chp 1: consumption- based model and overview 郑振龙厦门大学金融系个人网站： ...

Chp 1: Consumption-based model and overview

郑振龙厦门大学金融系个人网站：http://efinance.org.cn EMAIL: [email protected]

Structure

Basic Pricing Equation Stochastic Discount Rate Prices and Payoffs Classic Issues in Finance Discount Factors in Continuous time

Copyright © 2012 Zheng, Zhenlong

Investors’ Utility Function

Utility comes from consumption and is time seperable.

u(•)is increasing and concave. The curvature of u captures investor’s aversion to risk and to intertemporal substitution.

β is called subjective discount rate and it captures investors’ impatience.

1 1( , ) ( ) ( )t t t t tU c c u c E u c


Investor’s Objective

e—original consumption level.

--the amount of the asset he choose to buy.

pt—the price of the asset at time t.

1

1 1 1

( ) ( )

. .

,

t t t

t t t

t t t

Maxu c E u c

s t

c e p

c e x


First order condition

is the loss in utility if the investor buys another unit of the asset.

is the increase in (discounted, expected) utility he obtains from the extra payoff at t+1.

( )t tp u c

1 1( )t t tE u c x

1 1

11

( ) ( ) (1.1)

( )(1.2)

( )

t t t t t

tt t t

t

p u c E u c x

u cp E x

u c


Stochastic discount factor Define the Stochastic discount factor mt+1(1.3):

For every investor, m is the same for every asset.

mt+1 is stochastic because it’s not known at t.

mt+1 is also called the marginal rate of substitution.

mt+1 is also called the pricing kernel, change of measure or a state-price density.

11

( )

( )t

tt

u cm

u c


Basic pricing formula Putting (1.3) into (1.2) we can get (1.4):

Breaking (1.2) into (1.3) and (1.4) represents a deep and useful separation.

For example, notice p=E(mx) would still be valid if we changed the utility function, but we would have a different function connecting m to data.

1 1( )t t t tp E m x


Prices and payoffs: Stocks

The price pt gives rights to a payoff xt+1.

For stocks, the one-period payoff is: xt+1=pt+1+dt+1,so we get:

1 1 1( )t t t t tp E m p d


Prices and payoffs: Return

Defines gross return:

We can think return as a payoff with price one.

So we get: 1=E(mR)

11

tt

t

xR

p


Prices and payoffs: Price-dividend ratio

Dividing the stock payoff by dt, we can get a payoff of the form xt+1=(1+pt+1/dt+1)dt+1/dt corresponding to a price pt/dt.

So we get:

1 11

1

/ (1 )t tt t t t

t t

p dp d E m

d d


Prices and payoffs: Excess return If you borrow a dollar at Rf and

invest it in an asset with return R, you pay no money out-of-pocket today, and get the payoff R-Rf.

You can also short-sell stock b and invest the proceeds in stock a and get an excess return Re with zero price.

Se we get:

1 1

1 1 1

0 ( )

0 ( )

ft t t

a bt t t t

E m R R or

E m R R


Prices and payoffs: Managed portfolio

13:20Copyright © 2012 Zheng, Zhenlong 12

Prices and payoffs: Moment condition

Taking an unconditional expectation of ptzt=Et(mt+1xt+1)zt yielding E(ptzt)=E(mt+1xt+1zt ).

We can think this operation as creating a “security”with payoff xt+1zt, and “price” E(ptzt) represented with unconditional expectations.


Prices and payoffs: discount bond

The payoff of one-period discount bond is 1.

So we have:

pt=Et(mt+1)


Prices and payoffs: options

For an European call option, we have:

ct=Et[mt+1max(ST-X,0)]


Prices and payoffs: Real or nominal

If prices and payoffs are nominal, we should use a nominal discount factor. For example, if p and x denote nominal values, then we can create real prices and payoffs to write:

Obviously, it is the same as defining a nominal discount factor by

1 1

1

( )[ ]

( )t t t

tt t t

p u c xE

u c

11

1

( )[( ) ]

( )t t

t t tt t

u cp E x

u c


Prices and payoffs: risk-free rate

Because the risk-free rate is known ,so:

1=E(mRf)=RfE(m)

Rf=1/E(m) (1.6)

If a risk-free rate is not traded, we can define 1/E(m)as the “shadow” risk-free rate. It is sometime called “zero-beta” rate.


Economics behind risk-free rate Suppose: no uncertainty, and power utility

function:

Then we have:

Findings: 1.Rf is high when people are impatient(βis low). 2. Rf is high when consumption growth is high. 3. Rf is more sensitive to consumption growth if γ

is large.

