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Where recursive preferences meet equilibrium prices
I found the derivations that standard programmes never show — Epstein-Zin utility, stochastic discount factors, and long-run risk, built from first principles.
The models that explain equity premiums most convincingly — recursive utility, the Bansal-Yaron long-run risk framework, the Hansen-Jagannathan bound — are rarely explained with the mathematical care they require. This platform works through each one deliberately, separating risk aversion from intertemporal substitution, deriving equilibrium prices from preference fundamentals, and engaging honestly with the empirical debates that remain open. It is built for learners who already know that the subject is hard and want to understand it properly.
Recursive utility theory
Models that actually explain
Standard finance programmes describe the equity premium as a historical observation. Recursive utility theory derives it from preference fundamentals — separating how much an agent dislikes risk from how willing they are to substitute consumption across time. That separation, formalised by Epstein and Zin, changes the stochastic discount factor in ways that matter enormously for equilibrium asset prices.
Epstein-Zin
Recursive preferences allow the coefficient of relative risk aversion and the elasticity of intertemporal substitution to differ independently, resolving anomalies that standard expected utility cannot address without implausible parameter values.
2 Parameters separated
Bansal-Yaron
Module
Long-run risk models embed small, persistent components in expected consumption growth that, under recursive preferences, generate large equilibrium risk premiums — connecting the equity premium to macroeconomic dynamics in a disciplined theoretical framework.
3 Puzzles addressed
Hansen-Jagannathan
Module
The volatility bound on the stochastic discount factor provides a discipline for any asset pricing model — recursive utility models are assessed against this bound, revealing how much variation in the discount factor the preference structure actually generates.
1 Bound type
GMM methods
Module
Estimating and testing stochastic discount factor models requires the generalised method of moments — a tool well suited to the moment conditions implied by recursive preferences, and the primary empirical method covered in the final module.
1 Estimation method
Why recursive utility models matter
The theoretical frameworks explained here address questions that simpler models leave
unanswered — about risk premiums, discount factor volatility, and the structure of
equilibrium prices.
Separate risk from time preference
Standard expected utility ties the coefficient of relative risk aversion to the inverse of the elasticity of intertemporal substitution. Recursive preferences break that link, allowing each parameter to be set independently. That separation is not merely a technical refinement — it changes the stochastic discount factor, alters equilibrium risk premiums, and opens the door to models that fit observed asset prices without requiring implausibly high risk aversion. Understanding precisely how that separation works is one of the central achievements of modern asset pricing theory.
Trace equity premiums to their source
The equity premium has puzzled economists since Mehra and Prescott quantified it
in the 1980s. Recursive utility models — particularly the long-run risk
framework of Bansal and Yaron — offer a coherent theoretical account of how
persistent components in consumption growth, combined with a preference for
early resolution of uncertainty, generate large equilibrium risk premiums.
Working through that account carefully reveals both the power of the model and
the assumptions it requires, including the ones that remain contested in the
current literature.
Engage the empirical debates honestly
Recursive utility models make testable predictions about asset prices, return predictability, and the term structure of equity risk. The empirical evidence is substantial but not uniform — identification of the long-run risk component in consumption growth is difficult in finite samples, and alternative models such as habit formation and rare disasters offer competing explanations. This platform presents those debates as they are, without resolving them artificially, because genuine understanding requires knowing where the evidence is strong and where it remains genuinely uncertain.
Build understanding that holds under scrutiny
The goal of working through recursive utility theory is not familiarity with a set of results but the capacity to derive, question, and extend them. A learner who has followed the preference aggregator through to the equilibrium price-dividend ratio, and who understands why the log-linearisation approximation is used and what it costs, can read the primary literature critically. That kind of understanding is durable — it survives an encounter with a paper that modifies the framework — and it is what this platform is built to develop.
Curriculum overview
Five modules that build from preference foundations to empirical applications in recursive asset pricing
From utility axioms to equilibrium prices
The curriculum opens with the limitations of standard expected utility and
works forward through recursive preferences, stochastic discount factors,
long-run risk models, and empirical testing methods. Each stage builds
precisely on the one before, with no step assumed and every derivation shown
in full.
Epstein-Zin
Core framework
Explore module
Recursive preferences and equilibrium
The Epstein-Zin formulation is the theoretical centre of the curriculum. Working through the preference aggregator, the value function, and the optimality conditions that connect recursive utility to the stochastic discount factor reveals why this framework changed the direction of asset pricing research and what it implies for observed risk premiums.
A focused platform for serious engagement with recursive utility and asset pricing theory
Built by researchers and practitioners who encountered the same gap — rigorous, accessible
explanations of recursive preference models were simply not available in one place. The
platform has grown steadily since its founding, attracting learners from academic
institutions, asset management practices, and policy research organisations across the
Asia-Pacific region and beyond.
5 modules
Structured curriculum covering recursive utility and equilibrium pricing
From the foundations of dynamic programming and preference representation through to empirical testing of stochastic discount factor models, the curriculum is organised into five carefully sequenced modules. Each builds directly on the preceding material, with no assumed shortcuts and every key derivation worked through explicitly.
