01:640:103 Chap.11 Growth Factors and Compound Interest
Growth Factors and Compound Interest
Percent changes are multiplications
Raising a quantity by 100i percent multiplies it by the growth factor 1 + i: the 1 keeps the original amount and the i adds the growth.
A 6% rise multiplies by 1.06 and a 15% fall multiplies by the decline factor 0.85. Statements are made in percentages, but every calculation uses the decimal form, and the same arithmetic describes prices, tuition, savings and populations.
Successive changes
Two changes in a row multiply their factors; they never simply add.
Two years of 4% growth multiply by 1.0816, an 8.16% rise, because the second year's growth also applies to the first year's growth. A 25% rise followed by a 20% fall multiplies by exactly 1, no change at all, although adding the percentages suggests plus 5%.
Over n periods of the same rate the overall factor is the growth factor raised to the power n.
Compound interest
Interest left in an account earns interest itself. With present value PV, rate i per period and n periods, the future value is FV = PV(1 + i)^n.
In the guide's example $4,000 at 5% for 25 years reaches about $13,545 when the interest stays in the account, against $9,000 when each year's interest is withdrawn.
The formula can also run backwards, dividing a target by the growth factor to find how much to invest now.
Working backwards from a target
The compound interest formula can be solved for the present value by dividing the future value by the growth factor raised to the number of periods.
To have $8,000 in three years at 3.2% a year, divide 8,000 by 1.032 cubed, about 1.099, which gives roughly $7,278.65 to invest today. The same move answers questions about the past: a price that is $90.10 after a 6% rise was $85.00 before it.
Divide by a growth factor to go back in time; multiply by it to go forward.
Population uses the same arithmetic
Growth and decline factors apply to any quantity that changes by a fixed percentage per period, not only money. A population shrinking by 0.5% a year is multiplied by 0.995 each year, so after 20 years it keeps about 90.46% of its size.
The course notes that real populations speed up and slow down, so a constant rate is a modelling assumption.
A common exam trap is to add percentages over many years. Twelve years at 6% is not a 72% rise; the factor 1.06 to the twelfth power is about 2.012, slightly more than doubling.
What this chapter covers
- 01
Growth factors and decline factors
- 02
Successive percentage changes
- 03
Growth over many periods
- 04
Compound interest versus withdrawn interest
- 05
Present value and future value
Worked example · free
A price that rises and falls
- 1Growth factor 1.15, then decline factor 0.85.
- 2Overall factor 1.15 × 0.85 = 0.9775.
- 1Final price $80 × 0.9775 = $78.20.
Key terms
- Growth factor
- A growth factor is the number 1 + i that multiplies a quantity growing by 100i percent in one period.
- Decline factor
- A decline factor is the number 1 − i that multiplies a quantity falling by 100i percent.
- Compound interest
- Compound interest is interest earned on both the original deposit and all interest already credited.
- Future value
- Future value is the amount an investment or debt will be worth at a stated time ahead.
- Present value
- Present value is what an investment or a future payment is worth today.
- Simple interest
- Simple interest is paid only on the original deposit, as when each year's interest is withdrawn instead of left to grow.
Growth Factors and Compound Interest FAQ
Why is a 10% rise followed by a 10% fall not a return to the start?
The fall is 10% of the higher price, so it removes more than the rise added. The factors multiply to 1.10 × 0.90 = 0.99, a net loss of 1% whichever order the two changes come in.
What does the 1 in a growth factor like 1.04 stand for?
It represents keeping all of the original amount. Multiplying by 0.04 alone gives only the increase; adding the 1 gives the new total, 104% of the old amount.
How do I find the overall percentage change after several years?
Multiply the yearly growth factors together, or raise one factor to the number of years if the rate is constant, then subtract 1 and convert to a percent. Three years at 4% gives 1.04 cubed, about 1.1249, a rise of 12.49%.
Exam move
Rewrite every word problem as a chain of factors before touching the calculator, and keep full precision until the final rounded answer.
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