FM for Actuaries

64 CHAPTER 2
2.8 Summary
1. Annuities are series of payments at equal intervals. An annuity-immediate
makes payments at the end of the payment periods, and an annuity-due makes
payments at the beginning of the periods.
2. Algebraic formulas for the present and future values of annuity-immediate
and annuity-due can be derived using geometric progression. These formulas
facilitate the calculation of installments to pay off a loan, or installments
required to accumulate to a targeted amount in the future.
3. Perpetuity has no maturity and makes payments indefinitely. Deferred annuity
makes the first payment some time in the future.
4. Present- and future-value formulas can be derived for a general accumulation
function.
5. When the payment period and the interest-conversion period are not the
same, a simple approach is to compute the effective rate of interest for the
payment period and do the computations using this rate. Algebraic approach
can be used for both the cases of payments more frequent than interest conversion
and less frequent than interest conversion.
6. Computations for continuous annuity can be used as approximations to frequent
payments such as daily annuities.
7. Annuities involving simple step-up or step-down of payments in an arithmetic
progression can be computed using basic annuity formulas.
8. The equation of value can be adopted to solve for n or i given other information.
The calculation of i from the equation of value generally admits no
analytic solution. Numerical methods such as grid search may be used.
Exercises
2.1 Let the effective rate of interest be 5%. Evaluate
(a) s 10⌉
,
(b) ä 5⌉ ,
(c) (Is) 7⌉ ,
(d) (Ia) 7⌉
,
(e) a ∞⌉ .