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Stochastic discount factor

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from Wikipedia

The concept of the stochastic discount factor (SDF) is used in financial economics and mathematical finance. The name derives from the price of an asset being computable by "discounting" the future cash flow ${\tilde {x}}_{i}$ by the stochastic factor ${\tilde {m}}$ , and then taking the expectation.^[1] This definition is of fundamental importance in asset pricing.

If there are n assets with initial prices $p_{1},\ldots ,p_{n}$ at the beginning of a period and payoffs ${\tilde {x}}_{1},\ldots ,{\tilde {x}}_{n}$ at the end of the period (all xs are random (stochastic) variables), then SDF is any random variable ${\tilde {m}}$ satisfying

E({\tilde {m}}{\tilde {x}}_{i})=p_{i},{\text{for }}i=1,\ldots ,n.

The stochastic discount factor is sometimes referred to as the pricing kernel as, if the expectation $E({\tilde {m}}\,{\tilde {x}}_{i})$ is written as an integral, then ${\tilde {m}}$ can be interpreted as the kernel function in an integral transform.^[2] Other names sometimes used for the SDF are the "marginal rate of substitution" (the ratio of utility of states, when utility is separable and additive, though discounted by the risk-neutral rate), a "change of measure", "state-price deflator" or a "state-price density".^[2]

Properties

[edit]

The existence of an SDF is equivalent to the law of one price;^[1] similarly, the existence of a strictly positive SDF is equivalent to the absence of arbitrage opportunities (see Fundamental theorem of asset pricing). This being the case, then if $p_{i}$ is positive, by using ${\tilde {R}}_{i}={\tilde {x}}_{i}/p_{i}$ to denote the return, we can rewrite the definition as

E({\tilde {m}}{\tilde {R}}_{i})=1,\quad \forall i,

and this implies

E\left[{\tilde {m}}({\tilde {R}}_{i}-{\tilde {R}}_{j})\right]=0,\quad \forall i,j.

Also, if there is a portfolio made up of the assets, then the SDF satisfies

E({\tilde {m}}{\tilde {x}})=p,\quad E({\tilde {m}}{\tilde {R}})=1.

By a simple standard identity on covariances, we have

1=\operatorname {cov} ({\tilde {m}},{\tilde {R}})+E({\tilde {m}})E({\tilde {R}}).

Suppose there is a risk-free asset. Then ${\tilde {R}}=R_{f}$ implies $E({\tilde {m}})=1/R_{f}$ . Substituting this into the last expression and rearranging gives the following formula for the risk premium of any asset or portfolio with return ${\tilde {R}}$ :

E({\tilde {R}})-R_{f}=-R_{f}\operatorname {cov} ({\tilde {m}},{\tilde {R}}).

This shows that risk premiums are determined by covariances with any SDF.^[1]

References

[edit]

^ ^a ^b ^c Kerry E. Back (2010). Asset Pricing and Portfolio Choice Theory. Oxford University Press.
^ ^a ^b Cochrane, John H. (2001). Asset Pricing. Princeton University Press. p. 9.

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Stochastic discount factor

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from Grokipedia

The stochastic discount factor (SDF), also known as the pricing kernel, is a positive random variable in financial economics that adjusts future asset payoffs for both time value and risk under uncertainty, serving as the core mechanism in no-arbitrage asset pricing models.^[1]^[2] It is formally defined such that the price

p_t

of any asset at time

t

equals the conditional expectation of the product of the SDF

m_{t+1}

and the future payoff

x_{t+1}

, i.e.,

p_t = E_t[m_{t+1} x_{t+1}]

, where the expectation incorporates information available at time

t

.^[2] This framework unifies the pricing of diverse assets, from stocks and bonds to derivatives, by linking prices directly to economic fundamentals like consumption and investor preferences.^[1] In asset pricing theory, the SDF extends classical discounting by incorporating stochastic elements to account for state-contingent risk, ensuring that the law of one price holds and arbitrage opportunities are absent.^[1] It derives from the first-order conditions of investor optimization, representing the intertemporal marginal rate of substitution (IMRS), such as

m_{t+1} = \beta \frac{u'(c_{t+1})}{u'(c_t)}

in consumption-based models, where

\beta

is the subjective discount factor and

u'

is marginal utility.^[2] For power utility functions, this simplifies to

m_{t+1} = \beta (c_{t+1}/c_t)^{-\gamma}

, with

\gamma

denoting risk aversion, highlighting how consumption growth influences risk premia.^[2] The SDF's positivity is crucial, as it implies the existence of an equivalent martingale measure for risk-neutral valuation.^[1] Historically, the SDF concept emerged from no-arbitrage foundations in the 1970s, building on works by Ross (1976) and Harrison and Kreps (1979), and was integrated with risk preferences by Rubinstein (1976) and Lucas (1978).^[1] It provides a general representation that encompasses specific models like the Capital Asset Pricing Model (CAPM), where the SDF is linear in the market return, and multifactor models such as Fama-French.^[2] Empirically, the SDF framework facilitates testing via the Generalized Method of Moments (GMM), though challenges like the equity premium puzzle underscore the need for volatile yet mean-stable SDFs to match observed Sharpe ratios.^[2] Bounds such as the Hansen-Jagannathan inequality,

