Download Asset Pricing

Asset Pricing
Stochastic Discount Factor
Outline
1 Introduction
2 Traditional Linear Pricing Models
3 Empirical Asset Pricing 1
4 Uses of Pricing Models
5 Stochastic Discount Factor
6 Contingent Claims
7 Empirical asset pricing 2
46 / 171
Asset Pricing
Stochastic Discount Factor
Index
Hypothesis and derivation
asset prices: interpretation of prices and returns
link with the discounted cash flow model
linear pricing
link with CAPM
the mean-variance frontier
discounted expected value model and market efficiency
when return predictability does not imply inefficiency
47 / 171
Asset Pricing
Stochastic Discount Factor
SDF: strong points
general hypothesis about preferences and time horizon
it links pricing to some characteristic of the economy
it prices assets whatever the return distribution is
intertemporal model:
it explains intertemporal diversification (with only one
asset)
implications for the dynamics of prices and returns over
time.
48 / 171
Asset Pricing
Stochastic Discount Factor
The intertemporal problem of the representative
investor
two periods: savings and asset demand are equal to zero
at time t + 1.
0
the investor is non satiated (U > 0) and risk averse
00
(U < 0)
price-taking
short selling and borrowing are allowed
49 / 171
Asset Pricing
Stochastic Discount Factor
max Et [U(Ct , Ct+1 )] = U(Ct ) + δEt [U(Ct+1 )]
ω
(
Ct = et − Pt ω
Ct+1 = et+1 + xt+1 ω
(26)
where Ct consumption; et initial endowment; ω amount of
asset bought; xt+1 asset payoff; Pt asset price; Et conditional
expectation; δ subjective intertemporal discount factor, that
ranges from 0 to 1.
50 / 171
Asset Pricing
Stochastic Discount Factor
Ct and ω are the choice variables and depend on parameters
and realizations of random variables.
⇔ max U(et − Pt ω) + Et [δU(et+1 + xt+1 ω)]
ω
FOC:
0
0
Pt U (Ct ) = Et [δU (Ct+1 )xt+1 ]
(27)
The investor saves until the utility loss deriving from the
0
purchase of an additional unit of the asset, Pt U (Ct ), is equal
to the expected utility growth deriving from the additional
0
payoff at time t + 1, Et [δU (Ct+1 )xt+1 ].
51 / 171
Asset Pricing
Stochastic Discount Factor
Price as dependent variable
0
U (Ct+1 )
Pt = Et δ 0
xt+1
U (Ct )
(28)
Given the payoff distribution xt+1 and consumption choices Ct ,
Ct+1 , we can determine the price.
0
t+1 )
If we define the factor mt+1 = δ UU(C
, we can rewrite (28)
0
(Ct )
as follows:
Pt = Et [mt+1 xt+1 ]
(29)
This is the SDF pricing formula.
52 / 171
Asset Pricing
Stochastic Discount Factor
Traditional pricing formulas
Without uncertainty:
Pt =
1
xt+1
Rf
(30)
With uncertainty, we use discount factor that are not
stochastic, R1i , that account for asset i risk:
Pti =
1
i
]
Et [xt+1
i
R
with
Ri > Rf
(31)
(29) includes corrections for the risk of the different assets in
ONE COMMON discount factor, the same for every asset.
i
It is the correlation between mt+1 and xt+1
that
generates the appropriate correction for asset i.
53 / 171
Asset Pricing
Stochastic Discount Factor
Expected return as dependent variable
gross return= ratio between payoff and price.
We divide (29) by P:
P = E [mx] ⇒ 1 = E [mR]
(32)
In the case of a risk free asset we have:
1 = E [mR f ] ⇔ 1 = E [m]R f
R f increases with decreasing
0
U (Ct+1 )
,
U 0 (Ct )
i.e. with increasing
(Ct+1 )
:
(Ct )
it reflects that it is more expensive not to consume
today when future consumption is larger, in terms of utility.
Hence, savings should be remunerated more.
54 / 171
Asset Pricing
Stochastic Discount Factor
CCAPM and intertemporal diversification
An asset is worth more than its payoff discounted at
the risk free rate when it is a good hedge against future
consumption risk.
