The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
The stochastic discount factor and the CAPM
Pierre Chaigneau
[email protected]
November 8, 2011
Appendix
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
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Can we price all assets by appropriately discounting their
future cash flows?
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What determines the risk premium on any asset?
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What is the mean-variance frontier?
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How do we obtain the CAPM from first principles?
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Suppose all assets are expected to pay higher dividends in the
future. Should their prices rise?
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The consumer problem (1)
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Multifaceted problem: how much to save for next period
(intertemporal saving problem), and asset allocation.
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Standard problem in macroeconomics. The solution is given
by the Euler equation.
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The agent maximizes the discounted utility of this period (t)
and next period (t + 1) consumption (denoted by c). He is
endowed with et this period and et+1 the next (think about
labor income). Consider any asset with price pt and random
payoff x̃t+1 . Denote by α the quantity of this asset purchased.
We assume that utility is time-separable, with subjective
discount factor δ.
max u(ct ) + δEt [u(ct+1 )]
α
s.t.
ct = et − αpt
and
ct+1 = et+1 + αxt+1
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The consumer problem (2)
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Plug ct and ct+1 into the objective function and take the FOC
to get:
pt u 0 (ct ) = Et [δu 0 (ct+1 )xt+1 ]
(1)
pt = Et [δ
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u 0 (ct+1 )
xt+1 ]
u 0 (ct )
(2)
Interpretation of (1): equate marginal loss and marginal gain
at the margin, in utility terms and in present values.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The stochastic discount factor (asset prices)
The basic asset pricing formula
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We define the stochastic discount factor mt+1 as
mt+1 ≡ δ
u 0 (ct+1 )
u 0 (ct )
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It is the marginal rate of substitution, i.e., the rate at which
the investor is willing to substitute time t + 1 consumption for
time t consumption.
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We rewrite (2) as
pt = Et [mt+1 xt+1 ]
(3)
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The stochastic discount factor (asset prices)
Corporate Finance
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A firm takes some (real/physical) investment decisions
(capital budgeting), and raises money on the financial markets
to fund these projects.
What will be the equilibrium price (i.e., the maximum price
that investors will be willing to pay) of a claim on a random
sequence of future cash flows from a given set of projects?
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Market value of equity of a firm.
How much money can be raised by issuing equity.
It depends to what extent the cash flows generated by its
projects covary with aggregate consumption.
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It also depends on the risk preferences of investors.
Time-varying risk aversion? What is the optimal economic
context to fund a risky project? What about a riskfree project?
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The stochastic discount factor (returns)
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For any future payoff, define the realized gross return as
Rt+1 ≡ 1 + rt+1 ≡ xpt+1
.
t
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Plugging into the SDF formula in (3):
1 = Et [mt+1 Rt+1 ]
(4)
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The riskfree rate and the price of time
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Consider the future payoff equal to xt+1 with probability one
(no uncertainty). We know that the present value of this
payoff is
1
pt =
xt+1
(5)
1 + rf
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Applying (4) to the riskfree asset:
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The average value of m gives the value of time.
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The riskfree rate exists and can be calculated, based on
macroeconomic fundamentals (c), time preferences (δ) and
risk preferences (u 0 ), even if there is no riskfree asset available.
E [m] =
1
1+rf
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The price of risk (asset prices)
p = E [mx] = cov (m, x) + E [m]E [x]
p=
p=
(6)
E [x]
+ cov (m, x)
Rf
E [x]
cov (u 0 (ct+1 ), x)
+δ
Rf
u 0 (ct )
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The value of a payoff is the discounted expected value of the
future payoff plus an adjustment for risk. Assets whose future
payoffs positively covary with the SDF (i.e., with future
marginal utility) are more valuable, and conversely.
