The exact formula, the shortcut and why it drifts, and a forty-year savings pot restated in today's money at five inflation rates.
Divide, do not subtract: real return = (1 + nominal return) ÷ (1 + inflation) − 1. At an illustrative 7 per cent return and 2.5 per cent inflation that is 4.39 per cent a year, not 4.5. Over forty years the difference compounds: a pot of 738,846 buys what about 275,169 buys today, 37.2 per cent of the number on the statement.
Worked in full in Stock Market Investing for Beginners by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Every projection a beginner is shown is in nominal money, the figure the account will display. What matters is what that figure will buy. The conversion takes one division, and the shortcut most people use instead is close enough for a single year and wrong enough over a working life.
The case is the forty-year saver of the book's fee illustration: 3,600 a year (300 a month) paid in at the start of each year from 25 to 64, growing at 7 per cent before a low yearly cost of 0.15 per cent. All figures are illustrative, not a forecast.
| Input | Value |
|---|---|
| Return before costs, each year | 7.00% |
| Yearly cost | 0.15% |
| Inflation, each year | 2.50% |
| Contribution, each year | 3,600 |
| Years | 40 |
Start with 10,000. At 7 per cent it grows to 10,700. A basket of goods that cost 10,000 now costs 10,250. So the 10,700 buys 10,700 ÷ 10,250 of the original baskets, which is 1.0439 of them, or 10,439 in today's money. The real gain is 439, not 450.
Real return = (1 + r) ÷ (1 + i) − 1
= 1.07 ÷ 1.025 − 1 = 1.0439 − 1 = 4.39% (4.3902% unrounded)
The shortcut r − i gives 4.50%, which overstates it by 0.11 of a point.
Excel, with the return in B1 and inflation in B2: =(1+B1)/(1+B2)-1
The shortcut fails because inflation also erodes the year's gain, not just the starting capital. The error is roughly the real return times the inflation rate, small at low rates and larger when both are high.
To turn a nominal figure at the finish into today's money, divide by the price level after forty years: (1 + i)40 = 1.02540 = 2.6851. Something that costs 1 today costs 2.69 then.
Value in today's money = nominal value ÷ (1 + i)n
Nominal pot after forty years, each year (balance + 3,600) × (1 + 0.07 − 0.0015) = 738,846
738,846 ÷ 2.6851 = 275,169
Excel: =B3/(1+B2)^40
Inflation took 62.8 per cent of the headline. That is not a loss in the usual sense, since prices and probably wages will have risen too, but it is the right number to hold in your head when you ask whether a plan is enough.
Costs come off the nominal return before the division, so the exact figure is (1 + r − c) ÷ (1 + i) − 1 = 1.0685 ÷ 1.025 − 1 = 4.24 per cent, against 4.35 by subtraction. This is the rate to use when you project in today's money from the start.
Run the forty years at 4.24 per cent with the contribution held at 3,600 in today's money and the result is 377,817, also in today's money. It is higher than 275,169 because it describes a different saver: one who raises the contribution with inflation each year, as most people do when their pay rises. In cash that saver pays in 242,649 rather than 144,000, and the statement at 65 shows 1,014,464.
| Case | Paid in | Nominal pot | Today's money |
|---|---|---|---|
| Flat 3,600 a year, low cost | 144,000 | 738,846 | 275,169 |
| Flat 3,600 a year, cost one point higher | 144,000 | 567,961 | 211,526 |
| Contribution rising 2.5% a year, low cost | 242,649 | 1,014,464 | 377,817 |
Holding the 7 per cent return and the flat contribution, the nominal pot stays at 738,846. Only what it buys changes.
| Inflation | Exact real return | Shortcut | Price level after 40 years | Pot in today's money | Share of headline |
|---|---|---|---|---|---|
| 1.5% | 5.42% | 5.50% | 1.81 | 407,298 | 55.1% |
| 2.0% | 4.90% | 5.00% | 2.21 | 334,616 | 45.3% |
| 2.5% | 4.39% | 4.50% | 2.69 | 275,169 | 37.2% |
| 3.0% | 3.88% | 4.00% | 3.26 | 226,498 | 30.7% |
| 4.0% | 2.88% | 3.00% | 4.80 | 153,894 | 20.8% |
One and a half points of inflation either side of the base case moves the real value of the same pot from 407,298 to 153,894. Over long horizons, inflation is as large a lever as the return itself, which is why a projection quoted without it is a number rather than an answer.
Subtracting instead of dividing. One year, 0.11 of a point. Forty years, the growth factor becomes 1.04540 = 5.8164 instead of 1.043940 = 5.5769, an overstatement of 4.3 per cent. On the inflation-linked saver the shortcut (7 − 0.15 − 2.5 = 4.35 per cent) gives 387,888 in today's money against the correct 377,817.
Mixing real and nominal. Projecting at a real rate of 4.24 per cent but keeping the contribution flat in cash, or projecting at 7 per cent and comparing the result with today's cost of living. Pick one world: nominal return with nominal contributions, or real return with contributions in today's money.
Forgetting that cash can lose. A savings rate of 2 per cent with 3 per cent inflation is a real return of −0.97 per cent. The balance rises every year and buys less every year.
The same division underlies the companion figures for fees: one percentage point of cost turns 275,169 of today's money into 211,526. See what one percentage point of annual fees actually costs for that calculation in full.
Sheet 4 of the free workbook for this case does the adjustment both ways and lets you type your own inflation rate. For how long a real return takes to double money, see how accurate the rule of 72 is.
Only approximately. The exact relation divides: (1 + nominal) / (1 + inflation) minus 1. At 7 per cent and 2.5 per cent inflation that is 4.39 per cent against 4.50 by subtraction. The gap looks small in one year, but over forty years the shortcut overstates the growth factor by 4.3 per cent, 5.8164 against 5.5769.
Divide it by one plus inflation raised to the number of years. With 2.5 per cent inflation over forty years the divisor is 1.025^40 = 2.6851, so 738,846 at the finish is worth 275,169 today. In Excel: =amount/(1+inflation)^years. Use the same years as the projection, not the years left after the last contribution.
Before. Costs are charged on the nominal balance, so the exact real return after costs is (1 + return minus cost) / (1 + inflation) minus 1. With 7 per cent, a 0.15 per cent cost and 2.5 per cent inflation that is 4.24 per cent, against 4.35 if all three are simply subtracted.
This article is one calculation from Stock Market Investing for Beginners. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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