C hapter 5. Mean-variance frontier and beta representations

Asset Pricing Zheng Zhenlong

Chapter 5. Mean-variance frontier and beta representations

Asset Pricing Zheng ZhenlongMain contents

• Expected return-Beta representation• Mean-variance frontier: Intuition and Lagrangian

characterization• An orthogonal characterization of mean-variance

frontier• Spanning the mean-variance frontier• A compilation of properties of • Mean-variance frontiers for m: H-J bounds

*** ,, xRR e


5.1 Expected Return-Beta Representation


Expected return-beta representation•

....)( bibaiaiRE

... , 1, 2,...i a b it i ia t ib t tR t T

Asset Pricing Zheng ZhenlongRemark(1)

• In (1), the intercept is the same for all assets.• In (2), the intercept is different for different

asset.• In fact, (2) is the first step to estimate (1).• One way to estimate the free parameters

is to run a cross sectional regression based on estimation of beta

is the pricing errors

( ) ....iia a ib b iE R

,

i

Asset Pricing Zheng ZhenlongRemark(2)

• The point of beta model is to explain the variation in average returns across assets.

• The betas are explanatory variables,which vary asset by asset.

• The alpha and lamda are the intercept and slope in the cross sectional estimation.

• Beta is called as risk exposure amount, lamda is the risk price.

• Betas cannot be asset specific or firm specific.

Asset Pricing Zheng ZhenlongSome common

special cases• If there is risk free rate, • If there is no risk-free rate, then alpha is called

(expected)zero-beta rate.• If using excess returns as factors, (3)• Remark: the beta in (3) is different from (1) and (2).• If the factors are excess returns, since each factor has

beta of one on itself and zero on all the other factors. Then,

• 这个模型只研究风险溢酬，与无风险利率无关

( ) ..., 1, 2,...eiia a ib bE R i N

( ) ( ) ( ) ..., 1, 2,...ei a bia ibE R E f E f i N

fR


5.2 Mean-Variance Frontier: Intuition and Lagrangian Characterization

Asset Pricing Zheng ZhenlongMean-variance frontier

• Definition: mean-variance frontier of a given set of assets is the boundary of the set of means and variances of returns on all portfolios of the given assets.

• Characterization: for a given mean return, the variance is minimum.

Asset Pricing Zheng ZhenlongWith or without risk free

rate

tangency risk asset frontier original assets

( )R

)(RE

fR

mean-variance frontier


When does the mean-variance exist?• Theorem: So long as the variance-covariance

matrix of returns is non singular, there is mean-variance frontier.

• Intuition Proof:• If there are two assets which are totally

correlated and have different mean return, this is the violation of law of one price. The law of one price implies the existence of mean variance frontier as well as a discount factor.

Asset Pricing Zheng Zhenlong奇异矩阵的经济含义

•


Mathematical method: Lagrangian approach• Problem:

• Lagrangian function:])')([(

),(

11',',.,'min }{

ERERE

REE

wuEwtswww

1)'(' 'wuEwwwL

Asset Pricing Zheng ZhenlongMathematical method:

Lagrangian approach(2)• First order condition:

• If the covariance matrix is non singular, the inverse matrix exists, and

01EwwL

11'1'11''

,1)1('1'1

,)1(''

),1(

11

11

1

1

1

uE

EEE

Ew

uEEwE

Ew

Asset Pricing Zheng ZhenlongMathematical method:

Lagrangian approach(3)• In the end, we can get

1'1,1','

,2)var(

,)(1)(

,,

111

2

2

21

22

CEBEEABAC

ABuCuR

BACBuABCuEw

BACBuA

BACBCu

p

Asset Pricing Zheng ZhenlongRemark

• By minimizing var(Rp) over u,givingmin var 1 1/ , 1/(1' 1)u B C w


5.3 An orthogonal characterization of mean variance frontier

Asset Pricing Zheng ZhenlongIntroduction

• Method: geometric methods.• Characterization: rather than write portfolios as

combination of basis assets, and pose and solve the minimization problem, we describe the return by a three-way orthogonal decomposition, the mean variance frontier then pops out easily without any algebra.

Asset Pricing Zheng ZhenlongSome definitions

• *R *x

)()( 2*

*

*

**

xEx

xpxR

*eR

* (1| ),

{ , . . ( ) 0}

ee

e

R proj R

R x X s t p x


P=0(超额收益率）

Rf

P=1(收益率）

状态 1回报

状态 2回报

R*1

Re*

x*

pc

Asset Pricing Zheng ZhenlongTheorem:

• Every return Ri can be expressed as:

• Where is a number, and ni is an excess return with the property E(ni)=0.

