Mean-Variance Analysis: State-Space Approach

Question 1

Motivation

Answer 1

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)

Question 2

Set-up and goal

Answer 2

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

Question 3

Variance minimisation vs second moment minimisation

Answer 3

Question 4

Mean-Variance efficient payoffs

Answer 4

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

Question 5

The Payoffs x* and x hat

Answer 5

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]

Question 6

x* is MV efficient with proof

Answer 6

x∗ minimizes the second moment among all payoffs x ∈ X with the same price as x∗. In particular, x∗ ∈ Xmv

Question 7

x hat is MV efficient with proof

Answer 7

xˆ minimizes the second moment among all payoffs x ∈ X with the same expectation as xˆ. In particular, xˆ ∈ X mv

Question 8

Intermediary summary

Answer 8

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

Question 9

Idiosyncratic payoffs

Answer 9

Question 10

Systematic and Idiosyncratic components

Answer 10

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

Question 11

Intuitive Idea behind MV efficiency

Answer 11

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

Question 12

Properties of decomposition in systematic and idiosyncratic component

Answer 12

Question 13

Main theorem as the characterisation of the set X MV

Answer 13

Question 14

Proof, Direction Xmv ⊂ span{x∗,xˆ}

Answer 14

Both conditions can only hold simultaneously if ε = 0 and hence x ∈ span(x∗,xˆ)

Question 15

Proof, Direction span{x∗,xˆ} ⊂ Xmv

Answer 15

Question 16

Systematic component is best MV efficient approximation

Answer 16

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

Question 17

Portfolios of Mean-variance Efficient Payoffs Are again in Xmv

Answer 17

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

Question 18

Two Funds Separation for Payoffs

Answer 18

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

Question 19

Side Remark: Understanding dim X mv

Answer 19

Question 20

The Returns R∗ and Rˆ

Answer 20

Question 21

R∗ and Rˆ Span the Space Xmv

Answer 21

Question 22

Characterisation of the MV frontier

Answer 22

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

Question 23

Two funds theorem

Answer 23

Question 24

Special case with a risk free asset

Answer 24

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

Question 25

Two funds separation with Arbitrary frontier returns

Answer 25

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:

Question 26

Orthogonal decomposition of returns theory

Answer 26

Question 27

Orthogonal decomposition of returns proof

Answer 27

Question 28

Combining decomposition and Two funds theorems

Answer 28

Question 29

Illustration: Orthogonal Decomposition in the Payoff Space

Answer 29

Question 30

Illustration: Orthogonal Decomposition in the Mean-Variance Diagram

Answer 30

Question 31

Summary

Answer 31

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