\sigma(m)/E(m) \geq |E(R^e)| / (1 + R_f)

, quantify the required SDF volatility to explain risk premia.^[2]

Definition and Interpretation

Formal Definition

In the single-period asset pricing model, consider an economy where assets deliver payoffs

X

at time

t+1

, contingent on the realization of states of the world, with prices

p

determined at time

t

based on information available up to

t

. The stochastic discount factor (SDF), denoted

m_{t+1}

, is defined as a positive random variable such that the price of any payoff

X_{t+1}

is given by

p_t = \mathbb{E}_t \left[ m_{t+1} X_{t+1} \right],

where

\mathbb{E}_t[\cdot]

denotes the conditional expectation under the physical probability measure

\mathbb{P}

given the information filtration at time

t

.^[3] This formulation establishes

m_{t+1}

as the state-price density, pricing state-contingent claims directly; for instance, the price of an Arrow-Debreu security that pays one unit in a specific state

s

t+1

and zero otherwise is

\pi(s) = m_{t+1}(s) \mathbb{P}(s)

, where

\mathbb{P}(s)

is the physical probability of state

s

.^[4] For an asset with price

p_t = 1

at time

t

and payoff

X_{t+1}

t+1

, the gross return is

R_{t+1} = X_{t+1}/p_t = X_{t+1}

, yielding the one-period asset pricing equation

\mathbb{E}_t \left[ m_{t+1} R_{t+1} \right] = 1.

This holds for any traded asset return

R_{t+1}

, reflecting the no-arbitrage condition in expectation form.^[3] For the risk-free asset, which delivers a payoff of

R^f_{t+1}

in every state (assuming price 1 at

t

), the equation simplifies to

\mathbb{E}_t \left[ m_{t+1} \right] = \frac{1}{R^f_{t+1}},

providing the pricing of the risk-free rate through the unconditional expectation of the SDF.^[4] The SDF

m_{t+1}

also serves as the building block for the risk-neutral measure

\mathbb{Q}

, defined via the Radon-Nikodym derivative

\frac{d\mathbb{Q}}{d\mathbb{P}} = \frac{m_{t+1}}{\mathbb{E}_t[m_{t+1}]}

on the

t+1

-period sigma-algebra.^[3] To derive the pricing under

\mathbb{Q}

, consider any gross return

R_{t+1}

with price 1 at

t

. Under

\mathbb{P}

\mathbb{E}_t[m_{t+1} R_{t+1}] = 1

. Substituting the change of measure,

\mathbb{E}_t^{\mathbb{Q}} \left[ R_{t+1} \right] = \mathbb{E}_t \left[ \frac{m_{t+1}}{\mathbb{E}_t[m_{t+1}]} R_{t+1} \right] = \frac{\mathbb{E}_t[m_{t+1} R_{t+1}]}{\mathbb{E}_t[m_{t+1}]} = \frac{1}{\mathbb{E}_t[m_{t+1}]} = R^f_{t+1},

implying that all gross returns have expectation equal to the gross risk-free rate under

\mathbb{Q}

, or equivalently, discounted asset prices (divided by the cumulative risk-free return) are martingales under this measure.^[4] This equivalence highlights the SDF's role in transforming expectations from the physical to the risk-neutral measure for pricing purposes.

Economic Interpretation

The stochastic discount factor (SDF), often denoted as

m

, serves as a pricing kernel that encapsulates both intertemporal discounting and risk adjustment in economic decision-making. It represents the product of a deterministic time discount factor

