Merton's portfolio problem¶
Merton's portfolio problem is a problem in continuous-time finance and in particular intertemporal portfolio choice.
Core Idea¶
Merton's portfolio problem is treated here as the recurring crossdomainmodelsstructuresrepresentations identity summarized by this source-grounded definition: Merton's portfolio problem is a problem in continuous-time finance and in particular intertemporal portfolio choice. Merton's portfolio problem is a problem in continuous-time finance and in particular intertemporal portfolio choice. An investor must choose how much to consume and must allocate their wealth between stocks and a risk-free asset so as to maximize expected utility. The problem was formulated and solved by Robert C. Merton in 1969 both for finite lifetimes and for the infinite case.
Scope of Application¶
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Extensions. A utility function other than CRRA can be used.
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The wealth evolves according to the stochastic differen. The utility function is of the constant relative risk aversion (CRRA) form.
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Extensions. Although this cost structure seems unrepresentative of real life transaction costs, it can be used to find approximate solutions in cases with additional assets, for example individual stocks, where it becomes.
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The objective is. where E is the expectation operator, u is a known utility function (which applies both to consumption and to the terminal wealth, or bequest, W T ), ε parameterizes the desired level.
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Extensions. A numerical solution method was provided by Schroder in 1995.
Clarity¶
A clear use of Merton's portfolio problem names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is Merton's portfolio problem is a problem in continuous-time finance and in particular intertemporal portfolio choice. The strongest recognition evidence in the frozen account is: The Merton line represents portfolios having the stock/bond proportion derived by Merton in the absence of.
Manages Complexity¶
Merton's portfolio problem compresses multiple crossdomainmodelsstructuresrepresentations details into a stable diagnostic relation. The source shows both the central mechanism—for proportional transaction costs the problem was solved by Davis and Norman in 1990.—and the practical consequence—this requires a model for how r,\mu,\sigma change over time. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit.
Abstract Reasoning¶
- Type the carrier. Identify the crossdomainmodelsstructuresrepresentations entities to which the claim applies.
- State the relation. Use the source-grounded identity: Merton's portfolio problem is a problem in continuous-time finance and in particular intertemporal portfolio choice.
- Check operation and conditions. For a graphical representation, the amount invested in each of the two assets can be plotted on the x- and y-axes; three diagonal lines through the origin can be drawn: the upper boundary, the Merton line and the lower boundary.
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Merton's portfolio problem transfers literally when a new case preserves the same carrier type, relation, and recognition test. A utility function other than CRRA can be used. The utility function is of the constant relative risk aversion (CRRA) form. Beyond the home domain. No canonical parent is asserted for Merton's portfolio problem. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.
Relationships to Other Abstractions¶
Current abstraction Merton's portfolio problem Domain-specific
Parents (1) — more general patterns this builds on
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Merton's portfolio problem is a kind of Optimization Problem Domain-specific
Merton's portfolio problem satisfies the defining boundary of Optimization Problem: An optimization problem specifies decision variables, a feasible set determined by domains and constraints, and an objective function or preference ordering whose optimum is sought, optionally with uncertainty, multiple objectives, or approximation criteria.
Hierarchy path (1) — routes to 1 parentless root
- Merton's portfolio problem → Optimization Problem → Optimization
Neighborhood in Abstraction Space¶
Merton's portfolio problem sits in a crowded region of the domain-specific corpus (30th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Financial Indices & Trading Indicators (15 abstractions)
Nearest neighbors
- Value at risk — 0.90
- Kelly criterion — 0.89
- Portfolio (finance) — 0.89
- Elasticity of intertemporal substitution — 0.89
- Wicksell's theory of capital — 0.88
Computed from structural-signature embeddings · 2026-10-08