Interest and inflation · F00 · Sunflower Intelligence

Before you start

You will be able to

Calculate simple and compound interest, and explain why compounding makes balances grow faster over time.
Calculate future values with more than one compounding period a year, and convert a quoted annual rate into an effective annual rate.
Distinguish an APR (a quoted annual rate) from an effective annual rate, and compare two rates quoted on different bases.
Explain inflation and purchasing power, and calculate an inflation rate from a price index.
Calculate a real return exactly and by approximation, and recognise when the approximation fails.

Review first

What finance is: money, time and risk — This lesson builds on future value, FV = PV(1 + r)^n, over whole years.
Capital and liquidity — You need interest, principal and the terms of Lantern’s bank loan.
Percentages as decimals (6% = 0.06) and powers on a calculator (1.05 to the power 10).

Fictional case A fictional coffee business founded by Ana Lim. It starts as a catering and barista-training service, opens its own cafés and an online shop for beans, and plans a canned cold brew.

The idea

Fictional case

On 1 January 2026 Ana has $26,000 in a Coastline Bank savings account that pays 1.5% a year, credited once a year. It feels safe: the balance never falls. But the prices of the things she buys, from rent to coffee beans, are also rising, and she assumes they will rise by about 2.8% a year over the next two years. Two questions follow. How much does her money grow, and how fast? And is it growing faster than prices, so that it buys more next year than it does today, or more slowly, so that it buys less? The first question is about interest; the second is about inflation.

Interest: the price of time

Interest is what a borrower pays a lender for the use of money over time, usually quoted as a percentage of the principal per year. A saver who puts money in a bank account is lending to the bank, so the bank pays the saver interest. When Lantern borrows from Coastline Bank, Lantern pays the interest. The rate is the price of time: what it costs to have money now instead of later, or what you are paid for waiting.

There are two ways to calculate interest over more than one period, and the difference between them grows with time.

Simple interest: Interest is calculated on the original principal only. Each year’s interest is the same: principal × rate. Over $t$ years the interest is $I = P \times r \times t$ .
Compound interest: Interest is calculated on the principal and on the interest already added to the balance, so each year’s interest is larger than the last. Over $n$ whole years the balance becomes $P(1 + r)^n$ , the future value formula from lesson 1.

Lantern’s loan shows where simple interest arises in practice. Coastline Bank charges 6% a year on $10,000, and Lantern pays the interest every three months, $150 a time, rather than letting it build up. Because the interest is paid out as it falls due, the balance stays at $10,000 and the interest never earns interest. Over the two years Lantern pays $10{,}000 \times 0.06 \times 2 = 1{,}200$ dollars, the same as eight payments of $150.

A savings account usually works the other way: the interest is added to the balance and left there, so next year the bank pays interest on it too. Put $1,000 in an account paying 10% a year and leave it alone. After one year it holds $1,100. In the second year the bank pays 10% on $1,100, which is $110, not $100: the extra $10 is interest on interest. After three years the balance is $1{,}000 \times 1.1^3 = 1{,}331$ dollars, compared with $1{,}000 + 3 \times 100 = 1{,}300$ under simple interest.

$1,000 left for ten years at 10% a year, compounded annually. Each bar splits the balance into the principal, the interest a simple-interest account would have paid ($100 a year) and the interest earned on earlier interest. That last slice is $10 after two years, $110.51 after five and $593.74 after ten.

Over long periods the gap becomes large. At 5% a year, $10,000 grows to $15,000 in ten years with simple interest but to about $16,288.95 with compound interest; over thirty years the figures are $25,000 and about $43,219.42. Compounding is why starting to save early matters so much, and also why an unpaid credit-card balance can grow so quickly.

Compounding more than once a year

Interest is often added more often than once a year: every six months, every quarter, every month, even every day. The compounding frequency is how many times a year that happens. The bank quotes an annual rate, divides it by the number of periods to get the rate per period, and applies that rate each period. A 6% annual rate compounded monthly means 0.5% a month ( $6\% \div 12$ ), applied twelve times.

