Investing on an Infinite Time Horizon

Two weeks ago when my parents met with their retirement adviser, he recommended that they reallocate some of their stock portfolio into bonds as they are drawing ever closer to their retirement age. My father, familiar with my knowledge on finance and markets, consulted me about the move before making his final decision: I ended up agreeing with the adviser's assessment for allocation away from risk. However, it occurred to me that it is kind of silly that they should miss out on this potentially lucrative investment returns of the stock market simply because their age dictates they should seek low risk, capital sustaining investments. It got me wondering if it was possible to ignore the Life-Cycle Theory and achieve more optimal investment choices. Specifically, if the investment time horizon is infinite, the risk of holding stocks is essentially eliminated and "excess" returns can be made.

The idea of life-cycle investing can be described by Burton Malkiel in his book A Random Walk Down Wall Street. Malkiel, like most life-cycle investors, concludes:

It is this fundamental truth that makes a life-cycle view of investing so important. The longer the time period over which you can hold on to your investments, the greater should be the share of common stocks in your portfolio. In general, you are reasonably sure of earning the generous rates of return available from common stocks only if you can hold them for relatively long periods of time.

Malkiel also includes the following chart for those who would like to visualize this concept:

By defining investment "risk" as the standard deviation of average return, there seems to be some form of risk-mitigating that can come from holding stocks for a greater amount of time. The ability to hold stocks for 25, 50, possibly even 100 years would allow investors to continue to reduce the standard deviation of their average annual returns, potentially even eliminating it.

However, the definition of investment risk may not be appropriate, as the followers of the Fallacy of Time Diversification argue. Their argument holds that investors do not care about their "average returns" but on the ending value of their portfolio; and while the average return can be useful to calculate where a likely portfolio value will be, they prove that the variance inherent in stock returns actually increases the range of ending portfolio values. Peter Haggstrom uses math to show how, when total return is considered, time actually increases risk:

Unwittingly some proponents of the argument may be signing up to the following claim: standard deviation of the annualised return = standard deviation of the total return. For a time scale of more than 1 year, this equation is false...

       

This bit of math proves that as the time horizon increases (t increases), so too does the standard deviation of total return. However, while it may be true that investors seek to maximize their ending portfolio value given a finite time period, if investors were to consider their time horizon as extending infinitely into the future, this problem goes away. In a continuously compounding portfolio, where the ending value of the portfolio is defined as the sum of all returns on the portfolio between time 1 and time t, when t goes to infinity so too does the value of the portfolio. Not only is there not an "ending point" of the portfolio to maximize against, but there is no ability to maximize (since it is infinity).

The question now returns to what function that investors should seek to maximize, given that their "original" portfolio value function cannot be maximized. It now makes sense that investors should seek to increase the rate at which they are moving toward this infinite portfolio value; or, in other terms, they should seek to maximize the average sum of all returns on the portfolio between time 1 and time t. Once again, we have returned to the maximization of rate of return (which has already been proved mathematically have reduced, and even eliminated, risk when there is an infinite time horizon). Investors, while seeking to maximize this rate of return, should increase their holding of those assets which have historically produced the highest returns; if you constrain these investment options to only stocks and bonds, it would make most sense for these infinite time horizon investors to hold 100% in equities at all times.