How much to spend? What to do with what's left?

A planner that maximizes money is easy to write and almost always wrong. It will cheerfully trade a decade of scraping by for a fortune at ninety, because to it a dollar is a dollar whenever it turns up. What actually matters is the satisfaction that money buys each year.

The first $30k of a year’s spending buys somewhere to live, food, and heat. The thirtieth $30k buys a nicer holiday. Both are worth having; they are not worth the same. Economists call the gap between them diminishing marginal utility, and it has been the standard way to describe attitudes to risk since Pratt and Arrow put numbers on it in the mid-sixties^5,6.

At the risk aversion set above, this model prices that gap precisely. A dollar added to a year spending $20,000 is worth 16.0 times a dollar added to the $80,000 year the estimate above actually produces. Raising the risk-aversion dial grows that multiple; the curve below bends harder.

Two things follow from that bend:

The curve is steeper below the current level than above it. A fall in spending costs more than an equal rise gains, so an estimate built on this objective defends the downside hard without needing a separate rule telling it to. Worth being precise about what this is not: it is not the reference-dependent loss aversion of prospect theory⁷. Nothing in this model remembers where a household started or feels a loss as such. The asymmetry comes from the level of spending itself, which is exactly why one setting can describe a household at thirty-five and at eighty-five.

The curve flattens at the top. Past a point, more money stops buying much. That has a consequence people find genuinely surprising: a bet can multiply the average outcome and still be a bad bet.

The lottery ticket has by far the largest expected payout of the three, and it is by far the worst of them, because 999,999 times out of a million it halves spending for the rest of a life. Maximizing expected dollars would take it. Almost nobody would. That gap is the single best argument for optimizing the thing that actually matters instead of the thing that is easy to add up.

One honest caveat: those bets are about how the objective ranks risk, not about what this estimate simulates. This is not, strictly, an average path. As "What volatility actually costs" explains, the market swing is priced into it twice, once as a deterministic drag on growth that has nothing to do with anyone’s preferences, and once more by the stated risk aversion, so the balance shown compounds a little below what a true average of many markets would show, and lower still once that aversion is priced on top. But it is still one line, not a distribution: there is no bad decade in it, only its long-run cost, priced in advance.

Robert Merton settled this one in 1969⁸, and extended it into the full consumption-and-portfolio problem two years later⁹. Paul Samuelson reached the same answer in the same year from a discrete-time model¹⁰. The result is unusually clean: the right fraction of total wealth to hold in the risky asset is a constant. It rises with the extra return that asset is expected to earn over the safe one, and it falls with both how much the asset swings around and how much the household dislikes swings.

Notice what is not in that list. Age is not in it. Neither is how many years remain. The famous rules of thumb, a hundred minus age, and its descendants, have no support here at all.

At the returns, volatility and risk aversion entered above, that constant works out at 62% of total wealth. The estimate above holds 100% of the accounts in it today, which is a different number, and the gap between the two is the whole of the next paragraph.

The share applies to everything owned, which includes the earnings still ahead. Bodie, Merton and Samuelson made that concrete¹¹: a stable salary is a long bond already held and impossible to sell. It sits in total wealth whether it is counted or not, so the money that is actually invested is tilted to make the whole portfolio come out right. In this model, the tilt can only push the accounts to 0% or 100% risky, never past either end: there is no borrowing here against a paycheck that has not arrived yet, and no shorting the risky asset to lean the other way. Someone with a large bond-like income and modest savings hits the 100% end for years at a stretch. While that lasts, the tilt has run out of room: the accounts hold everything they are allowed to, and the portfolio is still short of the total-wealth share above, not because the arithmetic disagrees but because a real brokerage account will not lend against a job.

Now the glide path above reads as a consequence rather than a rule, and which way it moves depends on which answer given to Income Profile. Set to bond-like earnings, the orange band above dominates early on: a great deal of bond-like wealth is already on the books. The small invested balance can afford to be aggressive, and the risky-asset share starts high and glides down as that ballast runs out. Set to stock-like, the model already counts the orange band as risk already carried, so it holds less of that risk in the financial accounts while those earnings are still ahead. The risky-asset share starts low, often at zero, and climbs as the band runs out and stops offsetting the risk on its own. Either way the share converges on the same constant Merton's formula dictates.

