5. Mean-Variance Frontier and Beta Representations
Objective
Explanation of central concepts:

The Expected Return Beta Representation of a Factor Model

Traditional Construction Methods of the Mean-variance Frontier (MVF)

Orthogonal Decomposition and Spanning of the MVF

Mean-variance Frontier for Discount Factors
Contents
5.1
Expected Return-Beta Representation
5.2
Mean-Variance Frontier: Construction Methods
5.3
Orthogonal Characterization of the MVF
5.4
Spanning the MVF
5.5
A Compilation of Properties of the Payoff, Return, and Excess Return Minimum
Second Moment Return Portfolio
5.6
Mean-Variance Frontier for Discount Factors: The Hansen-Jagannathan Bounds
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MVF and Beta Representations
31
5.1 Expected Return Beta Representations
Frequently, we explain the expected return by linear beta-representations of the form
( )
E R i = γ + β ia λa + β ibλb …+ β if λ f …+ β ik λk
i = 1,2…N , f = a,b…k
(5.1)
The expected return is explained as a linear function of the beta-factors, β if . These betafactors are explanatory variables that vary from asset to asset in this cross-sectional relation.
The intersect γ grasps the factor-independent expected return. On the contrary, λ f represent
the slopes in the above linear relation, and β if λ f presents the expected return related to a
single factor f.
The beta-terms are determined as the coefficients of s multiple linear time-series regression of
the returns on the factors f = a,b…k :
Rt i = α i + β ia ft a + β ib ft b …+ β ik ft f + ε t i
t = 1,2…T
(5.2)
The regression explains to what extend the factor influences the rates of return of the assets.
The factors are proxies of the marginal utility of (consumption) growth. Canonical examples
of factors are consumption growth and market return.
β if is interpreted as quantity of the exposure of the asset i’s return to the factor f. λ f is the
price of this risk exposure. Thus, β if λ f represents the risk premium paid to asset i for the
exposure to the risk induced by factor f.
One way to estimate the free parameters γ and λ f used in the cross sectional model (5.1) is
( )
to run a cross sectional regression of the average returns of the assets E Ri on the betafactors β if ,
( )
E R i = γ + β ia λa + β ibλb …+ β if λ f …+ β ik λk + ηi .
(5.3)
The parameters γ and λ f stand for the intercept and the slope in this cross sectional
regression. The residuals α i are the pricing errors. Of course the regression model (5.3) is a
test of the relation (5.1). The model predicts ηi = 0 . The residuals ηi should be small and
insignificant.
Since the beta-factors can not be attended to asset-specific or firm-specific characteristics
such as small companies.
Some Common Special Cases
If there is a risk free rate, where all beta-factors are zero, we have
γ = Rf ,
and model (5.1) can be expressed as
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MVF and Beta Representations
32
( )
E R ei = β ia λa + β ibλb …+ β if λ f …+ β ik λk .
i = 1,2…N , f = a,b…k
(5.4)
Sometimes, the factors are also returns, e.g. in the case of CAPM. If all factors are returns we
can write the model (5.4) as
( )
( )
( )
( )…+ β E ( f )
E R ei = β ia E f a + β ib E f b …+ β if E f
f
k
ik
(5.5)
In this case the cross section regression is unnecessary because no free parameters are left in
the time regression analysis.
5.2 Mean-Variance Frontier
The MVF is the boundary of means and variances of the returns on all feasible portfolios.
E(R)
Elementary Asets
Rf
σ(R)
Figure 5.1: Mean-variance frontier
The MVF can be constructed in two ways. The more elegant way is the spanning method.
However, this method does not work if there are some short sales restrictions. The Lagrangian
works also in case of short sales restrictions. Of course, these restrictions has to be considered
explicitly. Our central assumption is that the covariance matrix can be inverted.
Assumption 5.1: Existence of the Inverse Covariance Matrix
The covariance matrix ∑ is non-singular.
5.2.1 Spanning Method
In a first step we determine two arbitrary portfolios of the MVF. In a second we span the
entire MVF as the convex hull of these MVF portfolios.
1. Construction of a MVF-Portfolio6
A MVF can be calculated as the normalized solution of a simple linear equation system. The
solution of this system represents an MVF-portfolio whose tangent owes a certain value of the
ordinate.
Theorem 5.1: Determination of an efficient portfolio
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MVF and Beta Representations
33
Given assumption 5.1 (non-singularity of the covariance matrix) for any arbitrary abscissa c
there exists a corresponding MVF-portfolio that satisfies the following equations:
(
)
(5.6)
zi
(5.7)
z = ∑ −1 µ − ce
wi =
zi
=
N
∑z
j=1
zT e
j
e is a N-dimension l unit vector. z is the solution of (5.6) for an arbitrarily given constant c.
