any of the above, shifts SML upwards and vice versa.
Parallel shift- changes in the above factors that affect all investmentsat all risk levels.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 39 / 45
CAPM and Single Period Valuation of Risky Cashflows
CAPM gives a quantifiable measure of risk for individual asset
Suppose an asset with a risky payoff, P̃τ at the end of the period,
Let the price today be P0, then the return is; r̃i = P̃τ−P0P0
CAPM can be written as
E (ri ) = rf + E (rm − rf )Cov(ri , rm)
Var(rm)(24)
= rf + λCov(ri , rm) (25)
where λ = E(rm−rf )Var(rm) , and is the market price per unit of risk
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 40 / 45
CAPM and Single Period Valuation
Substituting for r̃i into equation 25 we get
E (Pτ )− P0
P0= rf + λCov(ri , rm) (26)
Therefore P0 can be interpreted as the equilibrium price
P0 =E (Pτ )
1 + rf + λCov(ri , rm)or
E (Pτ )
1 + rf + β(E (rm)− rf )(27)
The above is called the risk-adjusted return valuation formula
If the asset is risk-free it will have zero covariance with the market
For any asset with positive risk, a risk premium is added to (1 + rf )
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 41 / 45
CAPM and Single Period Valuation
Note that the covariance between this risky asset and the market can
be written as; Cov(ri , rm) = Cov(P̃τ−P0
P0, rm)
Cov(ri , rm) = E
[(P̃τ − P0
P0− E
( P̃τ − P0
P0
))(rm − E (rm)
)](28)
=1
P0Cov(Pτ , rm) (29)
Substituting into equation 27
P0 =E (Pτ )
1 + rf + λP0Cov(Pτ , rm)
(30)
We can derive the certainty-equivalent valuation formula as
P0 =E (Pτ )− λCov(Pτ , rm)
1 + rf(31)
Note the above is obtained without any understanding of investorutility function
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 42 / 45
CAPM and Valuation of Risky Cash-flows- Example
Suppose a company is considering expanding its operation byinvesting in a project whose initial cost is R1bn today, it is expectingto receive R300m in the first year, R400m in the second year, andR500m in the third year. The average beta for companies thatoperate in similar lines of business is 0.8. The risk-free rate is 4% andthe average market risk premium is 5%. Advise the company on thisnew investment.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 43 / 45
CAPM and reality
Is the condition of zero alphas for all stocks as implied by the CAPMmet?
Is the CAPM testable?- we will look at this in the next session
CAPM is still considered the best available description of securitypricing and is widely accepted
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 44 / 45
Summary
Two efficient portfolios are enough to span the entire mean-varianceefficient frontier
In the presence of a risk-less asset, only one efficient portfolio(tangency portfolio) and the risk-less asset is required to span thefrontier.
We can use this to derive the risk-adjusted return valuation formula.
Godfrey Ndlovu (AERC) Financial Economics- CPP January 1, 2021 45 / 45