From (32), and as cov (m, x) = E [mx] − E [m]E [x]:
P = E [mx] = E [m]E [x] + cov (m, x) =
E [x]
+ cov (m, x)
Rf
0
E [x] cov [δU (Ct+1 ), xt+1 ]
P= f +
R
U 0 (Ct )
(33)
If the second term is positive, the asset ensures a negative
covariance between its payoff and future consumption.
55 / 171
Asset Pricing
Stochastic Discount Factor
Consumption is made more volatile by the asset whose
return covaries positively with consumption (negatively
with marginal future utility). Therefore, it has a return
higher than the risk free asset.
1 = [mR] ⇔ 1 = E [m]E [R] + cov (m, R)
⇒ E [R] =
1
cov (m, R)
−
E [m]
E [m]
(34)
Using the definition of m:
0
E [R] = R f −
cov [U (Ct+1 ), Rt+1 ]
E [U 0 (Ct+1 )]
(35)
56 / 171
Asset Pricing
Stochastic Discount Factor
Static portfolio choice and the CAPM
Static portfolio choice and the CAPM: the volatility of wealth
is reduced by diversification among assets.
CAPM: an asset risk premium is proportional to its amount of
systematic risk.
Intertemporal portfolio choice with one asset only and
CCAPM: the volatility of the marginal utility of consumption is
reduced by the intertemporal diversification.
CCAPM: an asset risk premium is proportional to its
contribution to consumption volatility.
57 / 171
Asset Pricing
Stochastic Discount Factor
Linearity and economic interpretation as in the
CAPM
From (34) E [R i ] = R f −
cov (m,R i )
E [m]
we have:
var (m)
cov (m, R i )
E [R ] = R + −
var (m)
E [m]
i
f
(36)
i.e.
i)
βi,m = − covvar(m,R
: R i regression coefficient over the
(m)
discount factor m (with negative sign);
(m)
λm = var
represents the price of risk.
E [m]
58 / 171
Asset Pricing
Stochastic Discount Factor
With an appropriate parametrization of the utility function and an approximation, we
can express:
beta as a function of the cov between return and consumption growth
the price of risk as a function of risk aversion and the consumption growth
volatility.
E [R i ] = R f + βi,∆c λ∆c
with
λ∆c = γvar (∆c)
59 / 171
Asset Pricing
Stochastic Discount Factor
Mean-variance frontier
We know from (34) that:
1 = E [m]E [R i ] + cov (m, R i ) = E [m]E [R i ] + ρm,R i σ(R i )σ(m)
⇒ E [R i ] = R f − ρm,R i
σ(m)
σ(R i )
E [m]
(37)
with −1 ≤ ρ ≤ 1. Equation 37 implies
|E [R i ] − R f | ≤
σ(m)
σ(R i )
E [m]
60 / 171
Asset Pricing
Stochastic Discount Factor
every expected excess return is included inside the frontier
the frontier is generated by: |ρm,R i | = 1. Returns on the upper part: perfectly
negative correlated with m, positively with c: maximum risk. Returns on the
lower part: perfectly negatively correlated with c: they are an insurance for
investors.
assets inside: have idiosyncratic risk, that is not correlated with consumption
and that is not remunerated by any excess return. Asset B is riskier than asset
A, but they have the same return.
61 / 171
Asset Pricing
Stochastic Discount Factor
Two−fund separation theorem holds!
All returns on the frontier are perfectly correlated with m,
hence they are perfectly correlated with each other.
By selecting the market portfolio (with return R m ) on the
frontier and the risk free asset (or two other efficient
portfolios), it is possibile to generate every return belonging to
the mean−variance frontier by combining the two:
R mv = R f + a(R m − R f )
(38)
62 / 171
Asset Pricing
Stochastic Discount Factor
Market efficiency and return time−series
EMH: excess returns are unpredictable if the market is using
available information at best.
