That person, who would seem to be a perfect age to benefit from a rising stock market in the post-World War II era, by dint of unlucky timing was in rough shape. A person who started saving in 1928, right on the cusp of the Great Depression, actually did better; the stock market returned a bit less in the 30 years starting in 1928 than in the 30 years starting in 1951, only 9.5 percent per year. But because the returns were backloaded, a person who started saving $10,000 a year then ended up with $1.96 million in the bank, almost four times as much as the person who started in 1951.

One other scenario year is interesting just for the dramatic disconnect it shows. It involves those fortunate enough at the outset to have a pile of cash to invest. The period from 1975 to 2005 was a stunning three decades for the American stock market, returning 13.7 percent per year. A person who socked away $300,000 in the S&P 500 at the start of 1975 and reinvested all dividends walked away with $14.2 million in 2005.

But the pattern of returns was distinctly unfavorable for someone who was a slow and steady saver throughout that period. Some of the years of the sharpest gains came early in that window, like the 38 percent return in 1975 and 24 percent in 1976. And then the last few years were weak, encompassing three straight years of sharp declines to start the 2000s.

As a result, our person who invested $10,000 each year walked away with $1.59 million come retirement — nothing terrible, certainly; it was almost impossible to hold stocks during that 30 years without making money. But hardly $14 million.

A final note. You can’t control any of this. We all happen to be born when we are born, and the future returns of the stock market are unknowable. As an investor, it is generally best to focus on the things you can control, like how much you save and whether you are putting money in investment vehicles with low fees that are tax efficient.

But the luck of when you are born may have a bigger impact on how much you have for retirement than you might like.