Before trying to optimize a portfolio, it seems like we should take a moment to figure out how to deal with risk tolerance. Closer to retirement, it is well-known that you should switch out of high-risk stocks and into low-risk bonds to ensure there will be enough to spend without having to go back to work. Of course, in the long term spending and savings are linked so it’s also possible to avoid losses by spending less. How do we actually settle on a spending and asset allocation strategy?

Optimal control theory provides one way to solve this problem. In a given year, \(t\), say you have known income, \( I_t \), a starting amount of assets, \(W_t\), and some fixed minimum costs of living, \(C^0_t\). You now have to decide how much total spending, \(C_t\), you can afford. Whatever is unspent will be invested and become \(W_{t+1}\) next year. For simplicity, assume it grows with fixed rate \(R_t\).

$$ W_{t+1} = e^{R_t} \, (W_{t} + I_t - C_t) $$

The catch is that each year you have to trade off future spending for present spending. If we measure discretionary spending by the utility function, \(u(C) = \log(C)\), then we are trying to optimize total utility over years 0 through T,

$$ V(\{C_t\}, \mu) = \sum_{t=0}^T u(C_t - C^0_t) + \mu W_T.$$

This objective function contains a Lagrange multiplier, \(\mu\) to enforce a fixed final wealth, \(W_T = 0\). In a continuous case, we can integrate

$$ dW/dt = R W(t) + I(t) - C(t) ,$$ to give $$ W(t) = e^{Rt} \left( W(0) + \int_0^t d\tau e^{-R\tau}(I(\tau) - C(\tau)) \right) .$$

The optimal spending profile is then,

$$ C_t = C^0_t + \mu^{-1} e^{R(t-T)} .$$

It says that discretionary spending should increase exponentially at the market rate! The constant, \(\mu\), can be worked out from \( W(T) \).

It turns out this type of analysis is standard in economics, where it is known as the ‘perfect foresight’ model. It has several known issues. First, if \(W(t)\) dips below zero, it means that we have to borrow at rate \(R\), which is not usually possible. Second, it does not take into account market risk, so we should do something statistical to fix that.

Third, and possibly most importantly for ordinary folks, future income is not known exactly. Losses in future income are also hard to insure. The monetarist Milton Friedman argued that uncertainty in future incomes should change spending vs. saving preferences significantly in A Theory of the Consumption Function. There’s a nice paper from Christopher Carroll that shows how this would work in practice. I’ll write more about it in the next post.