Exam FM: which formulas actually earn their keep
There are dozens of interest theory formulas and about a dozen that carry most of the exam. Here is the ranking, and what each one unlocks.
The formula sheet is longer than the exam
Interest theory looks enormous when you first meet it: annuities immediate and due, deferred and continuous, increasing and decreasing and geometric, loans, sinking funds, bonds, spot rates, forward rates, duration, convexity. Almost all of it collapses onto a handful of expressions, and the candidates who see that finish the exam with time to spare.
Tier one: learn these until they are reflexes
Four expressions carry an enormous share of the paper. If you know them cold and can move between them, most questions become bookkeeping.
- v = 1/(1+i) and d = i/(1+i) - every conversion question is one of these rearranged.
- aₙ = (1 − vⁿ)/i - and the fact that äₙ is just aₙ(1+i).
- P = L/aₙ for a loan payment, and Bₜ = P·a_(n−t) for the balance.
- P = Fr·aₙ + Cvⁿ for a bond price.
Tier two: the ones that turn a hard question into an easy one
These do not appear on every paper, but when they do they save several minutes each - and the alternative is summing a series by hand under time pressure.
- (Ia)ₙ = (äₙ − n·vⁿ)/i, and the check (Ia)ₙ + (Da)ₙ = (n+1)aₙ.
- The geometric annuity value (1 − ((1+g)/(1+i))ⁿ)/(i − g).
- 1/aₙ = i + 1/sₙ, the bridge between amortisation and sinking funds.
- D_Mod = D_Mac/(1+i), and ΔP/P ≈ −D_Mod·Δi.
Tier three: know what they are, look up the details
Makeham's formula, the callable-bond rules, convexity's exact expression and the swap rate formula are all worth recognising, but they reward understanding over memorisation. If you know that a swap rate is (1 − last discount factor) divided by the sum of the discount factors, you can rebuild it; memorising it as symbols without that sentence is fragile.
Where do these eight expressions actually get derived?
Ranking formulas is only half the advice; the other half is knowing where to go when one of them stops making sense. Every expression above is derived on a free reference page here, and the table maps them one to one. Memorising from the table is the wrong use of it - go to the derivation once, understand why the expression has the shape it has, and the memorising gets much cheaper.
| Expression | Tier | Where it is derived |
|---|---|---|
| v = 1/(1+i), d = i/(1+i) | One | Effective interest and discount rates |
| aₙ and äₙ | One | Annuity-immediate |
| P = L/aₙ and Bₜ = P·a_(n−t) | One | Loan amortisation |
| P = Fr·aₙ + Cvⁿ | One | Bond pricing |
| (Ia)ₙ and (Da)ₙ | Two | Increasing annuities |
| Geometric annuity value | Two | Geometrically increasing annuities |
| 1/aₙ = i + 1/sₙ | Two | Sinking funds |
| D_Mod and ΔP/P ≈ −D_Mod·Δi | Two | Modified duration and price sensitivity |
How many FM formulas do you actually have to hold?
Eight, in the sense that matters. The reference side of this site publishes twenty-nine FM topics because the syllabus genuinely has twenty-nine distinct things in it, and a wide exam will ask about any of them. But the eight expressions in the first two tiers are the ones you need available without thinking, because they are the ones a question is built on rather than the ones a question is about.
The other twenty-one are recognition work: you need to know that Makeham's formula exists and what it is for, not to be able to write it from memory under time pressure. The failure mode worth avoiding is treating all twenty-nine as equally urgent, which spreads recall practice so thin that none of it becomes automatic.
The discipline that beats memorisation
Every FM formula is a present value. When recall fails, write the cash flows on a time line and discount them one by one - slower, but never wrong. Candidates who practise that fallback can afford to forget a formula in the exam room; candidates who cannot are one memory lapse from losing a question they understood perfectly.
That is also the test for whether a formula has earned a place in tier one: if you could not rebuild it from a time line in ninety seconds, it needs to be memorised; if you could, it does not.