What those dollars will buy
A retirement plan is a promise about purchasing power, not dollars. Inflation converts one into the other, and tax on the nominal return makes it worse.
- Compute an exact real return, and say how far the shortcut is off
- Inflate today's spending to the year it is needed
- Find the nominal return a real target needs
- Compute an after-tax real return and say why it can be negative
The intuition
A client's portfolio earned 6% last year while prices rose 3%. They are better off, but not by 6%: the dollars they gained buy less than the dollars they started with. The real return is what is left after prices rise: 2.91%, a little under the 3% the shortcut gives, because inflation also eats part of the gain. At 3% inflation, prices double in about 23 years; a fixed pension will buy half as much by then.
The second step is the one clients miss. In a taxable account, tax is charged on the nominal return, including the part that only kept pace with inflation. A bond yielding 6% in a 32% bracket keeps 4.08% after tax; against 3% inflation that is about 1.05% real, and a smaller gap between yield and inflation turns it negative. A plan built in nominal dollars with today's spending goal is the most common way plans look better than they are.
Real return = (1 + nominal) ÷ (1 + inflation) − 1, which is nominal − inflation less a small correction. Future cost of today's spending = spending × (1 + inflation)ᵗ. Nominal needed for a real target = (1 + real)(1 + inflation) − 1. After-tax real = (1 + nominal × (1 − tax)) ÷ (1 + inflation) − 1.
Why it works
- The conventions here: constant inflation and returns, annual compounding. The real return uses the exact Fisher relation; the shortcut nominal − inflation is shown for comparison. Tax is charged on the nominal return each year at the stated rate, as interest income in a taxable account is.
- Divide the growth factors; do not subtract the rates. The error in the shortcut is (nominal − inflation) × inflation ÷ (1 + inflation): small at low rates, half a point a year at 15% nominal and 10% inflation, and it compounds.
- Work in real dollars. Real returns with today's-dollar goals keep the targets recognisable to the client. A nominal projection against a today's-dollar goal is the classic way a plan overstates itself.
- Prices double in ln 2 ÷ ln(1 + inflation) years; the rule of 72 is the quick check. A level, non-indexed pension halves in purchasing power over that span.
- Tax is charged on the inflation part too. The after-tax nominal return is nominal × (1 − tax); only then is inflation taken out. That is why a yield comfortably above inflation can still lose purchasing power once the bill arrives.
- Asset location is the fix: interest-paying assets in tax-deferred or exempt accounts where possible; inflation-linked or tax-exempt bonds in taxable ones. Which account holds which asset is often worth more than fine-tuning the mix.
| Exact real return: 1.06 ÷ 1.03 − 1 | 2.913% |
| Shortcut: 6% − 3% | 3.00%, 8.7 basis points too high |
| Spending in 20 years: $60,000 × 1.03²⁰ | $108,367 |
| Nominal needed for 3% real: 1.03 × 1.03 − 1 | 6.09% |
| Years for prices to double: ln 2 ÷ ln 1.03 | 23.4 (the rule of 72 says 24) |
| After-tax real: 6% × (1 − 32%) = 4.08%, then ÷ 1.03 | 1.049% |
Tax took the real return from 2.91% to 1.05%, because it was charged on the whole 6%, including the 3% that only kept pace with prices.
The formulas
Divide the growth factors; the shortcut is close at low rates.
Today's spending, inflated to the year it is needed.
The Fisher relation run backwards.
When a fixed pension buys half as much.
Tax first, on the whole nominal return; then inflation.
Worked example
Divide the growth factors for the exact real return, then compare with the shortcut. The follow-up says when the shortcut becomes dangerous.
See it move
Same client. Change the nominal return, inflation, the years until retirement, today's spending, the tax rate and the real return the plan assumes.
- Raise inflation. The real and after-tax real bars fall, the future cost rises, and prices double sooner; the nominal bar does not move.
- Raise the nominal return. Real and after-tax real rise; the future cost of spending does not move, because it depends on inflation alone.
- Raise the tax rate. Only the after-tax bar moves, and it falls. Push it until it goes below zero: a yield above inflation that still loses purchasing power.
- Raise the real return the plan assumes. The nominal return it needs rises by more than the change, because the two compound.
Run it backwards
The plan assumes a real return and inflation is known. What nominal return must the portfolio earn?
Multiply the growth factors: (1 + real) × (1 + inflation) − 1. It is a little more than real + inflation, for the same reason the shortcut is a little too generous the other way.
This is the number to compare with the portfolio's expected nominal return from the Portfolio Allocation lesson.
Traps
Say it in the interview
“This bond yields more than inflation. Is the client's money keeping its value?”
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.
- Real = (1 + nominal) ÷ (1 + inflation) − 1; divide, do not subtract.
- Build the plan in real dollars with real returns.
- Tax hits the nominal return, inflation part included: after-tax nominal first, then inflation.
- Prices double in ln 2 ÷ ln(1 + inflation) years.