1
Asset Prices: overview
• Euler equation
• C-CAPM
• equity premium puzzle and risk free rate puzzles
• Law of One Price / No Arbitrage
• Hansen-Jagannathan bounds
• resolutions of equity premium puzzle
2
Euler equation
• agent problem
max
∞ X
X
j=0
st
¡ ¡ ¢¢ ¡ ¢
β t u ct st Pr st
¡ ¢
¡ ¢
¡ ¢
¡ ¢
ct st + qta st · at+1 st ≤ Wt st
¡
¢
¡
¢ ¡
¡
¢
¡
¢¢
¡ ¢
Wt+1 st+1 = yt+1 st+1 + qta+1 st+1 + dt+1 st+1 at+1 st
• comment: at and qta are vectors of length equal to the number of assets
• Euler equation
¡
£
¢¤
u0 (ct ) qtai = βEt u0 (ct+1 ) qtai+1 + dit+1
£
¤
u0 (ct ) = βEt u0 (ct+1 ) Rti+1
∙ 0
¸
u (ct+1 ) i
1 = Et β 0
R
u (ct ) t+1
• transversality condition
(1)
(2)
£
¤
a
at+j = 0
lim β j E0 u0 (ct+j ) qt+j
j→∞
• pricing formula
repeated substitution of (1)
qta
=
∞
X
j
β Et
j=1
• no bubbles
1
∙
¸
u0 (ct+j )
dt+j
u0 (ct )
(3)
— transversality and st = 1
• complete markets consistency check
review A-D price with complete markets
t
qt+j
¢
¡
¡ j t¢
¡ t j¢
u0 cit+1 (st , sj )
Pr
s |s
s ,s = β
i
u0 (ct (st ))
→ (3)
3
CCAPM (Consumption Capital Asset Pricing Model)
• make (2) and (3) operational:
CCAPM ≡ use aggregate consumption in above equations
• justifications:
— equilibrium of representative agent economy (Lucas / Breeden)
— equilibrium with complete markets (Constantinides)
complete markets ⇐⇒ Pareto Optima ⇐⇒ representative consumer (weighted utility)
• back to Euler equation
∙ 0
¸
u (ct+1 ) i
1 = Et β 0
R
u (ct ) t+1
• Absence of arbitrage implies that there exists some mt+1 such that
£
¤
i
1 = Et mt+1 Rt+1
THE empirically testable condition (again)
• intuitive decomposition
1 = βEt
µ
µ 0
¶
¶
¡ i ¢
u0 (ct+1 )
u (ct+1 ) i
Et Rt+1 + βcovt
, Rt+1
u0 (ct )
u0 (ct )
→its the covariance that matters!
4
Equity Premium Puzzle
• Euler equations with data on Rstock market and Rbonds
2
• simple log-normal calculation
• preferences and consumption
u0 (c) = c−γ
½
¾
ct+1
1 2
= c̄∆ exp εc − σ c
ct
2
¡
¢
2
εc v N μc , σ c
¶
µ
ct+1
⇒ E
= μc
ct
• returns
R
i
εi
• Euler
"
1 = βE
Ri
½
¾
¡
¢
1 2
i
= 1 + r̄ exp εi − σ i
2
¡
¢
2
v N μc , σ c
¡ ¢
⇒ E Ri = Ri = 1 + r̄i
µ
ct+1
ct
¶−γ #
¶
µ
¡
¢
1 2
1 2
−γ
i
1 = β 1 + r̄ (c̄∆ ) Et exp εi − σ i − γεc + γ σ c
2
2
¶
µ
¢
¡
1 2
−γ
i
1 = β 1 + r̄ (c̄∆ ) Et exp (1 + γ) γ σ c − γσ ic
2
• taking logs...
• stocks and bonds:
¡
¢
1
log 1 + r̄i = − log β + γ log c̄∆ − (1 + γ) γ σ 2c + γσ ic
2
¡
¢
1
r̄f ≈ log 1 + r̄f = − log β + γ log c̄∆ − (1 + γ) γ σ 2c
2
1
r̄s ≈ log (1 + r̄s ) = − log β + γ log c̄∆ − (1 + γ) γ σ 2c + γσ sc
2
• premium:
¡
¢
r̄s − r̄f ≈ log (1 + r̄s ) − log 1 + r̄f = γσ sc
3
(4)
(5)
(6)
Table removed due to copyright restrictions.
Kocherlakota, Narayana R. "The Equity Premium Puzzle: It's Still a Puzzle."
Journal of Economic Literature 34, no. 1 (1996): 47 (Table 1).
• US data (from Mehra and Prescott):
r̄s = 7%
r̄f = 1%
σ rc = .219%
• Kocherlakota
• need γ = 27 to match (6)
equity premium puzzle
• to match (4) we need γ very high or very low risk free rate puzzle
4
Tables removed due to copyright restrictions.
Kocherlakota, Narayana R. "The Equity Premium Puzzle: It's Still a Puzzle."
