Tomorrow is March 14, or Pi Day (3.14), as it’s known in the Internet’s more mathematical precincts.

Pi allows us to design aircraft engines, pass a geometry midterm, and enjoy Thanksgiving dinner. It can also teach us something about investing.

Rational use of an irrational number

Pi is the most famous irrational number. It unspools in an infinite string of decimals, an abstraction that never hardens into a concrete number. If we want to calculate the area of a circle, we need to use an approximation of Pi, a rational number such as 3.14 or 22/7.

Investors must make a similar trade-off between precision and utility, between theoretical possibility and actionable reality.

The asset allocation parallel

Consider the most important investment decision, asset allocation. What is the right asset allocation for a particular investor with a particular goal such as a financially secure retirement?

Each of us has risk different preferences, suggesting that each of us has a different optimal portfolio. How can we derive the optimal portfolio associated with our unique goals and preferences?


We simply estimate the expected returns, variances, and covariances of the different asset classes and use mean-variance-optimization techniques to identify the best portfolios. As new information about valuations, interest rates, and earnings growth become available, we can re-estimate those estimates and derive new best portfolios.

In the meantime, of course, our preferences may have changed in response to the passage of time, events, or even our mood. So we need to make sure that the newly derived portfolio remains consistent with our preferences.

The complexities compound. The precisely right answer seems ever more elusive. And yet we still have to pick a portfolio.

Rational portfolios

Like users of Pi, we can rely on good approximations. What is the 3.14 of the asset allocation decision? For many investors, it’s a low-cost target-date fund pegged to their expected retirement date.

Maybe there’s a better, customized allocation for each investor. Maybe not. The uncertainty associated with any asset class forecast makes that just about impossible to know. What’s clear is that target-date funds and other professionally managed asset allocations have delivered better and less varied results than what investors have achieved without them.

As with the calculation of Pi, the journey from irrational to rational means the loss of some precision. But in math and investing, approximately right can help us meet our goals while liberating us from the ceaseless—and costly—pursuit of theoretical perfection.





  • Please remember that all investments involve some risk. Be aware that fluctuations in the financial markets and other factors may cause declines in the value of your account. There is no guarantee that any particular asset allocation or mix of funds will meet your investment objectives or provide you with a given level of income.
  • Investments in Target Retirement Funds are subject to the risks of their underlying funds. The year in the Fund name refers to the approximate year (the target date) when an investor in the Fund would retire and leave the work force. The Fund will gradually shift its emphasis from more aggressive investments to more conservative ones based on its target date. An investment in the Target Retirement Fund is not guaranteed at any time, including on or after the target date.