$10,000 for one year at a quoted 6% a year
Compounding	Periods a year	Rate per period	Balance after one year	Effective annual rate
Annual	1	6%	$10,600.00	6.00%
Semi-annual	2	3%	$10,609.00	6.09%
Quarterly	4	1.5%	$10,613.64	6.14%
Monthly	12	0.5%	$10,616.78	6.17%
Daily	365	about 0.0164%	$10,618.31	6.18%

Each balance is 10,000 × (1 + 0.06 ÷ m) raised to the power m, where m is the number of periods. Rounded to the cent and to two decimal places of a percentage.

More frequent compounding pays a little more, because interest starts earning interest sooner. The effect shrinks as frequency rises: going from annual to monthly adds $16.78, going from monthly to daily only $1.53 more. The last column, the effective annual rate (EAR), is the rate that, compounded once a year, would give the same result. It is the honest way to compare rates quoted with different compounding.

Inflation: what time takes away

Inflation is a sustained rise in the general level of prices: not one product becoming dearer, but prices across the economy rising on average. It is measured with a price index, which tracks the cost of a fixed basket of goods and services over time. In the United States the best known is the Consumer Price Index (CPI), which the Bureau of Labor Statistics describes as a measure of the average change over time in the prices urban consumers pay for a market basket of consumer goods and services.

Inflation reduces the purchasing power of money: what a given number of dollars can buy. If prices rise by 2.8% in a year, $26,000 next year buys what about $26{,}000 \div 1.028 \approx 25{,}291.83$ dollars buys today. The banknotes have not changed, but they stretch less far.

This is the point of Ana’s worry. Suppose she had kept all $26,000 in her account for a year. At 1.5% it would grow to $26{,}000 \times 1.015 = 26{,}390$ dollars. But with prices 2.8% higher, $26,390 buys only what $26{,}390 \div 1.028 \approx 25{,}671.21$ dollars buys today. In dollars she has gained $390; in what her money can buy, she has lost about $328.79.

Nominal and real returns

A nominal return is the percentage increase in the number of dollars: Ana’s 1.5%. A real return is the percentage increase in purchasing power: how much more, or less, the money can buy. The real return removes inflation from the nominal one:

1 + r_{\text{real}} = \frac{1 + r_{\text{nom}}}{1 + i} \qquad\Longrightarrow\qquad r_{\text{real}} = \frac{1.015}{1.028} - 1 \approx -1.26\%

Ana’s real return is negative: her savings are losing purchasing power at about 1.26% a year. A quick approximation subtracts inflation from the nominal rate: $1.5\% - 2.8\% = -1.3\%$ . The two answers are close here because both rates are small. The approximation is easy to remember, but it is only an approximation, and the theory section shows when it goes wrong.

In depth

Simple and compound interest: the formulas side by side

With principal $P$ , annual rate $r$ (as a decimal) and $n$ whole years, simple interest gives a balance of $P(1 + rn)$ , which grows by the same amount each year: a straight line. Compound interest gives $P(1 + r)^n$ , which grows by the same percentage each year: a curve that gets steeper. For one year the two agree. After that, compound interest is always ahead, and the gap widens every year, as the figure in “The idea” shows.

Simple interest appears wherever interest is paid out instead of added to the balance: Lantern’s loan, most bonds (which pay their interest to investors every six months or year) and some short-term loans. Compound interest appears wherever interest stays in: savings accounts, certificates of deposit that pay at maturity, unpaid credit-card balances and long-term investments whose income is reinvested.

A handy rule of thumb, the rule of 72, estimates how long compounding takes to double money: divide 72 by the percentage rate. At 6% a year money doubles in about $72 \div 6 = 12$ years; the exact answer is about 11.9 years. The same rule works for prices: at 2.8% inflation, prices double in about $72 \div 2.8 \approx 25.7$ years (exactly, about 25.1). It is an approximation that works well for rates between about 2% and 15%.

