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Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Portfolio Theory
Gorazd Brumen
Morgan Stanley
September 11-12, 2009
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Topics I will cover
1 Part I: Portfolio Selection in One Period
2 Part II: Portfolio Selection in Continuous Time
3 Part III: Advanced Topics in Portfolio Theory
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Requirements, prior knowledge
Basic linear algebra, optimization techniques.
Basic probability theory.
Stochastic integration, SDE.
Basic microeconomics.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Part I
Part I: Portfolio Selection inOne Period
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Historical perspective on portfolio selection
Even though the concept of diversification is firmly groundedin today’s economic thinking, this was not always the case.
Before Markowitz’s seminal contribution investors did notconsider portfolio diversification but rather stock picking: Ofall stocks in a market pick the one which brings you highestcombination of dividends (and capital gains).
Portfolio theory answers the question which risks are pricedand in what extent.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Framework and Notations
One period model. Vectors will be underlined, such as x , matricesare boldface, e.g. Γ. Begining of period at time 0, end of period at1. Return on an asset i = 1, . . . ,N in terms of its price Si in thisperiod is
Ri =Si(1) − Si(0)
Si(0)
(Ri is a random variable) Let the number of units of asset i be ni .The value of the portfolio X (t) at time t holding ni units is
X (t) = n′S(t). The return on such a portfolio is RX = X (1)−X (0)X (0) .
If xi(0) = xi = niSi (0)X (0) then x ′1 = 1 and
RX = x ′R.
(proof left as an exercise.)Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Analysis of mean and variance
We focus on the first two moments of R. Let µ = (µi) = E(Ri)and Γ = (σij) = (cov(Ri ,Rj )). Then
µX = E(RX ) = x ′µ
var(RX ) =∑
i ,j
xixjσij = xΓx
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Portfolio with risk-less asset
We introduce the riskless asset by S0(1) = S0(0)(1 + r) andadditionally
x0 = 1 −N∑
i=1
xi
Portfolio returns in this case are
RX = x0r +N∑
i=1
xiRi = r +N∑
i=1
xi(Ri − r)
RX − r = x ′(µ − r1),
i.e. portfolio excess return is a linear combination of excess returnsof individual stocks.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Simple relations
Definition
Assets i = 1, . . . ,N are redundant if there exists N scalarsλ1, . . . , λN such that
∑Ni=1 λiRi = k for some constant k. The
portfolio λ is risk-free.
Proposition
The assets i = 1, . . . ,N are not redundant if and only if Γ ispositive definite.
(exercise.)
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Efficiency criteria and optimization program
Definition
Portfolio (x∗,X ∗) is efficient if for every other portfolio y we havethat if σY < σX∗ then µY < µX∗ and σY = σX∗ impliesµY ≤ µX∗ .
Portfolio optimization problem:
maxx
E(RX ) s.t. x ′Γx = k x ′1 = 1.
The Lagrangian of this problem is
L(x ,θ
2, λ) = x ′µ −
θ
2x ′Γx − λx ′1
First order condition gives us
µ − θΓx∗ − λ1 = 0 (1)
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Optimization program (contd.)
Equivalently
µi = λ + θ
N∑
j=1
x∗
j σij .
FOC are neccessary and sufficient, since the second derivative isstrictly concave (Γ is positive definite).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Connection to utility theory
Criterium of portfolio efficiency is consistent with the economicagents with the following utility
u(x) = E(RX ) −θ
2var(RX ),
= x ′µ −θ
2x ′Γx .
where the Lagrange parameter θ now represents some degree ofrisk aversion, i.e. the higher θ is, the more averse the agent is wrt.(variance) risk.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Competitive economic equilibirum
A set of agents i = 1, . . . , I .
A set of assets Sj , j = 1, . . . ,N in net supply y .
Definition (Competitive equilibrium)
Portfolio x∗ and price system S is a competitive equilibrium if
x∗
i is the solution to the optimization problem
maxx i
ui (x) s.t. x ′
iS = Wi
Markets clear:
I∑
i=1
x∗
i = y
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Two funds separation (Black)
Consider any two efficient portfolios x and y . Then
Theorem
Any convex comination of x and y, i.e. ux + (1 − u)y isefficient.
Any efficient portfolio is a combination of x and y (notnecessarily convex).
The efficient frontier is a parabola in the expectedreturn-variance space (µ, σ2) and a hyperbola in the expectedreturn-standard deviation space (µ, σ).