• The three components are orthogonal,

* *i i e iR R R n

i

* * * *( ) ( ) ( ) 0e i e iE R R E R n E R n

Asset Pricing Zheng ZhenlongTheorem: two-fund theorem

for MVF•

* *mv eR R R

Asset Pricing Zheng ZhenlongProof: Geometric method

* *i e iR R R n

0

R=space of return (p=1)

Re =space of excess return (p=0)

R*R*+wiRe*

Re*

E=0 E=1 E=2

Rf=R*+RfRe*

NOTE:1 、回报空间为三维的。 2 、横的平面必须与竖的平面垂直。 3 、如果有无风险证券，则竖的平面过 1 点，否则不过，此时图上的 1 就是 1 在回报空间的投影。

等预期超额收益率线

1

Asset Pricing Zheng ZhenlongProof: Algebraic approach

• Directly from definition, we can get*

* *

2 2 * * 2

( ) ( ) 0

( ) ( ) ( )

( ) ( ) ( )

0

i e i

i i e

i i e i

i

E n E R n

E R E R w E R

R R w R n

n

＝

只有时，方差在收益率给定情况下才最小。


Decomposition in mean-variance space• *R

)()()()( 22*22*2 ie nEREwRERE


• The minimum second moment return is not the minimum variance return.(why?)

* 2 2 2 2 2 2( ) ( )OR E E E R E E R

( )R

R*

R*+wiRe*

Ri

E(R)


5.4 Spanning the mean variance frontier


Spanning the mean variance frontier• With any two portfolios on the frontier. we can

span the mean-variance frontier.• Consider

/

)1()(

,

,0,

*****

**

**

wy

yRRyRRwRwRR

RRR

RRR

e

e

e


5.5 A compilation of properties of R*, Re*, and x*

Asset Pricing Zheng ZhenlongProperties(1)

• Proof:,)(

1)( 2*2*

xERE

)(/1)()()(

,)(

,)(

2*2*

**2*

2*

**2*

2*

**

xExERxERE

xERxR

xExR


• Proof:)( 2*

**

RERx

)()(

,)(

2*

*2***

2*

**

RERxERx

xExR


• can be used in pricing.• Proof:

• For returns,

*R

)()()()( 2*

**

RExRExxExp

**

*2

* *2

( )1 ( ) ( )( )

( ) ( )

E R Rp R E x RE R

E R R E R


• If a risk-free rate is traded,

• If not, this gives a “zero-beta rate” interpretation.

*2

* *

1 ( )( ) ( )

f E RRE x E R


• has the same first and second moment.• Proof:

• Then * * * *2( ) ( ) ( )e e e eE R E R R E R

*eR

))(1)(()()()var( **2*2** eeeee RERERERER


• If there is risk free rate,

• Proof:** eff RRRR

*

* * *

*

* *

1 (1| ) (1| )11 (1| ) 1

e

e

f

f

ef f f

proj R proj R

R proj R RR

R R R R

（从上图的两个相似三角形可以看出）

注意R前面的1和R 都是标量，其余都是向量（这里的第二个R 是标量，其余都是向量）

Asset Pricing Zheng ZhenlongIf there is no risk free rate

• Then the 1 vector can not exist in payoff space since it is risk free. Then we can only use

*

*

* *

* *

**

*2

(1| ) ( (1| ) | ) ( (1| ) | )

(1| ) (1| )

(1| ) (1| )

(1| ) )

( )(1| )( )

e

e

e

proj X proj proj X R proj proj X R

proj R proj R

R proj X proj R

proj X R

E Rproj X RE R

＝－E(x

＝－


• Since

• We can get

* 1

* * *

' ( ')

( ) ( )

x p E xx x

p x E x x

pxxEpxxxEp

xpxR 1

1

*

**

)'('''

)(


• Following the definition of projection, we can get

• If there is risk free rate,we can also get it by:

* ' 1(1| ) ( ) ' ( )e e e e e eR proj R E R E R R R －

pxxEpxxxEp

RR

RR

ff

e1

1**

)'(')'('1111


5.6 Mean-Variance Frontiers for Discount Factors: The Hansen-Jagannathan Bounds


Mean-variance frontier for m: H-J bounds• The relationship between the Sharpe ratio of an

excess return and volatility of discount factor.