\beta

, which reflects investors' impatience or time preference for consumption today over tomorrow, and a stochastic marginal rate of substitution (MRS) that captures the variability in investor preferences across uncertain future states of the world.^[5] This decomposition highlights how the SDF adjusts asset prices not only for the passage of time but also for the economic risks inherent in different outcomes, providing a unified way to value contingent claims. In economic terms, the SDF plays a crucial role in determining state prices, where higher values of

m

in adverse states—such as those with low consumption growth or elevated marginal utility—lead to higher implied prices for assets that deliver payoffs precisely in those states. This mechanism embodies risk aversion, as investors demand greater compensation for bearing uncertainty in economically challenging scenarios, effectively making safe assets relatively more expensive and risky ones cheaper in expectation. The SDF thus facilitates intertemporal consumption smoothing by linking current pricing decisions to expected future utility trade-offs, motivating the conceptual form

m_{t+1} = \beta \frac{u'(c_{t+1})}{u'(c_t)}

, where

u

denotes the utility function and

c

consumption, illustrating how marginal utility ratios drive pricing under uncertainty.^[6] The concept of the SDF emerged as a unifying framework in asset pricing, with Ross (1976) formalizing its role in deriving linear pricing rules from no-arbitrage conditions via the Arbitrage Pricing Theory (APT) to address limitations in traditional models like the CAPM.^[7] Hansen and Richard (1987) further introduced the SDF in a dynamic setting, emphasizing its use in deducing testable restrictions from conditioning information, thereby providing a general tool to reconcile diverse asset pricing implications without relying on specific equilibrium assumptions.^[5]

Theoretical Foundations

No-Arbitrage Derivation

The no-arbitrage principle underpins the existence of the stochastic discount factor (SDF) in financial markets. The first fundamental theorem of asset pricing states that a market is free of arbitrage opportunities if and only if there exists at least one equivalent probability measure under which the discounted prices of traded assets are martingales.^[8] This measure, known as the risk-neutral or equivalent martingale measure

Q

, ensures that the price

p_t

of any traded asset at time

t

equals the expected value under

Q

of its discounted future payoff

X_{t+1}

, normalized by the risk-free return

R_f

p_t = \frac{1}{R_f} E^Q[X_{t+1}]

.^[8] In this framework, the SDF

m_{t+1}

emerges as a positive random variable that represents the Radon-Nikodym derivative linking the physical measure

P

Q

, scaled by the risk-free factor:

\frac{dQ}{dP} = m_{t+1} R_f

.^[1] To derive the SDF explicitly, consider a complete market with a finite discrete state space consisting of

S

possible states

s = 1, \dots, S

, each occurring with positive physical probability

p_s > 0

under measure

P

, where

\sum_{s=1}^S p_s = 1

. In such a setting, no-arbitrage implies the existence of unique state prices

\psi_s > 0

(also called Arrow-Debreu prices) for each state, such that the price

p

of any payoff

X

with realization

X_s

in state

s

is given by

p = \sum_{s=1}^S \psi_s X_s

.^[9] The SDF is then defined state-by-state as

m_s = \psi_s / p_s

, ensuring positivity since both

\psi_s

and

p_s

are positive.^[1] Substituting into the pricing equation yields the SDF representation:

p = \sum_{s=1}^S p_s m_s X_s = E[m X],

where the expectation is under the physical measure

P

. For the risk-free asset with payoff 1 and price

1/R_f

, this implies

1/R_f = E

, normalizing the SDF to reflect time value and risk adjustment.^[9] In incomplete markets, where not all contingent claims can be replicated by traded assets, the SDF is no longer unique. No-arbitrage still guarantees the existence of a positive linear pricing functional on the space of traded payoffs, but multiple SDFs may price these assets correctly, each corresponding to different extensions to non-traded payoffs.^[1] The set of valid SDFs forms a convex set consisting of all strictly positive random variables m satisfying

E[m X_j] = p_j

for all traded payoffs

X_j

with prices

p_j

.^[2] While the pricing functional is unique for payoffs within the span of traded assets (via the projection of any valid SDF onto that space), for non-traded payoffs outside this span, no-arbitrage imposes bounds on possible prices: the minimum and maximum values of

E[m X]

over all valid SDFs m.^[2]^[9] The SDF framework is equivalent to risk-neutral pricing, as the change-of-measure formula directly connects the two. Specifically, the pricing equation under the SDF becomes

E[m X] = E^Q \left[ \frac{X}{R_f} \right],

derived by substituting

m = \frac{1}{R_f} \frac{dQ}{dP}

into the left-hand side and using the definition of the expectation under

Q

. This equivalence holds in both complete and incomplete markets, with the SDF providing a unified representation that embeds the martingale property of discounted prices under

Q

.^[1]

Consumption-Based Formulation

In representative agent models of asset pricing, the stochastic discount factor emerges from the agent's optimization problem under expected utility maximization and consumption smoothing. The representative agent maximizes expected lifetime utility

\mathbb{E}_t \sum_{s=0}^\infty \beta^s u(c_{t+s})

