What a pot can pay out
A retirement portfolio is a race between what it earns and what the client takes out. Below the real return the money lasts forever; above it, the decline accelerates.
- Compute a withdrawal rate and the spending a rule allows
- Work out how long a pot lasts at a steady real return
- Size the portfolio a spending goal needs under a rule
- Find the most a pot can pay for exactly N years, and say why the 4% rule is lower
The intuition
A retiree has $1,500,000 and wants $60,000 a year, rising with inflation. That is a 4% withdrawal rate. If the portfolio earns 2% real, it makes $30,000 a year and the other $30,000 comes out of capital; next year the capital is smaller, earns less, and the gap widens. At a steady 2% real the money lasts about thirty-five years. Take $30,000 a year instead and it lasts forever.
The famous 4% rule is a rough answer to a harder question: how much can I take, rising with inflation, and survive a bad sequence of markets over thirty years? Constant-return arithmetic understates the danger, because it never has a crash in year two. But it is where planning starts, and it shows how sensitive the plan is to the return. The same formula run backwards gives the most a pot can pay for exactly thirty years, which is always more than a safe rule allows, for a reason worth being able to explain.
Withdrawal rate = spending ÷ portfolio. Portfolio needed under a rule = spending ÷ rule rate. If spending exceeds the real return on the pot, years it lasts = −ln(1 − r × V ÷ W) ÷ ln(1 + r); if not, forever. Most a pot can pay for N years = V × r ÷ (1 − (1 + r)^−N).
Why it works
- The conventions here: everything in real terms, so spending rises with inflation and the return is the real return, constant every year. Withdrawals come out at the end of each year. The safe withdrawal rate is a stated planning rule, 3%, 3.5% or 4% of the starting portfolio, not derived here. No taxes, fees or sequence of returns.
- The line that divides forever from finite is r × V. Spending at or below the real return leaves the capital intact. Every dollar above it is paid from capital, and the capital then earns less next year.
- Years it lasts is the annuity formula solved for n. It blows up as spending falls toward r × V, which is the formula's way of saying forever.
- The most a pot can pay for N years is a loan payment in reverse: the pot is the loan, the client is the lender, the payment is what runs it to zero on the last day.
- That is always more than the 4% rule, because it assumes the same return every year. Real returns arrive in a random order, and a bad run early, when withdrawals are selling assets at low prices, does damage an average hides. The rule was set to survive those sequences.
- Advisors manage sequence risk three ways: a cash or short-bond buffer for a few years of spending, a glide path most conservative at the start of retirement, and flexible spending rules that trim withdrawals after bad years.
| Withdrawal rate: 60,000 ÷ 1,500,000 | 4.00% |
| Spending a 3.5% rule allows: 3.5% × $1,500,000 | $52,500 |
| Portfolio a 4% rule needs for $60,000: 60,000 ÷ 4% | $1,500,000, 25 times spending |
| Years the money lasts at 2%: −ln(1 − 30,000 ÷ 60,000) ÷ ln 1.02 | 35.0 years |
| At 1% real instead | 28.9 years |
| Most it can pay for exactly 30 years: 1,500,000 × 2% ÷ (1 − 1.02⁻³⁰) | $66,975, a 4.46% rate |
One point less return costs six years. The 30-year maximum is above the 4% rule because it assumes every year returns exactly 2%.
The formulas
The first number every planner checks.
The rule inverted; 4% means 25 times spending.
The annuity formula solved for n; forever if W ≤ rV.
A loan payment in reverse.
No return, so the pot is simply divided.
Worked example
First check that spending exceeds the real return on the pot; otherwise it lasts forever. Then the annuity formula solved for n. The follow-up says why it blows up at r × V.
See it move
Same retiree. Change the portfolio, the spending, the real return, the years the money must last and the rule rate.
- Raise the real return. The most the pot can pay for the stated years rises; the rule's spending does not move, because the rule is a fixed share of the pot.
- Lengthen the years the money must last. Sustainable spending falls, and flattens: it can never fall below the real return on the pot, which lasts forever.
- Raise the spending wanted. The withdrawal rate rises; the other two bars do not move.
- Raise the rule rate. The rule's spending rises and the portfolio the rule needs falls.
Run it backwards
The pot, the return and the years are known; the money must run to zero on the last day. What is the most it can pay each year?
It is a loan payment in reverse: V × r ÷ (1 − (1 + r)^−N). The pot is the loan, the client is the lender, and the payment exhausts it exactly at N.
The follow-up asks why the 4% rule is lower than this: because this assumes the same return every year, and real sequences have bad runs early.
Traps
Say it in the interview
“This client wants to spend $60,000 a year from $1,500,000. Is that sustainable?”
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.
- Spending at or below r × V lasts forever; above it, years = −ln(1 − rV ÷ W) ÷ ln(1 + r).
- Portfolio needed under a rule = spending ÷ rule rate; 4% means 25 times spending.
- Most for N years = V × r ÷ (1 − (1 + r)^−N), a loan payment in reverse.
- The 4% rule is below that because of sequence risk.