Ha and Jeon's income purchases end 14.38 years before retirement

At a 2% interest rate, buying income for 12.5 per unit of income flow stops making sense 14.38 years before retirement in Ha and Jeon's model, regardless of risk aversion. Few life-cycle results can be put to a client without first asking about preferences. This one can. The paper's treatment of human capital in a portfolio, though, matters more than its purchase rule.

A career with income for sale

The investor has CRRA preferences, one stock and a riskless bond. Labor income continues until a fixed retirement date. She cannot borrow against that income, and liquid wealth must remain nonnegative. Before retirement, she can pay a fixed price p from her wealth to increase her income rate permanently. Degrees or licences are the intended analogy, with the extra income treated as riskless and unavailable for resale. Retirement leaves her with a plain Merton problem.

Kim and Jeon (2024), cited in the paper, studied the infinite-horizon version. The finite horizon is the addition here. As retirement nears, an extra unit of income loses its value, eventually reaching zero. Time remaining therefore governs the problem. In dual variables, the consumption-to-income ratio has two moving boundaries: wealth is zero below the lower boundary, and the investor buys income above the upper boundary.

Establishing the properties of these boundaries takes up most of the paper. The upper boundary emerges from infinity at a positive time, defeating the usual free-boundary methods. A partial hodograph transformation maps both boundaries to a fixed interval. The authors then solve the resulting quasilinear equation and join that solution to an obstacle problem. Both boundaries are infinitely differentiable. For computation, they derive and solve coupled Volterra integral equations. Their baseline is r = 0.02, θ = 0.30, β = 0.03, γ = 2, p = 12.5, with a 40-year career running from ages 25 to 65.

Where does the 14-year cutoff come from?

The argument works path by path. An income unit bought now pays until retirement and has present value φ(τ) = (1 − e^(−rτ))/r. Suppose φ(τ) is below p. Take any strategy that purchases income, remove the purchase, and keep its consumption and portfolio choices. Wealth under the second strategy is at least as high on every path. The borrowing constraint therefore remains satisfied.

The purchase window ends where φ(τp) = p, or τp = −(1/r)log(1 − rp). For rp = 0.25, the cutoff is 14.38 years before retirement: purchases cease after age 50.6. Ha and Jeon emphasize that this window depends only on r and p, "not on risk aversion, impatience or the equity premium." They prove the cutoff is exact by showing that the purchase region is nonempty before it. Their policy implication follows directly. Delaying retirement by a year shifts the last purchase date by exactly a year.

A constant purchase price makes this tidy result possible. The authors leave age-dependent prices open; with those prices, the no-purchase set could cease to be an interval. The 14.38 years belong to this pricing assumption. They should stay out of advice about actual mid-career training.

Salary gets a liquidity discount

The portfolio rule challenges how practitioners count labor income. The investor applies the Merton fraction θ/(γσ) = 0.75 to liquid wealth plus "effective human wealth" W per unit of income, the value of income that supports her stock position.

With 40 years left to retirement and liquid wealth equal to one year's income, W is 8.2 years of income. Compare that with a market value of 27.5 years for the income stream and a purchase price of 12.5. The paper proves that W cannot exceed p. In this state, its numerical results value effective human wealth at under a third of market value.

The income itself carries no risk. The borrowing constraint produces the discount. Buying more income at p also caps W at that price. Because the investor can neither sell the stream nor borrow against it, income supports stock exposure only by changing future liquid wealth. An adviser who books future salary as a bond at present value would overstate this investor's bond holding by a factor of about 3.4.

Stocks can still reach 6.9 times liquid wealth

In that same state, the risky position is 6.9 times liquid wealth, from 0.75 × (1 + 8.2). The restriction on borrowing against wages says nothing about stock margin; the model requires only that wealth remain nonnegative. Actual margin limits would bind at 6.9x. The paper leaves them outside the model.

The purchase threshold also shows where the proofs end and the numerical patterns begin. Expressed in years of income, the threshold diverges as τ falls to τp. Far from retirement, it converges to a stationary solution whenever one exists. Such a solution always exists for γ > 1, including the baseline. In the baseline calculations, the threshold falls to about 2.24 at τ ≈ 18.5, then rises to 2.53 at τ = 40. As the authors put it, "the U-shape is a property of the baseline parameters rather than a theorem." Expressed in years of consumption, the threshold declines monotonically.

At τ = 40, the purchase threshold is 2.53 for γ = 2, compared with 0.76 for γ = 5. Greater risk aversion brings purchases of riskless income forward to a lower wealth level. Raising θ from 0.3 to 0.4 instead lifts the threshold from 2.53 to 5.04, reflecting the equity premium earned by liquid wealth. Throughout, a single wealth-income cutoff remains.

An illustrative life cycle

The simulation follows 2×10^4 paths, each starting with zero wealth and income 1. The median investor first buys income at age 28.3. By 65, income has a median of 3.32 and a mean of 3.94. The median wealth-to-income ratio climbs from 1.6 at age 50.6 to 17.8 at 65. Retirement saving thus begins in earnest after income purchases end. At age 25, the option to buy income is worth 2.0 years of income without savings, or 4.7 years with three years' income already saved.

These are model results under illustrative parameters. The paper does not estimate p. Its 7.6% internal rate of return on income bought at 25 comes from the assumed p = 12.5, making that return an input in disguise. The numerical work is internally consistent: for τ ≥ 20, the Volterra boundaries agree with a finite-difference solution to five significant digits. Uniqueness of the Volterra system remains unproved by the authors.

We could not backtest this.

Doing so would require an instrument selling guaranteed income increases until a fixed date at a fixed price, plus a history connecting those purchases to later pay. Replacing the riskless bond with a bond ETF would introduce duration risk. Running the policy on simulated model paths would merely repeat the paper's figures.

The 8.2-versus-27.5 gap is worth taking away: within this model, a constrained young earner's salary contributes under a third of its present value to the equity decision.