, where

\beta \in (0,1)

is the subjective discount factor and

u(\cdot)

is a strictly increasing, strictly concave period utility function. The first-order condition for optimal consumption and portfolio choice yields the Euler equation: for any asset with price

p_t

and payoff

x_{t+1}

u'(c_t) p_t = \mathbb{E}_t [\beta u'(c_{t+1}) x_{t+1}]

. Rearranging gives the pricing relation

\mathbb{E}_t [m_{t+1} x_{t+1}] = p_t

, where the stochastic discount factor is

m_{t+1} = \beta \frac{u'(c_{t+1})}{u'(c_t)}

, representing the intertemporal marginal rate of substitution in consumption.^[10]^[11] Specific functional forms of the utility function determine the explicit expression for

m_{t+1}

. For constant relative risk aversion (CRRA) or power utility,

u(c) = \frac{c^{1-\gamma}}{1-\gamma}

with

\gamma > 0

, the marginal utility is

u'(c) = c^{-\gamma}

, so

m_{t+1} = \beta \left( \frac{c_{t+1}}{c_t} \right)^{-\gamma}

. This form links asset prices directly to consumption growth, with higher

\gamma

implying greater sensitivity to consumption risk. For constant absolute risk aversion (CARA) or exponential utility,

u(c) = -\frac{1}{\gamma} e^{-\gamma c}

with

\gamma > 0

, the marginal utility is

u'(c) = e^{-\gamma c}

, yielding

m_{t+1} = \beta e^{-\gamma (c_{t+1} - c_t)}

. This specification is less common in macroeconomic models due to its implications for absolute risk levels but useful in settings with additive separability. To separate risk aversion from the elasticity of intertemporal substitution, Epstein-Zin recursive preferences define utility via

V_t = \left[ (1-\beta) c_t^{\frac{1-\gamma}{1-1/\psi}} + \beta \mathbb{E}_t [V_{t+1}^{1-\gamma}]^{\frac{1-1/\psi}{1-\gamma}} \right]^{\frac{1-1/\psi}{1-\gamma}}

, where

\psi > 0

is the elasticity of substitution and

\gamma > 0

is risk aversion. The resulting SDF is

m_{t+1} = \beta \left( \frac{c_{t+1}}{c_t} \right)^{-\frac{1}{\psi}} \left( R_{w,t+1} \right)^{\theta - 1}

, with

\theta = \frac{1-\gamma}{1-1/\psi}

and

R_{w,t+1}

the return on the agent's wealth portfolio.^[2]^[12] These consumption-based formulations have significant implications for explaining asset return puzzles. In calibrating a Lucas exchange economy with power utility to U.S. data from 1889–1978, Mehra and Prescott found that matching the observed historical equity premium of approximately 6% requires a risk aversion parameter

\gamma

between 10 and 40. However, such high

\gamma

implies a low elasticity of intertemporal substitution, inconsistent with the observed volatility of consumption growth (standard deviation around 1–2% annually), highlighting the equity premium puzzle.^[13] In a general equilibrium setting with complete markets, such as an Arrow-Debreu economy, the SDF serves as the shadow price of consumption goods across states and time. It equals the representative agent's marginal rate of substitution between current and future consumption in each state, ensuring that asset prices clear the market by equating supply and demand for contingent claims.^[10]

Mathematical Properties

Positivity and Normalization

The stochastic discount factor

m

, also known as the pricing kernel, must satisfy the positivity condition

m > 0

almost surely to preclude arbitrage opportunities and ensure that state prices are positive across all possible states of the world. This requirement stems directly from the fundamental theorem of asset pricing, which equates the absence of arbitrage with the existence of a strictly positive linear pricing functional on the space of contingent claims. If

m

were negative with positive probability in some state, an investor could construct an arbitrage by short-selling assets with payoffs positively correlated with that state while investing in assets that pay off in other states, yielding a nonnegative payoff with a positive expected value and zero cost. Such positivity guarantees that no portfolio delivers a sure positive payoff without risk or cost, aligning the SDF with economically meaningful intertemporal marginal rates of substitution. A key normalization condition for the SDF is that its unconditional expectation equals the inverse of the gross risk-free return,

E = 1/R_f

, where

R_f

denotes the one-period gross risk-free rate. This arises from applying the general SDF pricing equation to the risk-free asset, which has a price of 1 and a certain payoff of

R_f

, yielding

1 = E[m R_f] = R_f E

. The normalization encapsulates the time value of money, where

1/R_f

represents the pure discount for time passage in a risk-free setting, while deviations in

m

across states adjust for risk aversion and economic uncertainty. This condition anchors the SDF to observable market data, facilitating derivations of the risk-free rate as