According to (5.7) w is a normalization of z.
Proof:
1. Assumption A.5.1 (non-singularity of the covariance matrix) guarantees the existence of
the solution of equation (5.6) for a given value of c. Thus, we receive the corresponding
efficient portfolio as the normalization of z.
2. The efficient portfolio satisfying (5.6) and (5.7) is the tangency point tangent line with
intersection c and the MVF.
• Any tangential portfolio maximizes or minimizes the risk premium per unit of risk of the
portfolio. In other words λ has to attain an extreme value (maximum or minimum):
λ=
(
wT µ − ce
)
(5.8)
w ∑w
T
The first order condition for such an optimal portfolio is
( µ − ce) − 2 ∑ wλ = 0 .
(5.9)
This follows from optimizing λ with respect to a portfolio element wh :
(
)
(
) (
)
T
T
T
T
∂λ w ∑ w ∂ ⎡⎣ w µ − ce ⎤⎦ ∂wh − ⎡⎣ w µ − ce ⎤⎦ ∂ w ∑ w ∂wh
=
= 0.
2
∂wh
wT ∑ w
(
)
(5.10)
As the denominator is positive the nominator has to be equal to zero and therefore the first
order condition (5.10) can be expressed as
(
)
(
)
∂ ⎡⎣ wT µ − ce ⎤⎦ ∂wh − λ ∂ wT ∑ w ∂wh = 0.
(5.11)
The derivation of (5.11) results the following first order condition for an optimal portfolio
element wh
(µ
h
)
− c − 2 ∑ h wλ = 0 ,
(5.12)
where µ h and ∑ h represent the lines h of µ and ∑ . If we apply this procedure to all
portfolio weights we receive the first order condition (5.9) for the optimal portfolio.
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MVF and Beta Representations
34
• In order to distinguish between maxima and minima we derive the second order condition
from
(
)
∂ ⎡⎣ µ h − c − 2 ∑ h wλ ⎤⎦ ∂ wh = −2σ hh λ
(5.13)
If λ > 0 , the extremum is a maximum, if λ < 0 , it is a minimum. λ > 0 , implies
wT µ − ce > 0 (ascending), and λ < 0 , implies wT µ − ce < 0 (descending tangent).
(
)
(
)
• Substituting 2λ w by z and normalizing z to one shows that any MVF portfolio that
maximizes or minimizes λ satisfies the equations (5.6) and (5.7).
For a different value of the constant c (intersection) we receive a different pair of MVFportfolios.
2. Spanning of the MVF
We will span the entire MVF from two MVF-portfolios.
Theorem 5.2: Spanning of the MVF
The MVF is the convex hull of two arbitrary (independent) elements of the MVF.
Proof:
For two arbitrary constants c X and cY we receive the solutions
( µ − c e) − ∑ z
( µ − c e) − ∑ z
X
Y
= 0 , and
(5.14a)
=0.
(5.14b)
X
Y
Moreover, the corresponding normalizations
wX =
wY =
•
zX
(5.15a)
zXT e
zY
(5.15b)
zY T e
For any real number a, there is a constant cP = acX + (1 − a )cY , and a portfolio P with the
composition zP = az X + (1 − a ) zY such that a is a solution of the equation (5.16):
(
)
(
)
µ − cP − ∑ z P = µ − ⎡⎣ acx + 1 − a c y ⎤⎦ − ∑ ⎡⎣ az x + 1 − a z y ⎤⎦ = 0
•
(5.16)
From this follows theorem 5.2.
This method allows a simple and computing time saving determination of the MVF.
However, the application of this method demands the absence of any short sales restrictions
and/or transaction costs.
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MVF and Beta Representations
35
5.2.2 Langrangian Approach to the MVF
This approach is based on the minimization of the portfolio variance under various
restrictions. We restrict our analysis on the simple standard case where we have only the
restriction, that we form a portfolio that guarantees a minimum expected return.
Theorem 5.3: Constrained minimum variance
A MVF portfolio X is obtained as a solution of the following program:
Min wX T ∑ wX
(5.17)
wX T e = 1
(5.18)
wX T µ = µ X
(5.19)
wX
s.t.
