Rising prices
Prices rise over time. A general rise in prices is called inflation, and it is measured as a percentage a year. If inflation is 4% a year, a basket of shopping that costs $1000 today costs 1000 × 1.04 = 1040 dollars a year from now.
So a dollar buys less as time passes. Suppose your savings earn 6% a year while prices rise by 4% a year. The balance grows, but so do the prices of the things it will be spent on.
After one year
Save $1000 at 6% a year. After a year the balance is 1000 × 1.06 = 1060 dollars, and the basket that cost $1000 a year ago now costs $1040. The balance buys the basket and has $20 left over, but that is $20 at the new, higher prices.
Divide out the price rise
Every price is now 1.04 times what it was, so a dollar now buys what 1 ÷ 1.04 of a dollar bought a year ago. To find what $1060 now is worth in last year's money, divide it by the price multiplier: 1060 ÷ 1.04 = 1019.23 dollars, to the nearest cent.
That is the real value of the balance: $1060 now buys what $1019.23 bought a year ago. The $1060 on the statement is called the nominal value. After n years of inflation at i% a year, prices have been multiplied by , so an amount A then has a real value of in today's money.
This is discounting, as in The Time Value of Money, with the rate at which prices rise in place of the interest rate.
The line runs from the $1000 saved to the $1060 balance, and 1040 marks what the basket now costs. Divided by 1.04, the balance is worth $1019.23 in last year's money: $19.23 more buying power than the $1000 had.
Real growth
In real terms the $1000 grew to $1019.23 in the year, a real growth rate of 1.92%. The real growth factor is the account's factor divided by the price factor: , for an interest rate of r% and inflation of i%. Here 1.06 ÷ 1.04 = 1.019231, to 6 decimal places, so the real growth rate is 1.92%, to 2 decimal places.
Subtracting the rates, 6% − 4% = 2%, gives a number close to 1.92% but not equal to it. r − i is an approximation to the real growth rate. It is close when both rates are small and further out when they are large: at 20% interest and 15% inflation, 1.20 ÷ 1.15 = 1.043478, a real growth rate of 4.35%, not 5%.
When the interest rate equals the inflation rate, the real growth factor is : the balance grows exactly as fast as prices, and it buys the same as before. When prices rise faster than the balance, the real value falls even though the balance grows. At 3% interest and 4% inflation, 1.03 ÷ 1.04 = 0.990385, so the real value falls by about 0.96% a year.
Over ten years
Over 10 years at these rates, $1000 grows to dollars, to the nearest cent. Over the same 10 years prices are multiplied by , to 6 decimal places, so the real value is 1790.85 ÷ 1.480244 = 1209.83 dollars, to the nearest cent.
The statement shows a gain of $790.85, but the gain in buying power is only $209.83. Subtracting the rates would give dollars, which is $9.16 too high.
$1000 at the start; the balance of $1790.85 after 10 years at 6%; and its real value, $1209.83, in the money of the start, after 10 years of 4% inflation.
The usual mistakes
Giving the balance on the statement as the real value. $1060 is the nominal value; the price rise still has to be divided out.
Subtracting the rates as if that were exact. The factors divide: the real growth factor is 1.06 ÷ 1.04 = 1.019231, not 1.06 − 0.04 = 1.02.
A price index
The second problem below measures prices with a price index: a number that follows the cost of a fixed basket of goods, set at 100 in a chosen base year. An index of 150 means the basket costs times what it cost in the base year, so prices have been multiplied by 1.5 since then.
Worked example: A Pay Rise Smaller Than the Rise in Prices
Question Mei earns $41600 a year. Her employer offers a rise of 3%. Over the same year prices rise by 4%. (a) Find her new salary, and what that salary is worth in the money of today. (b) Find the rise that would have left her exactly as well off, and say how far the offer falls short of it.
1.A rise of 3% multiplies the salary by 1.03, so the new salary is 41600 × 1.03 = $42848.
