Lecture 10: Discount Factor, Beta, Mean-variance Frontier

L10: Discount Factors, beta, and Mean-variance Frontier 1

Lecture 10: Discount Factor, Beta, Mean-variance Frontier

• The following topics will be covered:– Discount factor (Existence Theorem– Beta Representations– Mean-variance Frontier– Relation between Discount Factors, Betas, and

Mean-variance Frontiers (Equivalence Theorem)– Implications

Materials are from chapters 4, 5, & 6, JC.


Law of One Price

• Payoff space– The payoff space X is the set of all the payoffs that investors can

purchase, or it is a subset of the tradable payoffs that is used in a particular study.

– Payoff space includes some set of primitive assets, but investors can form new payoffs by forming portfolios of the primitive assets. This leads to:

• Portfolio formation (A1)– x1, x2 є X ax1 + bx2 є X for any real a, b.– i.e., $1 to get R, $2 to get 2R, -$1 to get -R– This assumption rules out short sell constraint, bid/ask spread, leverage

limitation, etc.– We can trade nonlinear functions of a basis payoff, e.g., call option


Law of One Price

• Law of one price (A2): p(ax1+bx2)=ap(x1) + bp(x2)– linearity– The law of one price says that investors cannot make

instantaneous profits by repackaging portfolios.– The law is meant to describe a market that has already

reached equilibrium. If there are any violations of the law of one price, trader will quickly eliminate them so they cannot survive in equilibrium


Theorem on the Existence of a Discount Factor

• Theorem: Given free portfolio formation A1, and the law of one price A2, there exists a unique payoff x* є X such that p(x)=E(x*x) for all x.– x* is a discount factor

• I.e., m=x* -- a special m• p(x) = E(mx) = E[proj(m|X)x] = E(x*x)

– x* = proj(m|X)• On the existence of m (the discount factor).• Unique in payoff space X.• See Figure 4.2• It can be shown that:

xxxEpx 1)'('*


What the Theorem Does Not Say


No Arbitrage and Positive Discount Factors• No Arbitrage (absence of arbitrage)

– Positive payoff implies positive price. I.e., x>0 then p>0.

• m>0 implies no arbitrage• No arbitrage and the law of one price imply

m>0.


What Does It Not Say?

• Discount factor m>0 exists, but it does not say that m>0 is unique.

• Does not say every m>0• We can extend the pricing function defined on X to all

possible payoffs Rs


Explicit Formula for x*, the discount factor

)]'()][(([)],(*)([1

xExxExEwherexExEpb

• Assuming the discount factor x* is the linear function of the shocks to payoffs: x* = E(x*) + (x – E(x))’b

• Finding b to ensure that x* prices the asset x:

p = E(xx*) = E(x*)E(x) + E[x(x – E(x))’]b

• We have:

• The convenient alternative formula:

)]([)]'(*)([*)(* 1 xExxExEpxEx


Expected Return-Beta Representation• Linear factor pricing model:

where β terms are defined as the coefficients in a multiple regression of returns on factors

Factors f are proxies for marginal utility growth. • The regression is not about predicting return from variables

seen ahead of time.• Its objective is to measure contemporaneous relations or

risk exposure

NiRE bbiaaii ,...,2,1...)( ,,

TtffaR it

btbi

ataii

it ,...,2,1...,,


Expected Return-Beta Representation

• Beta is the amount of exposure of asset i to factor a risks

• Expected return-beta relationship should be tested via cross-sectional regression

• Test the constraint:

NiRE ibbiaaii ,...,2,1...)( ,,

0i


Some Common Special Cases

• If there is a risk-free rate, we have• If there is no risk-free rate, then γ must be

estimated in the cross-sectional regression. It is called zero-beta rate.

• In the form of excess returns, we have:

• Also, it is often the case that the factors are returns or excess returns.

fR

NiRE bbiaaii ,...,2,1...)( ,,

NifEfERE bbi

aai

i ,...,2,1...)()()( ,,


Mean-Variance Frontier• Mean –variance frontier of a given set of assets is the boundary of the set of means

and variances of the returns on all portfolios of the given assets.


Lagrangian Approach to Get Mean-Variance Frontier

• Start with a vector of asset return R.

• mean return: E= E(R)

• variance-covariance matrix: ∑=E[(R-E)(R-E)’]

• we construct an optional portfolio and choose weight w

• Objective: choose a portfolio to minimize variance for a given mean:

11''..