Restriction (5.18) ensures that we form a portfolio of assets. (5.19) that the portfolio earns a
the minimum return µ X . The Lagrangean corresponding to this problem is
(
) (
L = wX T ∑ wX + ξ 1 − wX T e + ψ µ X − wX T µ
w,ξ ,ζ
)
The solution has to satisfy the following first order conditions:
Lw = 2 ∑ wX − ξe − ψµ = 0
(5.20a)
Lξ = 1 − wX T e = 0
(5.20b)
Lψ = µ X − wX T µ = 0
(5.20c)
Pre-multiplying (5.20a) by the inverse of the covariance matrix results
2 ∑ −1 ∑ wX = ξ ∑ −1 e + ψ ∑ −1 µ ,
(5.21)
or
ξ
ψ
wX = ∑ −1 e + ∑ −1 µ .
2
2
(5.21’))
Inserting (5.21’) into (5.20b) and (5.20c) results
ξ
ψ
1 = eT ∑ −1 e + µ T ∑ −1 e
2
2
(5.22a)
ξ
ψ
µ X = eT ∑ −1 µ + µ T ∑ −1 µ
2
2
(5.22b)
Equations (5.22a) and (5.22b) can be expressed by the following matrix equation
⎡ eT ∑ −1 e
⎢ T −1
⎣e ∑ µ
µ T ∑ −1 e ⎤ ⎡ ξ 2 ⎤ ⎡ 1 ⎤
⎥⎢
⎥= ⎢ ⎥.
µ T ∑ −1 µ ⎦ ⎣ψ 2 ⎦ ⎣ µ X ⎦
(5.23)
Using the abbreviations A = µ T ∑ −1 µ , B = µ T ∑ −1 e = eT ∑ −1 µ , and C = eT ∑ −1 e equation
(5.23) simplifies to
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MVF and Beta Representations
36
⎡C
⎢
⎣B
B⎤ ⎡ ξ 2 ⎤ ⎡ 1 ⎤
⎥=⎢ ⎥,
⎥⎢
A⎦ ⎣ψ 2 ⎦ ⎣ µ X ⎦
(5.23’)
Inserting the solutions
A − Bµ X
ξ 2=
AC − B 2
ψ 2=
B − Cµ X
AC − B 2
, and
(5.24a)
.
(5.24b)
into (5.21’) results the desired MVF-portfolio with the expected return µ X
e ⎡ A − Bµ X ⎤⎦ + µ ⎡⎣C µ X − B ⎤⎦
⎡ ξ
ψ⎤
.
wX = ∑ −1 ⎢ e + µ ⎥ = ∑ −1 ⎣
2⎦
AC − B 2
⎣ 2
(5.25)
Thus, the risk of this MVF-portfolio is
e ⎡ A − Bµ X ⎤⎦ + µ ⎡⎣C µ X − B ⎤⎦
wX T ∑ wX = wX T ∑ ∑ −1 ⎣
AC − B 2
⎡ A − Bµ X ⎤⎦ + µ X ⎡⎣C µ X − B ⎤⎦
=⎣
AC − B 2
=
C µ X 2 − 2Bµ X + A
(5.26)
AC − B 2
The expected return of the minimum variance portfolio can be determined by minimizing the
variance of the MVF-portfolio with respect to µ X . The first order condition demands an
expected return of
µ mv =
B µ T ∑ −1 e
=
.
C eT ∑ −1 e
(5.27)
Inserting µ mv into (5.25) in order to determine the minimum variance portfolio wmv results
wmv = ∑
=0 


⎡
⎤
⎡
⎤
B
B
e ⎢ A − B ⎥ + µ ⎢C − B ⎥
C⎦
⎣ C
⎦
−1 ⎣
AC − B 2
=
∑ −1 e
∑ −1 e
= T −1 .
C
e ∑ e
(5.28)
The corresponding risk is
σ 2 mv = wX T ∑ ∑ −1 e
1 1
1
= = T −1 .
C C e ∑ e
(5.30)
In order to derive the MVF we express (5.26) as
σ XX
⎡ 2
B
B2 B2 A ⎤
C ⎢µX − 2 µX + 2 − 2 + ⎥
C
C⎦
C
C
= ⎣
2
AC − B
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MVF and Beta Representations
37
1
C
= +
C AC − B 2
⎡
B⎤
⎢µX − C ⎥
⎣
⎦
2
(5.26’)
Obviously, the first summand represents the risk of the minimum variance portfolio. Solving
(5.26’) for µ X results the MVF as
AC − B 2 ⎡
1⎤
σ XX − ⎥ .
⎢
C
C⎦
⎣
B
µX = ±
C
(5.31)
B C represents the return of the minimum variance portfolio. Thus, we have a complete
characterization of the MVF.