R_f = 1/E

and ensuring consistency in multi-asset pricing frameworks. The Hansen-Jagannathan bound imposes a fundamental restriction on the second moments of the SDF, stating that

\frac{\sigma(m)}{E} \geq \sup \frac{|E[R^e]|}{\sigma(R^e)},

where the supremum is taken over all excess returns

R^e

(returns in excess of the risk-free rate), and equality holds if and only if the excess return is the mean-variance efficient portfolio achieving the maximum Sharpe ratio.^[14] This bound highlights the minimal volatility required for the SDF to price observed asset returns accurately, linking asset risk premia directly to the variability of intertemporal marginal rates of substitution. It serves as a benchmark for evaluating asset pricing models, as any valid SDF must satisfy this volatility bound. In complete markets, where spanning assets replicate any contingent claim, the SDF is unique and fully determined by no-arbitrage conditions. In contrast, incomplete markets admit a continuum of valid SDFs, all of which price the traded assets correctly but differ in their implications for untraded risks. Among these, the minimal relative entropy SDF—minimized subject to pricing constraints—provides a canonical reference, as it corresponds to the least informative prior consistent with observed prices and has desirable statistical properties for estimation.

Martingale and Projection Properties

In asset pricing theory, the stochastic discount factor (SDF), denoted as

m_{t+1}

, plays a central role in establishing the martingale property under the risk-neutral measure

Q

. Specifically, under the equivalent martingale measure

Q

, the discounted prices of traded assets form martingales, meaning that the expected value of the discounted future price equals the current price.^[4] This property ensures no-arbitrage opportunities, as any deviation would allow riskless profits. The SDF serves as the density process, or Radon-Nikodym derivative

\frac{dQ}{dP}

, that transforms the physical measure

P

into the risk-neutral measure

Q

, adjusting for risk preferences and thereby linking observed prices to risk-adjusted expectations.^[4] In incomplete markets, where not all risks can be hedged using traded assets, the SDF is not unique, leading to a projection interpretation for pricing. The true SDF can be projected onto the span of payoffs from traded assets, minimizing the mean-squared pricing errors for those assets. This projection yields the unique pricing kernel consistent with observed asset prices, while the orthogonal component captures unhedgeable risks.^[15] The Hansen-Jagannathan distance measures the extent of this misspecification by quantifying the minimal distance between a candidate SDF proxy and the space of valid SDFs that price the assets correctly, providing a bound on model inadequacy.^[15] The covariance representation connects expected excess returns to the SDF, forming the basis for beta pricing models. For an excess return

R^e

, the expected excess return satisfies

E[R^e] = -R_f \frac{\mathrm{Cov}(m, R^e)}{E},

where

R_f

is the risk-free rate. This equation implies that expected returns compensate for the covariance between the asset return and the SDF, with negative covariance indicating assets that pay off when marginal utility (reflected in the SDF) is high, thus bearing systematic risk.^[16] In dynamic settings, the time-series properties of the SDF ensure intertemporal consistency. The conditional expectation satisfies

E_t[m_{t+1}] = 1/R_{f,t}

, derived from the pricing equation for the risk-free asset, which guarantees that the one-period risk-free payoff is correctly discounted under the conditional information available at time

t

. This normalization links the SDF across periods, maintaining the martingale structure in multi-period models.^[16]

Asset Pricing Applications

Single-Period Pricing Equations

In a single-period asset pricing model, the stochastic discount factor (SDF)

m_{t+1}

prices any asset with payoff

x_{t+1}

at time

t

according to the fundamental equation

p_t = E_t \left[ m_{t+1} x_{t+1} \right],

where

p_t

denotes the time-

t

price. This relation derives from no-arbitrage conditions and holds universally for traded claims. For a risk-free bond with payoff

x_{t+1} = 1

, the price is

p_t = 1 / (1 + r_{f,t})

, yielding

E_t [ m_{t+1} ] = 1 / (1 + r_{f,t})

. For stocks, the payoff is

x_{t+1} = d_{t+1} + p_{t+1}

, where

d_{t+1}

is the dividend and

p_{t+1}

the ex-dividend price. For options, such as a European call, the payoff is

x_{t+1} = \max(S_{t+1} - K, 0)

, with

S_{t+1}

the underlying price and

K

the strike. Defining gross returns as

R_{i,t+1} = x_{i,t+1} / p_{i,t}

, the pricing equation simplifies to

E_t [ m_{t+1} R_{i,t+1} ] = 1

for any asset

i

. Taking expectations and using the risk-free pricing gives

E_t [ R_{i,t+1} ] = (1 + r_{f,t}) \left( 1 - \frac{ \operatorname{Cov}_t ( m_{t+1}, R_{i,t+1} ) }{ E_t [ m_{t+1} ] } \right),

or equivalently, the risk premium is

E_t [ R_{i,t+1} ] - (1 + r_{f,t}) = - (1 + r_{f,t}) \operatorname{Cov}_t ( m_{t+1}, R_{i,t+1} )

