Nominal vs. real returns: adjusting savings for inflation

A nominal return is how fast your balance grows in dollars. A real return is how fast its buying power grows once rising prices are taken into account. To convert one to the other, divide instead of subtracting: real return = (1 + nominal return) ÷ (1 + inflation) − 1. At a 7% return and 3% inflation, that is 3.88% a year, not 4%. The gap looks small, but at those rates over 30 years the subtraction overstates a today’s-dollars projection by about 3.4%.

What is the difference between a nominal and a real return?

A nominal return is the percentage change in the dollar value of an investment. It is the number on an account statement, and the rates quoted on savings accounts and loans are nominal. Nominal here means not adjusted for inflation. That is a different use of the word from the nominal rate in APR vs. APY, which means a yearly rate that leaves out compounding.

A real return is the percentage change in what that money can buy, after adjusting for price changes over the same period. In the U.S., price changes are usually measured with the Consumer Price Index (CPI), the government’s measure of how the cost of a typical basket of consumer goods and services changes over time.

A today’s-dollars figure (also called real or inflation-adjusted dollars) restates a future amount at today’s prices: “$190,306.38 in 30 years will buy what $78,403.71 buys now.”

Both figures are correct, but they answer different questions. The nominal figure is what your statement will show. The real figure is the one to compare with prices and goals you think of in today’s money.

What is the formula for real return?

Divide 1 plus the nominal return by 1 plus the inflation rate, then subtract 1. This relation between nominal return, real return and inflation is known as the Fisher equation, after the economist Irving Fisher.

• $r_{\text{nominal}}$ is the dollar return over a period, as a decimal (7% is 0.07).
• $i$ is the inflation rate over the same period (3% is 0.03).
• $r_{\text{real}}$ is the growth in purchasing power over that period.

Here is why it divides. Invest $25,000 at 7% for a year while prices rise 3%. You end with $26,750, but each dollar now buys 1 ÷ 1.03 of what it bought before. At today’s prices, $26,750 ÷ 1.03 = $25,970.87, so your buying power grew by $970.87, or 3.88%. Subtracting (7% − 3% = 4%) would claim $1,000.

Turned around, the formula gives the nominal return needed for a target real return: $1 + r_{\text{nominal}} = (1 + r_{\text{real}})(1 + i)$, which is real + inflation + real × inflation. The U.S. Treasury uses this form, applied to half-year rates, for Series I savings bonds. Their combined rate is the fixed rate, plus twice the semiannual inflation rate, plus the fixed rate times the semiannual inflation rate. The fixed rate plays the part of the real return, and the last term is the product the subtraction shortcut drops.

Why is subtracting inflation only an approximation?

Nominal minus inflation skips the division by $1 + i$. The error it adds is exactly inflation times the real return:

At 7% and 3%, that is 0.03 × 3.88% ≈ 0.12 percentage point a year. Irving Fisher, who set out the relation between money and real interest in an 1896 monograph, noted in The Theory of Interest (1930) that point-for-point subtraction is strictly true only for rates compounded continuously. You can check this: as continuously compounded rates, 7% and 3% a year are ln(1.07) = 6.77% and ln(1.03) = 2.96%. Their difference, 3.81%, converts back to $e^{0.0381} - 1$ = 3.88% a year, the exact real return.

The shortcut is close enough for a one-year comparison at low inflation. It drifts when inflation is high (the 15% and 10% row is Fisher’s own illustration, where he called the real rate “about 5 per cent”), when the real return is large, and when the error compounds over decades. When the real return is negative, the shortcut shows a slightly bigger loss than actually occurs.

Worked example: $25,000 over 30 years in future and today’s dollars

Example assumptions, not forecasts: $25,000 invested today, a 7% nominal return every year, 3% inflation every year, 30 years, returns compounded once a year, and no taxes, fees or withdrawals.

1. Nominal balance: $25,000 × $1.07^{30}$ = $25,000 × 7.6122550 = $190,306.38.
2. Price level: $1.03^{30}$ = 2.4272625, so prices rise 142.73% and a dollar in year 30 buys 41.20% of what it buys today.
3. Today’s dollars, method 1 (deflate): $190,306.38 ÷ 2.4272625 = $78,403.71.
4. Today’s dollars, method 2 (grow at the real rate): the real return is 1.07 ÷ 1.03 − 1 = 3.8835%, and $25,000 × $(1.07 \div 1.03)^{30}$ = $78,403.71. The two methods agree exactly only when the real rate comes from dividing.
5. The shortcut: $25,000 × $1.04^{30}$ = $81,084.94, which overstates the real value by $2,681.23 (3.42%).

The shortcut’s error grows with the horizon:

The inflation assumption matters more than the formula. With the 7% return held fixed, the same $190,306.38 is worth $105,062.61 in today’s dollars at 2% inflation, $78,403.71 at 3% and $58,675.01 at 4%. Each one-point change moves the answer by $19,728.70 to $26,658.90, about seven to ten times the shortcut’s error. Nominal returns may partly rise and fall with inflation, so treat this as a sensitivity check, not a forecast.

Should contributions rise with inflation in a today’s-dollars projection?