Compounding frequency and the effective annual rate

If a quoted annual rate $r$ is compounded $m$ times a year, the rate per period is $r \div m$ and there are $m \times n$ periods in $n$ years, so the future value is $P(1 + r/m)^{mn}$ . Over three years at 6%, $10,000 grows to $11,910.16 with annual compounding and to about $11,966.81 with monthly compounding ( $10{,}000 \times 1.005^{36}$ ).

The effective annual rate converts any quoted rate into the equivalent once-a-year rate: $\text{EAR} = (1 + r/m)^m - 1$ . Two rates can be compared fairly only once both are expressed as EARs. The comparison can reverse a first impression: worked example 2 shows a lower quoted rate that costs more.

APR, EAR and APY

The rate quoted on a loan is usually a nominal annual rate: the rate per period multiplied by the number of periods in a year, ignoring compounding. On loans and credit cards it is commonly called the APR (annual percentage rate). A card that charges 1.5% a month has an APR of $12 \times 1.5\% = 18\%$ , but because unpaid interest is added to the balance each month, its EAR is $1.015^{12} - 1 \approx 19.56\%$ . The APR understates what a balance left unpaid for a year really costs.

Standards: What the labels mean in US consumer finance

For closed-end consumer loans in the US, such as mortgages and car loans, the APR lenders must disclose under Regulation Z is broader than the interest rate: as the Consumer Financial Protection Bureau explains for mortgages, it also reflects points, broker fees and certain other charges, so it is usually higher than the interest rate. A credit card’s APR, by contrast, is its periodic rate times the number of periods in a year (Regulation Z, §1026.14), as in the 18% example above. For deposit accounts, US banks disclose an annual percentage yield (APY) under the Truth in Savings rules (Regulation DD): a percentage rate reflecting the total interest paid on an account, based on the interest rate and how often interest is compounded. An APY is therefore an effective annual rate. Other countries use other labels, such as AER (annual equivalent rate) for savings in the UK. Always check which rate a number is before comparing it with another.

This course uses the words this way: a quoted or nominal annual rate (an APR, in its simple sense) is the periodic rate times the number of periods; the effective annual rate includes compounding. F04 Mathematics of Finance develops both, including loans repaid in instalments, where APRs matter most.

Measuring inflation

A price index sets the cost of its basket in a chosen base period equal to 100 and expresses other periods relative to it. The inflation rate between two dates is the percentage change in the index, not the difference in index points. If a fictional index rises from 300.0 to 308.4 over a year, inflation is $(308.4 - 300.0) \div 300.0 = 2.8\%$ , although the index rose 8.4 points. OpenStax’s Principles of Economics warns against the shortcut of subtracting index numbers for exactly this reason.

Different indexes measure different baskets. The CPI tracks what urban consumers buy. The Federal Reserve states its goal as inflation of 2% over the longer run, measured by the annual change in a different index, the price index for personal consumption expenditures (PCE). No index matches any one person’s spending exactly: Ana, who rents and buys a lot of coffee, may face a different inflation rate from the average household. That is one reason her 2.8% is an assumption, not a forecast.

Over several years inflation compounds just as interest does. At 2.8% a year for ten years, prices rise by a factor of $1.028^{10} \approx 1.318$ , about 31.8%, so $10,000 in ten years’ time would buy only what about $10{,}000 \div 1.318 \approx 7{,}587$ dollars buys today. If Ana had left her whole $26,000 earning 1.5% for those ten years, it would have grown to about $30,174.06 in dollars but would buy only what about $22,893.00 buys today.

Real returns: the exact formula and the approximation

The exact relationship between nominal return, inflation and real return is often called the Fisher relation, after the economist Irving Fisher: $(1 + r_{\text{nom}}) = (1 + r_{\text{real}})(1 + i)$ . Multiplying out gives $r_{\text{nom}} = r_{\text{real}} + i + r_{\text{real}} \times i$ . The approximation $r_{\text{real}} \approx r_{\text{nom}} - i$ drops the last term, the product of two small numbers.