5.3 Orthogonal Characterization of the Mean-Variance Frontier
R* is the return corresponding to the payoff x* that can act as a discount factor (c.f. chapt. 4,
p. 7.). The price of x* is determined as p x* = E x* T x* . Thus the definition of R* is
( )
R* ≡
(
)
x*
x*
=
.
p x*
E x* 2
(5.32)
( )
( )
R e * is defined as the projection of 1 on the space of excess returns
( )
e
R e* = proj 1 R .
(5.33)
The space of the excess return is defined as
{
}
()
Re = Re x ∈ X , p x = 0 .
(5.34)
e
R e * is an excess return on R that represents the expected asset returns as an inner product
in the same way as the payoff x* in X represents the prices as an inner product. Since x* is a
discount in X it satisfies
()
( )
(
)
(
)
p x = E mx = E ⎡⎣ proj m X x ⎤⎦ = E x* x .
From this follows
( )
( )
( )
(
(5.35)
)
e
p R e = E 1R e = E ⎡ proj 1 R R e ⎤ = E R e * R e .
⎥⎦
⎣⎢
(5.36)
Moreover, R* and R e * have very nice properties.
Theorem 5.4: Orthogonal Decomposition of Assets Returns
Any asset return R i can be decomposed into the three orthogonal components R* , R e * ,
and ni
R i = R* +wi R e * +ni ,
(5.37)
( )
where wi ∈ℜ is a weight, and ni is an excess return with E ni = 0 .
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MVF and Beta Representations
38
Moreover, the three components are orthogonal to each other,
(
) (
) (
)
E R* R e * = E R* ni = E R e * ni = 0 .
(5.38)
Proof: Geometrical Construction
R=space of returns (p=1)
R*+wRe*
R*
Ri=R*+wRe*+ni
1
Re*
0
E=1
E=0
Re = space of excess returns (p=0)
Figure 5.2: Orthogonal decomposition and MVF
Algebraic Argument
From the definitions (5.32) and (5.33), and the orthogonality of R* and R e * we know
(
)
E R* R * =
e
(
E x* R e *
( )
E x*
2
) = 0.
(5.39)
Next we choose ni such that the decomposition exhausts R i . We pick any wi , and we define
ni such that
ni = R i − R* −wi R e * .
(5.40)
ni is an excess return, and thus orthogonal to R* ,
(
)
E R* ni = 0 .
Financial Theory, js
(5.41)
MVF and Beta Representations
39
( )
To show E ni = 0 and ni is orthogonal to R e * , we exploit the fact that ni is an excess
return,
( ) (
)
E ni = E R e * ni .
(5.42)
( )
Therefore, R e * is orthogonal to ni if we pick wi so that E ni = 0 . The value of wi has to
satisfy
w
/ =
i
( ) ( ),
E ( R *)
E R i − E R*
(5.43)
e
to make sure that the expected value of ni = R i − R* −wi R e * is zero
( )
( ) ( ) ( ) ( E)( R * ) E ( R * ) = 0 .
E n = E R − E R* −
i
i
E R i − E R*
e
(5.44)
e
Once we have constructed the decomposition, we can develop the mean-variance frontier.
Since E ni = 0 , and R* , R e * and ni are orthogonal the mean and the variance of
( )
R i = R* +wi R e * +ni are
(
) ( )
( )
E R mvf = E R* + wi E R e *
(
)
{
{
{
( )
var R mvf = E ⎡⎣ R i − E R i ⎤⎦
2
(5.45)
}
(
)
= E ⎡⎣ R* +wi R e * +ni − E R* +wi R e * +ni ⎤⎦
2
}
( )
( ) }
(5.46)
+2E { ⎡⎣ R* − E ( R* ) ⎤⎦ w ⎡⎣ R * − E ( R * ) ⎤⎦} + 2E { ⎡⎣ R* − E ( R* ) ⎤⎦ ⎡⎣ n − E ( n ) ⎤⎦}
+2E { ⎡⎣ R * − E ( R * ) ⎤⎦ ⎡⎣ n − E ( n ) ⎤⎦}
= Var ( R* ) + w Var ( R * ) + Var ( n )
( )
2
2
= E ⎡⎣ R* − E R* ⎤⎦ + wi2 ⎡⎣ R e * − E R e * ⎤⎦ + ⎡⎣ ni − E ni ⎤⎦
i
e
e
i2
e
i
e
2
e
i
i
i
i
The covariance terms disappear since R* , R e * , and ni are orthogonal to each other. Thus, for
each desired value of the mean, there is a unique wi . Returns with E ni = 0 minimize the
( )
variance for each mean. From this follows a further theorem.
