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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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (2)
Money, Bonds vs. Stocks
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SP500
US 1y Tsy
US 10Y Tsy
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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (3)
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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (4)
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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (5)
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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (6)
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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (7)
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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (8)
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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (9)
Hansen-Jagannathan Bound
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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (10)
expectedreturn
Rf
s
available portfolios
slope s (m) / E[m]
Hansen-Jagannathan Bound
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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (11)
Assuming Expected Utility
, , = , , U(c0,c1)
=
, ,
, ,
, ,
= , ,
,, ,
, ,
,
Stochastic discount factor
=MRS
=
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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (12)
Time-Separable Digression: if utility is in addition time-separable
, = Then
=
, ,
= ,
,, ,
,
,
And
=1 ,
=
,
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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (13)
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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (14)
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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (15)
Equity Premium Puzzle
Recall = cov ,
Now: = cov ,
Recall Hansen-Jaganathan bound
; =
1
1
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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (17)
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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FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (18)
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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