Theory of Interest Rates: Accumulation, Discounting, Annuities and Equations of Value
What CM1 tests
This domain tests whether you can move cashflows through time consistently and solve equations of value. Core tools are the effective rate i, discount factor v=1/(1+i), nominal rates convertible m times per year, and force of interest δ=ln(1+i). Accumulation and discount factors are reciprocals: a(t)=exp(integral from 0 to t of δ_s ds), v(t)=1/a(t). For constant i, accumulated value is C(1+i)^t and present value is Cv^t.
Annuities and equations of value
An annuity-immediate pays at period end; an annuity-due pays at period start. For level payments, a-angle-n=(1-v^n)/i and a-due-angle-n=(1-v^n)/d=(1+i)a-angle-n. Accumulated versions multiply by (1+i)^n. A deferred annuity is v^m times the annuity value at deferment. In an equation of value, choose a focal date, equate present values of inflows and outflows, and use one timing convention throughout. Irregular payments should be valued one by one.
Worked example
A fund pays 1,000 at time 0, receives 300 at the end of each year for 4 years, and pays 500 at time 4. At i=5%, PV inflows=300 a-angle-4=300(1-1.05^-4)/0.05=1,063.79. PV outflows=1,000+500(1.05)^-4=1,411.35. Net PV=-347.56, so the deal is loss-making at 5%. Accumulating to time 4 gives -347.56(1.05)^4=-422.49, which is consistent if every cashflow is accumulated at 5%.
Traps
Mixing due and immediate annuities; discounting a payment already at time 0; treating a nominal rate as effective; forgetting that δ and i are equivalent only when the force is constant; and using different focal dates for different cashflows.