Merton fraction
Quantitative Finance glossary · Quantitative Finance
Definition
Constant optimal proportion of wealth in the risky asset under CRRA utility with risk-aversion γ: π^* = (μ - r)/(γ,σ^2) — solution of the Merton portfolio problem via HJB. Striking property: under constant investment opportunity set, π^* is constant in wealth and time (constant-mix rebalancing). Generalises to stochastic-opportunity-set settings via the addition of a hedging-demand term (Merton 1971); empirical 'horizon effects' come from this hedging demand.
Merton fraction in plain English
If you had money to invest, what fraction should go into the risky, exciting stuff versus the safe stuff? The Merton fraction gives a clean answer, π^* = (μ - r)/(γ,σ^2): bet more when the extra reward μ - r is juicy, less when the ride is bumpy (big σ^2) or when you personally hate risk (big γ). The surprising punchline is that this ideal fraction stays the same whether you're rich or broke, young or old — so you just keep rebalancing back to that constant slice, like always keeping your plate 40% vegetables no matter how big the plate is. (Real life adds tweaks when tomorrow's opportunities themselves shift over time.)
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