Portfolio Theory Gorazd Brumen Morgan Stanley September 11-12, - - PowerPoint PPT Presentation

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part 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 Period Part II: Portfolio Selection in Continuous Time Part 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 Period Part II: Portfolio Selection in Continuous Time Part 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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Part I

Part I: Portfolio Selection in One Period

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Historical perspective on portfolio selection

Even though the concept of diversification is firmly grounded in today’s economic thinking, this was not always the case. Before Markowitz’s seminal contribution investors did not consider portfolio diversification but rather stock picking: Of all stocks in a market pick the one which brings you highest combination of dividends (and capital gains). Portfolio theory answers the question which risks are priced and in what extent.

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Framework and Notations

One period model. Vectors will be underlined, such as x, matrices are boldface, e.g. Γ. Begining of period at time 0, end of period at

• 1. Return on an asset i = 1, . . . , N in terms of its price Si in this

period 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 Period Part II: Portfolio Selection in Continuous Time Part 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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Portfolio with risk-less asset

We introduce the riskless asset by S0(1) = S0(0)(1 + r) and additionally 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 returns

• f individual stocks.

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Simple relations

Definition Assets i = 1, . . . , N are redundant if there exists N scalars λ1, . . . , λN such that N

i=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 is positive definite. (exercise.)

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Efficiency criteria and optimization program

Definition Portfolio (x∗, X ∗) is efficient if for every other portfolio y we have that if σY < σX ∗ then µY < µX ∗ and σY = σX ∗ implies µY ≤ µX ∗. Portfolio optimization problem: max

x

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 Period Part II: Portfolio Selection in Continuous Time Part 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 is strictly concave (Γ is positive definite).

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Connection to utility theory

Criterium of portfolio efficiency is consistent with the economic agents 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 of risk 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 Period Part II: Portfolio Selection in Continuous Time Part 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

max

xi

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 Period Part II: Portfolio Selection in Continuous Time Part 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 is efficient. Any efficient portfolio is a combination of x and y (not necessarily convex). The efficient frontier is a parabola in the expected return-variance space (µ, σ2) and a hyperbola in the expected return-standard deviation space (µ, σ). Due to the first bullet point above, any efficient portfolio can be described as a convex combination of just 2 portfolios. Proof of the first two bullet points left as an exercise.

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part 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 a parabola in this space.

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part 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 X lies on the straight line between 0 and X in the (µ, σ) space. The straight line between asset 0 and asset M is the efficient frontier called the Capital Market Line. (Tobin’s two fund separation) Any efficient portfolio is a combination of only 2 portfolios (e.g. 0 and M).

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Efficiency set with riskless asset (contd.)

Theorem (contd.) Any efficient portfolio satisfies x∗ = ˆ θΓ−1(µ − r1) Tangent (market) portfolio (m, M) is m = ˆ θMΓ−1(µ − r1) ˆ θM = 1 1′Γ−1(µ − r1) (Proof left as an exercise.)

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Capital Market Equilibirium

Since the market portfolio is efficient there exists scalars λ and θ such 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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

CAPM equilibrium

In particular for market portfolio it holds that µM = λ + θσ2

M

from where it follows that θ = µM−λ

σ2

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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

CAPM as a Pricing Model

Question: If a security delivers ˜ V (1) at time 1, what is the price V (0) of this security at time 0? Assuming that the risk-free asset exists then E( ˜ V (1)) V (0) = E(1 + R) = 1 + r + θcov ˜ V (1) V (0), RM

• where θ = µM−r

σ2

M

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 Period Part II: Portfolio Selection in Continuous Time Part 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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Part II

Part II: Portfolio Selection in Continuous Time

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Continuous time financial market

Financial market: Risk-free security: dBt = Btrt dt, process rt is progressively measurable (adapted with cadlag paths) and T

0 ru du < ∞.

d stocks with dynamics: dSt + Dt dt = IS(µt dt + 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 µt dt < ∞ and

T

0 σtσ′ t dt < ∞.

If σ is invertible, the financial market is complete. Market price of risk (Sharpe ratio): θt = σ−1

t (µ − r1).

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part 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 u(ct, t) dt

• where u(·, t) : R+ → R is strictly increasing, strictly concave and

twice 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 Period Part II: Portfolio Selection in Continuous Time Part 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) =

1 1−R (c + δ)1−R, R ≥ 0, δ > 0 an example of HARA

utilities.

