Compound interest, depreciation and real value

1000 invested at 5% a year, with inflation running at 3%. Run the years and watch the three lines separate.

10 years

1500.00simple

1628.89compound

1212.03real value

Compound beats simple. Inflation beats both.

The formula

FV = PV (1 + r100k)^kn where k is the number of compounding periods a year: 1 yearly, 4 quarterly, 12 monthly. Your calculator has this built in; the marks are for choosing k and n correctly, not for the arithmetic.

1000, 5%, 10 yr→ simple: 1000 + 500 = 1500 → compound yearly: 1628.89 quarterly→ 1643.62 monthly→ 1647.01 daily→ 1648.66 and it stops there

More frequent compounding pays, but it runs out of road. Yearly to monthly gains 18.12; monthly to daily gains 1.65; and no frequency ever beats 1648.72. Compounding more often is worth far less than people assume.

Depreciation is the same formula

60000 baht, −15%, 5 yr→ 60000 × 0.85⁵ = 26,622.32

Use 0.85, not −15. A 15% fall leaves 85% behind, and the multiplier is what goes in the power. Note also that it never reaches zero: percentage decline always leaves something.

Real value, which is the examinable part people skip

Divide by the inflation factor. After 10 years at 3%, prices are 1.03¹⁰ times higher, so your 1628.89 buys 1628.89 ÷ 1.03¹⁰ = 1212.03 of what you could have bought on day one.

The real growth rate is 1.051.03 − 1, about 1.94% a year, not 2%. Close, but it is a ratio and not a subtraction.

Worked example

A family puts 250,000 baht into an account paying 4.5% a year, compounded monthly. Inflation runs at 2.5% a year.

What is it worth after 7 years, and what can it actually buy?

k = 12, n = 7, so 84 periodsk is how often it compounds, n is years. The power is k × n, and putting 84 in both places is the classic wreck.
Monthly rate = 4.5 ÷ 12 = 0.375%Divide the annual rate by the number of periods. Never use 4.5% monthly.
FV = 250000 × (1.00375)⁸⁴One multiplier, applied eighty-four times.
= 342,364 bahtThe statement figure. It has grown by about 37%.
Real value = 342364 ÷ 1.025⁷ = 288,018 bahtPrices rose by a factor of 1.025⁷ = 1.189 over the same period. So the money grew by 92,364 baht and the family gained 38,018 baht of actual spending power. Both numbers are true, and only one of them is on the statement.

Your turn

1. 1000 at 5% compounded yearly for 10 years, to 2 decimal places.

2. A 60,000 baht motorbike depreciates 15% a year. Its value after 5 years, to 2 decimal places.

3. Interest is 5% and inflation is 5%. After ten years your money:

Has grown by about 63%. Buys exactly what it did at the start. Has doubled in real terms. Has lost half its value.

Where the marks go

Getting k and n right. Monthly for 10 years is k = 12 and n = 10, giving 120 periods; putting 120 in both places is the classic wreck.

Using the multiplier for depreciation, 0.85 rather than −15.

Dividing by the inflation factor when real value is asked for, and saying what the answer means: the money grew and still buys less.

Come back to this

These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.

Come back to these in a week, and again in a month.

Which method?

A motorbike worth 60,000 baht loses 15% of its value each year. What is it worth after 5 years?

Depreciation, the same formula with a multiplier under 1. 60000 × 0.85⁵ = 26,622.32 baht. Use 0.85, never −15.
Which method?

A sequence begins 2, 5, 8, 11. What is its 50th term?

Arithmetic. 2 + 49(3) = 149. Equal differences, so add; no repeated percentage anywhere.
Which method?

Interest is 5% and inflation is 5%. After ten years, has your money gained?

Real value. The two factors cancel exactly, so it buys precisely what it did at the start despite the number growing 63%.

The same idea elsewhere

1.7, amortization: A loan is this in reverse: interest added, payments taken off.
1.11, the sum to infinity: Both rest on rⁿ, and whether it grows or shrinks decides everything.

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