monthly vs annual inflation rates

Question

Explain why compounding a monthly inflation rate at 1% is NOT equivalent to an annual inflation rate of 12%.

Answer 1

Calculate the two and compare

A(1) = a₀(1 + 0.12)¹, for the yearly

A(12) = a₀(1 + 0.01)¹², for the monthly

Answer 2

Because more frequent compounding results in more desirable returns.

A = P(1 + r)^t

For annual interest of 12% on $1,000 (for example) you'd have 1000(1.12) = $1,120 at the end of the year.

For monthly interest A = P(1 + r/n)^nt where n = no. compounding periods/yr and t = no. years (1 for this example)

A = 1000(1 + .12/12)¹² = 1000(1.01)¹² = 1000(1.126825) = $1,126.83.

If you try the same thing with weekly (n = 52) or daily interest (n = 365), you'd see even more improvement in the balance at the end of the year.

Answer 3

Hi Lorena,

That's an excellent question.

If an amount increases at a simple annual rate of 12% and the end of the year 12% of the initial amount is added on. For instance, $100 becomes $112.

On the other hand, if that 12% annual rate is compounded as 1% monthly something different happens.

At the end of the first month $100 becomes $101. Now this is a small amount, but during the second month that extra dollar also earns 1%. At the end of the second month $101 becomes $102.01. Each month the balance at the beginning is bigger the month before and therefore 1% of that balance is more than the previous month. In other words, after the first compounding period, the interest earns interest.

By the end of the year, the initial amount would have been multiplied by 1.01 twelve times.

The final amount can be calculated as 100(1.01)¹² which equals $112.68.

Granted the difference in this example is only 68 cents, but larger investments, higher interest rates and longer durations could have huge impacts.

I hope this helps.