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Investor’s wealth dynamics

Progressively measurable portfolio process π generates investor’s wealth dynamics dXt = π′

t((IS)−1( dSt + 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 stock portfolio, the second the return on bonds and the third the consumption part. Definition (c, π) is admissible (belongs to A(x) iff Xt ≥ 0 for all t ∈ [0, T]. (c, π) is optimal (belongs to A∗(x) iff there does not exist (ˆ c, ˆ π) such that U(ˆ c) > U(c).

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Static approach

Let ηt = exp

• −

t θ′

v dW v − 1

2 t θ′

vθv dv

• .

Due to Novikov condition η is a martingale. Therefore we can change the measure dQ = ηT dP. This implies the following: It can be proven that ˜ W t = W t + t

0 θv dv is a Brownian

motion. Sv = Ev[Stξv,t + t

v ξv,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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Equivalence of approaches

Let B(x) =

• c : E

T ξtct 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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Proof for (a)

Let us have X ≥ 0, c ≥ 0, then ξtXt + t

0 ξvcv dv ≥ 0 for every

t ∈ [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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Proof of (b)

Let Et T

t

ξvcv dv

• =

Et T ξvcv dv

• −

t ξvcv dv = E0 T ξvcv dv

• +

t φv dWv − t ξvcv dv Choose φt = ξt[π′

tσt − Xtθ′ v]. Then by the equation () from

before we have that ξtXt + t ξvcv = t φv dWv + x = Et T

t

ξvcv dv

• − E0

T ξvcv dv

• + x

from where it follows that ξtXt ≥ Et T

t ξvcv 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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Static portfolio optimization

Constructing the Lagrangian: L = E T u(ct, t) dt

• − y
• E

T ξvcv dv

• − x
• Process c is optimal if there does not exist a process ∆t such that

L(c + ε∆) > L(c). The necessary condition is therefore

∂L ∂ε |ε=0 = 0 for every ∆. We have

∂L ∂ε |ε=0 = E T u′(ct, t)∆t dt

• − yE

T ξv∆v dv

• =

E T (u′(ct, t) − yξt)∆t dt

• from where it follows that u′(ct, t) = yξt, i.e. marginal utility

equals 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 Period Part II: Portfolio Selection in Continuous Time Part 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 ξvI(yξv, v) dv

• π∗

t

= Xt(σ′

t)−1θt + ξ−1 t (σ′ t)−1φ∗ t

φ∗

t

= Et[F ∗] − E[F ∗] F ∗ = T ξvc∗

v dv

X ∗

t

= Et T

t

ξt,vc∗

v dv

• Gorazd Brumen

Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Malliavin calculus (Stochastic calculus of variations)

Motivation: If F ∈ L2 then there exists by the martingale representation theorem a progressively measurable process φ such that F = E[F] + T φv dWv How to extract φ? The question is not important only in portfolio theory but also in derivatives pricing for hedging purposes.

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Hedging of derivative securities

Fundamental theorem of asset pricing states that the price of a derivative 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 ut dSt. Malliavin calculus gives us an answer to what is u.

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part 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 infinitely differentiable and of polynomial growth. Then the Malliavin derivative DF = {DtF : t ∈ [0, T]} is a d-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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Simple properties of Malliavin calculus

Malliavin derivative is the generalization of the Frechet derivative for stochastic processes. Malliavin derivatives are not adopted (anticipating processes). DtF = 0 if F ∈ Fs and s < t. Theory can be extended to appropriate spaces for stochastic processes called D2,1. If ST = S0 exp((µ − 1/2σ2)T + σWT) then DtST = STσ1[0,T](t).

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Properties of Malliavin calculus

Chain rule: Let g : Rm → R with bounded derivatives and F1, . . . , Fm ∈ D2,1. Then Dtg(F1, . . . , Fm) =

m

• i=1

∂g ∂Fi DtFi Dt(Ev[F]) = Ev(DtF) for v ≥ t Clark-Ocone formula: Let F ∈ D2,1. Then F = E(F) + T φv dWv where φv = Ev(DvF).

