How inflation factors work

Enter a sum, an annual inflation rate and a number of years. Two answers fall out of one factor.

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

$P_t$ is what the same basket of goods costs then. $B_t$ is what $P_0$ still buys then, measured in today's money. Prices going up and money buying less are the same event. Which version you want depends on the question in front of you.

Pricing something you will have to pay for later, a year of retirement spending, a course of study, a repair, 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.

Over 10 years at 3 percent the factor is $1.03^{10} = 1.34391638$. Multiply $50,000 and the future cost is $67,195.82. Divide $50,000 by the same factor and the buying power is $37,204.70. The share lost is 25.59 percent. No fee was charged and no transaction appeared on any statement. The money simply met higher prices.

The inflation calculator on this page returns both ends of that factor, the share lost, and the extra money needed just to stand still.

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 percentages drift further apart, because they are percentages of different starting points. Over 10 years at 3 percent, spending grows to $67,195.82, a rise of 34.39 percent, while cash of $50,000 falls to $37,204.70 of buying power, a fall of 25.59 percent. Same rate, same 10 years, two different percentages.

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.

Hold the rate at 3 percent and stretch the horizon to 40 years. The factor is $1.03^{40} = 3.26203779$. What costs $50,000 today costs $163,101.89. Cash of $50,000 buys what $15,327.84 buys now. That is 69.34 percent of the buying power gone. The share kept is the part that multiplies: 74.41 percent survives the first 10 years, and taking 74.41 percent of that three more times leaves 30.66 percent after 40.

Keep the 40 year horizon and raise inflation from 3 percent to 5 percent. The factor becomes $1.05^{40} = 7.03998871$. The basket costs $351,999.44. The cash buys $7,102.28. The loss goes from 69.34 percent to 85.80 percent, so what the money keeps drops from 30.66 percent to 14.20 percent, less than half as much. Two percentage points did that, because the forty years compound the rate rather than adding it up. Adding 5 percent forty times would multiply prices by 3. Compounding it multiplies them by 7.04.

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.

The inflation eroder holds a starting pile still and drags inflation across it, so you can see the buying power fall without choosing a nominal return first.

The factor on this page takes the inflation rate you type. A published consumer-price index is a basket someone else chose: rent weighted one way, fuel another, a phone contract a third. Your spending is a different basket. A year when fuel jumps and you do not drive will not match the index, and a year when rents jump and you rent will beat it.

That does not make the factor wrong. It makes the input a choice. Use a rate that covers the same period and the same kind of spending as the sum you are scaling. A monthly print annualised in your head is a different object from a ten-year average. The three sheets on this page hold the rate still so the years can be seen doing the work. They are not a forecast of the next published index.

Cash sitting still takes the whole loss. Money that earns something gives up the part inflation takes and keeps the rest, which is the real rate. How real returns work does that division and puts the answer on a balance.

The future cost minus today's cost is what has to be found from somewhere if the same basket is going to be bought later. Over 10 years at 3 percent that future cost is $67,195.82 against today's $50,000. Over 40 years at 3 percent it is $163,101.89. Over 40 years at 5 percent it is $351,999.44.

Those three gaps are not a savings plan. They are the hole the factor opens. Filling it is a deposit question, which is how savings goals work, or a return question, which is the real-rate page. This page stops at the hole.

The inflation eroder holds a starting pile still and drags inflation across it, so you can see the buying power fall without choosing a nominal return first. Nominal rate is the label on almost every quoted rate you will bring to that comparison.

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 the real rate, found by dividing rather than subtracting. How real returns work 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.

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.

This page is the one factor, the two percentages that are not mirrors, and the mistake of comparing a figure from decades ago with a figure from today as if they were the same unit. It is not a real-return conversion, not a forecast of the published index, and not a claim that cash is always expensive. The three sheets are $50,000 at 3 percent for 10 years ($67,195.82 / $37,204.70 / 25.59 percent lost), the same money for 40 years ($163,101.89 / $15,327.84 / 69.34 percent), and 5 percent for 40 years ($351,999.44 / $7,102.28 / 85.80 percent). This is educational material, not financial advice.