Time Value of Money

A rate is quoted at one compounding frequency (e.g., nominal 6% convertible monthly) and you need the effective annual rate or another nominal frequency to value a cash flow.

Convert the quoted rate to a per-period effective rate, then accumulate across the conversion period to get the target effective rate. Distinguish "nominal interest convertible m-thly" from "nominal discount convertible m-thly" — they accumulate differently. Verify the conversion direction by checking that more frequent compounding gives a higher effective rate.

(1 + i)^t = (1 + i^{(m)} / m)^{mt} = (1 - d^{(m)} / m)^{-mt} = e^{δt}.

A lump sum is moved through time using simple interest, compound interest, or a continuous force of interest, possibly with mixed conventions over different sub-periods.

For each sub-period, identify the convention and apply the matching accumulation factor: (1 + it) for simple, (1 + i)^t for compound, or e^{∫δ(s) ds} for continuous force. Multiply accumulation factors across sub-periods. Always anchor your time line to a consistent reference date before discounting back.

a(t) = e^{∫₀^t δ(s) ds}; PV = FV · a(t)^{-1}.

A force of interest δ(t) is given as a function of time and you must accumulate or discount a payment.

Integrate δ(s) over the relevant interval to get a(t) = e^{∫δ(s) ds}. For PV, divide by a(t); for FV from time s to t, use a(t) / a(s). Keep the integration limits straight — common error is integrating from the payment date when you should integrate from the valuation date.

δ(t) = a'(t) / a(t); a(t) = exp(∫₀^t δ(s) ds).