7+ frameworks
Theoretical frameworks examined across the full curriculum
The curriculum covers Epstein-Zin utility, the Bansal-Yaron long-run risk model, the Hansen-Jagannathan bound, habit formation models, rare disaster frameworks, the generalised method of moments, and cross-sectional regression approaches to risk premium estimation — each assessed on its own terms and in comparison with alternatives.
3 years
Years developing and refining the platform's academic content
Since the platform's founding, the team has worked continuously to ensure that
every explanation traces faithfully to primary literature, that mathematical
derivations are complete and correct, and that the empirical debates surrounding
recursive utility models are presented with the nuance and honesty that the evidence
actually warrants.
Reader perspectives
Finance professionals and researchers describe what working through recursive utility theory has meant for their understanding
Dr. Amara Osei
Quantitative Researcher
Asset Management
"I had encountered Epstein-Zin preferences in journal articles for years without fully understanding the derivation. Working through the stochastic discount factor module here was the first time the mathematics connected for me in a way that actually felt complete. The explanation does not skip the steps that matter."
Epstein-Zin utility
Discount factor derivation
Mathematical rigour
Marcus Lindqvist
Portfolio Analyst
Institutional Investor
"The long-run risk module gave me a framework for thinking about equity premiums that goes well beyond anything covered in my graduate programme. The honest treatment of the empirical evidence — including where the model struggles — is exactly what I needed to evaluate research papers critically rather than just accept their conclusions."
Long-run risk
Equity premium
Empirical evidence
Dr. Yuki Tanaka
Finance Lecturer
Research University
"I use the recursive preferences module as a reference when preparing graduate seminars. The derivation of the preference aggregator and the subsequent connection to equilibrium prices is clearer and more complete here than in several textbooks I have consulted. The treatment of the knife-edge expected utility case is particularly well done."
Graduate level content
Preference aggregator
Equilibrium prices
Benjamin Achebe
Risk Analyst
Sovereign Fund
"The Hansen-Jagannathan bound section resolved a confusion I had carried for a long time about what discount factor volatility implies for cross-sectional risk premiums. The platform's approach of building from the economics rather than the algebra first made the result feel intuitive even before the mathematics confirmed it."
HJ bound
Risk premiums
Cross-sectional pricing
Dr. Preethi Nair
Macroeconomics Researcher
Policy Institute
"As a macroeconomist rather than a finance specialist, I found the foundations module essential. It built the dynamic programming scaffolding I needed before the recursive preferences material made sense. The curriculum respects the learner's existing knowledge without assuming they have covered exactly the same prior material as a finance PhD student."
Dynamic programming
Interdisciplinary access
Preference foundations
What the platform covers
Five structured areas of content, each building on the one before, from preference theory foundations through to empirical asset pricing methods.
Recursive preferences and the Epstein-Zin specification
The central module develops Epstein-Zin-Weil preferences in full mathematical detail. The preference aggregator, the value function, and the first-order conditions are all derived explicitly. The critical separation of risk aversion from the elasticity of intertemporal substitution is examined carefully — not just stated — with attention to what different parameter configurations imply for equilibrium asset prices. The knife-edge case of standard expected utility emerges naturally from the recursive structure, making clear what is generalised and what is preserved. This module is the theoretical heart of the curriculum and the foundation on which everything that follows depends.
Long-run risk models and equilibrium asset prices
The Bansal-Yaron long-run risk framework is developed in full — its assumptions, its equilibrium solution via log-linearisation, and its calibration strategy. The module examines how small, persistent components in expected consumption growth generate large equilibrium risk premiums under recursive preferences, and what the model implies for the term structure of equity risk, return predictability, and the behaviour of the risk-free rate. Competing frameworks — habit formation and rare disaster models — are assessed alongside long-run risk in a structured comparison that identifies where each model succeeds and where the evidence remains genuinely mixed.
Empirical methods for recursive asset pricing models
The final area of content bridges theory and data, covering the generalised method of moments as the primary tool for estimating and testing stochastic discount factor models. Cross-sectional regression approaches to risk premium estimation are examined alongside their small-sample limitations. The particular identification challenges that arise in testing long-run risk — the difficulty of measuring persistent components in consumption growth in historical data — are treated honestly, with a survey of the evidence that researchers have brought to bear on both sides of the debate. Learners who complete this section are equipped to read and evaluate the current empirical literature with genuine critical understanding.
- AO
- ML
- YT
4.9 Average reader rating
Begin working through recursive utility and equilibrium asset pricing
The curriculum is here for learners who are ready to engage with the mathematics and equilibrium logic of recursive preferences — without shortcuts and without unnecessary simplification.
Where to start
The foundations module opens with the limitations of time-separable expected utility and
builds forward from there. Each subsequent module assumes only what has been established
in the one before, so the pathway is coherent whether a learner arrives from a finance
background, economics, or quantitative research. The research section provides
additional context on the primary literature referenced throughout.
View curriculum
Learners
Growing community
Serious theory, carefully explained
Working through recursive utility theory — from the preference aggregator through to the calibration of a long-run risk model — takes time and sustained attention. The curriculum is designed for exactly that kind of engagement, not for a quick overview.
The platform does not simplify what cannot be simplified. It builds the conceptual and mathematical scaffolding that lets a serious learner engage with recursive utility models on their own terms, with access to the full derivations and honest treatment of empirical debates.