. Assets covarying negatively with the SDF command positive risk premia, as low SDF states (high marginal utility) coincide with high payoffs.^[17] The capital asset pricing model (CAPM) arises as a special case when the SDF is linearly affine in the market return

R_{m,t+1}

, i.e.,

m_{t+1} = a - b R_{m,t+1}

, reflecting the market portfolio's mean-variance efficiency.^[17] Substituting into the risk-free pricing yields

a - b E_t [ R_{m,t+1} ] = 1 / (1 + r_{f,t})

.^[17] The general risk premium formula becomes

E_t [ R_{i,t+1} ] - (1 + r_{f,t}) = b (1 + r_{f,t}) \operatorname{Cov}_t ( R_{m,t+1}, R_{i,t+1} )

.^[17] Applying this to the market itself gives

E_t [ R_{m,t+1} ] - (1 + r_{f,t}) = b (1 + r_{f,t}) \operatorname{Var}_t ( R_{m,t+1} )

, so

b = [ E_t [ R_{m,t+1} ] - (1 + r_{f,t}) ] / [ (1 + r_{f,t}) \operatorname{Var}_t ( R_{m,t+1} ) ]

.^[17] Thus,

E_t [ R_{i,t+1} ] - (1 + r_{f,t}) = \beta_{i,t} [ E_t [ R_{m,t+1} ] - (1 + r_{f,t}) ]

, where

\beta_{i,t} = \operatorname{Cov}_t ( R_{m,t+1}, R_{i,t+1} ) / \operatorname{Var}_t ( R_{m,t+1} )

.^[17] The arbitrage pricing theory (APT) generalizes CAPM to multiple systematic factors by assuming a multifactor linear SDF

m_{t+1} = a - \mathbf{b}' \mathbf{f}_{t+1}

, where

\mathbf{f}_{t+1}

is a vector of factor returns or innovations.^[18] The risk premium is then

E_t [ R_{i,t+1} ] - (1 + r_{f,t}) = (1 + r_{f,t}) \mathbf{b}' \operatorname{Cov}_t ( \mathbf{f}_{t+1}, R_{i,t+1} )

.^[18] Defining the factor loadings

\mathbf{B}_{i,t} = \operatorname{Cov}_t ( \mathbf{f}_{t+1}, R_{i,t+1} ) \Sigma_{f,t}^{-1}

, where

\Sigma_{f,t} = \operatorname{Cov}_t ( \mathbf{f}_{t+1}, \mathbf{f}_{t+1} )

, and the factor risk premia

\boldsymbol{\lambda}_t = (1 + r_{f,t}) \Sigma_{f,t} \mathbf{b}

, yields

E_t [ R_{i,t+1} ] - (1 + r_{f,t}) = \mathbf{B}_{i,t}' \boldsymbol{\lambda}_t

.^[18] Pricing occurs through covariances with the factors, without requiring a complete market.^[18] These formulations imply key cross-sectional relations across assets. In CAPM, the security market line (SML) graphs expected returns linearly against betas, with steeper slopes indicating higher market risk premia.^[17] The mean-variance frontier consists of portfolios minimizing variance for given expected returns, equivalent to returns most correlated with the SDF's projection onto the space of marketed payoffs. In the multifactor case, the frontier generalizes to hyperplanes in factor-beta space, capturing diversified risks via SDF projections.

Multi-Period and Continuous-Time Extensions

In multi-period settings, the stochastic discount factor (SDF) framework extends to dynamic asset pricing through iterated conditional expectations, where the price at time

t

of a payoff

x_n

at future time

n

satisfies

p_t = E_t [m_{t,n} x_n]

, with

m_{t,n}

denoting the cumulative SDF from

t

n

.^[1] This cumulative SDF is constructed as the product of successive one-period SDFs,

m_{t,n} = \prod_{k=t}^{n-1} m_{t+k, t+k+1}

, enabling recursive valuation across horizons.^[1] Such formulations imply term structure effects, as the multi-period SDF incorporates compounding risk adjustments, leading to horizon-dependent pricing kernels that reflect long-run risk exposures in Markov environments.^[1] In continuous-time extensions, the SDF takes the form of an exponential martingale under diffusion models, expressed as

