Simple vs. Compound Interest

Both are choices of the accumulation function a(t). Under simple interest, only the original principal earns interest; under compound interest, interest itself earns interest.

By the end you'll be able to write both accumulation functions, compute either accumulated value, and know exactly where and why they diverge over a term.

Predict: for a time between 0 and 1 year, which is larger — simple or compound accumulation? Drag t below 1 and check.

The simple-interest line \(1+it\) and the compound-interest curve \((1+i)^t\) both start at 1 and both reach \(1+i\) at \(t=1\) — drag i and t and watch the shaded gap between them open and flip sides at \(t=1\).

At t = 0.50: simple 1.040 vs. compound 1.039

a(t) = 1+it vs. a(t) = (1+i)ᵗ — shaded gap shows who's ahead
compound (1+i)ᵗ simple 1+it shaded = |gap|

Simple and compound interest are the two standard choices of accumulation function you met in a(t) — the whole difference boils down to whether interest earns interest.

Formal

Under simple interest, \(a(t) = 1 + it\) — a straight line with constant slope i. Under compound interest, \(a(t) = (1+i)^t\) — exponential growth. Both agree at \(t=0\) (both equal 1) and at \(t=1\) (both equal \(1+i\)); for \(0 < t < 1\), \(1+it > (1+i)^t\), and for \(t > 1\), \((1+i)^t > 1+it\). Compound interest is the exam standard — assume it unless a problem says "simple."

Applied

This crossover matters in practice: a 90-day money-market note is often quoted with simple interest because over a fractional period it's the more generous — and simpler — convention, while a multi-year bond or loan is compounded because over whole periods compounding is what actually happens as credited interest joins the balance and starts earning its own interest.

Worked example

Invest 1000 at \(i=8\%\) for 3 years. Simple: \(1000(1+0.08\cdot3) = 1000(1.24) =\) 1240.00. Compound: \(1000(1.08)^3 = 1000(1.259712) =\) 1259.71. The 19.71 difference is the "interest on interest" that compounding captures over the 3rd year.

Your turn

Same setup, different numbers: invest 500 at \(i=10\%\) for 2 years. Simple: \(500(1+0.10\cdot2) = 500(1.20) = 600.00\). Compound: \(500(1.10)^2 = 500(1.21) =\) ____

Reveal the answer

\(500(1.21) =\) 605.00 — 5.00 more than the simple-interest value, matching the pattern that compound interest pulls ahead once \(t > 1\). Set i = 10% and drag t to 2 on the slider above to see it on the diagram.

More info — why the effective rate behaves differently

Recall from the effective rate \(i_n=(A(n)-A(n-1))/A(n-1)\): under compound interest \(i_n=i\) every period — level, because both numerator and denominator scale by \((1+i)\) each period. Under simple interest the numerator stays a flat i while the denominator \(1+i(n-1)\) keeps growing, so \(i_n = i/(1+i(n-1))\) actually decreases over time — the same absolute interest is being measured against an ever-larger base. See Dive deeper for more on accumulation and amount functions.

Check your understanding

Question 1 of 4

Invest 2000 at \(i=6\%\) for 4 years under compound interest. What is the accumulated value?

Question 2 of 4

Predict, then check: for a time strictly between 0 and 1 year, which accumulates more at the same rate \(i>0\) — simple or compound interest?

Question 3 of 4

At which values of \(t\) do simple interest \(1+it\) and compound interest \((1+i)^t\) give exactly the same accumulated value, for any \(i>0\)?

Question 4 of 4

Under compound interest, \(a(t)=(1+i)^t\). Using \(i_n=(A(n)-A(n-1))/A(n-1)\), what is the effective rate of interest \(i_n\) in period \(n\)?

Recap

  • Simple interest: \(a(t) = 1 + it\) — a line; only the original principal earns interest.
  • Compound interest: \(a(t) = (1+i)^t\) — exponential; interest earns interest. Compound is the exam-standard default.
  • The two agree at \(t=0\) and \(t=1\); simple is larger for \(0 < t < 1\), compound is larger for \(t > 1\).
  • Compound interest keeps the effective rate \(i_n\) level at i every period; simple interest's \(i_n\) decreases over time.

Dive deeper

Sources

  • Simple vs. Compound Interest