Due to the first bullet point above, any efficient portfolio can bedescribed as a convex combination of just 2 portfolios. Proof ofthe first two bullet points left as an exercise.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Proof of last bullet of two fund separation
We have shown in (1) that x∗ = θΓ−1(µ − λ1). From 1′x∗ = 1 we
get that λ =1′Γ
−1µ−θ
1′Γ1. Therefore x∗ = k1 + θk2 for appropriate
k1 and k2. The efficiency set is given by
ES = x∗ : x∗ = k1 + θk2, θ > 0
from where it follows that portfolio return µ′x∗ is linear and the
variance x∗′
Γx∗ is quadratic in θ. The efficiency frontier is aparabola in this space.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Efficiency set with riskless asset
Theorem
Asset 0 is efficient.
Any combination uRX + (1 − u)r of asset 0 and a portfolio Xlies on the straight line between 0 and X in the (µ, σ) space.
The straight line between asset 0 and asset M is the efficientfrontier called the Capital Market Line.
(Tobin’s two fund separation) Any efficient portfolio is acombination of only 2 portfolios (e.g. 0 and M).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Efficiency set with riskless asset (contd.)
Theorem (contd.)
Any efficient portfolio satisfies
x∗ = θΓ−1(µ − r1)
Tangent (market) portfolio (m,M) is
m = θMΓ−1(µ − r1)
θM =1
1′Γ−1(µ − r1)
(Proof left as an exercise.)
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Capital Market Equilibirium
Since the market portfolio is efficient there exists scalars λ and θsuch that
µi = λ + θcov(RM ,Ri ),
It follows that for any portfolio we have
E(RX ) =
N∑
i=1
xiµi
=
N∑
i=1
xi(λ + θcov(RM ,Ri))
= λ + θcov(RM ,RX )
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
CAPM equilibrium
In particular for market portfolio it holds that
µM = λ + θσ2M
from where it follows that θ = µM−λσ2
M
and therefore
µi = λ + θcov(RM ,Ri) = λ + (µM − λ)βi
where βi = cov(RM ,Ri )σ2
M
. If we set Ri = r the risk-less asset we get
E(Ri) = r + βi (µM − r).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
CAPM as a Pricing Model
Question: If a security delivers V (1) at time 1, what is the priceV (0) of this security at time 0? Assuming that the risk-free assetexists then
E(V (1))
V (0)= E(1 + R) = 1 + r + θcov
(
V (1)
V (0),RM
)
where θ = µM−r
σ2M
Solving for V (0) gives us
V (0) =E(V (1)) − θcov(V (1),RM)
1 + r.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Further topics and Relevant literature
Relevant literature:
Book by Huang/Litzenberger.
Mossin (Econometrica paper), Sharpe, Cass-Stiglitz.
Further topics:
Arbitrage pricing theory (Ross).
Behavioral portfolio theory. (Kahneman and Tversky)
Factor models. (partially given in the exercises)
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Part II
Part II: Portfolio Selection inContinuous Time
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Continuous time financial market
Financial market:
Risk-free security: dBt = Btrt dt, process rt is progressivelymeasurable (adapted with cadlag paths) and
∫ T
0 ru du < ∞.
d stocks with dynamics:
dS t + Dt dt = IS (µtdt + σt dW t)
where dS are stocks’ capital gains and D dividends.
IS = diag(S1,S2, . . . ,Sd).
(µ,σ) is a progressively measurable process such that∫ T
0 µtdt < ∞ and
∫ T
0 σtσ′
t dt < ∞.
If σ is invertible, the financial market is complete.
Market price of risk (Sharpe ratio): θt = σ−1t (µ − r1).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Investors
Progressively measurable consumption process c > 0, such that∫ T
0 ct dt < ∞ and U(c) < ∞ such that
U(c) = E
[∫ T
0u(ct , t)dt
]
where u(·, t) : R+ → R is strictly increasing, strictly concave andtwice continuously differentiable and satisfies the Inada condition:u′(0, t) = ∞, u′(∞, t) = 0 for every t ∈ [0,T ].
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Examples of utility functions considered
u(c , t) = ρtu(c) where ρt is subjective discount factor, e.g.ρt = exp(−
∫ t
0 βv dv).
u(c) = c1−R
1−R, R ≥ 0 is an example of CRRA utilities.
u(c) = 11−R
(c + δ)1−R , R ≥ 0, δ > 0 an example of HARAutilities.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Investor’s wealth dynamics
Progressively measurable portfolio process π generates investor’swealth dynamics
dXt = π′
t((IS)−1(dS t + Dt dt)) + (Xt − π′
t1)rt dt − ct dt
= π′
t [(µt− rt1)dt + σt dW t ] + (Xtrt − ct)dt,
where X0 = x given. The first term is the return on stockportfolio, the second the return on bonds and the third theconsumption part.