A rise of 3% multiplies the salary by 1.03: 41600 × 1.03 = $42848. 2.Prices are 1.04 times what they were, so a dollar of next year's money buys what 1 ÷ 1.04 of a dollar buys today. To value the new salary in today's money, divide it by 1.04.
Prices are 1.04 times what they were, so a dollar of next year’s money buys 1 ÷ 1.04 of what a dollar buys today. 3.(a) 42848 ÷ 1.04 = $41200. The new salary buys what $41200 buys today, so its real value has fallen by 41600 − 41200 = $400.
(a) 42848 ÷ 1.04 = $41200, so the real value of the salary has fallen by $400. 4.A rise that exactly keeps pace with prices multiplies the salary by 1.04, giving 41600 × 1.04 = $43264.
A rise that keeps pace with prices multiplies by 1.04, giving 41600 × 1.04 = $43264. 5.(b) The offer is 43264 − 42848 = $416 below that. Check: $416 of next year's money is 416 ÷ 1.04 = $400 of today's money, which is the fall found in part (a).
(b) That rise is worth $41600 in today’s money, exactly what she earns now, and the offer is $416 short of it.
Answer: (a) Her new salary is $42848, which is worth $41200 in today's money, a real fall of $400; (b) a rise of 4%, or $43264, would have kept her as well off, so the offer is $416 short
Common mistakes
- Subtracting the percentages, 3% − 4% = −1%, and calling the real loss 1% of $41600, or $416, in today's money. Growth is divided by prices, never reduced by them: the real multiplier is 1.03 ÷ 1.04 = 0.99038 to five decimal places, so the fall in today's money is $400. The $416 is a sum in next year's money.
- Undoing a rise of 4% in prices by taking 4% off, so multiplying by 0.96. That gives 42848 × 0.96 = $41134.08, which is $65.92 too low. A rise of 4% is undone by dividing by 1.04, not by multiplying by 0.96.
Worked example: A Pension Fixed in Dollars, Measured Against a Price Index
Question Mr Lim retired in 2010 on a pension of $2400 a month, and the amount has never changed since. The consumer price index, which stood at 100 in 2010, stands at 150 today. (a) Find what the pension is worth today in 2010 money, and how much of its buying power has gone. (b) Find the pension that would buy today what $2400 bought in 2010, and say how far the real pension falls short of it.
1.The index says that what cost $100 in 2010 costs $150 today, so today's prices are 150 ÷ 100 = 1.5 times the 2010 prices.
The index prices one basket at two dates: $100 in 2010 and $150 today, so prices are 1.5 times what they were. 2.To value today's dollars in 2010 money, multiply by 100150, which is 23.
To value today’s dollars in 2010 money, multiply by 100150 = 23. 3.(a) 2400 × 23 = $1600. The pension buys today what $1600 bought in 2010, so 2400 − 1600 = $800 a month of buying power has gone, which is a third of it.
(a) 2400 × 23 = $1600, so $800 a month of buying power has gone, a third of it. 4.To keep pace with prices, the pension would have to rise in the same ratio as the index, so multiply by 150100 = 1.5.
A pension that kept pace would have risen in the same ratio: 2400 × 150100 = $3600 a month. 5.(b) 2400 × 1.5 = $3600 a month. The pension is 3600 − 2400 = $1200 a month short of that. Check: $1200 of today's money is 1200 × 23 = $800 of 2010 money, which is the monthly loss found in part (a).
(b) The pension is $1200 a month short of that, and 1200 × 23 = 800 in 2010 money.
Answer: (a) The pension is worth $1600 a month in 2010 money, so $800 a month of buying power has gone, a third of it; (b) it would have to be $3600 a month, so the pension falls $1200 a month short
Common mistakes
- Reading a rise of 50% in prices as a loss of 50% in buying power, so $1200 a month. The pension still buys $1600 of 2010 goods, so a third of its buying power has gone, not a half. A rise of 50% in prices is undone by a fall of a third, because 1.5 × 23 = 1.
- Taking $2400 down by the index itself, as 2400 − 150 = $2250. An index number is not a sum of money. It is only the ratio 150100 that carries any meaning, and the pension is divided by that ratio.