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- Capital Asset Pricing Model and Factor Models
Capital market line (CML)
CML is the tangent line drawn from the risk free point to the feasible region for risky assets. This line shows the relation between rP and σP for efficient portfolios (risky assets plus the risk free asset).
The tangency point M represents the market portfolio, so named since all rational investors (minimum variance criterion) should hold their risky assets in the same proportions as their weights in the market portfolio.
- Every investor is a mean-variance investor and all have homo- geneous expectations on means and variances, then everyone buys the same portfolio. Prices adjust to drive the market to efficiency.
Based on the risk level that an investor can take, she will combine the market portfolio of risky assets with the risk free asset.
Example Consider an oil drilling venture; current share price of the venture = $875, expected to yield $1, 000 in one year. The standard deviation of return, σ = 40%; and rf = 10%. Also, rM = 17% and σM = 12% for the market portfolio.
Question How does this venture compare with the investment on efficient portfolios on the CML?
Given this level of σ, the expected rate of return predicted by the CML is
r = 0.10 +
× 0 .40 = 33%.
The actual expected rate of return =
−1 = 14%, which is well
below 33%. This venture does not constitute an efficient portfolio. It bears certain type of risk that does not contribute to the expected rate of return.
Sharpe ratio
One index that is commonly used in performance measure is the Sharpe ratio, defined as
ri − rf σi
excess return above riskfree rate standard deviation
We expect Sharpe ratio ≤ slope of CML.
Closer the Sharpe ratio to the slope of CML, the better the perfor- mance of the fund in terms of return against risk.
In previous example,
Slope of CML = 17%^ −^ 10% 12%
Sharpe ratio =
= 0. 1 < Slope of CML.
Proof
Consider the portfolio with α portion invested in asset i and 1 − α portion invested in the market portfolio M. The expected rate of return of this portfolio is
rα = αri + (1 − α)rM
and its variance is
σ α^2 = α^2 σ i^2 + 2α(1 − α)σiM + (1 − α)^2 σ M^2.
As α varies, (σα, rα) traces out a curve in the σ − r diagram. The market portfolio M corresponds to α = 0.
The curve cannot cross the CML, otherwise this would violate the property that the CML is an efficient boundary of the feasible region. Hence, as α passes through zero, the curve traced out by (σα, rα) must be tangent to the CML at M.
Tangency condition Slope of the curve at M = slope of CML.
Solving for ri, we obtain
ri = rf + σiM ︸ ︷︷ ︸^ σ^2 M βi
(rM − rf ) = rf + βi(rM − rf ).
Now, βi =
ri − rf rM − rf
expected excess return of asset i over rf expected excess return of market portfolio over rf
Predictability of equilibrium return
The CAPM implies in equilibrium the expected excess return on any single risky asset is proportional to the expected excess return on the market portfolio. The constant of proportionality is βi.
Alternative proof of CAPM
Consider (^) σ
iM = cov(ri, rM ) =^ eTi Ωw∗ M ,
where ei = (0 · · · 1 · · · 0) = ith^ co-ordinate vector is the weight of
asset i.
Recall w∗ M = Ω
− 1 (μ − r 1 )
b − ar
so that
σiM =
(μ − r 1 )i
b − ar
ri − r b − ar
, provided b − ar 6 = 0. (1)
Also, we recall μMP = c − br b − ar and σ P,M^2 = c − 2 rb + r^2 a (b − ar)^2 so that
μMP − r = c − br b − ar
− r = c − 2 rb + r^2 a (b − ar)^2
= (b − ar)σ^2 P,M. (2)
Eliminating b − ar from eqs (1) and (2), we obtain
ri − r = σiM σ M^2
(μMP − r).
Beta of a portfolio
Consider a portfolio containing n assets with weights w 1 , w 2 , · · · , wn.
Since rP =
∑^ n i=
wiri, we have cov(rP , rM ) =
∑^ n i=
wicov(ri, rM ) so that
βP = cov(rP , rM ) σ M^2
∑n i=1 wicov(ri, rM^ ) σ M^2
∑^ n i=
wiβi.
Some special cases of beta values
- When βi = 0, ri = rf. A risky asset (with σi > 0) that is uncor- related with the market portfolio will have an expected rate of return equal to the risk free rate. There is no expected excess return over rf even the investor bears some risk in holding a risky asset with zero beta.
- When βi = 1, ri = rM. A risky asset which is perfectly correlated with the market portfolio has the same expected rate of return as that of the market portfolio.
Extension
Let P be any efficient portfolio along the upper tangent line and Q be any portfolio. We also have
RQ − r = βP Q(RP − r), (A)
that is, P is not necessary to be the market portfolio.
More generally,
RQ − r = βP Q(RP − r) + ≤P Q (B)
with cov(RP , ≤P Q) = E[≤P Q] = 0.
The first result (A) can be deduced from the CAPM by observing
σQP = cov(RQ, αRM + (1 − α)Rf ) = αcov(RQ, RM ) = ασQM , α > 0
σ P^2 = α^2 σ^2 M and RP − r = α(RM − r).
Consider
RQ − r = βM Q(RM − r) =
σQM σ M^2
(RM − r)
σQP /α σ P^2 /α^2
(RP − r)/α = βP Q(RP − r).
Zero-beta CAPM
From the CML, there exists a portfolio ZM whose beta is zero. Consider the CML
rQ = r + βQM (rM − r),
since βM ZM = 0, we have rZM = r. Hence the CML can be expressed in terms of market portfolio M and its zero-beta counterpart ZM as follows
rQ = rZM + βQM (rM − rZM ).
In this form, the role of the riskfree asset is replaced by the zero- beta portfolio ZM. In this sense, we allow the absence of riskfree asset.
? The more general version allows the choice of any efficient (mean-variance) portfolio and its zero-beta counterpart.
Finding the non-correlated counterpart
Let P and Q be any two frontier portfolios. Recall that
w∗ P = Ω−^1 (λP 1 1 + λP 2 μ) and w∗ Q = Ω−^1 (λQ 1 1 + λQ 2 μ)
where
λP 1 = c − bμP ∆ , λP 2 = aμP − b ∆ , λQ 1 =
c − bμQ ∆ , λQ 2 =
aμQ − b ∆
a = 1 T^ Ω−^11 , b = 1 T^ Ω−^1 μ, c = μT^ Ω−^1 μ, ∆ = ac − b^2.
Find the covariance between RP and RQ.
cov(RP , RQ) = w∗
T
P Ωw
∗ Q =^
[
Ω−^1 (λP 1 1 + λP 2 μ)
]T
(λQ 1 1 + λQ 2 μ)
= λP 1 λQ 1 a + (λP 1 λQ 2 + λQ 1 λP 2 )b + λP 2 λQ 2 c
a ∆
( μP − b a
) ( μQ − b a
)
a
