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Deriving the Stochastic Discount Factor from Investor First-Order Conditions

Article Quant Q&A · Author: Absbert
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Summary

The document derives the stochastic discount factor (SDF) from an investor’s optimization problem. The investor chooses current consumption and holdings in risky assets, subject to a budget constraint today and a random consumption outcome in the future. Substituting future consumption into expected utility gives first-order conditions for current consumption and each asset position.

Dividing an asset’s marginal-utility condition by the current-consumption condition yields its price as the expected product of the SDF and its payoff. The SDF is therefore the ratio of expected marginal utility for future consumption to marginal utility for current consumption, as specified in the answer. This gives the pricing relation for the assets considered in the exercise. The explanation is a basic derivation rather than an empirical study; it assumes an optimizing investor and does not discuss existence conditions, market completeness, or how to estimate the SDF from data.

Key ideas

  • Investor first-order conditions connect marginal utility to asset prices.
  • The SDF is formed as a ratio of future and current marginal utility terms.
  • An asset price equals the expected SDF-weighted payoff under the stated setup.
  • The derivation relies on the investor’s optimization problem and its assumptions.

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Derivatives pricingStatistics
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# How to find the expression for the SDF and solve this exercise?


# How to find the expression for the SDF and solve this exercise?












I'm struggling to solve point a and b of this exercise, while in point c I got a very close result to the reciprocal of the relative risk aversion. If you can help me and explain how to do it, it would be really appreciated

## Answer by phdstudent (score 2, accepted)

https://quant.stackexchange.com/a/76972

Let me write this problem without the vector notation since it is easier:

$$ \max E[v(c_0,c_1)] $$ s.t. $$ \text { subject to } c_0+\sum_{i=1}^n \theta_i p_i=w_0 \text { and }\tilde{c}_1=Y+\sum_{i=1}^n \theta_i \tilde{x}_i \text {. }$$

Substituting in the second constraint, the Lagrangean for this problem is: $$ E\left[v\left(c_0, Y+\sum_{i=1}^n \theta_i \tilde{x}_i\right)\right]-\gamma\left(c_0+\sum_{i=1}^n \theta_i p_i-w_0\right),$$

and the first-order conditions are: $$ E\left[\frac{\partial}{\partial c_0} v\left(c_0, C_1\right)\right]=\gamma $$ $$ ( \forall i) E\left[\frac{\partial}{\partial c_1} v\left(c_0, C_1\right) \tilde{x}_i\right]=\gamma p_i.$$

Thus: $$ ( \forall i) E\left[\frac{\partial}{\partial C_1} v\left(c_0, C_1\right) \tilde{x}_i\right]=p_i E\left[\frac{\partial}{\partial c_0} v\left(c_0, C_1\right)\right]$$.

From the previous equation you can get:

$$ \frac{E\left[\frac{\partial}{\partial C_1} v\left(c_0, C_1\right) \tilde{x}_i\right]}{E\left[\frac{\partial}{\partial c_0} v\left(c_0, C_1\right)\right]} = p_i $$

Equivalently: $$ E[m \tilde{x}_i] = p_i$$

Where $m$ is the stochastic discount factor.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.