11( )

1t tu c c

11( )f t

t

cR

c


Risk-free rate under uncertainty Suppose c growth is lognormally distributed,

Start with

Using the fact that normal X means:

We have:

11/ ( )f tt

t

cR E

c

2( ) 0.5( )X E XE e e 1

2 21 1

1ln 1

( ln ) 0.5 ( ln )

2 21 1

1 1

1 1( )

( ln ) 0.5 ( ln )

: ln ; , ln ln ln

t

t t t t

c

c E c cft

ft t t

f ft t t

R E e e

r E c c

where r R e c c c


Risk-free rate under uncertainty(2)

Additional finding: Captures the effect of

precautionary savings. For the power utility function, the curvature

parameter γ simultaneously controls intertemporal substitution—aversion to a consumption stream that varies over time, risk aversion—aversion to a consumption stream that varies across states of nature, and precautionary savings.

2 210.5 ( ln )tc


Risk Corrections--price Using the definition of covariance cov(m,x)=E(mx)-E(m)E(x),

we can write p=E(mx) as:

The first term is the asset’s price in a risk-neutral world. The second term is a risk adjustment.

Marginal utility u’(c) declines as c rises. Thus an asset’s price is lowered if its payoff covaries positively with consumption.

It’s the covariance not the variance determines the riskiness.

1 1

( ) ( ) cov( , )

( ) / cov( , )

cov ( ),( ) /

( )

f

t tf

t

p E m E x m x

p E x R m x

u c xp E x R

u c

2 2 2 2( ) ( ) 2 cov( , ) ( )c x c c x x


Risk Corrections--Returns

From 1=E(mRi), we have:(1.11-13)

Assets whose returns covary positively with consumption must promise higher expected returns to induce investors to hold them.

1

1

1

1

1 ( ) ( ) cov( , )

( ) cov( , )

cov( ( ), ) ( )( )

( ) ( ( ))

cov( ( ), )( )

( ( ))

i i

i f f i

ii f t t

t t t

ii f t

t t

E m E R m R

E R R R m R

u c R u cE R R

u c E u c

u c RE R R

E u c


Idiosyncratic risk does not affect risk Only the component of a payoff perfectly correlated with

m generates an extra return. Idiosyncratic risk, uncorrelated with m, generate no premium.

We can decompose any payoff x into a part correlated with m and an idiosyncratic part uncorrelated with m by running a regression:

x=proj(x|m)+ε Projection means linear regression without a constant:

2

22 2

( )( | )

( )

( ) ( )Pr : ( ( | )) ( ) ( ) ( ) ( )

( ) ( )

E mxproj x m m

E m

E mx E mxoof p proj x m p m E m E mx p x

E m E m


Expected Return-Beta Representation

From (1.12) we have (1.14):

This is a beta pricing model. The λm is the price of risk and is the same for all assets, and βi,m is the quantity of risk in each asset and varies from asset to asset.

With m=β(ct+1/ct)-γ,we can get (1.15) by taking a Taylor approximation of (1.14):

,

cov( , ) var( )( )

var( ) ( )

( )

ii f

i fi m m

R m mE R R

m E m

E R R

,( )

: var( )

i fi c c

c

E R R

Where c


Mean-Variance Frontier

All assets priced by m must obey(1.17):

It’s because:

( )( ) ( )

( )i f im

E R R RE m

,

,

1 ( ) ( ) ( ) ( ) ( )

( )( ) ( )

( )

: 1.

i

i

i i i

m R

i f i

m R

E mR E m E R R m

mE R R R

E m

and


Classic Implications(1)

Means and variances of asset return must lie in the wedge-shaped region as Fig.1.1.The boundary is called the mean-variance frontier.

Rf

E(R)

σ(R)

Idiosyncratic riskRi

Slope σ(m)/E(m)

Some asset returns



All returns on the frontier are perfectly correlated with m: the frontier is generated by|ρm,R|=1. Returns on the upper part are perfectly negatively correlated with m and hence positively with consumption. They are “maximally risky” and thus get the highest expected returns.



All frontier returns are also perfectly correlated with each other. This fact implies that we can span or synthesize any frontier return from two such returns.

For example if you pick any single frontier return Rm then all frontier returns Rmv must be expressible as

Rmv= Rf + a (Rm -Rf )

for some number a.


Classic Implications(4) Since each point on the mean-variance

frontier is perfectly correlated with m, we must be able to pick constants a, b, d, e such that

m = a + bRmv

Rmv = d + em. Thus, any mean-variance efficient return

carries all pricing information. Given a mean variance efficient return and

the risk free rate, we can find a discount factor that prices all assets and vice versa.


Classic Implications(5) Expected returns can be described in a single - beta

representation using any mean-variance efficient return (except the risk free rate).

E(Ri) = Rf + βi,mv [E(Rmv) -Rf ].(Proof is in the next slide) The essence of the β pricing model is that, even though

the means and standard deviations of returns fill out the space inside the mean-variance frontier, a graph of mean returns versus betas should yield a straight line.