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Compare purchasing stocks to purchasing insurance.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The price of risk (expected excess return)
1 = E [mR] = cov (m, R) + E [m]E [R]
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Divide through by E [m], remember that Rf =
Rf =
1
E [m] :
cov (m, R)
+ E [R]
E [m]
E [R] = Rf −
E [R] − Rf = −
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(7)
cov (m, R)
E [m]
(8)
cov (u 0 (ct+1 ), R)
E [u 0 (ct+1 )]
An asset whose payoff covaries positively with consumption
(negatively with the marginal utility of consumption) will
demand a higher expected return, or equivalently a lower price.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
Idiosyncratic risk (1)
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The equations above show that if
E [x]
Rf
cov (m, x) = 0
then
p=
cov (m, R) = 0
then
E [R] = Rf
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If the payoff (or equivalently the return) of an asset is
uncorrelated with the SDF, then the risk premium is zero, its
expected return is equal to the riskfree rate, and the asset
price is equal to the expected payoff discounted at the riskfree
rate.
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Idiosyncratic risk does not matter in asset pricing.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
Idiosyncratic risk (2)
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Decompose any payoff x̃ into the part that is correlated with
the SDF, which we call the systematic component, and the
part that is not, which we call the idiosyncratic component:
x = proj(x|m) + E [mx]
m
E [m2 ]
E [mx] h E [mx] i
p(proj(x|m)) = p
m
=
E
m2 = E [mx] = p(x)
E [m2 ]
E [m2 ]
proj(x|m) ≡
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The asset’s beta
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Rewrite (8) as
E [Ri ] = Rf +
cov (m, Ri ) var [m] −
var [m]
E [m]
E [Ri ] = Rf + βi,m PR
(9)
(10)
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The price of risk: PR is the same for all assets (independent
of i).
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The quantity of risk in each asset: βi,m .
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This decomposition resembles the CAPM decomposition, but
it is different.1
1
In the CAPM, the price of risk is the equity premium (if the only risky
assets available are stocks), while β is defined as the value of the coefficient in
a univariate regression of asset i return on stock market returns.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The mean-variance frontier
Derived without the need to assume mean-variance preferences!
1 = E [mRi ] = E [m]E [Ri ] + ρ(m, Ri )σm σRi
σm
E [Ri ] = Rf − ρ(m, Ri )
σR
E [m] i
σm
|E [Ri ] − Rf | ≤
σR
E [m] i
(11)
(12)
(13)
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The set of possible mean and variance of returns is bounded.
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The boundary is called the mean-variance frontier.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The mean-variance frontier: idiosyncratic risk
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Assets which lie on the frontier are perfectly correlated with
the SDF, i.e., their idiosyncratic risk is nil.
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The (horizontal) distance to the frontier in the {σRi , E [Ri ]}
plane is a measure of idiosyncratic risk: intuitively, this part of
the standard deviation of returns does not generate a
compensation in the form of higher expected return.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The mean-variance frontier
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For any asset, equation (12) may be rewritten as
E [Ri ] = Rf −
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cov (m, Ri )
E [m]
Besides, given a point k on the mean-variance frontier which
is perfectly negatively correlated with the SDF
(ρ(m, Rk ) = −1) and satisfies σRk = σm . Then
1
E [Rk ] − Rf
=
2
E [m]
σm
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(14)
(15)
Plugging in (14):
E [Ri ] = Rf − βi (m, Ri ) E [Rk ] − Rf
E [Ri ] = Rf + βi (Rk , Ri ) E [Rk ] − Rf
(16)
(17)
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The CAPM
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The SDF involves consumption. Can we express the expected
returns on any asset as a function of the expected returns on
another asset or portfolio of assets?
This is the principle of the CAPM. It can be derived with
quadratic utility, or normally distributed asset returns.
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In any of these two cases, the expected utility associated with
any portfolio (a portfolio is a combination of assets) is fully
described by the mean and the variance of returns of this
portfolio.
In this case, any investor will select a portfolio among a set of
mean-variance efficient portfolios.
The set of mean-variance efficient portfolios is the set of
portfolios with the maximum expected return for a given
variance of return – or equivalently the set of portfolios with
the minimum variance of returns for a given expected return.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The CAPM
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If there are J assets, the optimal asset weights {ω1 , . . . ωJ } in
the investor’s portfolio are given by
J X
J
X
min
ωj ωi cov (rj , ri )
{ω1 ,...ωJ }
s.t.
J
X
j=1
j=1 i=1
ωj E [rj ] = E [r ]
and
J
X
ωj = 1
j=1
where rj is the return on asset j. Note that cov (rj , rj ) = var (rj ),
and E [r ] is the given required expected return on the portfolio.