When rates are small the dropped term is tiny and the approximation is good: at 8% nominal and 3% inflation the exact real return is $1.08 \div 1.03 - 1 \approx 4.85\%$ , against 5% by subtraction. When rates are large it fails badly. In an economy with 30% inflation, a deposit paying 40% has an exact real return of $1.40 \div 1.30 - 1 \approx 7.69\%$ , not 10%. Over long periods the annual error compounds: at 8% and 3% for 30 years the shortcut implies purchasing power multiplied by 4.32, against 4.15 exactly. Use the exact formula whenever inflation or interest rates are high, or the period is long.

Exact and approximate real returns
Nominal return	Inflation	Exact real return	Approximation (nominal − inflation)	Error (percentage points)
1.5%	2.8%	−1.26%	−1.30%	0.04
8%	3%	4.85%	5.00%	0.15
40%	30%	7.69%	10.00%	2.31

The first row is Ana’s savings account against her 2.8% assumption. The third is a fictional high-inflation economy.

The same logic gives the real interest rate on a loan, which is what borrowing really costs once inflation is taken into account. Inflation that turns out higher than expected when the rate was fixed helps borrowers: they repay in dollars that buy less than they expected. Inflation lower than expected helps lenders. Worked example 3 applies this to Lantern’s loan.

Formulas

Simple interest

I = P \times r \times t, \qquad FV = P(1 + rt)

$I$: Total interest earned or paid$
$P$: Principal: the amount lent or borrowed$
$r$: Annual interest rate, as a decimal (6% = 0.06)per year
$t$: Timeyears
$FV$: Balance at the end, if the interest is added to it rather than paid out$

Assumes. Interest is earned only on the original principal.

Fits loans and bonds whose interest is paid out as it falls due.

Compound interest with m periods a year

FV = P\left(1 + \frac{r}{m}\right)^{m n}

$FV$: Future value: the balance after n years$
$P$: Principal$
$r$: Quoted (nominal) annual rate, as a decimalper year
$m$: Compounding periods per year (1 annual, 4 quarterly, 12 monthly, 365 daily)
$n$: Number of yearsyears

Assumes. The rate stays the same and no money is added or withdrawn.

With m = 1 this is lesson 1’s FV = PV(1 + r)^n.

Effective annual rate

\text{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1

$\text{EAR}$: The equivalent rate compounded once a yearper year
$r$: Quoted (nominal) annual rate, such as an APR in its simple senseper year
$m$: Compounding periods per year

Assumes. Ignores fees; the APR disclosed on a US mortgage or car loan may include some.

Compare rates only after converting them to EARs. A US deposit APY is an EAR.

Inflation rate from a price index

i = \frac{\text{Index}_1 - \text{Index}_0}{\text{Index}_0}

$i$: Inflation rate over the periodper period
$\text{Index}_0$: Price index at the startindex points
$\text{Index}_1$: Price index at the endindex points

A percentage change, not a difference in index points.

Purchasing power in today’s dollars

\text{Real value} = \frac{\text{Future dollars}}{(1 + i)^{n}}

$\text{Future dollars}$: An amount of money n years from now$
$i$: Annual inflation rate, as a decimalper year
$n$: Number of yearsyears
$\text{Real value}$: What that amount will buy, measured in today’s prices$ of today

Assumes. Inflation is constant at i each year.

Real return (Fisher relation)

r_{\text{real}} = \frac{1 + r_{\text{nom}}}{1 + i} - 1 \;\approx\; r_{\text{nom}} - i

$r_{\text{real}}$: Real return: growth in purchasing powerper year
$r_{\text{nom}}$: Nominal return: growth in dollars (use the effective annual rate)per year
$i$: Inflation rateper year

Assumes. All three over the same period.

The approximation is close when both rates are small; use the exact form when they are large or the period is long.

Worked examples

A first example: Simple or compound?

Fictional figures

You lend $5,000 for five years at 4% a year. Compare the amount you have at the end if (a) the interest is paid to you each year and not reinvested (simple interest), and (b) the interest is added to the balance each year (compound interest).

