Inflation and purchasing power explained

Inflation is a rise in the average price of the things people buy. Turn that sentence around and it becomes a fall in what a unit of money buys, which is the version that matters to a saver, because it is your money on the wrong side of it. Purchasing power is the quantity of real goods a sum commands: a week of groceries, a month of rent, a year of retirement.

The two statements are not quite mirror images. A 3 percent rise in prices is a 2.91 percent fall in purchasing power, because the two are reciprocals rather than opposites. Money buys $1/1.03$ of what it did, which is 0.9709 of the old quantity, so the fall is always a little smaller than the rise that caused it. That small asymmetry is the reason real returns are found by dividing rather than subtracting, and it runs through everything below.

Inflation is also not the same thing as one item getting dearer. If coffee jumps while everything else holds, that is a relative price change and it tells you something about coffee. Inflation is the average across a wide basket of goods and services, so it tells you something about money itself.

Nothing in the ordinary machinery of a bank account reports any of this. A statement is a nominal document: it counts currency units, and currency units are exactly the thing whose value is moving. There is no line item labelled inflation, no fee and no deduction. The loss shows up in the shops instead, which is how it can run for years without being noticed.

The number a bank or a fund quotes is a nominal rate. It counts currency units and says nothing about what they buy. The number worth knowing is the real rate, which is what is left once inflation has been taken out of it.

The two are related by multiplication rather than addition:

$$1 + r_{\text{nominal}} = (1 + r_{\text{real}})(1 + i)$$

Rearranged for the figure you want:

$$r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + i} - 1$$

Here $i$ is inflation over the same period as the return. Both rates go in as decimals, so 6 percent is 0.06.

Division, not subtraction, and the reason is that inflation applies to the gain as well as to the money you started with. Earn 6 percent while prices rise 3 percent and the extra currency you gained is itself worth about 3 percent less by the time you hold it, so the whole ending balance has to be marked down rather than just the opening balance. Done properly, 6 percent against 3 percent inflation is a real 2.91 percent a year, not 3.

The size of the shortcut error is $(r - i) \times \frac{i}{1 + i}$, so it grows with the gap between the two rates and with inflation itself. At low single digits it is a fraction of a percentage point, which is exactly why it survives a glance, and it then compounds along with everything else in a long projection. The real return calculator at the top of this page does the division and puts the answer on a balance.

Here is the case that catches people. A savings account pays 1.5 percent while inflation runs at 3 percent. Both of those rates move around from one year to the next, so neither number is a permanent feature of anything, and the arithmetic below works the same way at whatever pair happens to be in front of you. The balance goes up every month, every figure the bank prints is accurate, and the saver is losing ground at about 1.46 percent a year.

Put it on money over a decade. $20,000 at 1.5 percent for 10 years reaches $23,210.82, and that balance buys what $17,271.03 buys today. Nothing was charged and no transaction appeared. The gap is prices moving faster than the interest, quietly, in the background.

Money that earns nothing is the same story with the disguise removed. Left alone for 25 years with inflation at 3 percent, $20,000 still reads $20,000 and buys $9,552.11 of what it originally would have. Prices a little more than doubled over that stretch, so the money ends up buying a little less than half as much.

Tax makes the arithmetic worse and it comes first in the order of operations. In the United States, interest in a taxable account is taxed on the nominal amount rather than on the real gain, and many other countries tax nominal interest the same way, so part of the bill falls on the portion of the return that only covered rising prices. Rates, thresholds and the treatment of sheltered accounts differ by country and change when the law changes, so what generalises is the order rather than any particular rate: work out the after-tax nominal rate first, then take inflation out of that.

None of this makes cash wrong to hold. Cash buys certainty over short horizons, and a small negative real return is what that certainty costs. What the horizon changes is not the rate of the loss but the number of years it is charged for, so the same holding that is cheap over one year is expensive over thirty. Which of those two applies is settled by when the money is actually needed.