SDF model: in an efficient market, from (35):
i
Et [Rt+1
− Rtf ] = −
i
cov (mt+1 , Rt+1
)
E [mt+1 ]
(39)
excess returns are unpredictable (constant) only if the second
term of the equation is equal to zero (constant). Predictability
− in an efficient market − depends on the dynamics of
h 0
i
δU (Ct+1 )
i
cov
,
R
i
0
t+1
cov (m, R )
U (C )
h 0t
i
=
t+1 )
E [m]
E U (C
0
U (Ct )
63 / 171
Asset Pricing
Stochastic Discount Factor
Source of asset return predictability that does not
contradict optimal exploitation of information:
predictable changes in the risk premium associated to
predictable changes in the asset exposure to risk, or
predictable changes in the aggregate risk, i.e. in the
market price of risk
Return predictability does not necessarily imply informational
inefficiency.
64 / 171
Asset Pricing
Stochastic Discount Factor
Implication of SDF on time-series of stock prices
EMH: stock prices
(adjusted by dividends) is a martingale:
Pt = Et [Pt+1 + Dt+1 ]
is equal to the flow of dividends, discounted by a constant
rate:
∞
X
Pt = Et
(δ j Dt+j )
(40)
j=1
65 / 171
Asset Pricing
Stochastic Discount Factor
Let us analyze the implication of SDF model:
In the case of a stock, [xt+1 ] = [Pt+1 + Dt+1 ]
Stock price is equal to:
Pt = Et [mt+1 (Pt+1 + Dt+1 )]
0
(41)
0
Only if U (Ct ) = U (Ct+1 ), then Pt = δEt [Pt+1 + Dt+1 ]
66 / 171
Asset Pricing
Stochastic Discount Factor
In order to obtain the price as a function of the future dividend flow only,
we can substitute in (41) future prices:
Pt = Et [mt+1 (Pt+1 + Dt+1 )]
Pt+1 = Et+1 [mt+2 (Pt+2 + Dt+2 )]
⇒ Pt = {Et mt+1 Et+1 [mt+2 (Pt+2 + Dt+2 )] + [mt+1 Dt+1 ]}
⇔ Pt = Et [mt+1 mt+2 Pt+2 + mt+1 Dt+1 + mt+1 mt+2 Dt+2 ] = Λ
Pt = Et
∞
X
mt+1,t+j Dt+j
(42)
j=1
where mt+1,t+j = mt+1 Λmt+j .
0
0
Only if U (Ct ) = U (Ct+1 ),
Pt = Et
∞
X
(δ j Dt+j )
(43)
j=1
67 / 171
Asset Pricing
Stochastic Discount Factor
Summary
SDF pricing accomodates any asset payoff, any well
behaved investor preferences
it nests DCF,CAPM-APT, CCAPM as special cases
it shows that return predictability need not imply market
inefficiency, contrary to the traditional EMH
68 / 171
Asset Pricing
Contingent Claims
Outline
1 Introduction
2 Traditional Linear Pricing Models
3 Empirical Asset Pricing 1
4 Uses of Pricing Models
5 Stochastic Discount Factor
6 Contingent Claims
7 Empirical asset pricing 2
69 / 171
Asset Pricing
Contingent Claims
Contingent Claims, Complete Markets and Risk
Sharing
1 Another pricing method (contingent claims/risk neutral probabilities) that can
be reconciled to the SDF.
2 Risk sharing among individuals
1 1. If we want to price an asset that is a perfect substitute, we apply the law of
one price.
APT: the same exposure to common risk factors, the same first and
second moment
HERE: perfect substitutes if they have the same payoff in the same
future states of nature.
2. If we want to price an asset that is not a perfect substitute?
APT: replicating its payoff with a portfolio of existing assets,
HERE: replicating its payoff with a portfolio of basic imaginary assets.
70 / 171
Asset Pricing
Contingent Claims
2
function of financial markets: sharing of risks among individuals
it explains the development of financial markets (Allen
and Gale, 1994)
by trading assets, individuals eliminate idiosyncratic
consumption fluctuations.
in a complete market, full insurance is provided
aggregate fluctuations, i.e. those that affect asset prices,
are undiversifiable and must ultimately be borne by
investors
71 / 171
Asset Pricing
Contingent Claims
Outline
1
The State Preference Approach
2
Contingent Claims
3
Pricing with contingent claims and SDF
Risk neutral probabilities
Equivalence between SDF and risk neutral probabilities.
Application Almeida and Philippon: how do bankruptcy
costs and optimal leverage decisions change if we use
risk neutral bankruptcy probability instead of the
historical one?