Theorem 5.5: Characterization of the Mean-Variance Frontier
R mv is on the mean-variance frontier iff
R mv = R* +wR e *
(5.47)
for a real number w .
( )
As you vary the number w, you sweep out the mean-variance frontier. Since E R e * ≠ 0 , we
add w changes of mean and variance to R mv .
Financial Theory, js
MVF and Beta Representations
40
We can interpret (5.47) as a „two-fund“ theorem, in the sense that every frontier return can be
interpreted as a portfolio of R* and R e * .
Decomposition in the Mean-Variance Space
The orthogonal decomposition can also be presented in the µ − σ space.
E(R)
R*+wiRe*
R*+wiRe*+ni
Elementary Asets
R*
σ(R)
Figure 5.3: Orthogonal decomposition of a return R i in the µ − σ space
In the mean-standard deviation space lines of constant second moments are circles. From
R i = R* +wi R e * +ni we can construct the minimum second moment returns as
( )
(
)
2
E R i2 = E ⎡ R* +wi R e * +ni ⎤
⎥⎦
⎣⎢
( )
( ) ( )
= E ( R* ) + w E ( R * ) + E ( n ) .
(
)
(
)
(
)
= E R* 2 + wi2 E R e * 2 + E ni2 + 2wi E R* R e * + 2E R* ni + 2wi E R e * ni (5.48)
2
i2
e
2
i2
Obviously the second moments of the returns are minimal iff wi = ni = 0 . The minimum
second moments return lies on the smallest circle that intersects the set of all portfolios on the
mean-standard deviation frontier.
R* minimizes the distance of an mean-variance-portfolio to the origin. The minimum second
moment return is not the minimum variance return. This can be seen from the following
formula:
OR* =
( )
E R mv
2
( )
+ σ 2 R mv =
( )
E R mv
2
(
) ( )
+ E R mv 2 − E R mv
2
=
(
E R mv 2
)
(5.49)
The second moment minimizing portfolio is inefficient because it lies on the decreasing
branch of the hyperbolic mean-standard deviation frontier. Nonetheless it can be used to span
the entire MVF.
( )
One can move along the frontier by adding some R e * . Since E ni = 0 , adding n does not
change the expected return of the portfolio. However, it increases the risk of the portfolio
because of var n = E ni2 > 0 . Therefore, by adding n moves us to the interior of the
()
( )
hyperbolic frontier.
Financial Theory, js
MVF and Beta Representations
41
5.4 Spanning the Mean-Variance Frontier
We can construct any point of the mean-variance frontier as a linear combination of R* and
R e * . In particular take any return
with γ ≠ 0 .
R a = R* +γ R e * ,
(5.50)
We can extract R e *
R a − R*
R*=
,
γ
e
(5.50’)
in order to substitute R e * in the returns of the mean-variance frontier:
R mv = R* +wR e* = R* +w
R a − R* γ − w
w
=
R* + R a
γ
γ
γ
(5.51)
It is convenient to use the risk free rate or any analogues in order to span the MVF.
When there is a risk free rate it is on the frontier representation
R f = R* + R f R e *
(5.52)
5.5 Compilation of Properties of R*, R e * and x*
The special returns that generate the mean-variance frontier have lots of interesting properties:
(1) Relation between x* and R*
( )
E R* 2 =
(
E x* R*
( )
E x* 2
)=
1
(5.53)
( )
E x* 2
( )
In order to derive (5.53) we multiply both sides of the definition R* = x* E x* 2 by R*
and take the expectation:
( )
E R* 2 =
(
E x* R*
( )
)
(5.53’)
E x* 2
(
)
Since R* is a return, we know that E x* R* = 1 .
(2) The reverse relation of R* and x*
( )
x* = R* E x* 2 =
R*
( )
E R* 2
.
(5.54)
( )
We derive x* from the definition of R* = x* E x* 2 and (5.53).
(3) Representation of prices by R*
R* can be used to represent prices just like x*. We start from the pricing equation and
substitute x* from (5.54). This results
Financial Theory, js
MVF and Beta Representations
42
()
(
(
E R* x
)
p x = E x* x =
),
for all x ∈ X .
( )
E R*
2
(5.55)
For returns we can express this relation as
1=
(
E R* R
),
( ) (
)
or E R* 2 = E R* R ,
( )
E R*
2
This is an alternative to the definition R* =
for all R ∈R .
