introduction to portfolio and stochastic discount...

Introduction to Portfolio and Stochastic Discount FactorMean Variance Frontiers

Enrique SentanaCEMFI

<[email protected]>

Center for Applied Economics and Policy ResearchIndiana UniversityOctober 3th, 2011

Enrique Sentana () Portfolio and SDF Frontiers 1 / 28

Outline of the presentation

1 Elements of financial decision theory under uncertainty

2 Digression on Hilbert spaces, projections and functionals

3 Portfolio selection

4 Mean-variance frontiers

5 The effects of considering additional assets


Elements of financial decision theory under uncertainty

Single consumption good

Discrete-state setting

States of nature (outcomes or elementary events): ωs, s = 1, . . . , SSample space: ΩEvent space: F (σ-algebra)

Probability measure: Π : πs ≥ 0,∑S

s=1 πs = 1Risky asset payoffs: xis (i = 1, . . . , N)Safe asset: x0s = 1 ∀s



Financial engineering: Combine existing assets to create new ones

Fixed weight portfolios ps = wixis + wjxjs, wi, wj ∈ RContingent claims (Arrow-Debreu securities): adrs = I(r = s),(r = 1, . . . , S)Market completeness

Felicity function u(xis; s, θ)If state independent, then we can regard asset payoffs as randomvariables xj defined on the underlying probability space (Ω,F ,Π)Preference ordering: xi xj , xi ≺ xj or xi ∼ xj

Expected utility rule: xi xj ⇔ E [u(xi; θ)]− E [u(xj ; θ)] > 0



Properties of expected utility1 Separable in outcomes2 Linear in probabilities:E [u(xi; θ)] =

∑Ss=1 πsu(xis; θ) =

∫u(x; θ)dFxi

(x)3 Defined up to affine transformations4 Risk aversion: E [u(xi; θ)] < u [E(xi); θ] (concavity of u(.))

Some commonly used utility functions:1 Linear utility (risk neutrality): u(xis; θ) = xis2 Quadratic utility: u(xis; θ) = xis + αx2

is3 Exponential utility: u(xis; θ) = − exp(−γxis)4 Power utility: u(xis; θ) = (1− φ)−1

(x1−φis − 1

)5 Logarithmic utility: u(xis; θ) = lnxis (limφ→1)



Asset prices: C(xi)Law of one price: C(wixi + wjxj) = wiC(xi) + wjC(xj) (Linear pricing)

Under complete markets C(xi) =∑S

s=1 xisC(ads)Stochastic discount factor: m

E(xim) =S∑

s=1

xismsπs = C(xi)

E(x0m) = E(m) = C(1)C(xi) = E(xi)E(m) + cov(xi,m)Under complete markets ms = C(ads)/πs for those s for whichπs > 0



Absence of arbitrage opportunities ms > 0 for all s for which πs > 0Also

C(ads)/C(adr) =πs · u′(xo

s; θ)πr · u′(xo

r; θ)

Under risk neutrality C(ads)/C(adr) = πs/πr ⇒ ms = k ∀s


Digression on Hilbert spaces, projections and functionals

A Banach space is a complete normed linear space

A Hilbert space is a Banach space whose norm arises from an innerproduct

Example: L2 : x : E(x2) <∞Mean square inner product: E(yx), x, y ∈ L2

Mean square norm:√E(x2), x ∈ L2

x = y ⇔ E(x− y)2 = 0Any closed linear subspace of L2, H say, is also a Hilbert space

If V (h) > 0 ∀h ∈ H, then H is also a Hilbert space with respect tothe covariance inner product, cov(h1, h2) and the associated standarddeviation norm V 1/2(h).



Examples:1 Conditional expectations: If G contains all possible Borel-measurable

functions of z with bounded second moments, then P (x|G) = E(x|z)2 Linear projections: If H = 〈y, z〉, where 〈y, z〉 = αy + βz, α, β ∈ R

is the linear span of y and z, then

P (x|H) =[E(xy) E(xz)

] [ E(y2) E(yz)E(yz) E(z2)

]−1(yz

)3 P (x| 〈1〉) = E(x|1) = E(x)



Linear functionalπ() : L2 → R is linear on H ⊆ L2 iffπ(a1h1 + a2h2) = a1π(h1) + a2π(h2) ∀a1, a2 ∈ R and ∀h1, h2 ∈ H.

