Financial Math FM/Annuities

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NOTE: This chapter assumes knowledge of capital-sigma notation for summations and some basic properties of summations and series, particular geometric series.

Geometric series[edit]

\sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\cdots+ x_{n-1} + x_n.

Level payment annuities[edit]

An annuity is a sequence of payments C_j made at equal intervals of time. We have n periods of times [0, 1], [1, 2], [2, 3], . . . [n - 1, n]. These periods could be days, months, years, fortnights, etc but they are of equal length. An annuity-immediate (also referred to an an ordinary annuity or simply an annuity) has each payment made at the end of each interval of time. That is to say, a payment of C_1 at the end of the first period, [0,1], a payment of C_2 at the end of the second period, [1,2] etc.

Contributions 0 C_1 C_2 C_3 \ldots C_n
Time 0 1 2 3 \ldots n

An annuity-due has each payment made at the beginning of each interval of time.

Contributions C_0 C_1 C_2 C_3 \ldots C_n 0
Time 0 1 2 3 \ldots n - 1 n


An annuity is said to have level payments if all payments C_j are equal. An annuity is said to have non-level payments if some payments C_j are different from other payments. Whether an annuity has level or non-level payments is independent of whether an annuity is an annuity-due or annuity-immediate. First we'll look at the present value of an annuity-immediate with level annual payments of one using accumulation function notation.

a_{\overline{n}|}= \frac{1}{a(1)} + \frac{1}{a(2)} +  \frac{1}{a(3)} +  \cdots + \frac{1}{a(n)} = \sum_{j=1}^n {\frac{1}{a(j)}}

The accumulated value of an annuity-immediate with level annual payments of one is

s_{\overline{n}|}= \frac{a(n)}{a(1)} + \frac{a(n)}{a(2)} +  \frac{a(n)}{a(3)} +  \cdots + \frac{a(n)}{a(n)} = \sum_{j=1}^n {\frac{a(n)}{a(j)}}
a_{\overline{n}|}= v + v^2 +v^3 + \cdots +v^n = \sum_{j=1}^n {v^j} = \frac{1-v^n}{i}
s_{\overline{n}|}= (1+i)^{n-1} + (1+i)^{n-2} + \cdots + (1+i) + 1 = \frac{(1+i)^n-1}{i}

Level payment perpetuities[edit]

Payable m-thly, or Payable continuously[edit]

Arithmetic increasing/decreasing payment annuity[edit]

A(t) = (P-Q)s(nbox) + Q(Ds)(nbox)

Geometric increasing/decreasing payment annuity[edit]

Term of annuity[edit]