What saving builds
Two engines run at once: the balance already there, compounding, and a stream of new contributions, each compounding for a shorter time than the one before.
- Compute the future value of a balance plus level contributions
- Split an ending balance into what was paid in and what compounding added
- Find the contribution a target needs
- Put a price on starting late
The intuition
A client has $50,000 saved and adds $10,000 at the end of every year for thirty years, earning 6%. Along the way they put in $350,000. They end with about $1,078,000. Two thirds of the final balance was never contributed; it was earned by money that had already been contributed. Early on the contributions do almost all the work; after a couple of decades the growth on the balance overtakes them.
That is why starting late costs so much more than the missing contributions. Skip the first ten years and the twin ends about $423,000 behind, having skipped $100,000 of deposits: the dollars skipped were the ones that would have compounded longest. The formula has two parts, the starting balance grown and the annuity factor that adds up a stream of level contributions, and both run backwards: the contribution a target needs, and the years a target takes.
FV = PV × (1 + r)ⁿ + PMT × ((1 + r)ⁿ − 1) ÷ r, with contributions at year end. Total contributed = PV + PMT × n; growth is the rest. PMT for a target = (target − PV grown) ÷ annuity factor.
Why it works
- The conventions here: a starting balance plus a level contribution at the end of each year, a constant return, annual compounding. Nominal throughout: no tax, fees or inflation (the next chapter handles inflation). The late-starting twin keeps the same starting balance invested for the full period and begins contributions later, ending on the same date.
- The annuity factor is a sum of growth factors. The first contribution compounds for n − 1 years, the last for none; ((1 + r)ⁿ − 1) ÷ r adds them all up. It is what turns a stream into a lump.
- Growth overtakes contributions, because each year's growth is earned on everything already there. Over a long career most of the final balance is growth.
- Start-of-year contributions earn one more year each, so the contribution part is (1 + r) times larger. Paying in on the first day rather than the last is a small, free improvement.
- Time is the only lever that costs nothing. The others are the return, which the client does not control, and the contribution, which they feel every month.
- Ten years early can beat thirty years late. Contributions made from 25 to 35 and then stopped can outgrow the same contributions made from 35 to 65, depending on the return, because they compound for thirty extra years.
| Growth factor: 1.06³⁰ | 5.7435 |
| Starting balance grows to: $50,000 × 5.7435 | $287,175 |
| Annuity factor: (5.7435 − 1) ÷ 6% | 79.058 |
| Contributions grow to: $10,000 × 79.058 | $790,582 |
| Future value | $1,077,756, of which $350,000 was paid in |
| Twin, 20 years of contributions: $287,175 + $10,000 × 36.786 | $655,030, so $422,726 behind for $100,000 skipped |
The shortfall is 4.2 times the contributions skipped. The skipped deposits were the earliest, the ones with the longest to compound.
The formulas
The balance grown, plus the stream of contributions added up.
What went in, and what compounding added.
What is left after the balance grows, spread over the annuity factor.
The formula solved for n with logs.
Worked example
Grow the starting balance, then add the future value of the stream of contributions through the annuity factor. The follow-up says what start-of-year contributions would change.
See it move
Same client. Change the starting balance, the yearly contribution, the return, the number of years, and how many years the late-starting twin waits.
- Raise the return. Growth and the future value rise; what was paid in does not move.
- Raise the contribution. Paid in and the future value both rise.
- Add years. Everything rises, and the growth bar rises faster than the paid-in bar: compounding overtakes contributions.
- Make the twin wait longer. The late-start balance falls and the cost of starting late grows, by far more than the contributions skipped.
Run it backwards
A target and a horizon are known. Take off what the starting balance will grow to; the contributions must fund the rest.
The starting balance grows to PV × (1 + r)ⁿ on its own. What remains of the target is divided by the annuity factor to give the level contribution.
The follow-up turns it into a monthly figure and asks what happens if the client can only manage less: the levers are time, return and contribution, and only time is free.
Traps
Say it in the interview
“How much will this client have at retirement, and what is the cost of starting late?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- FV = PV (1 + r)ⁿ + PMT × annuity factor; the factor adds up the stream.
- Growth overtakes contributions; most of a long career's balance is growth.
- Contribution for a target = (target − PV grown) ÷ annuity factor.
- Starting late costs several times the deposits skipped.