Continuous functional:π() : L2 → R is continuous on H ⊆ L2 ifflimn→∞E(hn − h)2 = 0⇒ limn→∞ π(hn) = π(h) for hn, h ∈ H.

Riesz representation theorem:If π() is a continuous linear functional mapping H on R, then∃!h# ∈ H such that π(h) = E(hh#) ∀h ∈ H.


Portfolio selection

Risky assets: x = (x1, . . . , xN )′, N <∞.

They could be primitive assets, such as stocks and bonds, or mutualfunds managed according to some active portfolio strategy.

Assume tr [E (xx′)] <∞, so xi ∈ L2

|E (xx′)| 6= 0 (no redundant assets) ⇒ Law of one price

V (x) = E (xx′)− E(x)E(x)′

|V (x)| 6= 0 ⇒ no non-trivial riskless portfolio of risky assets.

Thus, we rule out the existence of a safe asset x0 = 1Portfolio: p = w′x

Portoflio weights: w = (w1, . . . , wN )′

Expected value of a portfolio: E(p) = w′E(x)Variance of a portfolio: V (p) = w′V (x)wCost of a portfolio: C(p) = w′C(x)


Portfolio selection

Portfolio payoff space: P = 〈x〉 (linear span of x).

One important subset: R = p ∈ P : C(p) = 1, whose payoffs canbe directly understood as returns per unit invested.

If C(xi) 6= 0, gross returns: Ri = xi/C(xi), C(Ri) = 1Another important subset: A = p ∈ P : C(p) = 0, which is the setof all arbitrage (i.e. zero-cost) portfolios.

Partition x as (x1,x−1), where x−1 is of dimension n = N − 1A = 〈r〉, where r = x−1 −R1C(x−1) is the vector of payoffs on thelast n risky assets in excess of the first one.

In empirical work, excess returns are often computed by subtractingfrom gross returns a supposedly riskless asset, but the representationof A as the linear span of r remains valid irrespective of the orderingof the original assets as long as C(x1) 6= 0.


Uncentred representing portfolios over P

Since P is a closed linear subspace of L2, it is also a Hilbert spaceunder the mean square inner product, E(xy), and the associatedmean square norm

√E(x2), where x, y ∈ L2.

Therefore, we can formally understand C(.) and E(.) as linearfunctionals that map the elements of P onto the real line.

The expected value functional is always continuous on L2.

The cost functional is also continuous because of the law of one price.

Expected value representing portfolio:E(p) = E(pp) ∀p ∈ P ⇒ p = E(x′)[E(xx′)]−1x = P (1|P) sinceE(1 · x) = E(x)Cost representing portfolio:E(p) = E(pp∗) ∀p ∈ P ⇒ p∗ = C(x′)[E(xx′)]−1x = P (m|P)since E(m · x) = C(x)


Centred representing portfolios over P

Since we are assuming that no riskless asset exists, we can define analternative expected value representing portfolio such thatE(p) = Cov(p,þ) ∀p ∈ P.

Similarly, we can define an alternative cost representing portfolio suchthat C(p) = Cov(p,þ∗) ∀p ∈ P.

Not surprisingly,

þ∗ = C(x′)[V (x)]−1x = p∗ + [1− E(p)]−1C(p)p,þ = E(x′)[V (x)]−1x = [1− E(p)]−1p.


Representing portfolios over A

Unlike R, A is a closed linear subspace of P, and consequently aHilbert space itself.

Given that C(r) = 0, the centred and uncentred cost representingportfolios k∗ and a∗ are both 0 in this case.

But we can still define the mean representing portfolios.

Specifically,

a = E(r′)[E(rr′)]−1r,

k = E(r′)[V (r)]−1r = [1− E(a)]−1a.


