How this instrument works
P(t) = P₀(1+r)ᵗ multiplies a starting value P₀ by (1+r), the same factor, once for every period t. This is the growth-rate cousin of the general exponential function: instead of an arbitrary base b, the base here is always one plus a rate — 5% growth is a base of 1.05, a 3% decline is a base of 0.97 — which is why this shape, not a generic aᵇˣ form, is the one that shows up whenever something grows or shrinks by a fixed percentage of its current size.
The exponent stacks because each period's growth applies to the previous total, not to the original P₀. After one period the value is P₀(1+r); apply the same multiplier again and it becomes P₀(1+r)(1+r) = P₀(1+r)², and repeating that t times puts the factor (1+r) into the product t times, giving P₀(1+r)ᵗ. That repeated multiplication, rather than repeated addition, is exactly what separates compounding from ordinary linear growth.
Thomas Malthus used this same shape in 1798 to argue that population grows geometrically while food supply grows only arithmetically — his 'geometric' increase is this formula's exponential curve. A useful edge case: set r negative and the formula describes decline just as naturally as growth, since (1+r) drops below 1 for any rate between -100% and 0%. And because the exponent, not the base, carries the compounding, the time any P₀ takes to double depends only on r, never on the starting value itself.
- Enter the amount you're starting with into Starting value, P₀ — the quantity at t = 0, before any growth is applied.
- Enter the per-period rate into Growth rate per period (decimal) as a decimal, not a percentage: 5% growth is 0.05, a 3% decline is -0.03.
- Set Number of periods, t to how many times that rate compounds — years, quarters, generations, whatever period the rate is quoted for.
- Read P(t) for the total after all t periods of compounding have been applied to P₀.
Worked example — $1,000 growing at 5% for 10 periods
Start with P₀ = 1,000 and a growth rate of r = 0.05 (5%) per period, held for t = 10 periods. The formula multiplies the starting value by (1.05)^10 = 1.628894626777442, so P(10) = 1000 × 1.628894626777442 = 1,628.894626777442 — a figure worth carrying to full precision since it is the exact value this sheet checks itself against.
Compare that to what simple, non-compounding growth would give: 5% of the original 1,000 is 50 per period, and 10 periods of that adds up to only 1,000 + 500 = 1,500. The extra 128.89 comes entirely from later periods growing on top of earlier gains rather than on the fixed original amount — the entire point of the (1+r)ᵗ exponent instead of a flat (1+rt) multiplier.
Questions
What does the rate r mean in P(t) = P₀(1+r)ᵗ?
It's the fractional change per period, entered as a decimal rather than a percentage — 0.05 for 5% growth, -0.03 for a 3% decline. Because it's added to 1 before being raised to the tth power, a rate of exactly 0 leaves P(t) = P₀ unchanged no matter how many periods pass.
How is exponential growth different from linear growth?
Linear growth adds the same fixed amount each period; exponential growth multiplies by the same factor each period, so earlier gains keep earning more of their own. At 5% for 10 periods on 1,000, linear growth reaches 1,500 while this compounding formula reaches 1,628.89 — and the gap widens with every additional period.
Where does this specific growth-rate formula come from?
It comes from applying a percentage change repeatedly rather than once: after each period the new total becomes the base for the next period's growth. It's the mechanics behind compound-interest tables, population models going back to Thomas Malthus's 1798 essay, and decay written with a negative rate.
What mistake do people most often make with this formula?
Typing the rate as a whole number instead of a decimal — entering 5 for 5% instead of 0.05 turns modest growth into an absurd result raised to the tenth power. The second most common slip is pairing a rate quoted for one period, say annual, with a t counted in a different period, say months.
How does this differ from a generic exponential function f(x) = a·bˣ?
The generic form allows any base b and any coefficient a; this formula fixes the base to (1+r), tying it directly to a percentage rate, and swaps the continuous x for a whole number of periods, t. Every growth-rate calculation is an exponential function, but not every exponential function represents a growth rate.
Can the growth rate be negative, and what happens then?
Yes — a negative r between -1 and 0 makes (1+r) less than 1, so the formula describes decline instead of growth, using the exact same shape. A rate of -0.10, for instance, shrinks P₀ by 10% every period, compounding downward exactly as the positive case compounds upward.