Capital Asset Pricing Model and Arbitrage Pricing Theory

78 Pages·2014·0.63 MB·English

CapitalAssetPricingModel
Asummarizingdigression
ArbitragePricingTheory
Capital Asset Pricing Model
and
Arbitrage Pricing Theory
Nico van der Wijst
1 D.vanderWijst TI(cid:216)4146Financeforscienceandtechnologystudents
CapitalAssetPricingModel
Asummarizingdigression
ArbitragePricingTheory
1 Capital Asset Pricing Model
2 A summarizing digression
3 Arbitrage Pricing Theory
2 D.vanderWijst TI(cid:216)4146Financeforscienceandtechnologystudents
DerivationoftheCAPM
CapitalAssetPricingModel
InsightsfromtheCAPM
Asummarizingdigression
Undelyingassumptions
ArbitragePricingTheory
Empiricaltests
E[r ] Ind.1
p
C
Ind.2
M
B
r
f
A
σ
p
The Capital Market Line
3 D.vanderWijst TI(cid:216)4146Financeforscienceandtechnologystudents
DerivationoftheCAPM
CapitalAssetPricingModel
InsightsfromtheCAPM
Asummarizingdigression
Undelyingassumptions
ArbitragePricingTheory
Empiricaltests
Capital Asset Pricing Model CAPM
Capital Market Line only valid for e¢ cient portfolios
combinations of risk free asset and market portfolio M
all risk comes from market portfolio
What about ine¢ cient portfolios or individual stocks?
don(cid:146)t lie on the CML, cannot be priced with it
need a di⁄erent model for that
What needs to be changed in the model:
the market price of risk ((E(r ) r )/σ ),
m f m
(cid:0)
or the measure of risk σ ?
p
4 D.vanderWijst TI(cid:216)4146Financeforscienceandtechnologystudents
DerivationoftheCAPM
CapitalAssetPricingModel
InsightsfromtheCAPM
Asummarizingdigression
Undelyingassumptions
ArbitragePricingTheory
Empiricaltests
CAPM is more general model, developed by Sharpe
Consider a two asset portfolio:
one asset is market portfolio M, weight (1 x)
(cid:0)
other asset is individual stock i, weight x
Note that this is an ine¢ cient portfolio
Analyse what happens if we vary proportion x invested in i
begin in point I, 100% in i, x=1
in point M, x=0, but asset i is included in M with its market
value weight
to point I(cid:146), x<0 to eliminate market value weight of i
5 D.vanderWijst TI(cid:216)4146Financeforscienceandtechnologystudents
DerivationoftheCAPM
CapitalAssetPricingModel
InsightsfromtheCAPM
Asummarizingdigression
Undelyingassumptions
ArbitragePricingTheory
Empiricaltests
E[r ]
p
C
I(cid:146)
M
I
B
r
f
A
σ
p
Portfolios of asset i and market portfolio M
6 D.vanderWijst TI(cid:216)4146Financeforscienceandtechnologystudents
DerivationoftheCAPM
CapitalAssetPricingModel
InsightsfromtheCAPM
Asummarizingdigression
Undelyingassumptions
ArbitragePricingTheory
Empiricaltests
Risk-return characteristics of this 2-asset portfolio:
E(r ) = xE(r)+(1 x)E(r )
p i m
(cid:0)
σ = [x2σ2+(1 x)2σ2 +2x(1 x)σ ]
p i (cid:0) m (cid:0) i,m
q
Expected return and risk of a marginal change in x are:
∂E(r )
p
= E(r) E(r )
i m
∂x (cid:0)
∂σp = 1 x2σ2+(1 x)2σ2 +2x(1 x)σ (cid:0)12
∂x 2 i (cid:0) m (cid:0) i,m
h i
2xσ2 2σ2 +2xσ2 +2σ 4xσ
(cid:2) i (cid:0) m m i,m(cid:0) i,m
h i
7 D.vanderWijst TI(cid:216)4146Financeforscienceandtechnologystudents
DerivationoftheCAPM
CapitalAssetPricingModel
InsightsfromtheCAPM
Asummarizingdigression
Undelyingassumptions
ArbitragePricingTheory
Empiricaltests
First term of ∂σ /∂x is 1 , so:
p 2σp
∂σ 2xσ2 2σ2 +2xσ2 +2σ 4xσ
p = i (cid:0) m m i,m(cid:0) i,m
∂x 2σ
p
xσ2 σ2 +xσ2 +σ 2xσ
= i (cid:0) m m i,m(cid:0) i,m
σ
p
Isolating x gives:
∂σ x(σ2+σ2 2σ )+σ σ2
p = i m(cid:0) i,m i,m(cid:0) m
∂x σ
p
8 D.vanderWijst TI(cid:216)4146Financeforscienceandtechnologystudents
DerivationoftheCAPM
CapitalAssetPricingModel
InsightsfromtheCAPM
Asummarizingdigression
Undelyingassumptions
ArbitragePricingTheory
Empiricaltests
At point M all funds are invested in M so that:
x = 0 and σ = σ
p m
Note also that:
i is already included in M with its market value weight
economically x represents excess demand for i
in equilibrium M excess demand is zero
This simpli(cid:133)es marginal risk to:
∂σ σ σ2 σ σ2
p = i,m(cid:0) m = i,m(cid:0) m
∂x σ σ
(cid:12)x=0 p m
(cid:12)
(cid:12)
(cid:12)
9 D.vanderWijst TI(cid:216)4146Financeforscienceandtechnologystudents
DerivationoftheCAPM
CapitalAssetPricingModel
InsightsfromtheCAPM
Asummarizingdigression
Undelyingassumptions
ArbitragePricingTheory
Empiricaltests
So the slope of the risk-return trade-o⁄ at equilibrium point M is:
∂E(r )/∂x E(r) E(r )
p = i (cid:0) m
∂σ /∂x (σ σ2)/σ
p (cid:12)x=0 i,m(cid:0) m m
(cid:12)
(cid:12)
But at point M the slope o(cid:12)f the risk-return trade-o⁄ is also the
slope of the CML, so:
E(ri)(cid:0)E(rm) = E(rm)(cid:0)rf
(σ σ2)/σ σ
i,m(cid:0) m m m
Solving for E(r) gives:
i
σ
E(r) = r +(E(r ) r ) i,m
i f m (cid:0) f σ2
m
= r +(E(r ) r )β
f m (cid:0) f i
10 D.vanderWijst TI(cid:216)4146Financeforscienceandtechnologystudents