Since the beta model applies to every return including Rmv itself, and Rmv has a beta of one on itself, we can identify the factor risk premium as λ = E(Rmv-Rf ).


Proof (problem1.3) We know there are a,b, such that

m=a+bRmv. Determines a, b by pricing Rmv and Rf:

2

2

1 ( ) ( ) ( ) ( )

1 ( ) ( ) ( )

( ) ( )

var( )

( )

var( )

(1.12) : ( ) cov( , ) cov( , ) cov( , )

mv mv mv mv mv

f mv f f mv f

mv mv f

f mv

f mv

f mv

i f f i f f mv i f f mv i

E mR E a bR R aE R bE R

E mR E a bR R aR bE R R

E R E R Ra

R R

R E Rb

R R

From E R R R m R R R a bR R R bR R R

,

cov( , )( ( ) ) ( ( ) )

var( )

mv if mv f f mv f

i mvmv

R RR E R R R E R R

R



We can plot the decomposition of a return into a “priced” or “systematic” component and a “residual,” or “idiosyncratic” component as shown in Figure 1.

Asset inside the frontier or even on the lower portion of the frontier are not “worse” than assets on the frontier. You would not want to put your whole portfolio in one “inefficient” asset, but you are happy to put some wealth in such assets.


Mean-standard deviation frontier The slope of the mean-standard deviation

frontier is the largest available Sharpe ratio. From (1.17),the slope of the frontier is

Under power utility function, we have

( ) ( )( )

( ) ( )

mv ff

mv

E R R mR m

R E m

1

1

[( / ) ]( )

( ) [( / ) ]

mv ft t

mvt t

c cE R R

R E c c


Mean-standard deviation frontier(2) Suppose consumption growth is lognormal, we

have1 1

2 2 2 21 1 1 1

2 2 2 21 1 1

2 21 1

2 ln ln 21

2 ( ln ) 2 ( ln ) 2 ( ln ) ( ln )

( ln ) 0.5 ( ln ) ( ln )

ln ( ln ) 0.51

[( / ) ] ( ) ( )

1

[( / ) ] ( )

t t

t t t t

t t t

t t

c ct t

E c c E c c

E c c c

c E ct t

c c E e E e

e e

e e

E c c E e e

1

2 21

( ln )

( ln )1

1

[( / ) ]( )1

( ) [( / ) ]

t

t

c

mv fct t

mvt t

c cE R Re

R E c c


Mean-standard deviation frontier(3)

Using the approximation for small x that ex≈1+x, get(1.20)

2 21( ln )( )

1 ( ln )( )

t

mv fc

mv

E R Re c

R


Equity premium puzzle the slope of the mean-standard deviation frontier is

higher if the economy is riskier – if consumption is more volatile – or if investors are more risk averse.

Over the last 50 years in the U.S., real stock returns have averaged 9% with a standard deviation of about 16%, while the real return on treasury bills has been about 1%. Thus, the historical annual market Sharpe ratio has been about 0.5. Aggregate consumption growth has A mean and standard deviation of about 1%. Thus, we can only reconcile these facts with (1.20) if investors have a risk aversion coefficient of 50!

This is the “equity premium puzzle.”


Equity premium puzzle(2) Aggregate consumption has about 0.2 correlation

with the market return, If you add this fact, you need risk aversion of 250 to explain the market Sharpe ratio in the face of 1% consumption volatility!

Clearly, either 1) people are a lot more risk averse than we might have thought 2) the stock returns of the last 50 years were largely good luck rather than an equilibrium compensation for risk, or 3) something is deeply wrong with the model, including the utility function and use of aggregate consumption data.


Random walk From (1.1), we have (1.21)

It says that prices should follow a martingale after adjusting for dividends and scaling by marginal utility.

Since consumption and risk aversion don’t change much day to day, we might expect the random walk view to hold pretty well on a day-to-day basis.

However, more recently, evidence has accumulated that long-horizon excess returns are quite predictable.

1 1 1( ) ( )(t t t t t tp u c E u c p d ）


Time-varying Expected returns Writing our basic equation for expected return as:

It says that returns can be somewhat predictable. risk and risk aversion change over the business cycle, and this is exactly the horizon at which we see predictable excess returns.

1 11

1

11 1 1

1

1 1 1 1

cov ( , )( )

( )

( )( ) ( , )

( )

( ) ( ) ( , )

f t t tt t t

t t

t tt t t t t

t t

t t t t t t t t

m RE R R

E m

mR m R

E m

c R m R

－

－


Present-value statement(1) Suppose an investor can purchase a stream {dt+j} at

price pt . His long-term objective is: The first order condition gives (1.23):

From pt=Et[mt+1(pt+1+dt+1)], we can get

When there is no speculative bubbles, it will get (1.23).

