A (mathematically) sound investment strategy

It is common wisdom in the investment community that a long-term investor saving for his future would do well to invest in high-risk/high-return assets when he is young, slowly switching his portfolio over to low-risk/low-return assets as he grows older. I have never seen a mathematical demonstration of this, and would be interested in finding one.

This isn't obviously a mathematical question, but since econ.stackexchange.com doesn't exist yet, I'm asking it here.

We might begin to formulate the question along the following lines: for each year $t=0,1,\dots,T-1$ of his life an investor chooses to put a portion of his total wealth $W_t$ into either a risk-free bond, earning interest at rate $r$, or into a risky asset whose return is a random variable with mean $\mu>r$ and variance $\sigma^2$ (we could say it's a normal random variable, for simplicity). Let's say that at time $t$ he invests a fraction $\phi_t$ in the risky asset and $1-\phi_t$ in the riskless bond.

He is also able to invest an additional amount $P_t$ into his portfolio at time $t$ (which comes, for example, from his salary). We could take $P_t=P$ deterministically to begin with, and later generalize to non-homogeneous or stochastic $P_t$.

Then the question becomes: for a given level of risk-taking, what strategy $\{\phi_t\}$ maximises his expected wealth at time $T$? As proxies for the expected wealth and level of risk, we could take $E(W_T)$ and $\mathrm{Var}(W_T)$. A common approach is to introduce the Lagrange multiplier $\lambda$ and solve the unconstrained optimisation problem

$$\max_{\phi} \, E(W_T) - \lambda\mathrm{Var}(W_T)$$

to create an `optimal frontier' of strategies in $(E(W_T), \mathrm{Var}(W_T))$ space, although this isn't necessarily the best strategy for solving. I haven't taken this idea much further than this, and would welcome any comments or suggestions.

I have now given this some additional thought, and added my progress in an answer below. This is not complete though, and I'd welcome further input!

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It's not obvious to me that this "common wisdom" is directed towards maximizing expected wealth. Minimizing risk is also an issue as you get older, especially if you plan on supporting a family. (This is generally my issue with asking these kinds of questions on a mathematics site: it's never clear that you've found the right question to ask.) – Qiaochu Yuan May 9 '11 at 23:33
I agree, that is why I have phrased the question as "for a given level of risk..." - to try and capture some of the risk-averting behaviour. I don't know if the solution to the problem as stated will recover the "common wisdom" but I think it will be interesting whether it does or does not - either there is a simple mathematical argument that corroborates the known wisdom, or the model must leave put some important facet. – Chris Taylor May 9 '11 at 23:58
I feel you've modified the question in a suitable way for this site. It's very interesting, +1. – fdart17 May 10 '11 at 0:21
One element that hasn't been explicitly pointed out, though it influences the notion of risk-taking, is the nonlinearity of utility/value; a 10% chance of making a million dollars is substantially less useful for most people than a 100% chance of making a hundred thousand dollars is, even if both provide the same EV. – Steven Stadnicki May 10 '11 at 2:05
@Chris: I'm afraid that was me. I see that you have put a lot of thought into your question, and it is certainly formulated in mathematical language, but my concern is the same as Qiaochu's, namely that it is impossible to use only mathematics to "derive" what would be the best course of action for someone, because doing so necessitates deciding what "best" means (i.e., are we asking the right question), which is a non-mathematical (and IMHO essentially unanswerable) issue. I am sure your question is perfect for a hypothetical econ.stackexchange site, but I felt it was not a good fit here. – Zev Chonoles May 10 '11 at 19:52

I think a more sensible approach is to take a continuum approximation. To avoid confusion I will take $S_t$ to be the amount of money you have in your portfolio at time $t$, with $\phi_t$ invested in an asset paying a risky return with mean $\mu$ and volatility $\sigma$, and $1-\phi_t$ invested in a riskless bond paying return $r$. I also assume that you pay an amount $p_t$ per unit time into the portfolio. Then the portfolio process $S_t$ satisfies

$$dS_t = p_t dt + (1-\phi_t) r S_t dt + \phi_t \mu S_t dt + \phi_t \sigma dW_t$$

where $W_t$ is a standard Brownian motion, and the terms represent money paid in, interest from the bond, return from the stock and volatility from the stock (the risky term). Defining

$$\Phi_{s,t} = \exp \left( \int_s^t (r + (\mu-r)\phi_u + \tfrac{1}{2} \sigma^2 \phi_u^2) du + \int_s^t \sigma \phi_u dW_u \right)$$

we can write the solution for the value of the portfolio at time $t$ as

$$S_t = S_0\Phi_{0,t} + \int_0^t p_s \Phi_{s,t} ds$$

The question is now how to calculate the expectation and variance of such a portfolio, as a functional of the asset allocation process $\phi_t$, and then formulate and solve an Euler-Lagrange equation for $\phi_t$.

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You should try posting your question here: link Lots of financially-inclined mathematicians read/contribute to this site – ItsNotObvious May 11 '11 at 21:57