I Denoting the Lagrange multipliers associated to the first and
second constraint by λ and µ respectively, the first-order
conditions to this optimization problem are
J
X
2
ωj cov (ri , rj ) − λE [ri ] − µ = 0
for i = 1, . . . , J
j=1
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The CAPM
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Denote by rm =
Then
PJ
cov (rm , ri ) = cov
j=1 ωj rj
J
X
the return on the market portfolio.
J
J
X
X
ωj rj , ri =
cov (ωj rj , ri ) =
ωj cov (rj , ri )
j=1
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j=1
Using this equation, we can rewrite the FOC as
2cov (rm , ri ) − λE [ri ] − µ = 0
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j=1
for i = 1, . . . , J
This equation should hold for the market portfolio:
2var [rm ] − λE [rm ] − µ = 0
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This equation should also hold for the riskfree asset:
−λrf − µ = 0
(18)
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
The CAPM
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Combining these two equations:
λ=
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2var [rm ]
E [rm ] − rf
and
µ=−
2var [rm ]
rf
E [rm ] − rf
Plugging these values of λ and µ in (18),
2cov (rm , ri ) −
2var [rm ]
2var [rm ]
E [ri ] +
rf = 0
E [rm ] − rf
E [rm ] − rf
Or
E [ri ] − rf =
cov (ri , rm ) E [rm ] − rf
var [rm ]
Appendix
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The CAPM
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We define
βi ≡
cov (ri , rm )
var [rm ]
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The beta of any asset measures its contribution to the
riskiness of the market portfolio. Because of our assumptions,
the riskiness is measured by the variance of returns.
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The expected excess return on any asset i is
E [ri ] − rf = βi E [rm ] − rf
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The right-hand-side of this equation is the risk premium.
With the CAPM, the risk premium is given by the quantity of
risk in asset i, as measured with βi , multiplied by the price of
risk, as measured by the expected excess return on the market
portfolio of risky assets.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
In practice: correlations in 2010-2011
Important caveat: correlations change over time!
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T-bonds are negative beta assets over this time period.
Source:
http://www.assetcorrelation.com/user/correlations/366
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The discount rate effect and the payoff effect
With log utility
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The wealth portfolio is a claim to all future consumption.
Intuitively, if one owns all (respectively 1%) of the assets of all
types in the economy, he has a claim to 100% (resp. 1%) of
all future consumption.
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The price of the wealth portfolio is
h u 0 (c )
i
t+1
ptM = Et δ 0
ct+1
u (ct )
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With log utility (u(c) = ln(c)):
ptM = δct
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The discount rate effect and the payoff effect
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Strangely, the portfolio price ptM does not depend on future
consumption.
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This is because two effects exactly offset each other with log
utility. On the one hand, higher future consumption raises ptM
(direct payoff effect). On the other hand, higher future
consumption diminishes the SDF, which decreases ptM
(indirect discounting effect).
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Importantly, this only holds for the wealth portfolio. This
would not be verified for individual assets: any news of higher
future payoff would have a negligible effect on the SDF, so
that the payoff effect would be dominant.
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This effect must be kept in mind when there are news that
the economy is becoming more productive (think about the
late 1990s). Should this raise asset prices? Not necessarily!
The SDF
The risk premium
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The mean-variance frontier
The CAPM
The discount rate effect
Acknowledgements: Some sources for this series of slides
include:
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The slides of Martin Boyer, for the same course at HEC
Montreal.
Asset Pricing, by John H. Cochrane.
Finance and the Economics of Uncertainty, by Gabrielle
Demange and Guy Laroque.
The Economics of Risk and Time, by Christian Gollier.
Appendix
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The CAPM
As derived with CRRA utility
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The SDF involves consumption. Can we express the expected
returns of any asset as a function of the expected returns of
another asset or portfolio of assets?
I
This is the principle of the CAPM. It can be derived with
quadratic utility or normally distributed returns.
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Here, we will derive it with CRRA utility, which is a commonly
used utility function in finance, and we won’t assume anything
on the distribution of returns. However, in discrete time, we
will need to work with approximations.