4
Pareto-efficient risk diversification.
Brief digression on market incompleteness, financial innovation and
pricing
72 / 171
Asset Pricing
Contingent Claims
State Preference Approach
States of nature describe situation that investors cannot control.
Relevant states could be
good, poor or null orange harvest
success or failure of a join−venture
end or continuation of a war
Two or more states cannot occur together.
Once a state occurs, future wealth/consumption in that state becomes
certain
this guarantees that the simple mathematics used under certainty
can be used also under uncertainty.
73 / 171
Asset Pricing
Contingent Claims
Individual problem rewritten:
u(ct ) + βE [u(ct+1 )] = u(ct ) + βπ1 u(ct+1 , 1) + ... + βπs u(ct+1 , s)
ct = et − pt ξ
(44)
ct+1,s = et+1,s + xt+1,s ξ, ∀s
probability of state s is πs , with 0≤ πs ≤ 1,
least one s.
P
πs = 1 and πs > 0 for at
74 / 171
Asset Pricing
Contingent Claims
Contingent Claims
Asset: set of “contingent incomes” (conditional payoff).
Matrix representation of the payoff of two assets (J = 2) in two states of
the world (S = 2) is:
x11 x12
XSxJ =
(45)
x21 x22
first column: payoff of asset 1 in states 1 and 2.
75 / 171
Asset Pricing
Contingent Claims
Pricing with Contingent Claims
An Arrow−Debreu (imaginary) asset (AD): has a payoff of 1 in one state
only and zero otherwise. Two AD assets in two states of the world can
be represented as follows:
1 0
X =
(46)
0 1
the first column indicates the payoff of AD asset 1 in states 1 and 2.
AD asset: can be easly combined to replicate other assets.
To replicate asset 1: buy x11 of the first AD and x21 of the second AD:
1
0
x11
x11
+ x21
=
(47)
0
1
x21
To replicate asset 2:
x12
1
0
+ x22
0
1
=
x12
x22
(48)
76 / 171
Asset Pricing
Contingent Claims
Let the prices of the two assets be p1 and p2 respectively, the prices of
AD assets pc1 and pc2 .
If there are no arbitrage opportunities:
(
p1 = x11 pc1 + x21 pc2
(49)
p2 = x12 pc1 + x22 pc2
Pricing of a third, “redundant”:
given the payoff and prices of the assets, we derive the (imaginary)
prices of AD assets - provided the payoff matrix X is invertible, i.e.
rank(X ) ≥ S
given the payoff of a third asset x13 , x23 , we use state prices, (pc),
to price it.
77 / 171
Asset Pricing
Contingent Claims
With S states of the world and J assets (J number of columns of matrix
X ):
0
P = X PC ,
with
(J × 1) = (J × S)(S × 1)
(50)
With J = S linearly independent payoffs, X is invertible. We invert the
(sub) matrix of dimension S × S:
PC = X
0
−1
P,
with
(S × 1) = (S × (S ∈ N))((S ∈ N) × 1)
(51)
The market is complete if rank(X ) ≥ S
An asset is said to be redundant if the market is already complete before
including it.
78 / 171
Asset Pricing
Contingent Claims
Pricing with Contingent Claims and SDF
P
Let us generalize (49): p(x) = s pc(s)x(s)
Meaning: the value of a menu is equal to the sum of its components.
Multiply and divide
by the probability of each state:
P
p(x) = s π(s) pc(s)
π(s) x(s).
If we call m the ratio between state prices and probabilities:
pc(s)
π(s)
P
we can rewrite the equation as p = s π(s)m(s)x(s) = E [mx].
Note: if the market is complete, factor m in p = E [mx] exists.
m is often called “state price density”.
In the SDF section: the state price density coincides with consumption
marginal utility ratio.
m(s) =
79 / 171
Asset Pricing
Contingent Claims
Risk neutral probabilities
Let us define
π ∗ (s) = R f m(s)π(s) = R f pc(s) where
R f = 1/
X
pc(s) = 1/E [m]
s
(52)
Note: π ∗ (s) are positive, smaller or equal to 1 (pc(s) ≥ 1/R f ), the price
of a certain unit of consumption, and they add to 1. It is therefore
straightforward to interpret π ∗ (s) as probabilities.
