Asset Allocation in Continuous Time: Optimal Investing

In the one-period portfolio models like Markowitz and CAPM, you make your investment decision once and then sit and wait. You cannot change your mind. But in real life, you check your portfolio, see how the market moved, and rebalance. You do this every day, every hour, maybe every minute. Chapter 66 of Wilmott’s book, based mostly on Merton’s work, develops the theory of continuous-time investment and portfolio rebalancing. The results are surprisingly elegant and give genuine practical insight.

The Setup: One Risky, One Safe

Start simple. You have two assets. One is risk-free and earns a guaranteed rate r. The other is risky, following a lognormal random walk with drift mu and volatility sigma.

You invest a fraction w(t) of your wealth in the risky asset and the rest (1-w) in the risk-free one. You also choose how much to consume (spend) at a rate C(t). Your goal is to figure out the best w(t) and C(t) at every moment in time.

The first step is writing down how your wealth evolves. Three things change your wealth at each instant.

You lose money by consuming: that is just -C dt.

You earn interest on your cash position: that adds r(1-w)W dt.

Your risky asset position changes randomly: that contributes w(mu W dt + sigma W dX).

Notice something important. The stock price S does not appear explicitly in the wealth equation. Because you hold a proportion w of your wealth in the risky asset, and the asset follows a lognormal walk where drift and volatility scale with the price, everything scales with your wealth W instead. You get a self-contained stochastic differential equation for W.

What Are You Optimizing?

You want to maximize your expected utility, but there are two sources of utility. First, you enjoy consuming. The utility of consumption U(C(t)) over time, discounted by a rate rho that measures your impatience. Second, there is the bequest function B(W). If you have wealth left at the end of your time horizon (think retirement, or death), how much is that worth to you? How much do you want to leave to your kids?

The optimization problem is: choose w(t) and C(t) to maximize the expected sum of discounted consumption utility plus the bequest function.

The Bellman Equation

Merton’s approach uses stochastic control theory. You define a function J(W, t), called the derived utility of wealth. This function represents the maximum expected utility you can achieve starting from wealth W at time t, given that you make optimal decisions from now on.

Using Ito’s lemma and the principle of optimality (the idea that an optimal strategy must be optimal at every sub-interval), you arrive at the Bellman equation. This is a partial differential equation for J with conditions for the optimal C and w.

The optimization over consumption gives you: the marginal utility of consumption equals the marginal derived utility of wealth. In symbols, U’(C*) = dJ/dW. This makes intuitive sense. You consume until the happiness from spending one more dollar equals the happiness from having one more dollar invested.

The optimization over the asset weight gives you: the fraction in the risky asset equals (mu - r) / (sigma^2) times the ratio (-dJ/dW) / (d^2J/dW^2). This is a generalization of the classic Markowitz result, adjusted for the curvature of the derived utility function.

CRRA: The Clean Solution

When the utility function has constant relative risk aversion (CRRA), the equations simplify beautifully.

With U(C) = C^gamma / gamma for gamma < 1, and assuming no bequest (the “you can’t take it with you” condition), the Bellman equation has an explicit solution.

The results are remarkable. Consumption is proportional to wealth with a weak time dependence. If you have twice as much money, you spend twice as much. And a constant fraction of wealth should remain in the risky asset. That fraction is (mu - r) / ((1 - gamma) * sigma^2).

This means the optimal strategy is to maintain a fixed percentage in stocks and a fixed percentage in bonds, rebalancing continuously to keep the ratio constant. Your spending adjusts proportionally with your wealth. Rich or poor, you invest the same percentage in stocks.

This is the theoretical justification for the common financial advice of “keep X% of your portfolio in equities.” Of course, the optimal X depends on your risk aversion, the expected return and volatility of the market, and the risk-free rate. But the structure (constant fraction) is what comes out of the math.

CARA: A Different Story

With constant absolute risk aversion (CARA), the utility function is U(C) = -e^(-aC). The results are different in a telling way.

