Financial Economics- CPP

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


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


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.



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)



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


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 )


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


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


The Tangency

Therefore intercept of tangent to ω1 is expected return of itszero-covariance counterpart, ω2


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)


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


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


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.


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)


The Capital Asset Pricing Model (CAPM)


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


Recap from Mean-variance Analysis

We have shown that

σp =R̄p − Rf(

b − 2aRf + cR2f

) 12

As shown below

Source: Pennacchi


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)



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)



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



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.


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


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


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)





R̃i = Rf + βi (R̃m − ε̃m − Rf ) + ε̃i


(17)

Where εi = ε̃i − βi ε̃m

We can show that;




(18)


Var(R̃m)thus






R̃i = Rf + βi (R̃m − ε̃m − Rf ) + ε̃i


(17)

Where εi = ε̃i − βi ε̃mWe can show that;




(18)


Var(R̃m)thus



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



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).



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)



⇒ ∂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 ?


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



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.






So what if αi 6= 0?

I Over a long period alpha values should be zero.I If αi < 0 shares should be sold









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







So what if αi 6= 0?I Over a long period alpha values should be zero.I If αi < 0 shares should be sold





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



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.


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.


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 )


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.


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.


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.


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


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 )


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


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.


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


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.


Financial Economics- CPP

Documents

Transcript of Financial Economics- CPP