Hence, we can rewrite the equation for the “menu”:
X
1 X ∗
p(x) =
pc(s)x(s) = f
π (s)x(s)
R s
s
E ∗ [x(s)]
(53)
Rf
E ∗ indicates the use of “risk neutral probabilities”, rather than historical
ones, to evaluate the expected value,
⇒ p(x) =
80 / 171
Asset Pricing
Contingent Claims
Risk neutral probabilities: put more weight on those state with higher
marginal utility m. Risk aversion: is the same as paying more attention
to unfavorable states of the world, w.r.t. their real probability.
See Almeida on capital structure.
81 / 171
Asset Pricing
Contingent Claims
Equivalence between SDF and state price density
u(ct ) + δE [u(ct+1 )] = u(ct ) + δπ1 (ct+1 , 1) + ... + δπs u(ct+1 , s)
(∗)ct = et − pt ξ = wt
(54)
(∗∗)ct+1 = et+1,s + xt+1,s ξ = wt+1,s , ∀s
Multiply both sides of (∗∗) by pc: (∗∗)pcs ct+1,s = pcs wt+1,s
As thisP
result holds for P
every s, also the following equation holds:
(∗ ∗ ∗) s pcs ct+1,s = s pcs wt+1,s . Now, adding (∗) and (∗ ∗ ∗), we
obtain
constraint:
Pthe intertemporal budget
P
ct + s pcs ct+1,s = wt + s pcs wt+1 , s It shows that the individual can
move resources from today to any state of the world tomorrow - as AD
market is complete.
82 / 171
Asset Pricing
Contingent Claims
The solution of the problem stems from the maximization of the utility
w.r.t. {c, c(s)}, accounting for the intertemporal budget constraint:
X
max u(c) +
δπ(s)u[c(s)]
c,c(s)
s
subject − to
X
X
c+
pc(s)c(s) = w +
pc(s)w (s)
s
L = u(c) +
(55)
s
X
s
δπ(s)u[c(s)] − λ(c +
X
s
pc(s)c(s) − w
X
pc(s)w (s))
s
(56)
83 / 171
Asset Pricing
Contingent Claims
First order conditions:
 0

u (c) = λ


0


δπu (c(s)) = λpc(s)
0
u [c(s)]
pc(s) = δπ(s) u0 (c)



0


⇒ pc(s) = δ u [c(s)]
π(s)
u 0 (c)
(57)
SDF and state price density coincide in complete markets: if (57) does
not hold for some s, then it will be optimal to transfer resources between
today and s by using assets.
84 / 171
Asset Pricing
Contingent Claims
Moreover, for every pair of states the following must hold:
0
0
δπ(s2 )u [c(s2 )]
δπ(s1 )u [c(s1 )]
=
pc(s1 )
pc(s2 )
0
pc(s1 )
π(s1 )u [c(s1 )]
⇒
=
pc(s2 )
π(s2 )u 0 [c(s2 )]
(58)
0
⇔
m(s1 )
u [c(s1 )]
= 0
m(s2 )
u [c(s2 )]
Left−side: rate at which the investor can substitute consumption in state
2 with consumption in state 1, by buying and selling assets.
Right−side: rate at which the investor is willing to substitute.
If the two sides are not equal, the investor will trade assets and move
resources.
This clarifies the relation between SDF and state price density.
85 / 171
Asset Pricing
Contingent Claims
Risk sharing in a complete market
if the market is complete, individuals can eliminate the idiosyncratic
part of consumption risk by trading asset.
each obtains full insurance: the same consumption in every state of
the world, independently of individual bad/good luck
NO equal distribution of consumption among individuals
it explains why the only risk that matters in asset pricing is
systematic risk
Equation (57): marginal rate of substitution between c(s) and c for
individual i is equal to pc(s).
pc(s) is equal for individual j: hence, the marginal utility growth should
be equal for everyone.
