Merton's Return & Spending Calculator

Based on the Merton framework — calculates risk-adjusted return, optimal allocation, and spending rates using CRRA utility.

Return & Risk Inputs

All rates are annualised. μ is the expected excess return above the risk-free rate.

k

— Risk Allocation

Fraction of wealth in the risky asset

\mu

— Expected Excess Return

Risky asset drift above risk-free rate (e.g. equity risk premium)

\sigma

— Volatility of Risky Asset

Annual volatility (e.g. global equities ≈ 15–20%)

r_{rf}

— Risk-Free Rate

1-year government bond rate. Quick-fill:

\gamma

— Risk Aversion (3)

CRRA coefficient. Typically 2–3 for wealthy investors above subsistence; higher = more risk averse.

1 (risk tolerant)10 (very conservative)

Spending Inputs

Required to calculate optimal spending rates.

r_{tp}

— Rate of Time Preference

How much you value today vs future spending. Typical: 0%–4%, must be less than r_ce

T

— Time Horizon (years)

Planning horizon for finite-life spending

yrs

Fill in the inputs and click Calculate

Understanding the Merton Framework

Key insights from continuous-time portfolio theory

Risk-Adjusted Return ( $r_{ce}$ )

The certainty-equivalent return — what the risky portfolio is worth after accounting for the disutility of risk. It peaks at the optimal allocation $\hat{k}$ and falls on either side.

The Merton Share ( $\hat{k}$ )

Optimal allocation $= \dfrac{\mu}{\gamma \sigma^2}$ . It rises with expected excess return, falls with risk aversion and volatility. A useful starting point — adjust for human capital and constraints.

The 2× Rule

Allocating more than twice $\hat{k}$ makes $r_{ce}$ fall below the risk-free rate — you'd be better off in cash. This is why leverage and concentration are so dangerous even with good investments.

Optimal Spending

$\hat{c}_{\infty}$ balances $r_{ce}$ against your time preference $r_{tp}$ , scaled by $\gamma$ . The finite-life rate $\hat{c}_t$ is higher since you can also draw down capital — it annuitises wealth over $T$ years at rate $\hat{c}_{\infty}$ .