Chapter 6

Investing & Retirement

Compound interest, monthly contributions, and the accounts that hold them

Every other chapter here is about money leaving. This one is about the only mechanism that reliably works in the other direction, and about the single input that matters most to it, which is time.

Chapter 6 · The idea

Growth multiplies, it does not add

Interest earned becomes principal, and then earns interest itself. That is the whole of it — but it means growth is repeated multiplication rather than repeated addition, and repeated multiplication does something to a number over forty years that is genuinely hard to guess at.

A = P(1 + r)^t

Before running the numbers by hand, move the sliders. Watch what the years do on their own, with the deposit and the rate held still.

Figure 4The solid curve is a deposit made at 18. The faint one is the same deposit made ten years later — same money, same rate, ten fewer years. The gap between them at 65 is what the decade was worth.

Chapter 6 · Worked problem

Chloe's summer job, forty-seven years later

The problem

At 18, Chloe puts $3,000 from a summer job into a Roth IRA earning an average 8% a year. She never adds another dime. What is it worth when she retires at 65?

Working1 / 5

Compound interest, compounding once a year $A = P(1 + r)^t$

Chapter 6 · In your head

The rule of 72

Compound growth is hard to feel because the interesting part happens at the end. The rule of 72 makes it feelable: money doubles in about 72-divided-by-the-rate years. At 8% that is nine years; at 4%, eighteen; at 12%, six. It is an approximation, and a good one between about 4% and 12%.

Years to double, approximately

t_{\times 2} \approx \frac{72}{\text{rate in \%}}

The problem

Chloe's $3,000 at 8% for 47 years, without a calculator. How close does the rule get?

Working1 / 4

At 8%, one doubling every nine years $\frac{72}{8} = 9$

The exact answer was $111,696. The rule gets within a few hundred dollars in your head — close enough to notice that the last doubling, from $48,000 to $96,000, made more money than the first four together. That is the shape of compounding, and it is why the years at the end are the expensive ones to lose.

Chapter 6 · Every month

Adding to it every month

Nobody invests once at 18 and stops. The realistic version is a fixed amount every month, and each deposit compounds for however long it has left — so the total is a sum of forty years of separate deposits, each on its own curve. There is a closed form for that sum.

FV = m \times \frac{(1 + r)^n - 1}{r}

The monthly deposit

m

The monthly rate, and the number of months

r = \frac{\text{annual rate}}{12},\quad n = 12 \times \text{years}

The problem

$200 a month, from 25 to 65, at 8% a year. What does it come to, and how much of it was deposited?

Working1 / 5

Forty years of months, at a twelfth of 8% $n = 480,\quad r = 0.006667$

Chapter 6 · The wrappers

Three wrappers, one engine

A 401(k), a traditional IRA and a Roth IRA all hold the same investments and compound the same way. What differs is when the tax is paid — before the money goes in, or when it comes out — and, with a 401(k), whether an employer adds to it.

The wrappers

401(k)
Through an employer, pre-tax: the contribution comes off taxable income now and is taxed on withdrawal. Many employers match part of it.
Traditional IRA
Your own account, pre-tax on the same terms, with a lower annual limit.
Roth IRA
Your own account, post-tax: no deduction now, and no tax on any of the growth later. The right one for Chloe at 18 — her tax rate now is close to nothing, and forty-seven years of growth comes out untaxed.

The problem

Jordan's employer matches 50% of contributions, up to 6% of the $60,000 salary. If Jordan contributes the full 6%, what does the match alone become over forty years at 8%?

Working1 / 4

Six percent of the salary, per year $60000 \times 0.06 = 3600$