Definition
(c , π) is admissible (belongs to A(x) iff Xt ≥ 0 for allt ∈ [0,T ].
(c , π) is optimal (belongs to A∗(x) iff there does not exist(c , π) such that U(c) > U(c).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Static approach
Let
ηt = exp
(
−
∫ t
0θ′v dW v −
1
2
∫ t
0θ′vθv dv
)
.
Due to Novikov condition η is a martingale. Therefore we canchange the measure dQ = ηT dP. This implies the following:
It can be proven that W t = W t +∫ t
0 θv dv is a Brownianmotion.
Sv = Ev [Stξv ,t +∫ t
vξv ,sDs ds] where ξt = btηt and ξv ,t = ξt
ξv.
Arrow-Debreu prices are then ξT dP , i.e. a security paying 1ω
is ξT (ω)dP(ω).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Equivalence of approaches
Let
B(x) =
c : E
(∫ T
0ξtct dt
)
≤ x
.
Theorem
We have the following implications:
(a) If (c , π) ∈ A(x) then c ∈ B(x)
(b) If c ∈ B(x) then there exists π such that (c , π) ∈ A(x).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Proof for (a)
Let us have X ≥ 0, c ≥ 0, then ξtXt +∫ t
0 ξvcv dv ≥ 0 for everyt ∈ [0,T ]. LHS is a positive local martingale, which implies that it
is a supermartingale. Therefore E
[
ξTXT +∫ T
0 ξvcv dv]
≤ x and
therefore E
[
∫ T
0 ξvcv dv]
≤ x which implies that c ∈ B(x).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Proof of (b)
Let
Et
[∫ T
t
ξvcv dv
]
= Et
[∫ T
0ξvcv dv
]
−
∫ t
0ξvcv dv
= E0
[∫ T
0ξvcv dv
]
+
∫ t
0φv dWv −
∫ t
0ξvcv dv
Choose φt = ξt [π′
tσt − Xtθ′
v ]. Then by the equation () frombefore we have that
ξtXt +
∫ t
0ξvcv =
∫ t
0φv dWv + x
= Et
[∫ T
t
ξvcv dv
]
− E0
[∫ T
0ξvcv dv
]
+ x
from where it follows that ξtXt ≥ Et
[
∫ T
tξvcv dv
]
≥ 0, i.e.
Xt ≥ 0 for all t ∈ [0,T ]. This proves that (c , π) ∈ A(x).Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Static portfolio optimization
Constructing the Lagrangian:
L = E
[∫ T
0u(ct , t)dt
]
− y
E
[∫ T
0ξvcv dv
]
− x
Process c is optimal if there does not exist a process ∆t such thatL(c + ε∆) > L(c). The necessary condition is therefore∂L∂ε |ε=0 = 0 for every ∆. We have
∂L
∂ε|ε=0 = E
[∫ T
0u′(ct , t)∆t dt
]
− yE
[∫ T
0ξv∆v dv
]
= E
[∫ T
0(u′(ct , t) − yξt)∆t dt
]
from where it follows that u′(ct , t) = yξt , i.e. marginal utilityequals marginal costs. y is fixed by the condition
E
[
∫ T
0 ξvcv dv]
= x , y ≥ 0.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Summary of the portfolio optimization
Theorem
Optimal portfolio optimization gives us
c∗t = I (y∗ξt , t) where I = (u′)−1
y∗ : x = E
[∫ T
0ξv I (yξv , v)dv
]
π∗
t = Xt(σ′
t)−1θt + ξ−1
t (σ′
t)−1φ∗
t
φ∗
t = Et [F∗] − E[F ∗]
F ∗ =
∫ T
0ξvc∗v dv
X ∗
t = Et
[∫ T
t
ξt,vc∗v dv
]
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Malliavin calculus (Stochastic calculus of variations)
Motivation: If F ∈ L2 then there exists by the martingalerepresentation theorem a progressively measurable process φ suchthat
F = E[F ] +
∫ T
0φv dWv
How to extract φ? The question is not important only in portfoliotheory but also in derivatives pricing for hedging purposes.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Hedging of derivative securities
Fundamental theorem of asset pricing states that the price of aderivative security with payoff ϕ(ST ) at time T is given by
EQ[ϕ(ST )].