(a) Simple interest: each year earns $5{,}000 \times 0.04 = 200$ dollars, so five years earn $200 \times 5 = 1{,}000$ . Principal plus interest: $5{,}000 + 1{,}000 = 6{,}000$ .
(b) Compound interest: $FV = 5{,}000 \times 1.04^5$ . On a calculator $1.04^5 \approx 1.216653$ , so $FV \approx 6{,}083.26$ .
Difference: $6{,}083.26 - 6{,}000 = 83.26$ . That is the interest earned on earlier interest.

Answer. (a) $6,000; (b) about $6,083.26, which is $83.26 more.

What it means. Over five years at 4% compounding adds less than 1.4% to the final amount. Over longer periods and at higher rates the gap grows quickly, which is why compounding matters most for long-term savings and debts.

A harder example: The lower rate that costs more

Fictional figures

Ana’s friend is buying a van and compares two one-year quotes to borrow $20,000, with nothing repaid until the end of the year. A dealer offers 7.2% a year compounded monthly. A bank offers 7.4% a year compounded annually. Which is cheaper, and by how much?

Convert the dealer’s quote to an effective annual rate: $\text{EAR} = (1 + 0.072/12)^{12} - 1 = 1.006^{12} - 1 \approx 7.44\%$ .
The bank compounds once a year, so its EAR is its quoted rate: 7.4%.
Compare: 7.44% against 7.40%, so the bank is cheaper, although its quoted rate is higher.
In dollars: the dealer’s loan grows to $20{,}000 \times 1.006^{12} \approx 21{,}488.48$ ; the bank’s to $20{,}000 \times 1.074 = 21{,}480$ . The bank saves about $21{,}488.48 - 21{,}480 = 8.48$ dollars.

Answer. The bank’s 7.4% compounded annually is cheaper: its EAR is 7.40% against about 7.44% for the dealer, a saving of about $8.48 on $20,000 over the year.

What it means. The saving is small here, but the method matters: compare rates only after converting them to effective annual rates. With larger sums, longer periods or bigger differences in compounding, the quoted rate can mislead badly. Real loans are usually repaid in instalments, and fees can change the comparison; F04 covers both.

Applied: a business decision: What Lantern’s loan really costs

Fictional figures

Lantern borrows $10,000 from Coastline Bank on 1 March 2026 at 6% a year, interest paid quarterly, principal repaid on 1 March 2028. Using Ana’s assumption of 2.8% inflation a year, what is the real interest rate on the loan, and what is the $10,000 repayment worth in March 2026 dollars? Treat the 6% as an annual rate and ignore the timing of the quarterly payments.

Nominal cost: 6% a year, which is $10{,}000 \times 0.06 = 600$ dollars of interest a year, paid as $150 a quarter.
Real interest rate, exactly: $1.06 \div 1.028 - 1 \approx 3.11\%$ . The approximation gives $6\% - 2.8\% = 3.2\%$ .
Purchasing power of the repayment: two years of 2.8% inflation, so $10{,}000 \div 1.028^2 \approx 9{,}462.67$ dollars of March 2026 purchasing power.
Judgement: in real terms Lantern pays about 3.1% a year, not 6%, and repays less purchasing power than it borrowed, about $537 less. If inflation turns out higher than 2.8% the loan is cheaper still in real terms; if lower, dearer.

Answer. A real interest rate of about 3.11% a year (3.2% by the approximation); the $10,000 repayment is worth about $9,462.67 in March 2026 dollars.

What it means. Unexpected inflation transfers purchasing power from lenders to borrowers with fixed rates. That is why lenders build expected inflation into the rates they charge, and why an unexpected rise in inflation helps borrowers and hurts savers, Ana among them.

