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An in-depth analysis of the Capital Asset Pricing Model (CAPM), including its derivation, interpretation, and applications. topics such as the security market line, risk and return, efficient portfolios, and the generalization of CAPM under a utility framework.
What you will learn
Typology: Study notes
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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.
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 +
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
asset i.
b − ar
so that
σiM =
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
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
where
λP 1 = c − bμP ∆ , λP 2 = aμP − b ∆ , λQ 1 =
c − bμQ ∆ , λQ 2 =
aμQ − b ∆
Find the covariance between RP and RQ.
T
∗ Q =^
[
]T
a ∆
( μP − b a
) ( μQ − b a
)
a