'min

wEwts

www


Solving the System• Introduce Lagrange multiplier on the constrains, we have:

• Note w is a vector• We have the expression for the variance of the minimum

variance portfolio specified in 5.7 (p82)– The variance is a quadratic function of the mean.

• Minimum variance portfolio has minimum variance, weight specified on page 83

)11'(2)'(2'min,,

wEwwww

ww

ww

2'


Orthogonal Decomposition and Mean-Variance Frontier

• Alternative approach to derive mean-variance frontier• Hansen and Richard (1987) approach• We can describe any return by a 3-way orthogonal

decomposition, then the problem is solved.• Define R* -- the return corresponding the payoff x*

• Define Re*)*(

**)(** 2xE

xxp

xR

}0)(..{___

)|1(*

xptsXxreturnsexcessofspaceR

RprojRe

e


Orthogonal Decomposition and Mean-Variance Frontier

• Re* is an excess return that represents means on Re space with an inner product in the same way that x* is a payoff in X space that represents prices with an inner product.

• E(Re) = E[proj(1|Re) Re] = E(Re*Re)• Theorem: Every Ri can be expressed as Ri = R* + wiRe* + ni

where ni is an excess return with property E(ni) = 0 and the components are orthogonal.

• Theorem: Rmv is on the mean-variance frontier if and only if Rmv = R* + wiRe* for some real number w.


Mean-Variance Frontier in Payoff Space

Note: Re space is the space of excess returns, thus p=0


Orthogonal Decomposition in Mean Standard Deviation Space


Algebraic Proof

• With decomposition, E(ni)=0 and that the three components are orthogonal, we have:

• For each desired value of the mean return, there is a unique wi that minimize variance for each mean.

)()()(

)()()(2**22

**

ieii

eii

nRwRR

REwRERE


Spanning the Mean-Variance Frontier

• You can span the mean-variance frontier with any two portfolios that on the frontier

• Use risk free rate or its kind to span the space– Zero-beta return– Constant-mimicking portfolio return– Minimum variance return

• Span and diversification


R*, Re*, x*• See page 89 through 93.

*

*

*

2

*

2

2

22

*)7(

0)*()6(

)()*(

*)(1)5(

)()()4(

)*()*()*()()3(

)*(**)2(

)*(1)*()1(

emv

e

f

eee

wRRR

RRE

RERE

xER

RRERE

RExRExxExp

RERx

xERE


R*, Re*, x*(8) R* is the minimum second moment return(9) E(Re*) = E(Re*2)(10) If there is risk-free rate, Re*=1 - (1/Rf)*R*(11) Rf = R* + RfRe*

(12) If there is no risk-free rate, Proj(1|X)=proj(1|Re)+proj(1|R*)(13)

(14) pxxEpxxxEp

xpxR 1

1

)'(')'('

*)(**

pxxEpxxxEp

RR f

e1

1

)'(')'('11*


Hansen-Jagannathan Bounds)()()()()(0

,e

Rmee RmREmEmRE e

We have,

Hansen and Jagannathan (1991) read this as a restriction on the set of discount factors that can price a given set of returns

We need very volatile discount factors with mean near 1 (E(m)=Rf when risk free rate exists) to understand stock returns.

The higher the Sharpe ratio, the tighter the bound on the volatility of discount factor

)(|)(|

)()(

e

e

RRE

mEm


Explicit Form

)]([)]'()([)( 1 xExxEmEpmEm

)]([)]'()([)( 1 xExxEmEpm


P=E(mx) β

)(/1

))()var()(

)var(),cov(()( ,

mEwheremEm

mRmRE mmi

ii

**,)( xxiiRE

Beta representation using m

Beta representation using x* and R*

Theorem: 1=E(mRi) implies an expected return-beta model x*=proj(m|X) or R*=x*/E(x*2) as factors,

]*)([)( *, RERE Rii


More …

MV Frontier p=E(mx) and β (page 103)

Discount factors and beta models to mean-variance frontier (page 110)

fbamRE ii '')( (Page 105)


Factor-Mimicking Portfolios• When factors are not already returns or excess returns, it is convenient to

express a beta pricing model in terms of its factor-mimicking portfolios rather than factors themselves.

• Defining f*=proj(f|X)• m=b’f carries the same pricing implication on X as does m=b’f• p=E(mx)=E(b’fx)=E[b’(proj(f|X)x]=E[b’f*x]• Factor mimicking return

)]|([)|(*

XfprojpXfprojf

Lecture 10: Discount Factor, Beta, Mean-variance Frontier

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Transcript of Lecture 10: Discount Factor, Beta, Mean-variance Frontier