• 从经济意义上讲， m 的波动率不应该太大，所有夏普比率也不应太大。• If there is risk free rate,

)(|)(|

)()(

,1|)()()()(|||

,0)()()()()(

,

,

e

e

e

e

Rm

eRm

ee

RRE

mEm

RmREmE

RmREmEmRE

e

e

fRmE /1)(


• We need very volatile discount factors with a mean near one to price the stock returns.


The behavior of Hansen and Jagannathan bounds

• For any hypothetical risk free rate, the highest Sharpe ratio is the tangency portfolio.

• Note: there are two tangency portfolios, the higher absolute Sharpe ratio portfolio is selected.

• If risk free rate is less than the minimum variance mean return, the upper tangency line is selected, and the slope increases with the declination of risk free rate, which is equivalent to the increase of E(m).


The behavior of Hansen and Jagannathan bounds• On the other hand, if the risk free rate is

larger than the minimum variance mean return, the lower tangency line is selected,and the slope decreases with the declination of risk free rate, which is equivalent to the increase of E(m).

• In all, when 1/E(m) is less than the minimum variance mean return, the H-J bound is the decreasing function of E(m). When 1/E(m) is larger than the minimum variance mean return, the H-J bound is an increasing function.


Graphic construction

E(R)

( )m

)(R

)(/)( mEm

1/E(m)

E(m)

( ) / ( )e eE R R

Asset Pricing Zheng ZhenlongDuality

• A duality between discount factor volatility and Sharpe ratios.

{ } { }

( ) ( )min max( ) ( )e

e

eall m that price x X all excess returns R in X

m E RE m R

Asset Pricing Zheng ZhenlongExplicit calculation

• A representation of the set of discount factors is

• Proof:

0)(,0)(),',cov(

,)]([)]()([)( 1

xEExx

xExxEmEpmEm

pxEmEpxEmE

xxExExEmEpxEmE

xxExxEmEpmEEmxE

)()()()(

]))([()]()([)()(

)))]([)]()([)((()(1

1


An explicit expression for H-J bounds

• Proof:))()(())'()(()( 12 xEmEpxEmEpm

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

2'2 1 2

' 1 2

' 1

( )

( )

m p E m E x

p E m E x p E m E x

p E m E x p E m E x

s s e

s e

-

-

-

æ öé ù ÷ç= - +÷çê ú ÷çë ûè øé ù é ù= - - +ê ú ê úë û ë ûé ù é ù³ - -ê ú ê úë û ë û

å ååå


Graphic Decomposition of discount factor

* *m x e n

0

M=space of discount factors

E =space of m-x*

x*

x*+we*

e*

E()=0 E()=1 E()=2

NOTE: 横的平面必须与竖的平面垂直。

X=payoff space

1Proj(1lX)

Asset Pricing Zheng ZhenlongDecomposition of discount

factor•

* *m x we

nwexm **

* 1 (1| ) (1| )e proj X proj E

Asset Pricing Zheng ZhenlongSpecial case

• If unit payoff is in payoff space,

• The frontier and bound are just • And

• This is exactly like the case of state preference neutrality for return mean-variance frontiers, in which the frontier reduces to the single point R*.

*m x

* 1 (1| ) 0e proj X

)()( *22 xm

Asset Pricing Zheng ZhenlongMathematical construction

• We have got

( ) ( )( )( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

* * *

1

1 1

' 1

' 1*

' 12 *

' ' 1 1|' ' 1 ' '

''

cov '

m x we

p E xx x w proj X

p E xx x w E x E xx x

w p wE x E xx x

E m w p wE x E xx E x

m p wE x xx p wE xs

-

- -

-

-

-

= += + -

æ ö÷ç= + - ÷ç ÷è øé ù= + -ê úë û

é ù= + -ê úë ûé ù é ù= - -ê ú ê úë û ë û

Asset Pricing Zheng ZhenlongSome development

• H-J bounds with positivity. It solves

• This imposes the no arbitrage condition.• Short sales constraint and bid-ask spread is

developed by Luttmer(1996).• A variety of bounds is studied by Cochrane and

Hansen(1992).

2min ( ), . . ( ), 0,m s t p E mx m E m 固定


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C hapter 5. Mean-variance frontier and beta representations