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.

It is Winter break. Having recently finished reading Boyd and Vandenberghe’s Book on Convex Optimization I decided to take time out to read Hagen Kleinert’s paper on the Lévy distribution in finance. There’s a more complete description in Ch20. The paper shows that both the short- and long-time behavior of returns on the S&P500 index look like a Lévy distribution,

$$P(\Delta x|t) = \int_{-\infty}^\infty \frac{dp}{2\pi} e^{ip\Delta x - t H(p)}, \quad H(p) = -|c p|^\lambda $$ Here, \(\Delta x = x_t - x_0\) is the change in \(x\) over time \(t\).

This is the symmetric version of the general ‘Stable’ Lévy distribution implemented in scipy. Kleinert uses the alternate notation, \(\sigma^\lambda/2 = c^\lambda\).

It’s called stable because the characteristic function is obviously \( H(p) \), and adding random variables multiplies their characteristic functions together (the CF of the sum is the product of the CFs).

So, using \( dx(t) = \log(S(t+dt)/S(t)) \sim \) Lévy\((\lambda,c)\), \(\int_0^T dx \sim\) Lévy\((\lambda,T^{1/\lambda}c)\) is then the distribution of \(\log(S(t+T)/S(t))\)

It’s conceptually useful, and the graphs look nice, so what can we do with it?

Well, gains in the market from fancy prediction algorithms widely use Gaussian correlations (that is, Einstein’s Brownian motion on harmonic oscillators). Those correlations would allow you to ‘know’ the direction a stock is moving and then buy or sell to make a profit.

The Lévy distribution itself doesn't help with that directly. Instead, it is supposed to model the tails of the distribution better. Fits to real data show \(\lambda \sim 3/2\), and the tails fall off as \(\Delta x^{-5/2}\). That means the variance is infinite and Black Swan events are present in the model.

Using Bayesian methods to get correlations under the presence of Lévy distributions might help, but again the correlations are infinite, so we need to check other distributions (truncated Lévy, gamma, etc.).

Because it models fat tails, the Lévy distribution would be ideally suited for market risk analysis – for example to optimize a portfolio’s returns under a constraint that the probability of negative returns is less than one tenth of a percent or something. This brings us to Boyd and Vendenberghe’s book. Their ‘Example 4.8’ (p. 158) is a variant of the Markowitz portfolio optimization problem.

It might also help predict changes in a portfolio over time to help decide on a long-term allocation strategy. This latter problem is commonly thought of as retirement planning, and brings us to the next post – which will be on optimal control theory.