💰Compound Interest & Exponential Models

Doctor Kelvin keeps one rule chalked on the ice wall of Exponential Frost, and everything else in this craft hangs off it:

a percent change is worked out on the CURRENT amount, not the starting one

That single sentence is why interest, populations and fading light are all the same shape of problem, and why none of them is linear. If a vault grows by 4 % a year, the second year's growth is 4 % of a bigger number than the first year's was. Adding the same amount every year would be a different story altogether.

So the first move is always the same: turn the rate into one multiplier, and then multiply by it once per period.

• ◆Growth. Keep the whole 100 % and add the rate on top. A rise of 4 % keeps 104 %, so the multiplier is 1.04. In symbols, 1 + r, with r the rate written as a decimal.
• ◆Decay. Take the rate off the 100 %. A loss of 12 % leaves 88 %, so the multiplier is 0.88. In symbols, 1 - r.

Two conversions trip people up, so do them out loud. 4 % is 4/100 = 0.04, not 4. And 0.88 is the multiplier, not the rate — the rate is how far 0.88 sits from 1, which is 0.12, or 12 %.

Once you have the multiplier b, the model writes itself:

A(t) = A0 · b^t

with A0 the starting amount and t counting periods. 500 gold growing at 4 % a year is A(t) = 500(1.04)^t; 500 lumens fading at 12 % a year is A(t) = 500(0.88)^t. The exponent is a count of periods, so if the rate is monthly and t is measured in years, the exponent is 12t, not t. The prompt always tells you what t counts — read that line twice before you write anything.

A bank advertises a rate for a year but pays it in instalments. "6 % a year compounded monthly" does not mean 6 % every month. It means the year's rate is cut into 12 equal slices of 6/12 = 0.5 %, and each slice is paid onto whatever is in the vault at that moment. The formula bookkeeps exactly that:

A = P(1 + r/n)^{n t}

• ◆P is the starting deposit (the principal),
• ◆r is the advertised yearly rate as a decimal,
• ◆n is the number of compounding periods in a year,
• ◆t is the number of years, so n t is the number of periods.

Two slips live in that formula and nowhere else. The first is using r where r/n belongs — paying the whole year's rate every month. The second is putting t in the exponent where n t belongs — compounding monthly but only for as many periods as there are years. Say the two numbers out loud before you type: "the rate per period is r/n, the number of periods is n times t."

Why does more frequent compounding give more? Because the interest starts earning interest sooner. 1,000 gold at 6 % paid once a year becomes 1,060.00. The same 1,000 at 6 % paid monthly becomes 1,000(1.005)¹² = 1,061.68 — the January interest has eleven months of its own growth by December. That is also why "how much interest was earned" is a *different* question from "how much is in the vault": the interest is A - P, the growth only, with the deposit itself taken back out.

Cutting the year finer and finer keeps helping, but by less and less. Compare 6 % on 1,000 gold for a year:

once a year 1,060.00 quarterly 1,061.36 monthly 1,061.68 daily 1,061.83

The numbers are closing in on something rather than running away. That something is the next section.

Slice a year into n pieces and one gold coin at 100 % a year becomes (1 + 1/n)ⁿ. Slice it finer and that number climbs — 2 at n = 1, 2.25 at n = 2, 2.5937 at n = 12, 2.7146 at n = 365 — but it never reaches 3. It settles on e = 2.718281828..., an ordinary fixed number like pi, and the one every continuous-growth model is built out of. Nothing about e changes from problem to problem; it is simply the answer to "what if the compounding never paused at all?"

For a rate r, the same limit gives the continuous growth formula:

A = P e^{r t}

Compare it with A = P(1 + r/n)^{n t} and you can see what happened: the bracket and its exponent have collapsed into e^{r t}. The whole of r × t goes upstairs, in the exponent. Writing t × e^r instead is the classic wreck of this formula, and it gives an answer that grows in a straight line rather than a curve.

Continuous compounding is the ceiling, not a different world. On 10,000 gold at 5 % for 10 years, monthly compounding gives 16,470.09 and continuous gives 16,487.21 — seventeen gold apart after a decade. When a question asks how much MORE the continuous vault holds, work out both totals in full and subtract at the very end. Rounding each one to the nearest gold first would throw away most of the answer.

When you type a continuous model as an expression, the checker knows e, so 200e^0.05t is a legal answer and means exactly what it looks like.

Half-life is decay measured by the time it takes to halve, rather than by a percent. If h is the half-life, then

A = A0 · (1/2)^t/h

The exponent t/h is a count of half-lives, and it does not have to be a whole number. A 6-hour half-life over 15 hours is 15/6 = 2.5 half-lives, and (1/2)².5 = 0.1768, so 17.68 % of the sample is left. Two habits go wrong here. Counting periods off by one is the first: 15 hours is 2.5 half-lives, not 3. Taking half of the ORIGINAL amount each time is the second — each half-life removes half of what is left at that moment, which is a smaller amount every time, so the sample thins out forever and never reaches zero.

When the time is a whole number of half-lives, the fraction left is exact and worth knowing by sight: 1/2, then 1/4, then 1/8, then 1/16. Three half-lives leave (1/2)³ = 1/8 — not 1/3.

Two readings are enough to build a model. Suppose a colony numbers 1,200 in year 0 and 1,452 in year 2. Call the yearly multiplier b. Two years means b has been applied twice:

1200 · b² = 1452 b² = 1452 / 1200 = 1.21 b = sqrt(1.21) = 1.1

So the colony grows by 10 % a year and the model is P(t) = 1200(1.1)^t. Notice the last step: 1.21 is the multiplier for the whole two years, and one year's multiplier is its square root. Over n years you take the nth root. Reporting 1.21, or 21 %, answers a question nobody asked.

Two advertised rates cannot be compared until the compounding is folded in. The effective annual rate does that: it is the single yearly percent that would grow a deposit by the same amount.

effective rate = ((1 + r/n)ⁿ - 1) × 100 %

Follow one gold coin through one year and read the growth off the end. 6 % compounded monthly gives (1 + 0.005)¹² = 1.0617, so the effective rate is 6.17 %, not 6 %. That is what lets you see that 5.9 % compounded monthly (effective 6.06 %) beats a flat 6 % paid once a year, even though the advertised number is smaller. The two traps are quoting the nominal rate back (6 %) and quoting the multiplier (1.0617) instead of the growth on top of it.

Typing your answer at Exponential Frost

• ◆Money is a decimal to the nearest cent: 2309.46. No currency symbol is needed, though one is accepted, and the prompt always says "round to the nearest cent". Carry every decimal through the working and round only at the end.
• ◆A percent rate is a bare number, with no percentage sign: type 6.17, to the number of decimal places the prompt asks for. It says 1 or 2 decimal places every time.
• ◆A multiplier is a decimal to 2 decimal places: 1.1 or 0.88.
• ◆A fraction left after whole half-lives is typed in lowest terms: 1/8.
• ◆A model is an expression in t, with ^ for the power. 500(1.04)^t and 500*1.04^t are read the same way. A monthly rate over t years is 500(1.01)^12t — the brackets hold the whole exponent together. Continuous growth is 200e^0.05t.
• ◆A count of whole periods is a plain whole number: 8. These are answered by multiplying and counting, never by a logarithm — solving for the exponent itself is what Doctor Kelvin teaches next door, in Exponential & Logarithmic Equations.