One year of inflation is a rounding error next to a lifetime of it. The loss compounds in exactly the way interest compounds, so the rate and the horizon multiply rather than add: what money keeps is $1/(1+i)^t$, and doubling the years does about the same work as doubling the rate. Neither one is the bigger villain on its own, which is why a rate small enough to ignore over a year is not small over thirty. Here is what money keeps, as a share of what it buys today:

Read the 3 percent row. Ten years takes about a quarter of the buying power, twenty years takes nearly half, thirty years takes almost 60 percent of it. For a quick check without a table, money halves in buying power in roughly $70 / i$ years, where $i$ is inflation in percentage points: about 23 years at 3 percent, about 14 at 5 percent, about 35 at 2 percent. That is the doubling rule from compound interest, run in reverse on the same arithmetic.

This is why long projections have to be stated in real terms or they mislead. $20,000 invested at 6 percent for 30 years reaches $114,869.82, which is a satisfying number to write into a plan. With inflation at 3 percent it buys what $47,324.85 buys today, which is 41.2 percent of the face value and matches the thirty-year cell in the table. Both figures are correct. Only the second one is a quantity of goods.

Two things about that 6 percent are worth being strict about, because both of them flatter a projection. It is an assumption held fixed to make the arithmetic work, not a return anyone is owed, so the output reports what the assumption implies rather than what will happen. And it has to be a compound rate. The arithmetic average of a run of yearly returns is always at least as high as the compound rate that actually produced them, and strictly higher as soon as those returns vary at all, so an average dropped into a projection quietly overstates the ending balance before inflation is even considered.

The cleanest way to plan is to state the goal in today's money and use the real rate to reach it, so inflation never enters the projection twice. The compound interest calculator produces the nominal path; feed it the real rate instead and it produces the path in today's money.

Inflation is measured by pricing a basket. A statistics agency picks a large set of goods and services meant to stand in for what households actually buy, prices them repeatedly at the same outlets, and weights each one by how much of a typical budget it takes. The result is a consumer price index, and the inflation rate is the percentage change in that index over a period. In the United States the Bureau of Labor Statistics publishes the CPI, and most countries have an equivalent agency doing the same job.

Two versions circulate. The headline rate includes everything in the basket. A core rate takes out the items whose prices swing on weather, harvests and geopolitics, so it shows the underlying trend more steadily. What comes out is a national choice rather than a universal rule: the United States removes food and energy, the United Kingdom also removes alcohol and tobacco, and the core measure Japan quotes most often removes fresh food while leaving energy in. For working out what your money buys, headline is the honest figure, because you do buy food and fuel. For guessing where inflation is heading next, core carries less noise.

Your own rate is a third number again. The index weights a typical budget and nobody has a typical budget. If rent takes half of what you spend and rents are climbing faster than the index, your personal inflation rate is above the published one. If you own your home outright and mostly buy things whose prices are flat, it is below. Housing, medical care and education tend to move on schedules of their own, so a household heavily exposed to one of them can feel an inflation rate quite unlike the headline. For a projection that runs for decades, a long-run average matters far more than the latest month's reading, which is mostly noise.

Three habits make the real number useful rather than merely interesting.

• Compare the real rate to zero, not to the last rate you saw. Only the part of a return above inflation adds purchasing power. A rate below inflation still loses less ground than no rate at all, so 1.5 percent beats nothing; it is still a loss, so zero real is the line that decides whether the money is holding its own.
• State goals in today's money, then use a real rate. A goal written as a quantity of goods stays meaningful; a goal written as a nominal sum quietly shrinks while you save for it. Either inflate the target and use the nominal rate, or hold the target in today's money and use the real rate. Doing both counts inflation twice, and doing neither counts it once too few.
• Read the horizon before the rate. Over a year or two the real loss on cash is small and certainty is what the holder is buying with it. Over decades that same rate meets the compounding in the table above, so the cost of holding cash is a question about time before it is a question about rates.

Inflation runs on the other side of the balance sheet as well. A fixed-rate loan is a fixed number of currency units a month, so rising prices shrink what those payments are worth to the lender who receives them. Whether they also feel easier to the borrower is a separate question and turns on that borrower's own income, because wages do not track prices automatically and real wages often lag for years after a jump in prices. A variable rate carries none of that protection, since the payment itself moves. Lenders price the inflation they expect into the rate they quote, as far as they can forecast it, so it is inflation above what was expected that shifts value between the two sides. The loan payment calculator shows the nominal schedule that stays fixed while everything around it moves.

The measurement side of this sits in economics. How the price level is defined, and how a basket of goods becomes a single inflation number, is worked through here: inflation.