(5.56)
x*
x*
=
.
p x*
E x* x*
( )
(
)
(4) Valuation of excess returns
R e * represents the expected returns that is an element of the excess return space R e . We can
( )
determine E R e * as an inner product just as we represented x* ∈ X :
( ) (
E Re = E Re * Re
)
(5.57)
As x* is orthogonal to the planes of a constant price in X , R e * is orthogonal to the planes
()
(
)
of a constant mean in R e . (5.57) is the analogue to p x = E x* x , and it offers an
(
e
)
e
{
alternative to the definition R e* = proj 1 R , with R = x ∈ X
() }
p x =0 .
(5) Construction of the risk free rate of return
If there is a risk free asset we can construct R f as the inverse of the expected value of the
discount, and the expected value of (5.54) as
( )
E R* 2
1
1
.
R =
=
=
E m
E x*
E R*
f
( )
( )
(5.58)
( )
If there is no risk free asset, (5.58) gives a Zero-Beta-Characterization of the right hand side.
You can also derive this relation by applying (5.56) to R f .
(6) Orthogonality of R e * and R*
(
)
(
)
E R* R e * = cov R*,R e * = 0
(5.59)
Moreover, R* is orthogonal to any excess return.
(7) Construction of mean-variance frontier
Elements of the MVF are of the form
R mvf = R* +wR e * .
(5.60)
( ) ( )
(
) ( )
We proved this in section (5.3). From E R i2 = E R* 2 + wi2 E R e * 2 + E ni2 , and
( )
E ni = 0 follows the result.
Financial Theory, js
MVF and Beta Representations
43
(8) Construction of R*
R* is the minimum second moment return. Graphically R* is the return closest to the origin.
It lies on the smallest circle tangent to MVF. From E R i2 = E R* 2 + wi2 E R e * 2 + E ni2
( ) ( )
(
) ( )
R* satisfies w = n = 0 .
i
i
(9) Identical first and second moment of R e *
( ) (
E Re * = E Re * 2
)
(5.61)
We apply (5.57) to R e * . Therefore,
( ) (
) ( )
var R e * = E R e * 2 − E R e *
2
( )
( )
= E R e * ⎡⎣1 − E R e * ⎤⎦ .
(5.62)
(10) Alternative definition of R e *
If there is a risk free rate, then R e * can be defined as the residual of the projection of 1 on
R* :
(
)
R e* = 1 − proj 1 R* = 1 −
( ) R* = 1 −
E R*
( )
E R*
2
1
R*
Rf
(5.63)
(
)
Fig. 5.2 makes the first equation obvious. Note that R e* = proj 1 R e .
e
To proof it analytically, note that R* and R are orthogonal and span together X . Thus,
(
e
)
(
)
(
)
1 = proj 1 R + proj 1 R* = R e * + proj 1 R* .
(5.64)
The last equality in (5.64) follows from (5.58). In order to verify (5.63) analytically, notify
(
)
that R e * is a return defined on X with price zero, i.e. E R e * R* = 0 . Moreover, from
(
) ( )
(5.57) we have E R e * R e = E R e .
(11) Decomposition of the risk free rate of return
According to (5.63), R f has the decomposition
R f = R* + R f R e * .
(5.65)
Since R f > 1 , R* + R e * is typically located on the lower branch of the MVF.
(12) Further explanation of R e *
If there is no risk free rate of return, we can use
(
)
( ( ) R ) + proj ( proj (1 X ) R* )
proj (1 R ) + proj (1 R* )
proj 1 X = proj proj 1 X
e
e
(5.66)
to deduce the analogy to (5.63)
Financial Theory, js
MVF and Beta Representations
44
(
)
(
)
) E ((R* )) R* .
(
E R*
R e* = proj 1 X − proj 1 R* = proj 1 X −
(5.67)
2
(13) Derivation of R* from returns and prices
( )
According to the formula x* = pT E xx T
( )
( )
pT E xx T
x*
R* =
=
p x*
pT E xx T
( )
−1
−1
x
−1
x , we can construct R* from
.
(5.68)
p
( )
(
)
The denominator follows from p x* = E x* x* because
( )
E ⎡ pT E xx T
⎢⎣
−1
( )
xx T E xx T
−1
( ) E ( xx ) E ( xx )
p ⎤ = pT E xx T
⎥⎦
−1
T
T
−1
( )
p = pT E xx T
−1
p.
(14) Construction of R e * from basis assets
( )
Analogously to the construction of x* = pT E xx T
( ) (
R e* = E R eT E R e R eT
)
−1
−1
x , we can construct R e * as
Re ,
(5.69)
where R e is the vector of basis excess returns, e.g. R e = R − R* . Prices have to be substituted
by expected returns because R e * represents means of returns rather than prices.
