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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (1)

LECTURE 06: SHARPE RATIO, BONDS, &THE EQUITY PREMIUM PUZZLE

Markus K. Brunnermeier


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Money, Bonds vs. Stocks

$0

$5

$10$15

$20

$25

$30

$35

$40

$45

$50

1 9 7 2

1 9 7 5

1 9 7 8

1 9 8 1

1 9 8 4

1 9 8 7

1 9 9 0

1 9 9 3

1 9 9 6

1 9 9 9

2 0 0 2

2 0 0 5

2 0 0 8

2 0 1 1

SP500

US 1y Tsy

US 10Y Tsy


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Sharpe Ratios and Bounds Consider a one period security available at date

with payoff + . We have = + +

or = + + cov + , +

For a given + we let + =

Note that + will depend on the choice of + unless there exists ariskless portfolio + is the return from to 1 , typically measurable w.r.t. + . (An

exception is , which is measurable w.r.t. , but we stick withsubscript 1 .)


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Sharpe Ratios and Bounds (ctd.)

Hence

=

+

Expected PV

cov + , +Risk Adjustment

Positive correlation with the discount factor addsvalue, i.e. decreases required return


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in Returns

+ + = Divide both sides by and note that =

+

+ + = 1 Using + = 1/ + , we obtain

+ + + = 0 -discounted expected excess return for all assets

is zero.


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in Returns

Since + + + = 0

cov + , + += + + +

That is, risk premium or expected excess return

+ + = cov + , +

+ is determined by its covariance with thestochastic discount factor


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Sharpe Ratio Multiply both sides with portfolio

+ + = cov + , +

+

+ + = + , + + +

+

NB: All results also hold for unconditional expectations

Rewritten in terms of Sharpe Ratio = ... + +

+ , + = + +

+


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Hansen-Jagannathan Bound

Since 1,1 we have

+ + sup

+ + +

Theorem (Hansen-Jagannathan Bound):

The ratio of the standard deviation of astochastic discount factor to its mean exceedsthe Sharpe Ratio attained by any portfolio.


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Theorem (Hansen-Jagannathan Bound): The ratio of the standard deviation of a stochastic

discount factor to its mean exceeds the SharpeRatio attained by any portfolio. Can be used to easy check the viability of a

proposed discount factor Given a discount factor, this inequality bounds

the available risk-return possibilities

The result also holds conditional on date t info


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expectedreturn

Rf

s

available portfolios

slope s (m) / E[m]



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Assuming Expected Utility

, , = , , U(c0,c1)

=

, ,

, ,

, ,

= , ,

,, ,

, ,

,

Stochastic discount factor

=MRS

=


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Time-Separable Digression: if utility is in addition time-separable

, = Then

=

, ,

= ,

,, ,

,

,

And

=1 ,

=

,


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A simple example

= 2, = 3 securities with = 1,0 , = 1,0 , = 1,1

Let = , 1 , = = 1

Hence, = , = = = and

= 4,0 , = 0,2 , = ,

= 2, = 1, =


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Example: Where does SDF come from?

Representative agent with Endowment: 1 in date 0, (2,1) in date 1

Utility , , = ln ln , i.e. , , = ln ln , (additive) time separable u-

function

= , ,

, , =

,,

,=

, =

, 1

since endowment=consumption Low consumption states are high m -states Risk-neutral probabilities combine true probabilities and

marginal utilities.


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Equity Premium Puzzle

Recall = cov ,

Now: = cov ,

Recall Hansen-Jaganathan bound

; =

1

1


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Equity Premium or

Low Risk-free Rate Puzzle? Suppose we allow for sufficiently high risk

aversion s.t.

New problem emerges:

Strong force to consumption smooth over time(low intertemporal elasticity of consumption (IES))

due to concavity of utility function vNM utility function

smoothing over states = smoothing over time CRRA gamma = 1/ IES

Model predicts much higher risk-free rate


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Equity Premium or

Low Risk-free Rate Puzzle? Solution:

depart from vNM utility preference representation Found preference representation that allows split

Risk-aversion Intertemporal elasticity of substitution

Kreps-Porteus (special case: vNM) Epstein-Zin (special case: CRRA vNM)


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