Actuarial notation

Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables.

Traditional notation uses a halo system where symbols are placed as superscript or subscript before or after the main letter. Example notation using the halo system can be seen below.

Various proposals have been made to adopt a linear system where all the notation would be on a single line without the use of superscripts or subscripts. Such a method would be useful for computing where representation of the halo system can be extremely difficult. However, a standard linear system has yet to emerge.

${\displaystyle \,i}$ is the annual effective interest rate, which is the "true" rate of interest over a year. Thus if the annual interest rate is 12% then ${\displaystyle \,i=0.12}$.

${\displaystyle \,i^{(m)}}$ (pronounced "i upper m") is the nominal interest rate convertible ${\displaystyle m}$ times a year, and is numerically equal to ${\displaystyle m}$ times the effective rate of interest over one ${\displaystyle m}$^th of a year. For example, ${\displaystyle \,i^{(2)}}$ is the nominal rate of interest convertible semiannually. If the effective annual rate of interest is 12%, then ${\displaystyle \,i^{(2)}/2}$ represents the effective interest rate every six months. Since ${\displaystyle \,(1.0583)^{2}=1.12}$, we have ${\displaystyle \,i^{(2)}/2=0.0583}$ and hence ${\displaystyle \,i^{(2)}=0.1166}$. The ^"(m)" appearing in the symbol ${\displaystyle \,i^{(m)}}$ is not an "exponent." It merely represents the number of interest conversions, or compounding times, per year. Semi-annual compounding, (or converting interest every six months), is frequently used in valuing bonds (see also fixed income securities) and similar monetary financial liability instruments, whereas home mortgages frequently convert interest monthly. Following the above example again where ${\displaystyle \,i=0.12}$, we have ${\displaystyle \,i^{(12)}=0.1139}$ since ${\displaystyle \,\left(1+{\frac {0.1139}{12}}\right)^{12}=1.12}$.

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