Deriving a Pricing Kernel from Investor Consumption and Portfolio Choice
Summary
The document asks how to derive the stochastic discount factor, or pricing kernel, associated with a Cox–Ingersoll–Ross short-rate model and a stated bond-pricing equation containing a risk-factor term. The reply reframes the question as one of general equilibrium: the pricing kernel follows from an investor’s first-order conditions, rather than directly from the short-rate equation alone.
It illustrates this with a two-period consumption and investment problem. An investor chooses savings, leaving current consumption equal to the endowment less savings and receiving an uncertain investment return in the next period. The resulting first-order condition can be rearranged into an expectation involving the future-to-current ratio of marginal utility, time preference, and the asset return. This gives the standard consumption-based stochastic discount factor concept. The reply does not derive a CIR-specific kernel or connect the displayed risk-factor parameter to investor preferences; it notes that the notation and source framework matter. Thus, it provides conceptual grounding, while a full model-specific derivation requires additional assumptions.
Key ideas
- In a general-equilibrium model, the pricing kernel can arise from investors’ portfolio-choice first-order conditions.
- The example links savings to current and future consumption and uncertain returns.
- The stochastic discount factor includes time preference and a ratio of marginal utilities.
- The reply does not establish a CIR-specific kernel without the underlying model assumptions.
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Full text
# Stochastic Discount Factor of CIR bond pricing model
# Stochastic Discount Factor of CIR bond pricing model
The CIR model states $dr=\kappa(\theta-r)dt+\sigma dW$ and the corresponding bond pricing equation can be derived from the general equilibrium approach.
The equation is: $\frac{1}{2}\sigma^2rP_{rr}+[\kappa(\theta-r)-\lambda r]P_r-rP=P_{\tau} $
where $\lambda$ is the factor of risk.
What is the derivation of the corresponding stochastic discount factor and pricing kernel of CIR model?
## Answer by jd8 (score 2)
https://quant.stackexchange.com/a/35525
If you refer me to the exact text you are using I will be more specific - since your notation is not like their 1995 paper.
In any case, the derivation of the stochastic discount factor (also called the pricing kernel) in a general equilibrium model must come from the first order conditions of an investors portfolio choice. The investor will have a utility function over something he cares about (wealth, consumption, leisure, etc) and asset risk will be a function of the co-variance between the return on the asset and some function of the investors utility over something he cares about.
Consider the standard framework where an investor makes a consumption investment decision in a two period framework. Given some endowment today $e_0$ consumption is what is left after saving $c_1=e_1-s$, tomorrow you consume the uncertain return on investment $s\tilde{R}$
\begin{equation} \underset{s}{max}\quad u(e_1-s) + \beta\mathbb{E}[u(s\tilde{R})] \end{equation} The first order conditions of portfolio choice can be written as \begin{align*} u'(c_1) =& \mathbb{E}[\beta u'(c_2)\tilde{R}]\\ 1 =& \mathbb{E}[\beta\frac{u'(c_2)}{u'(c_1)}\tilde{R}]\\ \end{align*}
Anything else you see will be a variation on the same concept.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.