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Malliavin calculus rules

More rules: If F1 = T

0 φ1t dt then

DtF1 = T Dtφ1v 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 Period Part II: Portfolio Selection in Continuous Time Part 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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Optimal portfolios

Theorem We have π∗

t

= ξ−1

t Et

T

t

c∗

v

Rv ξv dv

• (σ′

t)−1θt

−ξ−1

t (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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Optimal portfolio (corollary)

In the case of deterministic opportunity set (meaning θt = σ−1(µ − r1) and r are constant) we have that Ht,v = 0 and we get π∗

t = Xt

Et[ T

t c∗

v

Rv ξv dv]

Et[ T

t ξvc∗ v dv]

(σ′

t)−1θt

In case when u(c, t) = ρ log c and ρ deterministic we get R = 1 and 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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Asset pricing

We follow Lucas (1978) model in continuous time. Let stocks have dividends that follow dDj

t = Dj t(γj t dt + λj t dW t)

where j = 1, . . . , d. We also assume that the aggregate consumption C = d

j=1 Dj follows

dCt = Ct[µC

t dt + σC t dW t]

where µC

t

=

d

• j=1

Dj

t

Ct γj

t

σC

t

=

d

• j=1

Dj

t

Ct λj

t

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part 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 time 0. 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 and X − π′1 = 0.

Gorazd Brumen Portfolio Theory

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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) + RtµC

t − 1

2RtPt(σC

t )(σC t )′

Pt = −u′′′(ct, t) u′′(ct, t) ct Prudence coefficient θt = Rt(σC

t )′

πt = St Xt = S′

t1

(Proof is left as an exercise.)

Gorazd Brumen Portfolio Theory

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Remarks and Corollary

rt = − Et[ dξt/ dt]

ξt

is the expected growth rate of SPD. θt = − σξ

ξt growth rate volatility of SPD.

We have the following: St = Et T

t

ξt,vDv dv

• =

EQ

t

T

t

bt,vDv dv

• =

Et T

t

mt,vDv dv

• Gorazd Brumen

Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Consumption CAPM (Breeden (1979))

From before we get that θt = Rt(σC

t )′ = σ−1 t (µt − rt1)

from where it follows that µt − rt1 = RtσtσC ′

t

Applying this to the market portfolio we get µm − rt = Rtσm′

t σC ′ t

from where it follows that Rt = µm

t −rt

σm′

t σC′ t . We get

µt − rt1 = βC

t (µm t − rt)

βC

t = σtσC ′ t

σm′

t σC ′ t

.

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Equity premium puzzle

From the CCAPM (d = 1) we have that µm

t − rt = Rtσm t σC t

• r equivalently

µm

t − rt

σm

t

= RtσC

t

Usual values for θm

t ≈ 0.37, σC t ≈ 0.036 and Rt ≈ 10.27. Mehra

and Prescott (1985) obtained that in this case µm

t − rt ≈ 0.4% for

levels 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 Period Part II: Portfolio Selection in Continuous Time Part 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 from

before rt = βt + RtµC

t − 1

2RtPtσC

t 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 Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Volatility of stocks

We have that Sj

tu′(ct, t) = Et

T

t

u′(cv, v)Dj

v dv

• for j = 1, . . . , d. Using Ito formula on both sides of the equation

gives (equating the volatility terms) u′′ctσC

t Sj t + u′Sj tσj t

= Et T

t

u′′(cv, v)DtcvDj

v dv +

T

t

u′(cv, v)DtD Further we have that Dtcv = cv(σC ′

t

+ HC

t,v)

where HC

t,v

= v

t

Dt(µC

u − 1

2σC

u σC ′ u ) du +

v

t

(DtσC

u ) dWu

DtDj

v

= Dj

v(λj′ t + HD,j t,v )

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Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Stock volatility (II)

After rearranging (exercise) we get that σj

t = λj t + hedging terms.

Empirically, the σm

t ≈ 0.2 while λj m ≈ 0.036. This is the volatility

puzzle (Schiller; Grossman and Schiller).

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Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part 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 continuous time portfolio theory. Mathematical aspects: Forward-Backward stochastic differential equations.

