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!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 {$$ \ddot{a}_{\overline{n}|} = \frac{1-v^n}{d}; \qquad \ddot{s}_{\overline{n}|} = \frac{(1+i)^n-1}{d}$$} *Relationships {$$ \ddot{a}_{\overline{n}|}= a_{\overline{n}|}(1+i); \qquad \ddot{s}_{\overline{n}|}= s_{\overline{n}|}(1+i)$$} ->Annuity-dues are a little bigger than annuity-immediates. >>red<< ->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. >>normal<< !!!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 {$$ (D\ddot{a})_{\overline{n}|} = \frac{n-a_{\overline{n}|}}{d}; \qquad (D\ddot{s})_{\overline{n}|} = \frac{n(1+i)^n - s_{\overline{n}|}}{d}$$} *Relationships {$$ (Da)_{\overline{n}|} + (Ia)_{\overline{n}|} = (n+1)a_{\overline{n}|}; \qquad (Ds)_{\overline{n}|} + (Is)_{\overline{n}|} = (n+1)s_{\overline{n}|};$$} ->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). |