The 7 Most Important Equations for Your Retirement

For a math guy like me, Moshe Milevsky’s The 7 Most Important Equations for Your Retirement was a must-read. Milevsky devotes a chapter to each of the equations most important to retirement planning and does his best to explain the equation to non-specialists as well as tell the interesting history of the people behind the development of these equations. This book is ideal for readers interested in what lies behind retirement planning software.

The equations Milevsky includes answer questions like how long will my money last, how long will I live in retirement, what is a pension annuity worth, and is my current plan sustainable? There are no precise answers to these questions, but the equations take into account this uncertainty.

For very mathematical readers, the book tends to gloss over fine details. For example, the first equation that computes how long a pot of money will last with a fixed interest rate and fixed withdrawals is a continuous approximation whose interest rate parameter is actually an instantaneous interest rate. However, these finer points will have little effect on any practical retirement plan.

An interesting fact I learned was that “people who voluntarily purchase life annuities actually live longer than the rest of the population.” The gap is about 5 years. It’s not clear whether people actually have some sense of their longevity or whether having an annuity somehow makes you live longer. I’d imagine that some annuity salespeople would make this part of their sales pitch.

One equation that I didn’t find very useful is one that calculates your asset allocation percentages for stocks and fixed-income investments. One of the inputs into this equation is your human capital, which is the present value of future earnings. This figure has a high degree of uncertainty and cannot be distilled into a single number, particularly for young people.

In addition to the uncertain human capital, the asset allocation equation makes the problematic assumption that you can borrow at the risk-free rate. You can’t, particularly for large loans. Another built-in assumption is that stock prices follow a normal curve. They don’t, and this severely limits the amount of leverage that makes sense. I’m also leery of the “isoelastic marginal utility assumption,” which is mathematically convenient, but hard to justify.

A result of these problems is that the asset allocation equation gives some wild answers. For example, even an extremely risk-averse young person in one example shown in the book is recommended to be leveraged 4:1 in stocks. This means taking $50,000 in savings and borrowing another $150,000 to invest $200,000 in stocks. A less risk-averse young person in the same situation has optimal leverage of 31.5:1!

Despite these criticisms of one of the equations, I quite enjoyed the book and recommend it to anyone with some mathematical curiosity and interest in planning for retirement.