Explore

Common mistakes

Using simple interest where interest is reinvested
If interest stays in the account, it earns interest. Multiplying the yearly interest by the number of years (5% × 10 years = 50%) understates the growth: at 5% compounded, money grows by about 62.9% in ten years. Use simple interest only when the interest is paid out.
Comparing quoted rates with different compounding
A 7.2% rate compounded monthly costs more than a 7.4% rate compounded annually. Convert both to effective annual rates before comparing, and check whether a figure is an APR, an APY or something else.
Dividing the annual rate by 12 but forgetting to multiply the periods
With monthly compounding, use the monthly rate (r ÷ 12) and the number of months (12 × years). Using r ÷ 12 with the number of years, or r with the number of months, gives an answer that is far too small or far too large.
“My balance went up, so I am better off.”
Only in dollars. If inflation exceeds the interest rate, as with Ana’s 1.5% account and 2.8% inflation, the money buys less each year. Judge savings by their real return.
Subtracting index points to get inflation
An index rising from 300 to 308.4 has risen 8.4 points but 2.8%. Inflation is the percentage change in the index.
Trusting nominal − inflation when rates are high
The shortcut drops the product of the two rates. At 40% nominal and 30% inflation it gives 10% instead of the true 7.69%. Use (1 + nominal) ÷ (1 + inflation) − 1 whenever rates are large or the period is long.

Guided practice

Work through these with hints when you need them. Every hint and every worked solution is there to be used.

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On your own

No hints this time. Answer, check, and read the feedback; the solution appears once you have answered or if you ask for it. Your first answers count toward mastery.

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In practice

Interest and inflation run through ordinary financial decisions, at home and at work.

Setting a savings target in real terms. A financial planner helping a fictional teacher save for a home deposit in eight years starts by asking what the deposit is in today’s prices. She then plans in real terms: she projects the savings with a real return (the expected interest rate adjusted for expected inflation) and compares them with the deposit in today’s dollars, rather than mixing nominal savings growth with today’s house prices. If the account’s rate is below inflation, she points out that the teacher is losing ground every year and suggests looking at better-paying accounts, while keeping enough in easy-access savings for emergencies.

Pricing in a business. A fictional hotel signs a three-year contract to host a company’s training courses at a fixed $180 a room-night. If the hotel’s costs rise with inflation at about 3% a year, the fixed price is worth about $180 \div 1.03^2 \approx 169.67$ dollars of today’s purchasing power at the start of the third year. The hotel’s sales manager therefore asks for a clause that raises the rate each year in line with a named price index, a common feature of long contracts, so that inflation does not quietly cut the real price.

Borrowing. The finance manager of a fictional battery-storage installer compares a fixed-rate loan with a variable-rate one. The fixed rate locks in the nominal cost; if inflation rises unexpectedly, the real cost falls, but if inflation falls, the business is left paying a high rate in real terms. She compares the offers on an effective-annual-rate basis, including fees, and then asks what the real rate would be under several inflation assumptions. Lenders do the same arithmetic from the other side, which is why rates tend to rise when expected inflation rises.

Fictional case

Ana now has the tools to judge her own savings. Her 1.5% account protects her dollars but, on her assumption of 2.8% inflation, loses about 1.26% of purchasing power a year. For the $6,000 she keeps as an emergency fund, she will have to balance three things: interest (how much the money earns), inflation (whether it keeps its purchasing power) and liquidity (whether she can reach it quickly). The F00 project asks you to make that comparison for her two choices, a certificate of deposit and a high-yield savings account.

Knowledge check

Six questions, one attempt each, no hints. Score 80% or more to complete the lesson; you can take it again at any time.

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Summary

Interest is the price of using money over time. Simple interest is paid on the principal only (I = P × r × t); compound interest is paid on the principal and on earlier interest (FV = P(1 + r)^n).
Compounding makes balances grow faster every year; the gap from simple interest is small at first and large over long periods.
With m compounding periods a year, FV = P(1 + r/m)^(mn). More frequent compounding pays slightly more, with diminishing effect.
A quoted annual rate (an APR in its simple sense) ignores compounding; the effective annual rate includes it. Compare rates only as EARs. US deposit APYs are EARs; a US card’s APR is its periodic rate × periods, while APRs on US mortgages and car loans also include certain fees.
Inflation is a sustained rise in the general price level, measured as the percentage change in a price index such as the CPI. It erodes purchasing power.
Real return = (1 + nominal) ÷ (1 + inflation) − 1. Ana’s 1.5% against 2.8% inflation is about −1.26% a year: her balance rises while its purchasing power falls.
Nominal − inflation is a good approximation only when both rates are small; at 40% and 30% it gives 10% instead of 7.69%.

