Financial Math FM/Annuities

Learning objectives
The Candidate will be able to calculate present value, current value, and accumulated value for sequences of non-contingent payments.

Learning outcomes
The Candidate will be able to:
 * Define and recognize the definitions of the following terms: annuity-immediate, annuity due, perpetuity, payable m-thly or payable continuously, level payment annuity, arithmetic increasing/decreasing annuity, geometric increasing/decreasing annuity, term of annuity.
 * For each of the following types of annuity/cash flows, given sufficient information of immediate or due, present value, future value, current value, interest rate, payment amount, and term of annuity, calculate any remaining item.
 * Level annuity, finite term.
 * Level perpetuity.
 * Non-level annuities/cash flows.
 * Arithmetic progression, finite term and perpetuity.
 * Geometric progression, finite term and perpetuity.
 * Other non-level annuities/cash flows.

Geometric series formulas
Recall the following formulas, which are useful for deriving the formulas for different types of annuities.
 * $$a+ar+ar^2+\cdots\overset{\text{ def }}=\sum_{k=0}^{\infty}ar^k=\frac{a}{1-r},\quad |r|<1 $$;
 * $$\underbrace{a+ar+ar^2+\cdots+ar^n}_{n+1\text{ terms }}\overset{\text{ def }}=\sum_{k=0}^n ar^k=a\left(\frac{1-r^{n+1}}{1-r}\right),\quad r\ne 1$$.

Annuity-immediate
Time diagram: ↓    ↓           ↓ 0    1     2    ...    n      t

Annuity-due
Time diagram: ↓   ↓     ↓           ↓ 0    1     2    ...   n-1      t

Annuities payable $m$thly
Sometimes, annuities are not payable annually, and can be payable more or less frequently than annually.

To calculate the present value of these kind of annuities, we can simply change the measurement period, and calculate the interest rate during that period that is equivalent to the given interest (or discount) rate (or force of interest), and the new term of the annuity in the new measurement period.

Using new terms and new interest rates, we can calculate the present value of these kind of annuities by applying previously discussed method.

Annuities payable continuously

 * Recall from the motivation of force of interest that "payable continuously" is essentially "payable $$\infty$$thly (abuse of notation)"
 * If an annuity pays 1 in each "infinitesimal" time interval, the present value of payments in a measurement period will be infinite, since there are infinitely many such time intervals during arbitrary measurement period.
 * Because of this, it does not make sense to say an annuity has payments of xxx payable continuously.
 * Instead, we should use the notion of to describe the behaviour of continuous payment.

Perpetuities
Time diagram: ↓    ↓    ...     0    1     2    ...           t


 * For perpetuities payable $$m$$thly, the approach for calculating their present values is the same as that for annuities payable $$m$$thly, namely adjusting the measurement period and calculating new interest rate correspondingly.
 * Also, for perpetuities payable continuously, the approach for calculating their present values is the same, except that we are dealing with improper integrals (upper limits of integrals involved are $$\infty$$).

Non-level annuities

 * In general, the present value of non-level annuities can be calculated by summing up the present value of each payment.
 * However, this approach can take a lot of time, and thus may not be the most efficient approach.
 * We will discuss several special cases of non-level annuities, for which the present value can be calculated in an efficient way.

Arithmetic varying annuities

 * For some annuities, payments vary (increase or decrease) in arithmetic sequence.
 * We will develop a formula for calculating their present values in this subsection.

Geometric varying annuities

 * Since the expression for present value of annuity is essentially geometric series, even with payments varying in geometric sequence, the expression is still geometric series, and thus we can use the geometric series formula to calculate the present value.
 * So, in general, for geometric varying annuities, we use "first principle" to calculate their present value, in the sense that we use geometric series formula to evaluate the expanded form of the present value.