Simple and compound interest, and the principle that rules out one of them

CM1 · Actuarial Mathematics · 7 min read

1The accumulation factor, before any formulae

Everything in this part of CM1 is built on one object. Write A(t₁, t₂) for the amount that one rupee invested at time t₁ grows to by time t₂. That is the accumulation factor, and the whole of the compound-interest syllabus is the study of what it can and cannot look like.

Two rules follow immediately, and both are worth stating because questions are built on them.

A(t, t) = 1. Money that has not been invested for any time has not grown.

A rupee is a rupee. The factor depends on the two dates, not on the amount, so ₹50,000 invested over the same period grows by the same factor as one rupee. Everything scales.

2The two candidate rules

Simple interest

Interest is calculated on the original sum only, every period. Interest earned is never itself invested.

Compound interest

Interest is added to the sum, and from then on earns interest in its own right.

Both are plausible-looking rules. Both are used in the real world. Only one of them survives the next section.

3The mechanism: the principle of consistency

Here is the idea the page exists for. Suppose money is invested from time 0 to time 3. There are two ways to describe the same journey.

Leave it alone for three years, and it grows by A(0, 3). Or take it out at time 1 and immediately put it back, and it grows by A(0, 1) and then by A(1, 3).

Nothing real has happened differently. The money sat in the same place for the same three years. So the two descriptions must give the same answer:

That is the principle of consistency, and it is not an assumption bolted on for convenience. It is a statement that the answer cannot depend on how you choose to describe the period — and the worked example shows that simple interest fails it while compound interest satisfies it exactly.

4Why failing it is fatal rather than untidy

If breaking the period gives a bigger number, then everybody breaks the period. A depositor offered simple interest for three years would withdraw and redeposit every year, and would be right to. The bank's quoted three-year rate would then be a fiction, because nobody would ever accept it on its own terms.

Put the other way round: the moment you allow money to be withdrawn and immediately reinvested, consistency forces compounding. It is not that compound interest is a more sophisticated choice. It is the only rule that survives contact with a market in which people can move their money.

And consistency does more than rule out simple interest. Applied to a constant rate it forces the exponential form: a factor that depends only on the length of the period and multiplies across adjoining periods has to be (1 + i)t for some i. That is why the same expression appears everywhere in the paper.

5So why is simple interest used at all?

Because over a short period the difference is small and the arithmetic is easier, and because some instruments are defined that way by contract rather than by theory. Three honest cases:

Periods under a year. A 90-day deposit quoted at a simple rate loses very little to the approximation, and the convention is universal in money markets.

Interest between compounding dates. Where a rate compounds annually and a payment falls mid-year, the fraction of a year is often handled simply as a matter of contract.

Penalties and statutory interest. A rule in a contract or a statute may prescribe a simple rate. That makes it the right calculation for that purpose and still not a model of how money grows.

The examinable point is to know which you have been given and never to mix them inside one calculation.

A worked example

An invented example, built for this page. A deposit of ₹1,00,000 at 10% a year for three years. Both rules applied to the same money, with every figure chosen so the arithmetic is exact.

1. 1Year by year, side by side

   End of year	Simple: balance	Interest that year	Compound: balance	Interest that year
   1	₹1,10,000	₹10,000	₹1,10,000	₹10,000
   2	₹1,20,000	₹10,000	₹1,21,000	₹11,000
   3	₹1,30,000	₹10,000	₹1,33,100	₹12,100

   Check each total against the interest columns above. Simple: 10,000 × 3 = ₹30,000, and 1,00,000 + 30,000 = ₹1,30,000. Compound: 10,000 + 11,000 + 12,100 = ₹33,100, and 1,00,000 + 33,100 = ₹1,33,100. Both tie.

   The two formulae give the same two numbers. Simple: 1,00,000 × (1 + 0.10 × 3) = 1,00,000 × 1.30 = ₹1,30,000. Compound: 1,00,000 × 1.103 = 1,00,000 × 1.331 = ₹1,33,100. The gap is ₹3,100, which is 3.1% of the principal after only three years.

   Notice where the gap comes from. The compound interest in year 2 is ₹11,000, and ₹1,000 of that is interest on year 1's interest. In year 3 it is ₹12,100, of which ₹2,100 is interest on earlier interest. Those two amounts are the whole difference: 1,000 + 2,100 = ₹3,100.

2. 2Now test consistency: split the three years at time 1

   Compound interest. Using the accumulation factors,

   A(0, 1) × A(1, 3) = 1.10 × 1.21 = 1.331 = A(0, 3)

   Split it somewhere else instead, at time 2:

   A(0, 2) × A(2, 3) = 1.21 × 1.10 = 1.331 = A(0, 3)

   Both splits give 1.331, the same factor as leaving the money alone. Consistency holds, and it would hold wherever the split were made, because the exponents simply add.

   Simple interest. Now the same two splits, where a withdrawal and immediate redeposit means the second period's interest is calculated on the new, larger principal.

   A(0, 1) × A(1, 3) = (1 + 0.10 × 1) × (1 + 0.10 × 2) = 1.10 × 1.20 = 1.32

   But leaving the money alone for three years gives

   A(0, 3) = 1 + 0.10 × 3 = 1.30

   which is not 1.32. Split at time 2 instead and the answer changes again: 1.20 × 1.10 = 1.32 as well, so both one-break routes give 1.32 against 1.30 for no break — and breaking twice, at times 1 and 2, gives 1.10 × 1.10 × 1.10 = 1.331, which is the compound answer.

3. 3What that is worth, in rupees

   On the ₹1,00,000 deposit, using the factors printed above:

   What the depositor does	Factor	Proceeds
   Leaves it for three years	1.30	₹1,30,000
   Withdraws and redeposits once	1.32	₹1,32,000
   Withdraws and redeposits every year	1.331	₹1,33,100

   So a bank offering 10% simple interest for three years is offering something a depositor can improve on by ₹2,000 with one withdrawal, and by ₹3,100 with two — and the second figure is exactly the compound answer from the first table. That is the argument of this page in one line: breaking the period cannot be allowed to change the answer, and under simple interest it does.

   One last check that compounding really is the fixed point of this process. Break the three years into n equal pieces and apply simple interest within each: at n = 1 the factor is 1.30, at n = 3 it is 1.331. Pushing n higher keeps raising it, towards e0.3; the rate at which it stops rising is what the force of interest measures, and that is a different page. What matters here is that the sequence does not stay at 1.30, and so 1.30 was never the right answer for a period anybody could interrupt.

Where this fits in CM1

This page explains one idea from CM1 Actuarial Mathematics. The full CM1 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CM1 topics explained

• Present value, the discount factor, and why d is not i
• Nominal rates payable p times a year, and the loan that costs more than it says
• Force of interest explained, and how it relates to i, v and d
• When the interest rate changes: steps, and a force that varies
• Continuous payment streams, and why they are worth more than they look
• Annuity functions in CM1: immediate, due, deferred and temporary
• Perpetuities and increasing annuities: payments that never stop, or never stay level
• Money rates, real rates and inflation: the exact link, not the subtraction
• Real yields from an inflation index, and index-linked payments
• Loan schedules: the prospective and retrospective methods
• Life table functions explained: l(x), d(x), q(x) and p(x)
• Net premiums and the equivalence principle, worked from first principles
• Reserves in CM1: prospective, retrospective, and why they agree
• Thiele's differential equation, explained term by term

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.