Journal of Economic Literature 34, no. 1 (1996): 42-71. (Tables 2 and 3).
5
Discount Factors: LOP and NA
I follow Cochrane and Hansen (1992) closely — great paper to read
• two periods "now" and "then" (t and t + 1 if you prefer)
• J “fundamental” assets:
— xj payoff “then”
— q j “now” price
→ stack into x and q (column) vectors
• payoff space for "then"
P ≡ {p : p = c · x for some c ∈ R}
• pricing function π (p) : P → R
then π (x) = q
5
• definition: Law of One Price (LOP) holds if the pricing function is linear
π (c · x) = c · π (x) = c · q
⇒ c· x = c0 · x then c · q = c0 · q
1
• definition: discount factor y ∈ P
π (p) = E (yp)
• Riesz representation Theorem
LOP ⇔ ∃ (stochastic) discount factor y ∈ P
• Let Y be the set of all discount factors
• note: y may be negative
• example:
−1
y ∗ = x0 (Exx0 )
q
note: if Exx0 is non-singular then remove assets from x until it is!
a non-singular Exx0 means that (a) there is a risk-free asset (b) there are two ways of getting
the same payoff
• Definition: No Arbitrage (NA) holds
p ≥ 0 ⇒ π (p) ≥ 0
p > 0 (with positive prob.) ⇒ π (p) > 0
• result NA ⇔ ∃ strictly positive discount factor y > 0
Let Y ++ be the set of all discount factors that are positive
• examples
m=
1
β t u0 (cthen )
u0 (cnow )
proof:
π (c · x) = π (c0 · x)
cπ (x) = c0 π (x)
cq = c0 q
6
6
Hansen-Jagannathan bounds
• all theories:
q = E (mp)
m = f (data, parameters)
(see Cochrane’s book)
• note pi /q i is rate of return
• H-J bounds:
diagnostic tool for models of m
• special case:
data on a single excess return relative to some baseline asset
r = p/q − p0 /q 0
then π (r) = 0 so that
0 = Emr = EmEr + cov (m, r)
= EmEr + σ m σ r corr (m, r)
−1 ≤
EmEr
= corr (m, r) ≤ 1
σmσr
¯
¯
¯ EmEr
¯
¯
¯
¯
σ m σ r ¯
≤ 1
σr
σm
≤
|Er|
Em
intuition: need volatile σ m
• note: Em = 1/Rf if there is a risk free rate Rf
• lets generalize:
for any random vector x we can consider the population regression:
m = a + x0 b + e
which just defines e uniquely as having Ee = 0 and cov(x, e) = 0
• by definition cov (e, x) = 0
⇒ var (m) ≥ var (x0 b)
7
• idea compute x0 b and var (x0 b) to get lower bound
→ check whether theories for y pass this test
b = [cov (x, x)]−1 cov (x, y)
a = Ey − Ex0 b
• How to compute b?
idea: if x = p then theory helps...
• assume x = p note that
cov (x, y) = q − E (y) E (x)
so:
b = [cov (x, x)]−1 [q − E (y) E (x)]
¢
¡
var x0 [cov (x, x)]−1 [q − E (y) E (x)] = var (x) [var (x)]−2 E (y)2 E (x)2
• if we knew E (y) we have a lower bound
otherwise ⇒ feasible region for pair (E (y) , var (y))
• convenient
— no need to recompute lower bound for each theory
— helps see where the theory fails
• 3 cases:
— risk-less return
→ E (y) pinned down and risky return
— one excess-return q = 0 Sharpe ratio and market price of risk (what we did before!)
— general case→very flexible, see CH paper
• figures 2.1: excess
• 2.2, 2.3, 2.4 from CH paper
8
7
Resolutions (?)
7.1
Exotic Preferences
• Risk Aversion vs. IES
(Weil / Epstein-Zin)
• first-order risk aversion
(Epstein-Zin)
• habit persistence e.g. u(ct − αct−t )
(Abel / Campbell-Cochrane)
• loss-aversion
7.2
Heterogenous Agent Incomplete Markets
• uninsured idiosyncratic risk
(Mankiw / Constantinides-Duffie)
• borrowing constraints (Euler with inequality)
(Luttmer / Heaton-Lucas)
• constrained optima with limited commitment
(Alvarez-Jermann)
7.3
Knightian Uncertainty
• risk vs. uncertainty
• fear of not understanding returns / uncertainty over probability distribution / desire for robust
decisions (Hansen and Sargent)
7.4
No risk premium!
• no risk premium to explain...
• historical returns on stocks were unexpected (McGratten-Prescott)
• bonds are money → low return
• stocks more risky than sample (low probability of a crash)
(see Reitz, Cochrane, Weitzman and Barro)
9
8
Conclusions
Risk premium puzzle
• great example of interplay between theory and data
• no strong consensus on resolution yet
many new ideas
• new models should explore
• revisit the welfare costs of BCs
(Alvarez and Jermann)
10