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).