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The market portfolio includes all existing assets in the
economy (stocks, bonds, real estate, commodities, etc.).
Cochrane talks about a wealth portfolio which is a claim to all
future consumption.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
The CAPM
As derived with CRRA utility
CAPM with consumption
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M the return on the wealth portfolio.
Denote by Rt+1
Wt+1
M
For agents with infinite horizons, ct+1
ct = Wt = Rt+1 .
−γ
M )−γ .
It follows that m = δ ct+1
= δ(Rt+1
ct
Use a first order Taylor expansion to rewrite this term as
δ(RM )−γ ≈ δ(RM )−γ + δ(−γ)(RM )−γ−1 (RM − RM )
Appendix
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The CAPM
As derived with CRRA utility
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Equation (14) rewrites as
E [Ri ] − Rf = −
E [Ri ] − Rf = −
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cov (m, Ri )
E [m]
cov (−δγ(RM )−γ−1 (RM − RM ), Ri )
E [m]
For small time intervals, E [m] = Rf−1 ≈ 1 and δRM
E [Ri ] − Rf = γcov (RM , Ri ) = γ
−γ−1
≈1
cov (RM , Ri )
var (RM )
var (RM )
E [Ri ] − Rf = γβ(Ri , RM )var (RM )
(19)
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
The CAPM
As derived with CRRA utility
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Setting i = M in (19) gives:
γvar (RM ) = E [RM ] − Rf
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Substituting in (19):
E [Ri ] − Rf = βi E [RM ] − Rf
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Remember that we used approximations to reach this result.
Appendix
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The CAPM
As derived with CRRA utility and consumption instead of portfolio returns
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Use a first order Taylor expansion to rewrite the SDF as
m=δ
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c −γ t+1
ct−γ
≈δ
c
t+1
−γ
ct
− γδ
On a short time interval, E (m) =
−γ−1
δ ct+1
≈ 1. Finally,
ct
m ≈1−γ
−γ−1 c
ct+1 t+1
−
ct
ct
ct
c
t+1
1
Rf
≈ 1 and
ct+1 ct+1 −
ct
ct
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
The CAPM
As derived with CRRA utility and consumption instead of portfolio returns
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Equation (14) rewrites as
E [Ri ] − Rf = −
cov (m, Ri )
E [m]
ct+1 ct+1 −
, Ri )
ct
ct
c
t+1
E [Ri ] − Rf ≈ γcov
, Ri
ct
c cov ct+1
,
R
i
ct
var t+1
E [Ri ] − Rf ≈ γ
ct
var ct+1
E [Ri ] − Rf ≈ −cov (−γ
ct
E [Ri ] − Rf ≈ γβ(∆c, Ri )var
c
t+1
ct
Appendix
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
The CAPM
Some lessons, and some perspective
ct+1
ct
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γvar
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It involves risk aversion and the variability of aggregate
consumption: risk preferences and macroeconomic
fundamentals.
The intuition as to why risk can be measured by the variance
of consumption (or returns) in continuous time is because
preferences are locally (for small variations) mean-variance.
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is the price of risk in the economy.
Continuous time models often assume a Brownian motion
process, which is continuous: for a very small time interval,
stock price variations are “small”.
Individuals are assumed to make portfolio choice decisions for
an arbitrarily small time interval, and re-optimize continuously.
Jumps introduce a discontinuity in the stock price process. In
this case, even in continuous time and continuous portfolio
choice, preferences are not mean-variance.
The SDF
The risk premium
The mean-variance frontier
The CAPM
The discount rate effect
Appendix
More on the wealth portfolio
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With log utility, the price of the wealth portfolio with an
infinity of future periods (instead of one) is
ptM = Et
∞
hX
τ =1
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∞
δτ
hX
u 0 (ct+τ )
ct
δ
c
=
E
δτ
ct+τ = ct
t+τ
t
0
u (ct )
ct+τ
1−δ
τ =1
The return on the wealth portfolio is
δ
M +c
pt+1
t+1
1−δ + 1 ct+1
=
δ
ct
ptM
1−δ
0
1 ct+1
u (ct+1 ) −1
−1
=
= δ 0
= mt+1
δ ct
u (ct )
M
Rt+1
≡