A tenure, a salary, or a pension behaves like a "risk-free" asset while commissions, equity compensation, or business income behaves more like a risky asset. The Income Profile choice is also, honestly, a coarser choice than most real income actually offers¹²: this page offers two positions, all bond or all stock, and most paychecks sit somewhere between. Steadier than a commission check, but more exposed than a government pension. There is no dial here for the middle. And whichever position is chosen, the dollar figure in the chart above is discounted the same way either way, at the safe rate. Only the portfolio tilt reads that choice; the valuation does not.

One detail worth keeping straight, because it is easy to conflate. Volatility enters this model three ways, and only one of them is free of the household’s preferences. It makes the risky asset less attractive, and the volatility impacts the share above. Whatever share is held then drags on compound growth for a reason that has nothing to do with anyone’s preferences. A wobbly return builds less than a steady one at the same average, full stop. And risk aversion adds a third discount on top of that drag, sized by exactly how much swings are disliked. This section explains the first impacts of volatility while "What volatility actually costs" puts numbers on the last two.

A wobbly return builds less than a steady one, even when the two average the same. Up 50% one year and down 50% the next does not get back to even. It leaves you at 75% of the starting point. Economists call the gap volatility drag, and the estimate already subtracts it: it is the difference between the top bar below and the middle one.

Drag is not the whole cost of risk, it is what the wobble does to the balance. There is a second, separate cost: what the same uncertainty costs in satisfaction. A smaller sure amount beats a larger unsure one, and how much smaller is exactly the risk-aversion dial set at the top of the page. So the estimate compounds at a rate lower still, referred to as the Certainty Equivalent Return.

At the numbers above: the money averages 7.00% a year. The wobble in it leaves 5.00% actually compounding, a 200 basis-point drag, present regardless of anyone’s appetite for risk. The estimate assumes 3.00%: 200 basis points lower still. That last step down is the safety margin, sized by the same risk aversion preference that bends the curve in "Why this maximizes satisfaction, not dollars".

What comes out of this framework is spending as a dynamic fraction of wealth. If the market falls 30% this year, next year’s number falls with it automatically. The estimate was never a fixed amount to begin with.

Franco Modigliani and Richard Brumberg proposed in 1954 that people do not spend their paycheck, they spend their lifetime¹³. Ando and Modigliani put the idea to work on real data a decade later¹⁴, Milton Friedman arrived at a close cousin of it from another direction¹⁵, and Modigliani’s Nobel lecture is still the clearest short statement of it¹⁶. The claim is that saving is not a virtue and not a goal. It is plumbing: the mechanism that carries money from years with a lot of income into years with none.

That is what the estimate above is doing, and it is easiest to see with the two flows side by side. One of them climbs and then stops dead. The other barely moves.

It also explains why the headline figure above is not simply a slice of the balance. The estimate holds $60,000 in accounts. The estimate has separately valued the earnings still ahead at $2,479,592, and it is choosing this year’s $80,000 against the total of the two.

Three limits of this particular version, stated plainly. It has no mortality in it: it models to the age entered as though it were certain to be reached and certain not to be passed, and it aims to spend the balance down to nothing on the way. Modigliani’s own framing left room for a bequest; this one does not. A full run can weight each year by the chance of being alive for it or be set for perpetual spending.

It also has no sequence of returns in it, only their long-run effect on growth as described by "What volatility actually costs". The short version: it does carry a precautionary motive, the impulse to hold a little back specifically because the future is uncertain rather than merely because later years need funding too, but priced into the compounding rate itself rather than modeled as fear of any particular bad year. What it genuinely cannot do is protect against which year a downturn lands in. A real portfolio does not earn the same return every year, and a bad run early in retirement does more damage than the same bad years averaged in later, because there is less left to recover with.

And the “minimum acceptable spending” figure in the form constrains nothing. It is reported against, never enforced. The only thing holding spending up in a thin year is the curvature from "Why this maximizes satisfaction, not dollars".

The dial the form calls risk aversion sets how sharply the curve above bends. It is the coefficient of constant relative risk aversion, and the words carry the assumption: what stays constant is the reaction to a change measured in proportions. Losing a fifth of spending is supposed to feel the same at forty thousand a year or four hundred thousand^5,6.

That is a strong claim about people, and it is the reason a single number can describe a whole life instead of needing a fresh one at every wealth level. Most estimates put people between 2 and 3; this form accepts anything from 0.1 to 4.

Raising it does three things at once. It lowers the willingness to hold the risky asset, which lowers the return the estimate can count on. At whatever share is held, it also lowers the rate the estimate assumes that share compounds at, the Certainty Equivalent Return. And it lowers the willingness to accept an uneven path, which pulls later spending back toward earlier spending. The first two effects both shrink the growth the estimate is willing to bank on; the third is worth perhaps a tenth of spending across a lifetime, and pulls in the opposite direction on the very first year. So the chart below shows the risky-asset share and the table shows the spending it pulls toward. "What volatility actually costs" puts a number on the second effect specifically.