The replicating portfolio is given by the process u such that
ϕ(ST ) = EQ[ϕ(ST )] +
∫ T
0ut dSt.
Malliavin calculus gives us an answer to what is u.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Malliavin calculus definitions
Definition
Let S be the space of smooth Brownian functionals, i.e.
S = f (Wt1 , . . . ,Wtn) : f ∈ C∞
p (Rdn)
and where C∞
p is the space of functions on Rdn which are infinitelydifferentiable and of polynomial growth.Then the Malliavin derivative DF = DtF : t ∈ [0,T ] is ad-dimensional stochastic process defined by
Di ,tF =
n∑
j=1
∂f
∂xij· 1[0,tj ](t)
for every i = 1, . . . , d (i corresponds to rows).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Simple properties of Malliavin calculus
Malliavin derivative is the generalization of the Frechetderivative for stochastic processes.
Malliavin derivatives are not adopted (anticipating processes).
DtF = 0 if F ∈ F s and s < t.
Theory can be extended to appropriate spaces for stochasticprocesses called D2,1.
If ST = S0 exp((µ − 1/2σ2)T + σWT ) thenDtST = STσ1[0,T ](t).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Properties of Malliavin calculus
Chain rule: Let g : Rm → R with bounded derivatives andF1, . . . ,Fm ∈ D2,1. Then
Dtg(F1, . . . ,Fm) =
m∑
i=1
∂g
∂FiDtFi
Dt(Ev [F ]) = Ev (DtF ) for v ≥ t
Clark-Ocone formula: Let F ∈ D2,1. Then
F = E(F ) +
∫ T
0φv dWv
where
φv = Ev (DvF ).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Malliavin calculus rules
More rules:
If F1 =∫ T
0 φ1t dt then
DtF1 =
∫ T
0Dtφ1v dv
=
∫ T
t
Dtφ1v dv
If F2 =∫ T
0 φ2t dWt then
DtF2 =
∫ T
t
Dtφ2v dWv + φ2t
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Malliavin Calculus of Stochastic Processes
Let
dSt = µ(t,St)dt + σ(St , t)dWt
then
ST = St +
∫ T
t
µ(Sv , v)dv +
∫ T
t
σ(Sv , v)dWv .
Applying the rules from before we get that
DtST =
∫ T
t
∂µ
∂S(Sv , v)DtSv dv +
∫ T
t
∂σ
∂S(Sv , v)DtSv dWv + σ(St , t)
from where it follows that
dDtSv =∂µ
∂S(Sv , v)DtSv dv +
∂σ
∂S(Sv , v)DtSv dWv
with initial condition DtSt = σ(St , t).Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Optimal portfolios
Theorem
We have
π∗
t = ξ−1t Et
[∫ T
t
c∗vRv
ξv dv
]
(σ′
t)−1θt
−ξ−1t (σ′
t)−1Et
[∫ T
t
c∗v (1 −1
Rv)ξvHt,v dv
]
where
Rt = −u′′(c∗t , t)
u′(c∗t , t)c∗t relative risk aversion
Ht,v =
∫ v
t
Dtrs ds +
∫ v
t
Dtθ′
v(dW v + θv dv)
Proof left as an exercise.Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Optimal portfolio (corollary)
In the case of deterministic opportunity set (meaningθt = σ
−1(µ − r1) and r are constant) we have that Ht,v = 0 andwe get
π∗
t = Xt
Et [∫ T
tc∗vRv
ξv dv ]
Et [∫ T
tξvc∗v dv ]
(σ′
t)−1θt
In case when u(c , t) = ρ log c and ρ deterministic we get R = 1and π∗
t = Xt(σ′
t)−1θt showing that the logarithmic utility function
exhibits myopic behavior.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Asset pricing
We follow Lucas (1978) model in continuous time. Let stocks havedividends that follow
dD jt = D j
t(γjt dt + λj
t dW t)
where j = 1, . . . , d . We also assume that the aggregateconsumption C =
∑dj=1 D j follows
dCt = Ct [µCt dt + σC
t dW t ]
where
µCt =
d∑
j=1
D jt
Ctγjt
σCt =
d∑
j=1
D jt
Ct
λjt
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Asset pricing (II)
Bonds are in zero-net supply (no exogenous supply of bonds).
Stocks are in unit supply.
Single (representative) investor with endowment (1, 0) at time0.
Definition
Equilibrium is the set of S0, µ, σ, r and (c , π) such that
(c , π) ∈ A∗(x0) given S0, µ, σ, r .