S_t = \exp\left( -\int_0^t r_s \, ds - \int_0^t \lambda_s \, dW_s - \frac{1}{2} \int_0^t \lambda_s^2 \, ds \right),

where

r_s

is the instantaneous risk-free rate,

\lambda_s

represents the market price of risk, and

W_s

is a Brownian motion.^[19] This structure arises from stochastic differential equations governing state variables, ensuring the SDF remains a local martingale and facilitates pricing via change of measure techniques in incomplete markets.^[19] The market price of risk

\lambda_s

captures the compensation for diffusion risks, linking intertemporal preferences to asset dynamics over infinitesimal intervals. Affine term structure models further specify the SDF in continuous time as

\exp(a_t + b_t' y_t)

, where

y_t

is a vector of state variables following an affine diffusion process, and

a_t

b_t

are deterministic functions.^[20] Bond prices under this SDF admit closed-form solutions of the form

\exp(A(t) + B(t)' y_t)

, with coefficients

A(t)

and

B(t)

solved via ordinary differential equations known as generalized Riccati equations, such as

\frac{d}{dt} \psi_Y(t, u) = R_Y(\psi_Y(t, u), e^{\beta_Z t w}),

with initial conditions tied to maturity

u

.^[20] Yields are then affine in the states,

y_t(u) = -\frac{A(t,u) + B(t,u)' y_t}{u}

, enabling tractable analysis of interest rate dynamics and risk premia in term structure modeling.^[20] The long-run risks model of Bansal and Yaron integrates predictability into the SDF by incorporating persistent shocks to expected consumption growth and time-varying volatility.^[21] The one-period SDF is given by

m_{t+1} = \theta \log \delta - \frac{\theta}{\psi} g_{t+1} + (\theta - 1) r_{a,t+1},

where

\theta = \frac{1 - \gamma}{1 - 1/\psi}

reflects Epstein-Zin preferences with risk aversion

\gamma

and intertemporal substitution

\psi

g_{t+1}

is consumption growth, and

r_{a,t+1}

is the return on the consumption claim.^[21] Consumption growth follows

g_{t+1} = \mu + x_t + \sigma_t \eta_{t+1}

, with

x_t

a persistent predictor (

\rho = 0.979

) and

\sigma_t^2

fluctuating volatility (

\nu_1 = 0.987

), allowing the multi-period SDF to price assets by amplifying compensation for long-horizon uncertainties.^[21] This setup explains equity premia and return volatility through correlated shocks to growth and uncertainty.^[21]

Empirical Analysis and Testing

Estimation Techniques

One prominent econometric approach to estimating the stochastic discount factor (SDF) involves the generalized method of moments (GMM), which exploits moment conditions derived from asset pricing equations. In this framework, parameters θ of a parameterized SDF m(θ) are estimated by minimizing the sample analogue of the population moment conditions E[g(θ)] = 0, where g(θ) incorporates the pricing restrictions E[m(θ) R_t - 1] = 0 for a set of gross asset returns R_t spanning the payoff space.^[22] This minimization is achieved via a quadratic form g_N(θ)' W g_N(θ), with weighting matrix W typically chosen as the inverse of the asymptotic covariance of g_N(θ) for efficiency; the resulting estimator is consistent and asymptotically normal under standard regularity conditions.^[22] Overidentification tests, such as the J-statistic J = n g_N(\hat{θ})' \hat{S}^{-1} g_N(\hat{θ}) \sim \chi^2(K - L), assess model specification, where n is the sample size, K the number of moment conditions, and L the number of parameters.^[22] Nonparametric methods offer flexibility by avoiding parametric assumptions on the SDF form, often relying on the cross-section of returns to infer m directly. Kernel estimation techniques approximate the SDF by smoothing the empirical distribution of returns to satisfy the pricing conditions E[m R] = 1, typically minimizing a distance measure like the conditional Hansen-Jagannathan bound while estimating the joint density of returns via kernel density estimators.^[23] For instance, local polynomial or Nadaraya-Watson kernels can recover the pricing kernel from high-dimensional return data, providing insights into its shape without imposing linearity.^[23] Complementing this, projection-based approaches infer the SDF as its minimum-variance projection onto the space spanned by the returns and the constant 1, equivalent to the return on the mean-variance efficient tangency portfolio scaled to satisfy normalization; this yields the unique admissible SDF that prices the observed assets with the lowest second moment.^[24] Cross-sectional regression methods, such as the Fama-MacBeth procedure, indirectly estimate the implied SDF by linking observed risk premia to factor exposures. In the first pass, time-series regressions estimate betas of test assets on common factors, followed by a second-pass cross-sectional regression of average excess returns on these betas to recover factor risk premia λ; the SDF is then inferred as m ≈ 1 - ∑ λ_j β_j, where the linear structure reflects the factor model's pricing implications.^[25] This two-step approach accounts for cross-sectional variation in expected returns and provides standard errors via time-series variability of the λ estimates, enabling tests of whether premia align with economic risk measures.^[25] Time-series approaches using vector autoregression (VAR) models capture the conditional nature of the SDF by incorporating predictability in returns and higher moments. A VAR system on observables like dividends, consumption, or wealth variables generates conditional expectations and variances, from which the conditional SDF is estimated—often as m_{t+1} = E_t[β u'(C_{t+1}) / u'(C_t)], with parameters fit via GMM on conditional moment conditions derived from the VAR innovations.^[26] For example, innovations from a VAR on consumption-wealth ratios can proxy conditional market prices of risk, allowing the SDF to vary with business-cycle predictors like the cay variable.^[26] This method highlights time-varying risk compensation but requires careful specification to avoid omitted variable bias in the conditioning information.^[27] Recent advances as of 2025 incorporate machine learning and alternative data sources to enhance SDF estimation. For instance, deep learning frameworks like NewsNet-SDF use pretrained language model embeddings from news articles combined with adversarial networks to estimate SDFs, achieving substantial improvements in pricing performance—such as a 471% better Sharpe ratio than the CAPM and a 74% reduction in pricing errors compared to the Fama-French five-factor model—on U.S. equity data from 1980 to 2022.^[28] Nonparametric methods leveraging delta-hedged option portfolios recover state-dependent SDF shapes, revealing heterogeneity across volatility regimes in S&P 500 options as of 2025.^[29] Production-based approaches derive SDFs from firm-level investment Euler equations, improving out-of-sample pricing for equity portfolios.^[30]