Terms introduced

Simple interest: Interest calculated on the original principal only, so it is the same every period: principal × rate × time.
Compound interest: Interest calculated on the principal and on interest already added to the balance, so the balance grows by the same percentage each period.
Compounding frequency: How many times a year interest is calculated and added to the balance: annually, semi-annually, quarterly, monthly or daily.
Effective annual rate: The interest rate that, compounded once a year, gives the same result as a quoted rate with more frequent compounding: (1 + r/m)^m − 1. Often shortened to EAR.
APR: Annual percentage rate. In its simple sense, a periodic rate multiplied by the number of periods in a year, ignoring compounding; a US credit card’s APR is calculated this way. For closed-end US consumer loans, such as mortgages and car loans, the disclosed APR is broader and also reflects certain fees.
Annual percentage yield: The rate US banks disclose on deposit accounts under the Truth in Savings rules, reflecting the total interest paid on the account including compounding. An effective annual rate. Often shortened to APY.
Inflation: A sustained rise in the general level of prices, measured as the percentage change in a price index.
Price index: A number tracking the cost of a fixed basket of goods and services over time, set to 100 in a base period.
Consumer Price Index: The best-known US price index, published by the Bureau of Labor Statistics: a measure of the average change over time in the prices urban consumers pay for a basket of goods and services. Often shortened to CPI.
Purchasing power: What a given amount of money can buy. Inflation reduces it.
Nominal return: The percentage change in the number of dollars an investment is worth, before allowing for inflation.
Real return: The percentage change in the purchasing power of an investment: (1 + nominal return) ÷ (1 + inflation) − 1.
Real interest rate: An interest rate adjusted for inflation: what borrowing really costs, or lending really earns, in purchasing power.

Sources

US Bureau of Labor Statistics. Consumer Price Index (programme home page). bls.gov/cpi
Board of Governors of the Federal Reserve System. Why does the Federal Reserve aim for inflation of 2 percent over the longer run? FAQ, last updated 22 August 2025. federalreserve.gov/faqs/economy_14400.htm
Greenlaw, S. A., Shapiro, D. & MacDonald, D. (2022). Principles of Economics, 3rd edition, section 22.1, “Tracking Inflation”. OpenStax. openstax.org/books/principles-economics-3e/pages/22-1-tracki…
Federal Deposit Insurance Corporation. Consumer Compliance Examination Manual, chapter VI-3, “Truth in Savings” (Regulation DD). Last updated May 2023. fdic.gov/consumer-compliance-examination-manual/vi-3-truth-s…
Consumer Financial Protection Bureau. What is the difference between a mortgage interest rate and an APR? consumerfinance.gov/ask-cfpb/what-is-the-difference-between-…
Consumer Financial Protection Bureau. Regulation Z (Truth in Lending), 12 CFR §1026.14: Determination of annual percentage rate. consumerfinance.gov/rules-policy/regulations/1026/14

Before you start

You will be able to

Review first

The idea

Interest: the price of time

Compounding more than once a year

Inflation: what time takes away

Nominal and real returns

In depth

Simple and compound interest: the formulas side by side

Compounding frequency and the effective annual rate

APR, EAR and APY

Measuring inflation

Real returns: the exact formula and the approximation

Formulas

Simple interest

Compound interest with m periods a year

Effective annual rate

Inflation rate from a price index

Purchasing power in today’s dollars

Real return (Fisher relation)

Worked examples

Explore

Common mistakes

Using simple interest where interest is reinvested

Comparing quoted rates with different compounding

Dividing the annual rate by 12 but forgetting to multiply the periods

“My balance went up, so I am better off.”

Subtracting index points to get inflation

Trusting nominal − inflation when rates are high

Guided practice

On your own

In practice

Knowledge check

Summary

Key formulas

Terms introduced

Sources