Market clearing conditions: c = C = D ′1, π = S andX − π′1 = 0.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Asset pricing (III)
Theorem
Rational expectations equilibrium exists and the following holds:
ξt = m0,t =u′(ct , t)
u′(c0, 0)
rt = −∂u′(ct ,t)
∂t
u′(c0, 0)+ Rtµ
Ct −
1
2RtPt(σ
Ct )(σC
t )′
Pt = −u′′′(ct , t)
u′′(ct , t)ct Prudence coefficient
θt = Rt(σCt )′
πt = S t
Xt = S ′
t1
(Proof is left as an exercise.)Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Remarks and Corollary
rt = −Et [ dξt/ dt]ξt
is the expected growth rate of SPD.
θt = −σξ
ξtgrowth rate volatility of SPD.
We have the following:
St = Et
[∫ T
t
ξt,vDv dv
]
= EQt
[∫ T
t
bt,vDv dv
]
= Et
[∫ T
t
mt,vDv dv
]
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Consumption CAPM (Breeden (1979))
From before we get that
θt = Rt(σCt )′ = σ
−1t (µ
t− rt1)
from where it follows that
µt− rt1 = Rtσtσ
C ′
t
Applying this to the market portfolio we get
µm − rt = Rtσm′
t σC ′
t
from where it follows that Rt = µmt −rt
σm′
t σC ′
t
. We get
µt− rt1 = βC
t (µmt − rt) βC
t =σtσ
C ′
t
σm′
t σC ′
t
.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Equity premium puzzle
From the CCAPM (d = 1) we have that
µmt − rt = Rtσ
mt σC
t
or equivalently
µmt − rtσm
t
= RtσCt
Usual values for θmt ≈ 0.37, σC
t ≈ 0.036 and Rt ≈ 10.27. Mehraand Prescott (1985) obtained that in this case µm
t − rt ≈ 0.4% forlevels of risk aversion R = 2, 3, 4 whereas in reality this is appx.6 − 8%.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Risk-free rate puzzle (Weil (1989))
In case u(c , t) = ρu(c) where ρt = exp(−∫ t
0 βv dv) we have frombefore
rt = βt + RtµCt −
1
2RtPtσ
Ct σC ′
t .
Empirically r ≈ 6 − 7% contrary to model prediction of 0.8%although this is questionable with the new data.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Volatility of stocks
We have that
S jtu
′(ct , t) = Et
[∫ T
t
u′(cv , v)D jv dv
]
for j = 1, . . . , d . Using Ito formula on both sides of the equationgives (equating the volatility terms)
u′′ctσCt S j
t + u′S jtσ
jt = Et
[∫ T
t
u′′(cv , v)DtcvD jv dv +
∫ T
t
u′(cv , v)DtD
Further we have that
Dtcv = cv (σC ′
t + HCt,v )
where
HCt,v =
∫ v
t
Dt(µCu −
1
2σC
u σC ′
u )du +
∫ v
t
(DtσCu )dWu
DtDjv = D j
v (λj ′
t + HD,jt,v )
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Stock volatility (II)
After rearranging (exercise) we get that
σjt = λj
t + hedging terms.
Empirically, the σmt ≈ 0.2 while λj
m ≈ 0.036. This is the volatilitypuzzle (Schiller; Grossman and Schiller).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Further topics
Multiple agents equilibrium does not resolve the puzzles.
Habit formation and connection to Forward-Backward SDE(Constantinides (1990), Detemple and Zapatero (1991)).
Incomplete and asymmetric information in the continuoustime portfolio theory.
Mathematical aspects: Forward-Backward stochasticdifferential equations.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Part III
Part III: Advanced Topics inPortfolio Theory
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Risk Measures
In one-period portfolio optimization, variance is taken as ameasure of risk and there is no reason to do so.
In practice, the popular measure of risk is VaR(Value-at-Risk):
VaRα(X ) = − infx : P(X ≥ x) ≤ 1 − α,
i.e. it is a quantile, e.g. VaR99%(X ) = 100M says that theprobability of a 100M loss over a certain time horizon is lessthan 1%. This risk measures was mandated in the Basel IIdocument for bank risk management.