If there is a risk free rate we can calculate
( )
( )
T
T
1
1 p E xx
e
R * = 1 − f R* = 1 − f
R
R pT E xx T
−1
−1
x
.
(5.70)
p
If there is no risk free rate we can use (5.67) to construct R e * . Therefore, we use
(
) ( ) E ( xx )
proj 1 X = E x
T
T
−1
x.
(5.71)
5.6 Mean-Variance Frontiers for Discount Factors: The Hansen-Jagannathan Bound
In chapter 1 we showed that the relation between the standard deviation and the expected
value of the discount factor puts an upper bound to the Sharpe Ratio of an excess return:
( ) ≥ E(R )
E ( m) σ ( R )
σ m
e
(5.72)
e
This follows immediately from the covariance decomposition
(
) ( ) ( )
( ) ( )
E mR e = E m E R e + ρm,Re σ m σ R e = 0 ,
(5.73)
(5.73) can be expressed as
Financial Theory, js
MVF and Beta Representations
45
ρm,Re = −
( ) ( ) ≤1
σ ( m) σ ( R )
E m E Re
e
( )
since by definition ρm,Re ≤ 1 . Moreover, if a risk free rate exists we have E m = 1 R f .
Hansen and Jagannathan interpreted the relation (5.72) as a restriction on the set of discount
factors that can price a given set of asset returns, as well as a restriction on the set of asset
returns given a specific discount. From this calculation follows that we need very volatile
discount factors with a mean closed to one.
We derive a bound that is valid if there is no risk free rate and prices a large number of assets.
We derive the tuple ⎡⎣ E m ,σ m ⎤⎦ that is consistent with a given set of asset prices and
( ) ( )
payoffs, and moreover is consistent with the mean-variance frontier of discount factors.
( )
From equation (5.72) we see, the higher the Sharpe Ratio the tighter the bound on σ m . For
( )
any value of 1 E m the portfolio with the highest Sharpe Ratio is the tangency portfolio.
( ) σ ( R ) = σ ( m) E ( m) . As we increase the
The slope of the tangency portfolio equals E R e
( )
e
value of 1 E m the slope becomes lower, and the Hansen Jagannathan bound decreases. At
the mean that corresponds to the minimum variance portfolio the Hansen Jagannathan bound
reaches a minimum. If we increase the value of 1 E m further the tangent touches the mean-
( )
variance frontier in its lower part. In this case the Sharpe Ratio and the Hansen Jagannathan
bound both increase.
There is duality between the discount factor and the Sharpe Ratios:
{all
( )=
} E ( m) {
σ m
min
m that price x∈X
max
e
all R in X
( )
} σ (R )
E Re
(5.74)
e
The relation between the Sharpe Ratio and the discount factor is illustrated in the following
graphical representation:
E(R)
1/E(m)
σ(m)
α
σ(m)
tgα=E(Re)/σ(Re)
σ(R)
E(m) E(m)
Figure 5.4 Graphical construction of the Hansen-Jagannathan bound
Financial Theory, js
MVF and Beta Representations
46
For an explicit calculation of the Hansen-Jagannathan bound we need a formula of the
discount. Following the derivation of formula (4.12) we derive discount factors that price a
given set of payoffs as p = E mx , as
( )
( )
( ) ()
()
T
m = E m + ⎡⎣ p − E m E x ⎤⎦ ∑ −1 ⎡⎣ x − E x ⎤⎦ + ε ,
(
)
()
(5.75)
( )
where ∑ = cov x,x T and E ε = E ε x = 0 . We can think of (5.75) as a projection of a
discount factor on the space of payoff.
In order to construct (5.75) we postulate a discount that is a linear function of the shocks of
the payoffs
( )
()
m = E m + bT ⎡⎣ x − E x ⎤⎦ .
(5.76)
( )
( ) ()
( )
We choose b such that that m prices assets, i. e. p = E mx = E m E x − cov m,x . Given
the postulated linear discount this implies
(
( ) ()
()
) ( ) ()
()
T
p = E m E x − cov b ⎡⎣ x − E x ⎤⎦ ,x = E m E x − E ⎛ b ⎡⎣ x − E x ⎤⎦ x ⎞ .
⎝
⎠
(5.77)
This in turn implies
( ) ()
T
b = ⎡⎣ p − E m E x ⎤⎦ ∑ −1 .
(5.78)
Inserting (5.78) into (5.76) results the desired (5.75).