Consumption now has a part that is independent of wealth. Even a broke CARA investor would want to consume some minimum amount. And the fraction of wealth in the risky asset decreases as wealth increases. The richer you get, the more conservative you become in percentage terms, even though your absolute dollar exposure to risk stays constant.

This is less realistic for most people. CRRA, where you scale your risk-taking with your wealth, matches observed behavior better. But CARA has its uses in theoretical models because of its mathematical tractability.

Many Assets

The extension to N risky assets is surprisingly straightforward. You hold a proportion w_i in each risky asset and the remainder in the risk-free asset. The wealth process looks similar, with the single-asset variance replaced by a quadratic form involving the covariance matrix.

The optimal weights are given by a formula involving the inverse of the covariance matrix applied to the vector of excess returns (mu minus r). This is Merton’s multi-asset result, and it looks a lot like the one-period Markowitz solution but derived in continuous time with an explicit accounting for dynamic rebalancing.

For CRRA utility, the weights are again constant. You invest a fixed fraction in each risky asset and rebalance continuously.

Maximizing Long-Term Growth

There is a special portfolio that deserves its own discussion: the growth-optimum portfolio. Here you do not consume anything. All gains are reinvested. The goal is to maximize the expected growth rate of your wealth.

This turns out to be equivalent to maximizing expected logarithmic utility, which is CRRA with gamma approaching zero. The optimal weight is:

w* = (mu - r) / sigma^2

for a single risky asset. For multiple assets, it involves the inverse covariance matrix applied to the excess return vector.

This is the continuous-time version of the Kelly criterion from gambling theory. If you recall the Blackjack example from the one-period chapter, it is the same idea: bet the fraction that maximizes the expected log of your wealth. The only difference is that in Blackjack, there is no risk-free rate earning interest while you sit at the table.

The growth-optimal weight is independent of current wealth and time. It does not matter if you are rich or poor, early or late. The optimal fraction is always the same.

Transaction Costs: The No-Trade Region

Everything above assumes costless continuous rebalancing. In reality, trading costs money. Incorporating transaction costs into the model is hard, but there is one elegant result for the growth-optimum portfolio.

Assume each rebalance costs a fraction epsilon of the total portfolio value. This is not the most realistic cost structure (normally you pay proportional to the trade size, not the portfolio size), but it allows an analytical solution.

The optimal strategy becomes: maintain your portfolio weights inside an ellipsoid centered on the Merton point (the optimal weights without costs). As long as the current weights are inside this ellipsoid, do nothing. When they hit the boundary, rebalance back to the center.

The shape and size of this ellipsoid depend on the covariance matrix, the excess returns, and the cost parameter. It is defined by a matrix equation that can be solved iteratively.

Wilmott shows a two-dimensional example with two risky assets. The no-trade region is an oval in weight space, centered on the Merton point. The asymptotic approximation matches the exact numerical solution remarkably well.

One surprising finding: with realistic parameters, the expected time between rebalances is on the order of a year or two. This is much longer than what most fund managers actually do. In practice, rebalancing happens more often because of perceived changes in drift and volatility, not because of transaction costs pushing you to the boundary.

Key Takeaways

Four things from this chapter. First, with CRRA utility, the optimal strategy is to keep a constant fraction of your wealth in each risky asset and rebalance continuously. Consumption scales proportionally with wealth. This is the math behind “keep 60% in stocks.”

Second, the growth-optimal portfolio maximizes expected log utility and gives you the Kelly criterion for continuous-time investing. The optimal weight is (mu - r) / sigma^2, independent of wealth and time.

Third, transaction costs create a no-trade zone around the optimal point. Do not rebalance until you drift to the boundary of this zone. The zone is an ellipsoid in weight space.

Fourth, Merton’s continuous-time theory has been around since the 1970s but is less well known and less used than one-period models like CAPM. That is a shame because it gives genuinely useful insights about dynamic portfolio management that static models miss.

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