Inflation calculator and buying power

Enter a sum, an annual inflation rate and a number of years. The calculator returns what the same basket of goods costs at the end, what the original sum still buys by then measured in today's money, the share of buying power that has gone, and the extra money needed just to stand still.

The first two figures are one fact seen from opposite ends. Prices going up and money buying less are the same event, and which version you want depends on the question in front of you. Pricing something you will have to pay for later, such as a repair, a course of study or a year of retirement spending, calls for the cost. Looking at money that is sitting still, in cash or in a fixed sum somebody has promised to hand you later, calls for the buying power. A promise that rises with a published index is the exception: there the payer carries the loss rather than the holder.

The rate to put in is an annual one, compounded once a year, and a long-run average rather than the latest published reading, which is mostly noise. A projection that runs for decades is only as good as the average behind it, so it is worth running twice at two plausible rates and treating the pair as a range.

Prices multiply and buying power divides, so a rise of one size does not produce a fall of the same size. One year at 3 percent makes a basket cost $1.03$ times as much, and it leaves the same money buying $1/1.03$ of what it did, which is $0.9709$. A 3 percent rise in prices is a 2.91 percent fall in what money buys.

Over $t$ years the two run in opposite directions from the same factor:

$$P_t = P_0 (1 + i)^t \qquad B_t = \frac{P_0}{(1 + i)^t}$$

One multiplies by $(1 + i)^t$ and the other divides by it, so the two percentages drift further apart the longer the horizon. Over 10 years at 3 percent, $50,000 of spending grows to $67,195.82, a rise of 34.39 percent, while $50,000 of cash falls to $37,204.70 of buying power, a fall of 25.59 percent. Same rate, same 10 years, two different percentages, because they are percentages of different starting points.

That is worth holding on to when one person quotes a rise in prices and another quotes a loss in buying power and the two numbers do not match. Both can be right at once. They are answers to different questions about a single factor.

3 percent a year is a rate that would pass without comment in any single year. Run it for the length of a career and it becomes the largest single effect in the projection.

Read the middle row. Somebody who starts work at 25 and stops at 65 watches a fixed sum lose just over 69 percent of what it buys, at a rate that never once looked alarming in any single year. This is the same compounding that makes a savings balance grow, pointed the other way, and 40 years of it is 40 years of it whichever direction it runs.

The bottom row changes only the rate. Two extra percentage points take the loss from 69.34 percent to 85.80 percent, because the rate sits in the base $(1 + i)$ that the years raise to a power. It is compounded forty times over, not added forty times: adding 5 percent forty times would multiply prices by 3, while compounding it multiplies them by 7.04. A wrong guess at the rate costs far more over a long horizon than a wrong guess at the sum.

None of this says cash is a mistake. Cash buys certainty over short horizons, and a small loss of buying power 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, which is why the same holding is cheap for one year and expensive for thirty.

Money sitting still takes the whole loss above. Money that earns something gives up the part inflation takes and keeps the rest, so the figure worth tracking is neither the growth rate nor the inflation rate on its own. It is the real rate, the growth left once inflation has been removed, and it is found by dividing rather than subtracting. The real return calculator does that division and puts the answer on a balance.

Almost every rate you are quoted is a nominal rate. A savings rate, a bond coupon, a projected return on a fund: all of them count currency units and none of them says anything about what those units buy. Two accounts paying the same rate in countries with different inflation are not paying the same thing.

One rule keeps a plan honest: pick a side and stay on it. Either state the goal in today's money and grow the savings at the real rate, or inflate the goal to what it will cost then and grow the savings at the quoted rate. Doing both counts inflation twice, and doing neither counts it once too few. The compound interest calculator handles the growth side either way, and inflation and purchasing power covers why the real figure is a division.