The other number the form asks for, time preference, is doing a related but different job. Risk aversion prices an uncertain dollar against a certain one; time preference prices a dollar available now against the same dollar a year from now, with nothing uncertain about either side of that trade. The form calls it δ, and it is the guaranteed return needed to justify putting off a dollar of spending by a year rather than spend it today.

This page defaults it to 3%, and that number comes from how people actually spend across a lifetime rather than from a lab gamble over a few weeks⁴. Raise δ and the estimate tilts spending earlier without changing the risk it takes; lower it and more gets carried forward into later years.

Neither of these two numbers is something most people can simply introspect, which is why both fields carry a short questionnaire behind them in the form above^2,3. The questionnaire narrows a handful of forced choices down to a bracket and a representative value. A handful of questions is a neighborhood, not a measurement, but it beats guessing at a number with no natural units and nothing to compare it against.

Where the ideas above come from. The model on this page is a direct implementation of Merton’s and Samuelson’s results, sitting inside Modigliani’s frame, solved numerically by Bellman’s method¹⁷. For the objective function stated formally, and what the paid tier’s fuller version of this model adds, see the methodology page.

1. 1. Haghani, V. & White, J. (2023) The Missing Billionaires: A Guide to Better Financial Decisions. Wiley.
2. 2. Barsky, R. B., Juster, F. T., Kimball, M. S. & Shapiro, M. D. (1997) Preference parameters and behavioral heterogeneity: an experimental approach in the Health and Retirement Study. Quarterly Journal of Economics 112(2), 537-579. Link
3. 3. Frederick, S., Loewenstein, G. & O'Donoghue, T. (2002) Time discounting and time preference: a critical review. Journal of Economic Literature 40(2), 351-401. Link
4. 4. Laibson, D., Lee, S. C., Maxted, P., Repetto, A. & Tobacman, J. (2026) Estimating discount functions with consumption choices over the lifecycle. Review of Financial Studies 39(6), 1580-1610. Originally circulated as NBER Working Paper 13314 (2007), by three of these five authors. Link
5. 5. Pratt, J. W. (1964) Risk aversion in the small and in the large. Econometrica 32(1/2), 122-136. Link
6. 6. Arrow, K. J. (1965) Aspects of the Theory of Risk-Bearing. Yrjo Jahnssonin Saatio, Helsinki.
7. 7. Kahneman, D. & Tversky, A. (1979) Prospect theory: an analysis of decision under risk. Econometrica 47(2), 263-291. Link
8. 8. Merton, R. C. (1969) Lifetime portfolio selection under uncertainty: the continuous-time case. Review of Economics and Statistics 51(3), 247-257. Link
9. 9. Merton, R. C. (1971) Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3(4), 373-413. Link
10. 10. Samuelson, P. A. (1969) Lifetime portfolio selection by dynamic stochastic programming. Review of Economics and Statistics 51(3), 239-246. Link
11. 11. Bodie, Z., Merton, R. C. & Samuelson, W. F. (1992) Labor supply flexibility and portfolio choice in a life cycle model. Journal of Economic Dynamics and Control 16(3-4), 427-449. Link
12. 12. Viceira, L. M. (2001) Optimal portfolio choice for long-horizon investors with nontradable labor income. Journal of Finance 56(2), 433-470. Link
13. 13. Modigliani, F. & Brumberg, R. (1954) Utility analysis and the consumption function: an interpretation of cross-section data. In Post-Keynesian Economics, Rutgers University Press. Reprinted in The Collected Papers of Franco Modigliani, MIT Press (2005). Link
14. 14. Ando, A. & Modigliani, F. (1963) The 'life cycle' hypothesis of saving: aggregate implications and tests. American Economic Review 53(1), 55-84. Reprinted in The Collected Papers of Franco Modigliani, MIT Press (2005). Link
15. 15. Friedman, M. (1957) A Theory of the Consumption Function. Princeton University Press / NBER. Link
16. 16. Modigliani, F. (1986) Life cycle, individual thrift and the wealth of nations. Science 234(4777), 704-712. The Nobel Prize lecture, delivered in 1985 and published the following year. Link
17. 17. Bellman, R. (1957) Dynamic Programming. Princeton University Press. Link