Practical Finance: An Approximate Solution to Lifecycle Portfolio Choice

by Choi, Liu & Liu

Merton (1969) solved the portfolio problem for an investor with only financial wealth: hold a constant fraction in equities, set by the equity premium, volatility, and risk aversion. The formula works because the state space is one-dimensional — all wealth is investable, and the optimal share never changes.

Add labor income and the problem breaks. Future paychecks act as a bond-like asset that depletes over time. The optimal equity share of financial wealth must change with age, the state space expands, closed-form solutions vanish, and the standard approach — backward induction on a discretized grid — requires serious computation. Cocco, Gomes, and Maenhout (2005), hereafter CGM, is the canonical numerical treatment. It works. It also takes hundreds of lines of code, hours of runtime, and yields solutions only for the specific parameter values the researcher chose.

Choi, Liu, and Liu bridge this gap with what they call practical finance: an analytic approximation to the optimal solution that anyone can compute in a spreadsheet, parameterized by inputs an investor can estimate. They solve the full CGM model across 5,103 parameter combinations, then regress the resulting human capital discount rates on model inputs. The output is two linear equations — one for working years, one for retirement — that map preferences and circumstances to a discount rate. Apply that rate to projected income, compute human capital $H$, and the Merton formula does the rest.

The Framework

Three steps:

Step 1 — Merton share ($\alpha^{\ast}$): the fraction of total wealth (financial + human) allocated to equities. Equation (8) gives $\alpha^{\ast} = \max\{0, \min\{1, (r - r_f + \frac{1}{2}\sigma_r^2) / (\gamma \sigma_r^2)\}\}$, where $r - r_f$ is the log equity premium, $\sigma_r = 0.185$ is stock return volatility, and $\gamma$ is relative risk aversion. For the paper's baseline ($\gamma = 7$, $r - r_f = 0.02$), $\alpha^{\ast} = 0.155$.

Step 2 — Human capital ($H$): the present value of all future labor income. Each year's expected income is discounted at a one-year-ahead rate derived from regression on the model parameters. Pre-retirement (Table 1, column 3):

$$r_{y,t} = 0.087 \cdot \frac{\gamma}{10} - 0.267 \cdot (r - r_f) + 1.132 \cdot r_f + 4.332 \cdot \sigma_u^2 + 0.028 \cdot \sigma_\varepsilon^2 + 0.010 \cdot \lambda - 0.149 \cdot \frac{\text{age}}{100} + 0.142 \cdot \left(\frac{\text{age}}{100}\right)^2 - 0.020$$

Post-retirement (Table 2, column 3) drops the income risk terms — retirement income is deterministic:

$$r_{y,t} = 0.0003 \cdot \frac{\gamma}{10} - 0.217 \cdot (r - r_f) + 0.893 \cdot r_f + 0.476 \cdot \frac{\text{age}}{100} - 0.295 \cdot \left(\frac{\text{age}}{100}\right)^2 - 0.166$$

Step 3 — Financial allocation: $\hat{\alpha}_t = \max\{0, \min\{1, \alpha^{\ast}(1 + H/W)\}\}$, where $W$ is current financial wealth. When $H/W$ is large — young, high income, low savings — the equity allocation pushes toward 100%.

What the Discount Rates Reveal

The discount rate for labor income rises with risk aversion (+0.9 pp per unit of $\gamma$), the risk-free rate (+1.1 pp per 1 pp of $r_f$), and permanent income shock variance (+2.8 pp moving from a high school dropout to a college graduate). It falls with the equity premium (−0.3 pp per 1 pp of $r - r_f$). Temporary income shocks barely matter.

Discount rates peak in early working life — when liquidity constraints bind and income growth steepens — then flatten through mid-career and drop at retirement as income becomes risk-free. Higher replacement rates push discount rates up near retirement by raising human capital's value, which raises the implicit bond holding and demands a larger equity offset.

Accuracy

The approximation matches the full numerical solution with $R^2 = 0.990$ across 80 million observation pairs (5,103 parameter sets × 78 ages × 201 wealth grid points). Regressing actual optimal equity shares on approximate shares yields a slope of 0.946 and intercept of 0.054, with RMSE of 3.66 percentage points. Sixty-six percent of parameter sets have RMSE below 4 pp; 95% fall below 5 pp.

The worked example in Section 3.3: a 55-year-old college graduate with $\gamma = 7$, log risk-free rate of 2%, log equity premium of 2%, replacement rate of 40%, \$100,000 expected income through age 66, and \$40,000 thereafter. The approximation yields $H = \$924{,}805$. With a \$1,000,000 financial portfolio, $\alpha^{\ast} = 15.5\%$, and $1 + H/W = 1.92$, the recommended equity share is 30%. The exact numerical solution gives 33% — an out-of-sample test, since this earnings trajectory was not in the regression training data.

Welfare Costs

Table 4 compares the lifetime certainty-equivalent consumption loss of common strategies versus the exact numerical optimum:

The approximation's 0.06% cost vanishes in practice. The "100 minus age" rule costs 33× more, penalized for ignoring income level, education, and risk preferences. A static 60/40 allocation costs 62× more, blind to the age-varying ratio of human to financial capital. Full equity non-participation and full equity commitment are the costliest — both ignore the fundamental tradeoff the model captures.

Correlated Income Extension

Section 5 allows permanent income shocks to correlate with stock returns. When income co-moves with equities (positive $\beta_\text{perm}$), human capital is less bond-like and the optimal equity allocation falls. The modified formula subtracts the implicit equity exposure embedded in human capital:

$$\hat{\alpha}_t = \max\{0, \min\{1, \alpha^{\ast}(1 + h_t/w_t) - \beta_\text{perm} \cdot h_t/w_t\}\}$$

The same discount rate regressions apply — only the allocation formula changes. With $\beta_\text{perm} = 0.3$, the downward shift in equity share is nearly as large at age 65 as at age 30, because the CGM model ties retirement benefits entirely to final permanent income. The extension's RMSE is 3.67 pp and welfare loss drops to 0.02%.

Temporary income correlations barely matter: $\phi = 0.0045$, close enough to zero that the paper recommends ignoring them.

Parameterization

The numerical grid spans ages 22–100, retirement at 66, risk aversion from 4 to 10, three education levels (no high school, high school, college) with PSID-calibrated cubic income trajectories, replacement rates of 40%–80%, log risk-free rates of 0%–2%, and log equity premia of 2%–4%. Stock volatility is fixed at $\sigma_r = 0.185$, the CRSP value-weighted index standard deviation from July 1926 to July 2024. The discount factor $\delta = 0.96$ and mortality follows the 2019 NCHS life table.

The equity premium range of 2%–4% sits well below the 6.5% historical realization. At 2%, the model already produces frequent 100% equity allocations. Any approximation accurate at 4% extends cleanly above it.

Limitations

Single risky asset, single riskless asset. No housing, pensions, stock options, or variable annuities. CRRA utility with no habit formation or loss aversion. Exogenous labor supply — career decisions do not respond to portfolio outcomes. No parameter uncertainty: the investor knows the equity premium and risk-free rate. The income process excludes unemployment spells as a separate state.

These are CGM's assumptions, not new ones. The contribution is not to relax them but to make the existing framework usable without writing code. The authors' Google Sheets implementation puts it in reach of anyone who can fill in six cells.

• Choi, Liu & Liu (2025)
• NBER Working Paper No. 3416