Exam FM formula sheet - Financial Mathematics
Free reference pages for every SOA Exam FM formula: interest rates, annuities, loans, bonds, spot and forward rates, duration, convexity and immunisation.
29 pages, all free to read. Each one carries the formulas as they are actually written, a sketch of where they come from, a worked example whose answer is recomputed by the test suite, the memory hook that regenerates the formula when recall fails, and the specific traps that cost marks. 180 original practice questions drill this syllabus in the question bank.
“Exam FM” is the descriptive name for the Society of Actuaries’ Financial Mathematics exam and is used here nominatively. The official syllabus is at soa.org; this site is not affiliated with or endorsed by the SOA.
Measurement of interest
Interest and discount
The effective rate i measures interest paid at the end of a period and the effective discount rate d measures it at the start; v = 1 − d = 1/(1+i) converts between them.
Nominal rates
A nominal rate i⁽ᵐ⁾ is an annual label for a rate applied m times a year; only the effective rate compounds, so every problem converts to effective first.
Force of interest
The force of interest δ(t) is the instantaneous rate of growth of an accumulation function - the continuous-compounding limit that makes every other rate a special case.
Equations of value
Every FM problem is one equation: money in equals money out, with both sides valued at the same date.
Annuities
Annuity-immediate
n payments of 1 at the END of each period: the workhorse annuity, and the form every other annuity is written in terms of.
Annuity-due
n payments of 1 at the BEGINNING of each period: an annuity-immediate valued one period later, which is where the (1 + i) factor comes from.
Deferred annuities
An annuity whose first payment is delayed: value it as usual, then discount the whole block back by the deferral period.
Perpetuities
Payments forever. The present value is finite because discounting beats accumulation, and the formulas are the cleanest in the syllabus.
m-thly annuities
One unit per year paid in m instalments: same total, paid sooner, so the value sits between the annual immediate and due forms.
Continuous annuities
Payment flows at a constant rate rather than in instalments; the annuity factor is the same numerator over the force of interest.
Increasing annuities
Payments 1, 2, 3, …, n. The closed form (äₙ − n vⁿ)/i turns an arithmetic series into two annuity factors you already know.
Decreasing annuities
Payments n, n−1, …, 1. The closed form (n − aₙ)/i, and the identity (Ia) + (Da) = (n+1)a that lets you check either one.
Geometric annuities
Payments growing by a fixed PERCENTAGE each period - inflation-linked cash flows. One substitution turns it back into a level annuity.
Loans
Amortisation
A level-payment loan is an annuity seen from the lender's side; the payment is L/aₙ and every later question is about splitting one payment into interest and principal.
Outstanding balance
Two ways to find what is still owed: look forward at the payments left, or look back at what has been paid. They must agree.
Sinking funds
The borrower pays the lender interest only and saves separately to repay the principal in one lump - two rates, two cash flows, one total outlay.
Bonds
Bond pricing
A bond is an annuity of coupons plus a lump at redemption; price it with the basic formula and every other bond formula follows.
Premium and discount
The gap between price and redemption value is written off over the bond's life; the write-down each period is a geometric series like a loan's principal repayments.
Book value
Book value is the price of the same bond with fewer coupons left; Makeham's formula prices the whole thing from the redemption value alone.
Bond yield
Given the price, the yield is the root of the price equation; for a callable bond you price to the WORST call date for the buyer.
Cash flow analysis
Net present value
Discount every cash flow to today at the required rate and add them up; positive NPV means the project beats the alternative.
IRR
The rate that makes NPV zero. Unique for a conventional project, and possibly not unique when the sign of the cash flows changes more than once.
Spot rates
The t-year spot rate is the single rate applying to one payment t years away; a set of them is the term structure, and each cash flow gets its own.
Forward rates
The rate locked in today for borrowing between two future dates, implied by no-arbitrage from the spot curve.
Interest rate risk
Macaulay duration
The present-value-weighted average time to the cash flows - the single number that says how long a bond really is.
Modified duration
The elasticity of price with respect to the interest rate: the first-order estimate of how much a portfolio loses when rates rise.
Convexity
The second-order term. Duration alone always underprices a bond after a rate move; convexity is the correction, and it is always favourable.
Immunisation
Match present value and duration, and make the assets more convex than the liabilities: small rate moves in either direction then leave a surplus.
Swaps
Exchange a floating stream for a fixed one; the swap rate is the fixed rate that makes the exchange worth nothing at inception.