Heath, Artzner, Delbaen and Eber postulated axioms that anyrisk measure should fulfil.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Axioms of coherence
Fix some probability space (Ω,F ,P) and a time horizon ∆.Denote by L0(Ω,F ,P) the set of rvs which are almost surely finiteand a convex cone M ⊂ L0 which we interpret as portfolio lossesover time period ∆: If L1, L2 are in M then alsoL1 + L2, λL1 ∈ M for λ > 0. Risk measures are real valuedfunctions ρ : M → R. ρ(L) is the amount of capital that should beadded to the position to become acceptable.A function ρ : L → R is coherent if it satisfies the following set(HADE) of axioms:
1 Monotonicity: If Z1,Z2 ∈ L and Z1 ≤ Z2 then ρ(Z2) ≤ ρ(Z1).2 Sub-additivity: If Z1,Z2 ∈ L then ρ(Z1 +Z2) ≤ ρ(Z1)+ ρ(Z2).3 Positive homogeneity: If α ≥ 0 and Z ∈ L then
ρ(αZ ) = αρ(Z ).4 Translation invariance: If a ∈ R and Z ∈ L then
ρ(Z + a) = ρ(Z ) + a.
Gorazd Brumen Portfolio Theory
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Rationale behind Subadditivity Axiom
Risk can be reduced by diversification. Non-subadditive riskmeasures can lead to very risky portfolios.
Breaking a firm into subsidiaries would reduce regulatorycapital.
Decentralization of risk-management system: Trading desksL1 and L2. Risk manager wants to ensure thatρ(L1 + L2) < M. It is enough to ensure that ρ(L1) < M1 andρ(L2) < M2 with M1 + M2 = M.
Positive homogeneity insures there is no diversification ofmultiplying a portfolio.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
VaR is not Coherent
VaR is not in general a coherent risk measure - it does notrespect the sub-additivity, which implies that VaR mightdiscourage diversification. An example: Let X1,X2, . . . ,Xn berevenues from different business lines, which can be equitytrading desk, interest rate trading desk, etc. Let us assumethat the capital requirements for operating a business line Xi
are exactly VaRα(Xi). Then the capital requirement fromoperating all business lines is greater than operating each oneseparately. This is in direct contradiction to the diversificationprinciple.
VaR is coherent for the class of elliptically distributed losses(e.g. normally distributed).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Risk measures as generalized scenarios
Denote by P the set of probability measures on the underlyingspace (Ω,F). Let MP = L : EQ(L) < ∞ for all Q ∈ P andρP : MP → R such that
ρP(L) = supEQ(L) : Q ∈ P.
Theorem
(a) For any set P of probability measures (Ω,F) the risk measureρP is coherent on MP (Exercise.)
(b) Suppose that Ω = ω1, . . . , ωd is finite and letM = L : Ω → R. Then for any coherent risk measure ρ onM there is a set P of probability measures on Ω such thatρ = ρP.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Examples of Coherent Risk Measures
Expected shortfall, defined as
ESα(X ) = mine
E(X − e|X ≥ VaRe(X ))
is coherent.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Mean-VaR portfolio optimization
Instead of taking variance as a risk measure one could consider thefollowing optimization problem:
maxπ
E(X ) − θVaRα(X ).
This problem was considered in, for example Basak (2001).
Further reading (a lot):
Literature on convex risk measures where the subadditivityaxiom is replaced by the convexity axiom (Foellmer, Schied).
Dynamic risk measures (Delbaen, Cheridito, El-Karoui,Ravanelli).
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Statistical Arbitrage
Understanding that arbitrage in a financial market is impossible toachieve, statistical arbitrage tries to be as close to it. Let usconsider two different stocks
S1t = ρ1Mt + ε1
t
S2t = ρ2Mt + ε2
t
where Mt is a market factor and εit is a market residual for this
stock. Constructing a portfolio
ρ2S1t − ρ1S
2t = ρ2ε
1t + ρ1ε
2t
we only have the residual risk. Notice that arbitrage does not existhere.
Gorazd Brumen Portfolio Theory
Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory
Trading strategy in this example
Assuming that the process Ut = ρ2ε1t − ρ1ε
2t follows an
Ornstein-Uhlenbeck process
dUt = −ρUt dt + σ dWt
a trading process can for example optimize the following: Select abuy-time τ1 and a sell-time τ2, such that τ1 < τ2 and
maxτ1,τ2
E(−e−rτ1Uτ1 + e−rτ2Uτ2).
This was solved and there exists boundaries A and B such that
τ1 = inft : Ut ≤ −A
τ2 = inft : t > τ1,Ut ≥ B
where A,B solve integral equations. The strategy is then a simplebuy-and-hold strategy.
Gorazd Brumen Portfolio Theory
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