How Much Should You Put in Stocks? Merton's Answer Is a Single Fraction

Markowitz solved a one-period problem: given a single investment horizon, what portfolio should you hold? That is useful, but it is not the problem a real person has. A real person invests over a lifetime, keeps consuming along the way, watches their wealth go up and down, and gets to change their mind every day.

Robert Merton took on that harder problem in 1969, in continuous time, and got an answer so clean that it is still the reference point for every asset allocation debate today.

The problem: investing is not a one-shot decision

The realistic version of the question has several moving parts at once. How much do I spend versus save? How do I split my savings between a risky asset and a safe one? And how should both of those answers change as I get richer, get poorer, or get older?

Intuitively you would expect a horribly complicated answer, one where every decision depends on your wealth, your age, and the current state of the market, all interacting.

Merton set the problem up carefully. Assume the risky asset's price follows a random walk with drift (technically, geometric Brownian motion), so returns are unpredictable and the investment environment does not change over time. Assume the investor gets utility from consumption and dislikes risk in a specific way, with a constant relative risk aversion: they care about percentage losses, so losing 10 percent of your wealth stings the same whether you have a thousand dollars or a million. Then solve for the strategy that maximizes lifetime happiness from consumption.

The key idea via analogy: the thermostat, not the schedule

You might expect the optimal strategy to be a schedule: a plan that says do this in year one, that in year two, adjusting as you go. Merton found something much simpler. Under his assumptions, the optimal strategy is a thermostat: a rule that says maintain this setting, always.

Specifically:

Hold a constant fraction of your wealth in the risky asset. Not a constant dollar amount. A constant fraction. If your portfolio doubles, you double your dollar exposure to keep the fraction the same. If it halves, you cut your exposure in half.

Consume a constant fraction of your wealth each year. Same logic. Get richer, spend proportionally more. Get poorer, tighten the belt proportionally.

And here is the striking part: the fraction does not depend on your wealth, and it does not depend on your age or how many years you have left.

The fraction itself has a beautifully interpretable form. It is proportional to:

the risky asset's excess return over cash, divided by (your risk aversion times the asset's variance).

Read that as three intuitive dials.

• Higher expected excess return, hold more. Obviously.
• Higher volatility, hold less. Note that risk enters as variance, meaning the square of volatility. Doubling an asset's volatility cuts your allocation by a factor of four, not two. Risk is punished hard.
• Higher risk aversion, hold less. Exactly proportionally.

That formula is the Merton share, and it should look extremely familiar to anyone who knows the Kelly criterion. Kelly, for a log-utility investor, is the special case where risk aversion equals one. Merton's result is Kelly generalized to any level of risk aversion, embedded in a full lifetime consumption-and-investment problem. The famous "half Kelly" heuristic that practitioners use is, in Merton's language, simply an investor with risk aversion of two.

Why it mattered

• It gave asset allocation a formula. Before Merton, "how much should I have in stocks" was a matter of judgment and rules of thumb. After Merton, there was a defensible answer with each input having a clear economic meaning. To this day, when a quant argues about sizing, the Merton share is the starting point they are arguing away from.
• The horizon result is deeply counterintuitive and deeply important. The conventional advice, then and now, is that young people should hold more stocks because they have time to recover from a crash. Merton's math says: under these assumptions, no. Your investment horizon does not enter the formula at all. A 25-year-old and an 80-year-old with the same risk aversion should hold the same fraction in equities. Samuelson made the same point in the discrete-time companion paper published in the very same issue. This result is what killed the naive "time diversifies risk" argument, and it forced everyone who wanted to justify age-based glide paths to explain which of Merton's assumptions they were breaking, which turned out to be a very productive question.
• It created continuous-time finance as a field. The mathematical machinery Merton used here, stochastic control and dynamic programming applied to Brownian motion, is the same toolkit he brought to option pricing a few years later. This paper is a founding document of the whole approach.
• It gave us the framework for everything after. Every subsequent model of lifetime investing, including those with labor income, mean-reverting returns, transaction costs, or changing risk aversion, is a modification of Merton's setup. His answer is the benchmark that the complications are measured against.

The honest limitations

Merton knew these were strong assumptions. Most of the last fifty years of portfolio choice research consists of relaxing them one at a time.

• The investment opportunity set is assumed constant. Expected returns and volatilities never change. In reality, valuations predict returns somewhat, and volatility clusters. Once returns are predictable, the horizon does matter again, and Merton himself later worked out the extra "hedging demand" term that appears when opportunities vary over time.
• No labor income. Real people have a stream of future earnings, which for most young workers is a large, relatively bond-like asset. Include it and the standard result flips back toward "young people should hold more stocks," not because of time diversification, but because their total wealth (financial plus human capital) is already heavily weighted toward something safe, so their financial portfolio should tilt aggressively to compensate. This is the modern, correct justification for glide paths, and it only became visible because Merton's clean result forced the question.
• No frictions. Continuous rebalancing with zero transaction costs is a mathematical convenience. Rebalancing constantly is expensive, and the real optimal policy involves no-trade regions.
• A very specific form of risk preference. Constant relative risk aversion is convenient and not crazy, but real people's attitudes to risk are messier, depend on their goals, and change with circumstance.
• Returns are not lognormal. Markets jump, crash, and have fat tails. Merton's smooth Brownian world has none of that, and tail risk changes the optimal allocation.

The one-line takeaway

Merton solved the lifetime consumption and investment problem and found that the optimal policy is startlingly simple: hold a constant fraction of your wealth in stocks, given by expected excess return divided by risk aversion times variance, and consume a constant fraction of your wealth, with your investment horizon playing no role at all.