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Measurement of interest - Exam FM formula sheet

Every Exam FM formula in measurement of interest on one printable sheet, condensed from the 4 reference pages that teach them.

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Measurement of interest

Interest and discount

Accumulation: $a(t) = (1+i)^t$ value at time t of 1 invested now
Discount factor: $v = \dfrac{1}{1+i} = 1 - d$
Discount rate: $d = \dfrac{i}{1+i} = iv$
Interest from discount: $i = \dfrac{d}{1-d}$
The identity worth memorising: $i - d = id$

Nominal rates

Effective from nominal: $1 + i = \left(1 + \dfrac{i^{(m)}}{m}\right)^{m}$
Nominal from effective: $i^{(m)} = m\left[(1+i)^{1/m} - 1\right]$
Nominal discount: $1 - d = \left(1 - \dfrac{d^{(m)}}{m}\right)^{m}$
The chain that links them: $\left(1 + \dfrac{i^{(m)}}{m}\right)^{m} = 1+i = (1-d)^{-1} = \left(1 - \dfrac{d^{(p)}}{p}\right)^{-p} = e^{\delta}$

Force of interest

Definition: $\delta_t = \dfrac{a'(t)}{a(t)} = \dfrac{d}{dt}\ln a(t)$
Accumulation: $a(t) = \exp\left(\int_0^{t} \delta_s\, ds\right)$
Constant force: $\delta = \ln(1+i), \qquad i = e^{\delta} - 1$
Discounting: $v^{t} = e^{-\delta t}$

Equations of value

General form: $\sum_{k} C_k\, v^{t_k} = \sum_{k} D_k\, v^{s_k}$
Comparison date is free: $\text{multiply both sides by } (1+i)^{T}$
Method of equated time: $\bar t = \dfrac{\sum_k t_k C_k}{\sum_k C_k}$ approximate single date replacing several payments

Measurement of interest - Exam FM formula sheet. Formulas condensed from the free reference pages on this site. Not affiliated with or endorsed by the Society of Actuaries.