Model Evaluation and Bounds

Model evaluation in stochastic discount factor (SDF) frameworks primarily assesses whether a candidate SDF correctly prices a set of test assets, meaning it satisfies the condition

E[m R] = 1

for gross returns

R

on those assets. This is typically tested using the generalized method of moments (GMM), where the overidentifying restrictions implied by the pricing equation yield a chi-squared statistic under the null hypothesis of correct specification; rejection indicates pricing errors and model misspecification.^[31] To quantify the extent of misspecification, the Hansen-Jagannathan distance measures the minimal pricing error as the distance between the mean-variance frontier of admissible SDFs and the projection of the candidate SDF onto that frontier, providing a metric for how far the model deviates from exact pricing. A key tool for evaluating SDF models without full specification is the Hansen-Jagannathan (HJ) bound, which imposes a lower limit on the volatility of any valid SDF capable of pricing observed asset returns. Specifically, the bound requires that

\frac{\sigma(m)}{E} \geq \max \frac{E[R^e]}{\sigma(R^e)},

where

\sigma(m)

is the standard deviation of the SDF

m

E

is its unconditional mean (often normalized near 1), and the right-hand side is the maximum Sharpe ratio across excess returns

R^e

; this tests whether a model's implied SDF volatility is sufficient to rationalize empirically observed Sharpe ratios, such as those from equity and bond markets.^[32] Extensions of the HJ bound incorporate joint distributional features of returns and the SDF, such as conditioning on information sets or higher moments, to derive tighter restrictions on feasible SDFs that match both means and covariances of asset payoffs. To address potential misspecification robustly, generalized HJ bounds employ moment inequalities rather than equalities, allowing for pricing errors within a tolerance while bounding the SDF's admissible region; this approach, developed in the context of approximate pricing, evaluates models by checking if their SDF lies within inequality-constrained mean-variance sets derived from asset data. These bounds are particularly useful in GMM-based tests of consumption-based models, where exact pricing often fails. Empirical applications of these evaluation techniques have highlighted significant challenges for consumption-based SDF models, such as the consumption capital asset pricing model (CCAPM). In CCAPM, the SDF derived from consumption growth exhibits volatility far below the HJ lower bound required to explain historical equity Sharpe ratios around 0.4-0.5, implying that consumption risk alone cannot account for the observed equity premium of approximately 6% annually over the risk-free rate.^[33]^[32] This discrepancy, known as the equity premium puzzle, is compounded by the risk-free rate puzzle, where the model predicts a risk-free rate of 7-10% but data show only about 1%, violating HJ volatility bounds even after adjusting for reasonable risk aversion and time preference parameters.^[34]^[35]

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Stochastic discount factor

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Definition and Interpretation

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Consumption-Based Formulation

Mathematical Properties

Positivity and Normalization

Martingale and Projection Properties

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Estimation Techniques

Model Evaluation and Bounds

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