Exam FM Theory
Basic formulas
Accumulating and discount factors
These are {$1+i$} and {$v$}, respectively. The relationships are
{$$ v= \frac{1}{1+i}, \qquad i=\frac{1}{v}-1. $$}
Effective rates of interest and discount
These are denoted {$i$} and {$d$}, respectively. The relationships are
{$$ v=1-d=\frac{1}{1+i}, \qquad d=iv=\frac{i}{1+i}, \qquad i=\frac{d}{v}=\frac{d}{1-d}. $$}
Annuity formulas
Constant annuity
- Present and accumulated values of a constant annuity-immediate for {$n$} years
{$$ a_{\overline{n}|} = \frac{1-v^n}{i}; \qquad s_{\overline{n}|} = \frac{(1+i)^n-1}{i}$$}
- Present and accumulated values of a constant annuity-due for {$n$} years
{$$ \ddot{a}_{\overline{n}|} = \frac{1-v^n}{d}; \qquad \ddot{s}_{\overline{n}|} = \frac{(1+i)^n-1}{d}$$}
{$$ \ddot{a}_{\overline{n}|}= a_{\overline{n}|}(1+i); \qquad \ddot{s}_{\overline{n}|}= s_{\overline{n}|}(1+i)$$}
Annuity-dues are a little bigger than annuity-immediates.
Be careful that for annuity-dues, the comparison date for future value calculations is at one year after the last payment. For annuity-immediates, the comparison date for present value calculations is at one year before the first payment.
Increasing annuity
- Present and accumulated values of an increasing annuity-immediate
{$$ (Ia)_{\overline{n}|} = \frac{\ddot{a}_{\overline{n}|}-nv^n}{i}; \qquad (Is)_{\overline{n}|} = \frac{\ddot{s}_{\overline{n}|}-n}{i} $$}
- Present and accumulated values of an increasing annuity-due
{$$ (I\ddot{a})_{\overline{n}|} = \frac{\ddot{a}_{\overline{n}|}-nv^n}{d};\qquad (I\ddot{s})_{\overline{n}|} = \frac{\ddot{s}_{\overline{n}|}-n}{d}$$}
Decreasing annuity
- Present and accumulated values of an decreasing annuity-immediate
{$$ (Da)_{\overline{n}|} = \frac{n-a_{\overline{n}|}}{i}; \qquad (Ds)_{\overline{n}|} = \frac{n(1+i)^n - s_{\overline{n}|}}{i}$$}
- Present and accumulated values of an decreasing annuity-due
{$$ (D\ddot{a})_{\overline{n}|} = \frac{n-a_{\overline{n}|}}{d}; \qquad (D\ddot{s})_{\overline{n}|} = \frac{n(1+i)^n - s_{\overline{n}|}}{d}$$}
{$$ (Da)_{\overline{n}|} + (Ia)_{\overline{n}|} = (n+1)a_{\overline{n}|}; \qquad (Ds)_{\overline{n}|} + (Is)_{\overline{n}|} = (n+1)s_{\overline{n}|};$$}
similarly for annuity-dues.
Other annuity formulas
- For a continuous annuity, change the denominator in all formulas to {$\delta$}.
- For a compound increasing annuity (where payments form a geometric series), write out the present or future value as a finite geometric sum and compute directly.
Non-annual time periods
- Denote {$\displaystyle i^(p)$} the nominal rate of interest convertible {$p$}-thly, and {$\displaystyle d^(m)$} the nominal rate of discount convertible {$m$}-thly. Here are the relationships
{$$ \left( 1+\frac{i^(p)}{p}\right)^p =1+i =\left( 1-\frac{d^(m)}{m}\right)^{-m} =(1-d)^{-1}.$$}
If we compound more than once a year, then {$i$} is a little bigger than {$i^(p)$}, and {$d$} is a little smaller than {$d^(p)$}.
- For an annuity of any kind consisting of payments of 1 every year, payable {$p$}-thly for {$n$} years (so there are payments of {$1/p$} per payment period), change denominators in all formulas to {$i^(p)$} or {$d^(p)$}. Alternatively, change everything to the effective {$p$}-thly rate of interest (or discount).