Portfolio mean-variance frontiers

minw V (w′x) s.t. E(w′x) = ν ∈ R, C(w′x) = γ ∈ R+

For any p ∈ P, p = P (p| 〈p∗, p〉) + uC(u) = E(up∗) = 0, E(u) = E(up) = 0Hence, if we require E(p) = ν and C(p) = γ, then the solution is

pMV (ν, c) = P (p| 〈p∗, p〉)= P (p| 〈p∗〉) + P [p− P (p| 〈p∗〉)| 〈p − P (p| 〈p∗〉)〉]= γR∗ + ρ(ν)a

where R∗ = p∗/C(p∗),

a = p − P (p| 〈p∗〉) = p − [C(p)/C(p∗)] · p∗

ρ(v) =ν − γE(R∗)E(a)


Portfolio mean-variance frontiers

Mean-variance frontier can be generated by p∗ and p alone.

Alternatively, we can generate it from þ∗ and þ:Two-fund spanning

Exceptions:

1 a = 0 (“risk neutrality”) The frontier collapses to γR∗

2 γ = 0 one-fund spanning with a (C(a) = 0)

Usually, γ = 1⇒ RMV (ν) = pMV (ν, 1)This is the RMVF, which is a parabola in mean-variance space, and ahyperbola in mean-standard deviation space.

But sometimes, γ = 0⇒ rMV (µ) = pMV (µ, 0)This is the AMVF, which is also a parabola in mean-variance space,but a straight line reflected at 0 in mean-standard deviation space.


Stochastic discount factor mean-variance frontiers (SMVF)Hansen-Jagannathan volatility bounds

minm∈L2 V (m) s.t. E(m) = c ∈ R+, E(mx) = C(x).

If there were a riskless asset with gross return 1/c:mMV (c) = P (m|P+) = p∗ + α(c)(1− p) = α(c) + þ∗ − cþwhere P+ = 〈1,x〉 and

α(c) =c− E (p∗)1− E(p)

= c [1 + E(þ)]− E(þ∗),

If no riskless asset exists, then we simply repeat this calculation for allpotential values of c, which implies that mMV (c) will be spanned by aconstant, p∗ = P (m|P) and (1− p) = 1− P (1|P) (or þ∗ and þ).

This frontier is also a parabola in mean - variance space for SDF’s,and a hyperbola in mean - standard deviation space.


Duality

Relationship with mean-variance analysis:mMV (c)− α(c) = p∗ − α(c)p so mMV (c) can generally written asan affine transformation of RMV (ν) for some ν

There are two exceptions, which correspond to the asymptotes of theRMVF and SMVF.

If we call RMVc the gross return associated to mMV (c),

|E (R)− 1/c|σ (R)

≤∣∣E (RMV

c

)− 1/c

∣∣σ (RMV

c )=σ[mMV (c)

]E [mMV (c)]

≤ σ (m)E (m)

Absence of arbitrage opportunities: mMV A(c) = maxmMV (c), 0


SMVF from arbitrage portfolios

In the case in which we only have access to arbitrage portfolios, then

mMV (c) =c

1− E(a)(1− a) = c1− [ð − E(k)],

so in this case the frontier SDF’s are spanned by a single “fund”.

Still, this frontier will continue to be a parabola in mean - varianceSDF space, but a straight line that starts from the origin in mean -standard deviation space.

The slope of this line is precisely the maximum (square) Sharpe ratioportfolio attainable over A.


The effects on the SMVF and RMVF of includingadditional assets

x1 : original assets (N1 × 1)

x2 : new assets (N2 × 1)

x =(

x′1 x′2)′ : expanded set of assets (N × 1, N = N1 +N2)

RMVF will shift to the left because our investment opportunitiesimprove.

SMVF will rise because there is more information in the data aboutm.

However, this is not always the case:x1 spans the RMVF and/or SMVF generated from x when the oldand new frontiers are the same.

A third and last possibility, which we call tangency, is that thefrontiers share a single point.

In the case of arbitrage portfolios, tangency is equivalent to spanning.


Frontiers with a countably infinite number of primitiveassets

P: closure of the set of payoffs from all possible portfolios of theprimitive assets

⋃∞N=1 PN

Problems: Even if E(xx′) has full rank for all N , there may exists

1 Limiting riskless unit-cost portfolios of risky assets2 Limiting arbitrage opportunities

Once we rule out those possibilities, our previous analysis in terms ofrepresenting portfolios applies.


introduction to portfolio and stochastic discount...

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