Lesson 05 · 4 min · 7 things to do
Small rates, long years
Two per cent a year sounds like nothing. Over a working life it is not nothing, and there is a way to see that in your head.
- Start£100After year one£110
A basket that cost £100. Prices rise 10% a year for two years. By how much have they risen in total?
- Yes.The second year's 10% is worked out on £110, not on £100, so it adds £11. The rises stack on each other rather than adding up.
- Not quite.Ten plus ten. That would be right if the second rise were worked out on the original £100. It is worked out on £110.
- Not quite.It would, for a real household. Here the figure is given: 10% a year on the whole basket, so the only question is how the two years combine.
- Start£100After year one£110After year two£121
Prices rise 10% a year for three years. What does the £100 basket cost at the end?
£Add 10% of £121, not 10% of £100.
Yes.£110, then £121, then £133.10. Each year's rise is worked out on last year's price, not the original one. That is all compounding is. At 2% a year, how long until prices have doubled?
Commit to one. Nothing is scored — this is here so you have a number in your head before you see the real one.
At 2% a year35 yearsAt 5% a year14 yearsAt 7% a year10 yearsAt 10% a year7 yearsYears until prices double. About 35 years.Not fifty, because each year's 2% lands on a bigger number than the last. Divide 70 by the rate and you have the doubling time: 70 ÷ 2 is 35. The rule works for any rate up to about ten.A £100 basket, forty years, and one inflation rate. Change the rate.
1% a yearAfter forty years the basket is £149. Slow, and it never stops.
2% a yearAfter forty years the basket is £221: more than doubled, at a rate that sounds like nothing.
4% a yearAfter forty years the basket is £480. Doubling the rate did far more than double the rise.
7% a yearAfter forty years the basket is nearly £1,500. Fifteen times the start, from seven per cent a year.
Doubling the rate from 2% to 4% did what to the rise after forty years?
- Yes.The rises stack, so twice the rate is far more than twice the rise. The gap between 2% and 4% is bigger than the gap between 0% and 2%, and it keeps growing.
- Not quite.That is what doubling the rate would do if each year's rise were on the original £100. It is on last year's price, and last year's price is bigger at 4%.
- Not quite.Two extra per cent times forty years. Adding the rates is the first mistake in this lesson, and it undercounts more every year.
Move the control to see what changes.
A country has 7% inflation. Using the rule of 70, how many years until prices double?
yearsYes.Seventy divided by seven. Prices double every decade at 7%. A house price, a pint, a bus fare: whatever it costs now, it costs twice that in ten years.At 7% a year a £100 basket doubles to £200 in ten years. Slide the marker to what it costs after TWENTY years.
Slide the marker to your guess and lock it. Nothing is scored — the point is to have your own number before you see the real one.
Start · £100After ten years · £200you said · £100actually · £387After twenty years: —
Doubling again is times two, not plus a hundred. The second decade starts from £200, so it ends near £400, not £300. Every ten years it doubles from wherever it is.Small rates over long years are not small. Divide 70 by the rate and you have the years until prices double, and then they double again.
- The pension, year one£1,000The pension, year twenty£1,000
The number on the statement never changes. A pension pays a fixed £1,000 a month for life, with no rises ever. Inflation runs at 3.5%. Roughly when does it buy half of what it does today?
Seventy divided by the rate.
- Not quite.The pounds never change. What they buy does, every single month, and the pension's owner shops in what they buy.
- Yes.Seventy divided by 3.5. The statement says £1,000 every month for twenty years, and by the end that £1,000 buys half a trolley.
- Not quite.That is 100 divided by 3.5, which adds the rate up year by year. The rises stack, so halving comes sooner than that.
Lesson complete
Divide 70 by the rate: that is the years until prices double, and then double again.
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