The problem the force of interest solves
A bank quotes 7%. A bond quotes 7%. A loan quotes 7%. They are not the same 7%, and the first job of the interest-theory half of CM1 is to make you precise about which one is meant. The force of interest is the most precise of all of them, and once you have it every other version of the rate is one line of algebra away.
Start with what everyone agrees on. If ₹1 grows to ₹1.07 after exactly one year, the annual effective rate of interest is i = 0.07. Effective means measured over a whole year, with interest credited once at the end. That is the benchmark, and every other rate in this topic is a different way of describing the same growth.
The five symbols, in plain words
i — the annual effective rate. Interest paid once, at the end of the year.
v — the discount factor, v = 1 / (1 + i). What ₹1 a year from now is worth today. Everything in CM1 that involves a cash flow in the future involves a power of v.
d — the annual effective rate of discount, d = i / (1 + i) = iv. The same growth seen from the far end: interest taken off the front rather than added at the back. A bill that pays ₹1 in a year and is bought for ₹1 − d.
i(p) — a nominal rate payable p times a year. This one is a trap because it is not a rate for any period that exists. The rate actually earned each period is i(p)/p; the nominal rate is that figure multiplied up to look annual.
δ — the force of interest. The rate of growth at an instant, compounding continuously.
The two equations worth memorising
Everything else follows from these.
eδ = 1 + i, so δ = ln(1 + i)
The first says: earn i(p)/p p times over, and you must end up where the effective rate says you end up. The second is the same statement with p sent to infinity. That is all the force of interest is — the nominal rate when interest is credited not twelve times a year, or 365, but continuously.
Accumulating for t years is then either form you prefer, and they are identical:
The right-hand form is the one to reach for when the force of interest is not constant. If δ is a function of time, the accumulation factor from time a to time b is exp(∫ab δ(s) ds), and the whole of the time-varying-force section of the syllabus is that integral with a different δ(s) in it. Nothing new is being introduced; the exponential is simply being fed a sum instead of a product.
Why they are never equal, and the order they come in
For a positive interest rate, these five always sit in the same order:
There is one idea behind the whole chain. The more often interest is credited, the more of it earns interest on itself, so a smaller nominal rate is enough to reach the same effective rate. Credit twice a year and you need slightly less than i; credit monthly and less again; credit continuously and you need the least of all, which is δ. The rate of discount d sits below even that, because it is charged at the start of the year rather than earned at the end.
This ordering is the cheapest check in the paper. Work out a nominal rate, see where it has landed relative to i and δ, and an answer that has come out above i or below δ is wrong before you have looked at the algebra.
Where marks are lost
Treating a nominal rate as a rate. If a question gives 9% convertible monthly, the monthly rate is 0.75%, not 9%. Multiplying by twelve to get back to something annual is the definition of the nominal rate, not an accumulation. Reading i(12) = 0.09 and then accumulating at 9% per month is the single most expensive slip in this part of the syllabus, because everything after it is arithmetically correct and still worth nothing.
Using δ where the question asked for a rate per period. δ is dimensionless in the sense that matters: it belongs inside an exponential, not inside (1 + x)n. If you find yourself writing (1 + δ)n, stop.
Rounding δ too early. δ = ln(1.07) = 0.0676586… Round it to 0.068 and a thirty-year accumulation is out by about 0.4% — small on one policy, not small on a reserve. Carry at least six decimal places through the working and round once, at the end.
Questions students ask
Is the force of interest an interest rate at all?
Not in the sense of "the amount you get". Nobody is ever paid δ. It is the instantaneous rate of growth — the limit of a nominal rate as the crediting gets infinitely frequent. It exists because it makes the algebra clean: discounting becomes multiplication by an exponential, and an exponential is far easier to integrate than a power. That is the entire reason it is in the course.
Why does CM1 bother with the rate of discount d?
Because real instruments work that way. A Treasury bill does not pay you interest; it is sold to you below its face value and redeemed at face value. The quoted figure is a discount rate, and converting it to an effective rate before comparing it to anything else is a genuine piece of actuarial work rather than an exam device.
Which form should I use in the exam?
Whichever makes the cash flows easiest to write down, then convert once. Annuities and loan schedules are cleanest in i and v; anything described as continuous, and anything where the rate changes with time, is cleanest in δ. Switching mid-question is how sign and power errors get in.
A worked example
Take an annual effective rate of i = 7% and find every other version of it. Work to seven decimal places and round at the end.
The force of interest
Nominal rates
Rearranging (1 + i(p)/p)p = 1.07 gives i(p) = p[(1.07)1/p − 1].
Payable quarterly, p = 4:
i(4) = 4 × 0.0170585 = 0.0682341 → 6.8234% p.a.
Payable monthly, p = 12:
i(12) = 12 × 0.0056541 = 0.0678497 → 6.7850% p.a.
The monthly rate actually earned is 0.0056541, or 0.56541% a month — not 6.785% a month, and not 7%/12.
The discount factor and the rate of discount
d = 0.07 / 1.07 = 0.0654206 → 6.5421% p.a.
The order check
| Symbol | Value |
|---|---|
| d | 6.5421% |
| δ | 6.7659% |
| i(12) | 6.7850% |
| i(4) | 6.8234% |
| i | 7.0000% |
In order, as they must be. Five numbers, all describing the same growth, differing by up to 46 basis points.
Using it: accumulate ₹50,000 for three and a half years
A non-integer term is where the exponential form earns its keep.
which is ₹63,359.72.
Check it the other way, through i: 1.073 = 1.2250430 and 1.070.5 = 1.0344080, and 1.2250430 × 1.0344080 = 1.2671943. The same factor to seven places, because (1 + i)t and eδt are the same thing written twice.
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
- Simple and compound interest, and the principle that rules out one of them
- 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
- 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.