0

( )jt t jj

E u c

1 1

( )

( )t jj

t t t j t t j t jj jt

u cp E d E m d

u c

1

( ) ( )[ ]

( ) ( )limt jj T t Tt t t j t t T

Tj t t

u c u cp E d E p

u c u c


Risk adjustment

From (1.23) we have:

,1 1,

1, ,

cov ( , )

: ( )

t t jt t t j t t jf

j jt t j

ft t j t t t j

E dp d m

R

Where R E m


Continuous time Suppose an asset’s price is pt, and it pays a

dividend stream Dt. The Utility function is

Define discount factor in continuous time as: Then we can write the pricing equation as(1.28):

( )

0

0

({ }) ( ) ( ) ( )

. . : ,

: ( ) ( )

s t st t s t t t ss t s

t t t t s t s t s

st t t t s t ss

U c E e u c ds u c dt E e u c ds

s t c dt e dt p c ds e ds D ds

FOC p u c E e u c D ds

( )s

t s t se u c

0t t t t s t ssp E D ds


One-period pricing equation From (1.28) we have:

For small Δ we get(1.29):

(1.29)says that after adjusting for dividends, marginal utility-weighted price should follow a martingale.

0[ ]t t t t s t s t t ts

p E D ds E p

[ ] [ ]

0 [ ]

0,

0 [ ( )]

t t t t t t t t t t t t t t t t

t t t t t t t

t t t t t

p D E p D E p p p

D E p p

As

D dt E d p


One-period pricing equation

Using Ito lemma ,we have

The one period pricing equation become

( )d p pd dp dpd

0 ( )t

D d dp d dpdt E

p p p


Pricing risk-free asset

We can think of a risk-free asset as having price 1 and paying rf

t as a dividend. i.e. p=1,Dt=rft. Or as a

asset pays no dividend but whose price climbs deterministically at a rate:

dpt/pt=rftdt

Applying (1.29) we get (1.34):

It’s equivalent to Rft=1/Et(mt+1).

0 [ ]

( )

f tt t

t

f tt t

t

dr dt E

dr dt E


More intuitive version

Using (1.34), we can rearrange (1.33) as(1.35):

It’s analogue to E(R)=Rf-Rfcov(m,R).

( ) [ ]ft t t tt t t

t t t t

dp D d dpE dt r dt E

p p p


Relationship between asset return and consumption risk From we have:( )t

t te u c 2

2 2

2

2

2

( ) ( ) 0.5 ( )

( ) ( )0.5

( ) ( )

( ) ( ): ,

( ) ( )

0.5

t t tt t t t t t

t t t t t t t

t t t t t

t t t tt t

t t

t t tt t

t t t

d e u c dt e u c dc e u c dc

d c u c dc c u c dcdt

u c c u c c

c u c c u cDefine

u c u c

d dc dcdt

c c


Ignoring the orders higher than dt, we have:

(1.35)becomes (1.38)

( ) ( )t t t tt t t

t t t t

d dp dc dpE E

p c p

( ) [ ]ft t t tt t t t

t t t t

dp D dc dpE dt r dt E

p p c p

Relationship between asset return and consumption risk(2)


Sharp Ratio

(1.38)can be rewritten as

Using ρ≤1 and

We have:

2 2 2 2( / ), [( / ) ], [( / ) ]p t t t p t t t c t t tE dp p E dp p E dc c

( ) ft tt t t t c p

t t

dp DE dt r dt

p p

ftp t

tt c

p

Ddt r dt

p


Problem 1.8 Suppose the utility function includes leisure. Derive the

pricing function. It’s a multifactor model. ( )

0

0

({ , }) ( , ) ( , ) ( , )

. . : ,

: ( , ) ( , )

: ( , )

[

s t st t t s s t t t t s t ss t s

t t t t s t s t s

st c t t t c t s t s t ss

tc

tcc

U c l E e u c l ds u c l dt E e u c l ds

s t c dt e dt p c ds e ds D ds

FOC p u c l E e u c l D ds

Defining e u c l

d dt e u dc

2 2

2 2

0.5 0.5 ]

[ 0.5 0.5 ]

( ) [ ] [ ] [ ]

, ( )

cl ccc cll ccl

cc cl ccc cll ccl

c c c c c

ft t t t t cc t t t cl t tt t t t t

t t t t c t t c t t

i f tt

u dl u dc u dl u dcdl

u u u u uddt dc dl dc dl dcdl

u u u u u

dp D d dp c u dp dc l u dp dlE dt r dt E E E

p p p u p c u p l

cor E R R

, ,i c i lcc t clt t

c c

u l u

u u


5213:20


chp 1: consumption- based model and overview 郑振龙 厦门大学金融系 个人网站： ...

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chp 1: consumption- based model and overview 郑振龙厦门大学金融系个人网站： ...