86 / 171
Asset Pricing
Contingent Claims
pc(s) is equal for individual j: hence, the marginal utility growth should
be equal for everyone:
0
0
j
i
)
u (cs,t+1
u (cs,t+1
)
j
δ
=
δ
0
0
j
i
u (ct )
u (ct )
i
(59)
where i and j refers to two different investors that can have different
patience and utilities.
If they have the same utility function, consumption of investor i and j
should move in parallel between state s (at time t + 1) and time t:
j
i
cs,t+1
cs,t+1
=
cti
ctj
(60)
shocks to consumption: perfectly correlated among individuals. In a
complete market, all individuals share the risks.
87 / 171
Asset Pricing
Contingent Claims
Pareto efficiency
This allocation is Pareto efficient, i.e. a benevolent planner - who maximizes consumer
i expected utility without altering that of consumer j - would come to the same
allocation of resources. S/he maximizes:
max
X
δ it u(cti )
subject − to
X
δ jt u(ctj ) = k
cti +
(61)
ctj
= cta
X
X
L=
δ it u(cti ) + λ
δ jt u(ctj )
subject − to
cti + ctj = cta
X
X
L=
δ it u(cti ) + λ
δ jt u(cta − cti )
where c a indicates the total available resources.
88 / 171
Asset Pricing
Contingent Claims
The FOC of this problem is:
0
0
δ it u (cti ) = λδ jt u (ctj ),
hence the same risk shared as in the previous case, equation
(59).
This is the counterpart of the welfare theorem in the case of
uncertainty, proved by Arrow and Debreu.
Why don’t we observe full insurance in practice?
89 / 171
Asset Pricing
Contingent Claims
Are Markets Complete
Individuals cannot obtain full insurance against idiosyncratic
risks at fair prices
insurance with deductible
absence of insurance against longevity risk, or against the
obsolescence of one’s human capital.
Shiller (1993): some new assets can make these risks insurable.
However, it is difficult to insure every risk, as individuals’
incentives change with risk hedging.
Moral hazard and adverse selection do not allow for full
insurance.
90 / 171
Asset Pricing
Contingent Claims
Assume to the contrary that an individual has full insurance at fair prices, that
reflect the observed probability of a car theft
S/he does not close her/his car properly, as this produces effort that
reduces her/his utility: this is an example of “moral hazard”
the probability of a car theft increases above the observed average and
the insurance company goes bankrupt
keeping this in mind, insurance companies do not offer full insurance at
fair prices.
Jane smokes, Elsa does not. Jane is willing to buy an health insurance at fair
prices, whereas Elsa is not - this is an example of adverse selection. Then the
probability of being ill among the pool of insured people is higher than the
observed one and the company goes bankrupt. Clauses as bonus/malus or
deductible allow to reach an equilibrium (Rothschild and Stiglitz, QJE, 1976)
91 / 171
Asset Pricing
Contingent Claims
The same situations may happen in financial markets
the majority shareholder of a company has a loan with interest rate reflecting
the average risk of investment projects
S/he is willing to increase the risk, as stock prices will increase. This is a
case of moral hazard, that leads to credit rationing (Stiglitz and Weiss,
1981).
Assume a market maker is ready to buy at bid price that reflects the probability
of a profit decrease, and to sell at ask price that reflects the probability of a
profit increase, conditional on publicly available information.
An insider is willing to sell when knowing the profits will decrease. A trader that
is not informed is not willing to trade stocks.
Adverse selection of the counterparts make the market maker reduce the bid
price conditional to traders’ sell orders, as those orders are associated to a
positive probability of privileged negative information. This is one of the sources
of market illiquidity (Glosten and Milgrom, 1985).
92 / 171
Asset Pricing
Contingent Claims
A similar intuition holds also in auction markets: liquidity is reduced, i.e.
the price impact of orders increase (Kyle, Econometrica 85; Grossman
and Stiglitz, AER 80).
In every example above, information asymmetry between insured and
insurance underwriter, funded and funder, and among traders, exists and
is associated to market incompleteness and incomplete risk diversification.
The degree of asymmetry among traders depends on market
micro−structure, particularly on negotiation mechanism (O’ Hara, 95)
and on the abidance to insider trading and transparency regulation
(Daouk and Bhattarcharya, 2002).
Financial innovation is trying to complete markets, thus improving on risk
diversification.
93 / 171