Only if you plan to raise them. A fixed deposit buys less every year, so “$6,000 a year” means very different things depending on whether the amount stays fixed in dollars or keeps pace with prices.

Example assumptions: $6,000 deposited at the start of each year for 30 years, with the same 7% return and 3% inflation.

With level deposits, each deposit buys less than the one before. The 30th deposit, made at the start of year 30, is worth $6,000 ÷ $1.03^{29}$ = $2,546.08 in today’s dollars.

With deposits that rise with inflation at the start of each year, every deposit is worth exactly $6,000 at today’s prices. The projection in today’s dollars is then an annuity due (payments at the start of each year) at the real rate, with deposit $C$ and $n$ years:

Here that is $6,000 × 57.141967 = $342,851.80. The shortcut rate of 4% would give $349,970.01, which is $7,118.21 too high. The gap between the two deposit plans, $93,007.27 in today’s dollars, is about 13 times the shortcut’s error.

If you project a fixed $6,000 at the real rate, you have assumed deposits rise with inflation without saying so. To show level deposits in today’s dollars, project in nominal terms and deflate the result.

What happens when inflation is higher than your return?

The real return turns negative: the balance grows in dollars but buys less. Example assumptions: $10,000 in a savings account paying 3% a year for 10 years while prices rise 4% a year. The balance reaches $13,439.16, but at today’s prices that buys what $9,079.02 buys now, 9.21% less than you started with. The real return is 1.03 ÷ 1.04 − 1 = −0.96% a year. Cash earning nothing loses 2.91% of its buying power a year at 3% inflation (1 ÷ 1.03 − 1).

What does a real-return projection still leave out?

• Taxes on nominal gains. U.S. income tax on interest is charged on the dollars received, with no adjustment for inflation, so tax takes a larger share of the real return than of the nominal one. Suppose the whole 7% were taxed every year at 25% (an example rate; your rate and account type set the actual figure). The after-tax return is 5.25%, and the after-tax real return is 1.0525 ÷ 1.03 − 1 = 2.18%. Tax takes a quarter of the nominal return but 43.75% of the real return, and the $25,000 example ends at $47,806.44 in today’s dollars instead of $78,403.71.
• Fees. A fee charged on the balance comes out of the real return, which is smaller than the nominal one, so it takes a bigger share. At 7% and 3% inflation, a 1% annual fee cuts the real return from 3.88% to 2.84% (1.07 × 0.99 ÷ 1.03 − 1), more than a quarter of it. How investment fees compound works through fee drag over decades.
• Your own inflation rate. The CPI measures average price changes for a broad population. The Bureau of Labor Statistics notes that it may not reflect the experience of people in rural areas, and for Americans 62 and older it publishes only a research index, not an official inflation rate. If your spending leans toward items that rise faster, test a higher rate.
• Returns and inflation that vary. A constant-rate projection hides the order in which good and bad years arrive, which matters once withdrawals start. See why average returns can mislead in retirement.

Common mistakes

• Deflating a return that is already real. Some projections and rules of thumb quote returns after inflation. If the 7% in the example were already real, $190,306.38 would already be in today’s dollars. Dividing by $1.03^{30}$ again would report $78,403.71, understating it by a factor of 2.43.
• Comparing figures in different units. A goal set in today’s money belongs next to the today’s-dollars projection, not the nominal balance.
• Reading the today’s-dollars figure as a balance. Your statement in year 30 shows the nominal amount. The today’s-dollars figure is what that amount will buy.
• Mixing periods. A return of 0.5% a month is $1.005^{12} - 1$ = 6.17% a year. Convert it to a yearly rate before dividing by a yearly inflation rate: 1.0617 ÷ 1.03 − 1 = 3.08%.
• Subtracting over long horizons or at high inflation. Use the division; the shortcut-error table shows where the shortcut drifts.

Try it

1. Compound interest calculator: enter an initial deposit of $25,000, a regular contribution of $0, an annual interest rate of 7%, compounding annually and 30 years to get $190,306.38. For the deposit plans, set the initial deposit to $0 and add a $6,000 yearly contribution at the start of each period: $606,438.25. Then set the yearly increase to 3%: $832,191.31.
2. Inflation calculator: under Calculate, choose today’s value of future dollars, then enter $190,306.38, 3% and 30 years to get $78,403.71 (cumulative inflation 142.73%). Repeat with $606,438.25 to get $249,844.53, and with $832,191.31 to get $342,851.80.
3. Retirement savings calculator: shows both columns at once. Enter current age 35, retirement age 65, current savings $0, a $6,000 yearly contribution at the start of each period, a 7% expected annual return and a 3% inflation rate. A yearly increase of 0% gives $606,438.25 and $249,844.53; 3% gives $832,191.31 and $342,851.80. For the lump sum, enter $25,000 of current savings and a $0 contribution to get $190,306.38 and $78,403.71.
4. FIRE calculator: projects in today’s dollars and converts your nominal return with the division formula, so enter spending and contributions at today’s prices. Checking “Raise contributions with inflation” matches the rising plan above; leave it unchecked for level deposits.