The variance of the discount can be determined as
( )
{
( )
σ 2 m = E ⎡⎣ m − E m ⎤⎦
2
}
( ) ()
T
( ) ()
T
()
()
( ) ()
T
()
= ⎡⎣ p − E m E x ⎤⎦ ∑ −1 ⎡⎣ x − E x ⎤⎦ ⎡⎣ x − E x ⎤⎦ ∑ −1 ⎡⎣ p − E m E x ⎤⎦ + σ 2 ε
( ) ()
(5.79)
()
= ⎡⎣ p − E m E x ⎤⎦ ∑ −1 ∑ ∑ −1 ⎡⎣ p − E m E x ⎤⎦ + σ 2 ε
()
Since σ 2 ε  0 , we have immediately an explicit formula for the Hansen-Jagannathan
bound,
( )
( ) ()
T
( ) ()
σ 2 m  ⎡⎣ p − E m E x ⎤⎦ ∑ −1 ⎡⎣ p − E m E x ⎤⎦ .
(5.80)
( ) ( )
As all asset returns must lie in a hyperbolic region in the ⎡⎣ E R ,σ R ⎤⎦ space all discounts
must lie in the hyperbolic region in the ⎡⎣ E m ,σ m ⎤⎦ space as illustrated in the right-hand
panel of figure 5.4.
( ) ( )
We can decompose any discount factor similar to the decomposition of the returns in section
5.3. Any discount factor is an element of the set M . The latter is perpendicular to the payoff
space X and goes through x* which is a discount that satisfies x* ∈ X . As illustrated in
figure 5.5 any discount m can be decomposed as
m = x* +we* +n .
Financial Theory, js
(5.81)
MVF and Beta Representations
47
In this presentation the residual ε of the traditional presentation m = x* +ε has been
decomposed into two components, we* and n. e* is defined as the residual from the
projection of 1 onto the payoff space X , or equivalently the projection of 1 onto the space E
which consists of the “excess m’s”, i.e. random variables of the type m − x* :
( )
( )
e* = 1 − proj 1 X = proj 1 E
(5.82)
e* generates means of m as R e * did for returns:
(
)
(
( )(
)
)
E m − x* = E ⎡⎣1 × m − x* ⎤⎦ = E ⎡⎣ proj 1 E m − x* ⎤⎦ .
()
Finally, n covers the residuals, with E n = 0 since it is orthogonal to 1 and orthogonal to X .
Corresponding to the returns, the MVF of the discount is given by
m = x* +we* .
(5.83)
( )
If the unit (sure) payoff is an element of X , then we know E m , and the bound and frontier
is just m = x* , and σ
2
( m) = σ ( x* ) . This corresponds to the risk neutral case of the MVF
2
for returns.
M
_ = space of discount factors
_
X payoff space
x*+we*
m=x*+we*+n
x*
1
proj(1|X)
_
e*
0
E=1
E=0
E
_ = space of m - x*
Figure 5.5: Decomposition of any discount factor m = x* +we* +n
Financial Theory, js
MVF and Beta Representations
48
In order to derive formulas for the construction we apply the formula for x*, i.e. the price of
any portfolio cT x ∈ X :
( )
x* = pT E xx T
−1
x.
(5.84)
Moreover, we get a formula for e*. The projection of 1 onto X is defined as
( ) ( ) E ( xx )
proj 1 X = E x
T
T
−1
x.
(5.85)
Thus, e* is defined as
( ) E ( xx )
e* = 1 − E x
T
T
−1
x.
(5.86)
Consequently, we can derive the variance minimizing discount factors
( )
m* = x* +we* = pT E xx T
−1
( ) E ( xx )
x + w ⎡1 − E x
⎢⎣
T
T
−1
x⎤ ,
⎥⎦
(5.87)
or
()
( )
T
m* = w + ⎡⎣ p − wE x ⎤⎦ E xx T
−1
(5.88)
x
As w varies we span the MVF for discounts m* as
( )
()
T
( ) E ( x) ,
E m* = w + ⎡⎣ p − wE x ⎤⎦ E xx T
−1
(5.89)
and
( )
()
T
()
σ 2 m* = ⎡⎣ p − wE x ⎤⎦ ∑ −1 ⎡⎣ p − wE x ⎤⎦ ,
(5.90)
( )
where ∑ = cov xx T . Obviously the Hansen-Jagannathan types of MVF for discounts are
equivalent to ordinary MVF for returns.
Finally, Hansen and Jagannathan derived a bound with strictly positive discounts. It follows
from the program below:
( )
min σ 2 m
w
s.t.
(5.91)
( )
p = E mx ,
(5.92)
m>0,
(5.93)
( )
(5.94)
E m = m.
The bound is tighter than the ordinary MVF for discounts because of the additional restriction
(5.93).
Financial Theory, js
MVF and Beta Representations
49