Mean-Variance Analysis: State-Space Approach Flashcards
Motivation
we describe the mean-variance frontier as a set in the payoff space X
in this formulation, the frontier turns out to be a straight line
it contains precisely the returns whose idiosyncratic component is zero
any return can be decomposed in a frontier return and an orthogonal idiosyncratic payoff
We characterize the frontier in terms of two natural returns that are always mean-variance efficient:
- the return R∗ associated with the SDF payoff x∗
- the return Rˆ associated with the constant-mimicking portfolio proj(1 | X )
(this equals the risk-free rate if a risk-free asset exists)
Set-up and goal
X is a payoff space that satisfies free portfolio formation (linearity)
p is a price function that satisfies the law of one price
x∗ denotes the unique SDF that is contained in the payoff space
Variance minimisation vs second moment minimisation
Mean-Variance efficient payoffs
When we define Rmv, we hold really two things fixed while minimizing second moments
not just the expected return E[R]
but also the price because we restrict attention to returns, that is p(R) = 1
For symmetry reasons, it is insightful to generalize to mean-variance efficient payoffs:
NotethatR∈Rmv ifandonlyifR∈Xmv andR∈R
The Payoffs x* and x hat
Any x ∈ Xmv minimizes E[x2] holding the values of two linear functions fixed
- the price function x → p(x)
- the mean function x → E[x]
We have seen in Lecture 2 that we can represent p as an inner product with the payoff x∗
∀x ∈X :p(x)=⟨x∗,x⟩=E[x∗x]
x∗ is the projection of any SDF m onto the payoff space, x∗ = proj(m | X)
The mean function is trivially represented as an inner product with the constant 1
∀x ∈X :E[x]=⟨1,x⟩=E[1·x]
but 1 may not be a payoff
so let’s use also here the projection trick and define the constant-mimicking payoff xˆ := proj(1 | X)
this is a payoff (xˆ ∈ X by construction) and it also represents the mean function ∀x ∈X :E[x]=⟨xˆ,x⟩=E[xˆx]
x* is MV efficient with proof
x∗ minimizes the second moment among all payoffs x ∈ X with the same price as x∗. In particular, x∗ ∈ Xmv
x hat is MV efficient with proof
xˆ minimizes the second moment among all payoffs x ∈ X with the same expectation as xˆ. In particular, xˆ ∈ X mv
Intermediary summary
x∗ and xˆ play a special role because they represent the functions we hold fixed
x∗ represents the price function p
xˆ represents the expectation E
Previous results: x∗,xˆ ∈ Xmv
We will show next that x∗ and xˆ in fact generate Xmv
Xmv =span{x∗,xˆ}
in words: any mean-variance efficient payoff turns out to be a portfolio of two payoffs
the SDF payoff x∗ and
the constant-mimicking payoff xˆ
Idiosyncratic payoffs
Systematic and Idiosyncratic components
Lecture 1: can decompose any payoff in systematic and idiosyncratic component
definition of systematic component of x there: E[x] + proj(x − E[x] | m) = proj(x | 1, m)
this only makes sense as a payoff if 1 ∈ X
was fine there because we have assumed existence of risk-free asset in Lecture 1
When there is no risk-free asset: use instead constant-mimicking payoff xˆ in the regression
For any x ∈ X, we can decompose
x =proj(x |x∗,xˆ)+ε
then both proj(x | x∗,xˆ) and ε are again payoffs
E[x∗ε] = E[xˆε] = 0, hence p(ε) = E[ε] = 0, so ε ∈ E
proj(x | x∗, xˆ) ∈ span{x∗, xˆ}
We call proj(x | x∗,xˆ) the systematic component of x, ε the idiosyncratic component
Intuitive Idea behind MV efficiency
An idiosyncratic component of a payoff has zero mean and price … but it may positively contribute to the second moment/variance
To minimize the second moment for given mean and price, the idiosyncratic component should
be as small as possible
→ Mean-variance efficient payoffs should have idiosyncratic component ε = 0
→ Mean-variance efficient payoffs should be contained in span{x∗,xˆ}
… we try to make this intuition precise in the following
Properties of decomposition in systematic and idiosyncratic component
Main theorem as the characterisation of the set X MV
Proof, Direction Xmv ⊂ span{x∗,xˆ}
Both conditions can only hold simultaneously if ε = 0 and hence x ∈ span(x∗,xˆ)
Proof, Direction span{x∗,xˆ} ⊂ Xmv
Systematic component is best MV efficient approximation
Xmv = span{x∗,xˆ} implies
systematic component of x = proj(x | x∗,xˆ) = proj(x | Xmv)
In other words: the systematic component is the payoff in Xmv that best approximates x
(in the sense of minimizing the mean-squared approximation error)
This also tells us: any payoff x can be decomposed x = xmv + ε
into a sum of a mean-variance efficient payoff xmv and an idiosyncratic payoff ε
xmv and ε are orthogonal (E[xmvε] = 0)
x is mean-variance efficient if and only if ε = 0
Portfolios of Mean-variance Efficient Payoffs Are again in Xmv
Theorem implies: Xmv is a linear space
This means: like X, the subspace Xmv satisfies free portfolio formation
Any portfolios formed from mean-variance efficient payoffs are again mean-variance efficient
Two Funds Separation for Payoffs
Theorem also implies: dim X mv ≤ 2
This means: any payoff in Xmv is a linear
combination of at most two given payoffs
(e.g. the payoffs x∗ and xˆ)
More precisely:
we can pick two payoffs x1,x2 ∈ Xmv
(linearly independent if dim X mv = 2, arbitary otherise)
then any mean-variance efficient payoff is a
portfolio that combines just two funds
1 a (scaled) investment into payoff x1
2 a (scaled) investment into payoff x2
Side Remark: Understanding dim X mv
The Returns R∗ and Rˆ
R∗ and Rˆ Span the Space Xmv
Characterisation of the MV frontier
Conclusion from previous slide:
the mean-variance frontier is given by Rmv ={R∗+w(Rˆ−R∗)|w∈R}
Geometric interpretation:
if Rˆ ̸= R∗ (regular case), Rmv is a straight line in the payoff space
if Rˆ = R∗ (risk-neutral case), Rmv is a single
point in the payoff space
Portfolio interpretation: two funds separation
Two funds theorem
Special case with a risk free asset
Suppose there is a risk-free return Rf ∈ R
Then1∈X ⇒xˆ=1⇒Rˆ=Rf
The two funds theorem tells us that any frontier return can be obtained from a portfolio that
combines
1 the risk-free asset
2 a (risky) fund with return R∗
But note: this does not mean that R∗ is the return on the tangency portfolio in fact, it is usually different from the tangency portfolio return
a portfolio yielding R∗ has itself a weight in the risk-free asset different from zero
Two funds separation with Arbitrary frontier returns
Two fund separation does not just work with R∗ and Rˆ
We can use any two nonidentical returns R1 ̸= R2 that are on the frontier:
Orthogonal decomposition of returns theory
Orthogonal decomposition of returns proof
Combining decomposition and Two funds theorems
Illustration: Orthogonal Decomposition in the Payoff Space
Illustration: Orthogonal Decomposition in the Mean-Variance Diagram
Summary
Two special mean-variance efficient payoffs x∗, xˆ
x∗ minimizes the second moment among all payoffs with the same price
xˆ minimizes the second moment among all payoffs with the same expectation together,
they span the set of all mean-variance efficient payoffs
Translation to return space
returns R∗, Rˆ span the mean-variance frontier
two funds separation theorem
orthogonal decomposition of returns
