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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: $v = \frac{1}{1 + i} = 1 - d$
Discount rate: $d = \frac{i}{1 + i} = i v$
Interest from discount: $i = \frac{d}{1 - d}$
The identity worth memorising: $i - d = i d$

Nominal rates

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

Force of interest

Definition: $δ_{t} = \frac{a ^{'} ( t )}{a ( t )} = \frac{d}{d t} ln a (t)$
Accumulation: $a (t) = exp (\int_{0}^{t} δ_{s} d s)$
Constant force: $δ = ln (1 + i), i = e^{δ} - 1$
Discounting: $v^{t} = e^{- δ t}$

Equations of value

General form: $\sum_{k} C_{k} v^{t_{k}} = \sum_{k} D_{k} v^{s_{k}}$
Comparison date is free: $multiply both sides by (1 + i)^{T}$
Method of equated time: $\overset{ˉ}{t} = \frac{\sum _{k} t _{k} C _{k}}{\sum _{k} C _{k}}$ 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.