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Part III

Part III: Advanced Topics in Portfolio Theory

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Risk Measures

In one-period portfolio optimization, variance is taken as a measure of risk and there is no reason to do so. In practice, the popular measure of risk is VaR (Value-at-Risk): VaRα(X) = − inf{x : P(X ≥ x) ≤ 1 − α}, i.e. it is a quantile, e.g. VaR99%(X) = 100M says that the probability of a 100M loss over a certain time horizon is less than 1%. This risk measures was mandated in the Basel II document for bank risk management. Heath, Artzner, Delbaen and Eber postulated axioms that any risk measure should fulfil.

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Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part 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 finite and a convex cone M ⊂ L0 which we interpret as portfolio losses

• ver time period ∆: If L1, L2 are in M then also

L1 + L2, λL1 ∈ M for λ > 0. Risk measures are real valued functions ρ : M → R. ρ(L) is the amount of capital that should be added 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.

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Rationale behind Subadditivity Axiom

Risk can be reduced by diversification. Non-subadditive risk measures can lead to very risky portfolios. Breaking a firm into subsidiaries would reduce regulatory capital. Decentralization of risk-management system: Trading desks L1 and L2. Risk manager wants to ensure that ρ(L1 + L2) < M. It is enough to ensure that ρ(L1) < M1 and ρ(L2) < M2 with M1 + M2 = M. Positive homogeneity insures there is no diversification of multiplying a portfolio.

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VaR is not Coherent

VaR is not in general a coherent risk measure - it does not respect the sub-additivity, which implies that VaR might discourage diversification. An example: Let X1, X2, . . . , Xn be revenues from different business lines, which can be equity trading desk, interest rate trading desk, etc. Let us assume that the capital requirements for operating a business line Xi are exactly VaRα(Xi). Then the capital requirement from

• perating all business lines is greater than operating each one
• separately. This is in direct contradiction to the diversification

principle. VaR is coherent for the class of elliptically distributed losses (e.g. normally distributed).

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Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Risk measures as generalized scenarios

Denote by P the set of probability measures on the underlying space (Ω, F). Let MP = {L : EQ(L) < ∞ for all Q ∈ P} and ρP : MP → R such that ρP(L) = sup{EQ(L) : Q ∈ P}. Theorem (a) For any set P of probability measures (Ω, F) the risk measure ρP is coherent on MP (Exercise.) (b) Suppose that Ω = {ω1, . . . , ωd} is finite and let M = {L : Ω → R}. Then for any coherent risk measure ρ on M there is a set P of probability measures on Ω such that ρ = ρP.

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Examples of Coherent Risk Measures

Expected shortfall, defined as ESα(X) = min

e E(X − e|X ≥ VaRe(X))

is coherent.

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Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Mean-VaR portfolio optimization

Instead of taking variance as a risk measure one could consider the following optimization problem: max

π

E(X) − θVaRα(X). This problem was considered in, for example Basak (2001). Further reading (a lot): Literature on convex risk measures where the subadditivity axiom is replaced by the convexity axiom (Foellmer, Schied). Dynamic risk measures (Delbaen, Cheridito, El-Karoui, Ravanelli).

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Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Statistical Arbitrage

Understanding that arbitrage in a financial market is impossible to achieve, statistical arbitrage tries to be as close to it. Let us consider two different stocks S1

t

= ρ1Mt + ε1

t

S2

t

= ρ2Mt + ε2

t

where Mt is a market factor and εi

t is a market residual for this

• stock. Constructing a portfolio

ρ2S1

t − ρ1S2 t = ρ2ε1 t + ρ1ε2 t

we only have the residual risk. Notice that arbitrage does not exist here.

Gorazd Brumen Portfolio Theory

Part I: Portfolio Selection in One Period Part II: Portfolio Selection in Continuous Time Part III: Advanced Topics in Portfolio Theory

Trading strategy in this example

Assuming that the process Ut = ρ2ε1

t − ρ1ε2 t follows an

Ornstein-Uhlenbeck process dUt = −ρUt dt + σ dWt a trading process can for example optimize the following: Select a buy-time τ1 and a sell-time τ2, such that τ1 < τ2 and max

τ1,τ2 E(−e−rτ1Uτ1 + e−rτ2Uτ2).

This was solved and there exists boundaries A and B such that τ1 = inf{t : Ut ≤ −A} τ2 = inf{t : t > τ1, Ut ≥ B} where A, B solve integral equations. The strategy is then a simple buy-and-hold strategy.

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