1.5 Useful Formulas from Chapter 1
2.1.1 Annuities & Actuarial Notation
The general present value formulas above will now be specialized to the case of constant (instantaneous) interest rate δ(t) ≡ ln(1 + i) = δ at all times t ≥ 0, and some very particular streams of payments sj at times tj, related to periodic premium and annuity payments. The effective interest rate or APR is always denoted by i, and as before the m-times-per-year equivalent nominal interest rate is denoted by i(m). Also, from now on the standard and convenient notation
will be used for the present value of a payment of $1 in one year.
(i) If s0 = 0 and s1 = · · · = snm= 1/m in the discrete setting, where m denotes the number of payments per year, and tj = j/m, then the payment-stream is called an immediate annuity, and its present value Gn
is given the notation a(m)ne and is equal, by the geometric-series summation formula, to
This calculation has shown
a(m)ne = 1 − vn
i(m) (2.1)
All of these immediate annuity values, for fixed v, n but varying m, are roughly comparable because all involve a total payment of 1 per year.
Formula (2.1) shows that all of the values a(m)ne differ only through the factors i(m), which differ by only a few percent for varying m and fixed i, as shown in Table 2.1. Recall from formula (1.4) that i(m) = m{(1 + i)1/m− 1}.
If instead s0 = 1/m but snm = 0, then the notation changes to ¨a(m)ne , the payment-stream is called an annuity-due, and the value is given by any of the equivalent formulas
¨a(m)ne = (1 + i(m)
m ) a(m)ne = 1 − vn
m + a(m)ne = 1
m + a(m)n−1/me (2.2)
The first of these formulas recognizes the annuity-due payment-stream as identical to the annuity-immediate payment-stream shifted earlier by the time 1/m and therefore worth more by the accumulation-factor (1+i)1/m = 1 + i(m)/m. The third expression in (2.2) represents the annuity-due stream as being equal to the annuity-immediate stream with the payment of 1/m at t = 0 added and the payment of 1/m at t = n removed. The final expression says that if the time-0 payment is removed from the annuity-due, the remaining stream coincides with the annuity-immediate stream consisting of nm − 1 (instead of nm) payments of 1/m.
In the limit as m → ∞ for fixed n, the notation ane denotes the present value of an annuity paid instantaneously at constant unit rate, with the limiting nominal interest-rate which was shown at the end of the previous chapter to be limm i(m) = i(∞) = δ. The limiting behavior of the nominal interest rate can be seen rapidly from the formula
i(m) = m³
(1 + i)1/m − 1´
= δ · exp(δ/m) − 1 δ/m
since (ez− 1)/z converges to 1 as z → 0. Then by (2.1) and (2.2), ane = lim
m→∞¨a(m)ne = lim
m→∞a(m)ne = 1 − vn
δ (2.3)
Table 2.1: Values of nominal interest rates i(m) (upper number) and d(m) (lower number), for various choices of effective annual interest rate i and number m of compounding periods per year.
i = .02 .03 .05 .07 .10 .15
m = 2 .0199 .0298 .0494 .0688 .0976 .145
.0197 .0293 .0482 .0665 .0931 .135
3 .0199 .0297 .0492 .0684 .0968 .143
.0197 .0294 .0484 .0669 .0938 .137
4 .0199 .0297 .0491 .0682 .0965 .142
.0198 .0294 .0485 .0671 .0942 .137
6 .0198 .0296 .0490 .0680 .0961 .141
.0198 .0295 .0486 .0673 .0946 .138
12 .0198 .0296 .0489 .0678 .0957 .141
.0198 .0295 .0487 .0675 .0949 .139
A handy formula for annuity-due present values follows easily by recalling that
1 − d(m)
m = ³
1 + i(m) m
´−1
implies d(m) = i(m) 1 + i(m)/m Then, by (2.2) and (2.1),
¨a(m)ne = (1 − vn) · 1 + i(m)/m
i(m) = 1 − vn
d(m) (2.4)
In case m is 1, the superscript (m) is omitted from all of the annuity notations. In the limit where n → ∞, the notations become a(m)∞e and
¨a(m)∞e , and the annuities are called perpetuities (respectively immediate and due) with present-value formulas obtained from (2.1) and (2.4) as:
a(m)∞e = 1
i(m) , ¨a(m)∞e = 1
d(m) (2.5)
Let us now build some more general annuity-related present values out of the standard functions a(m)ne and ¨a(m)ne .
(ii). Consider first the case of the increasing perpetual annuity-due, denoted (I(m)¨a)(m)∞e , which is defined as the present value of a stream of payments (k + 1)/m2 at times k/m, for k = 0, 1, . . . forever. Clearly the
Here are two methods to sum this series, the first purely mathematical, the second with actuarial intuition. First, without worrying about the strict justification for differentiating an infinite series term-by-term,
X∞
for 0 < x < 1, where the geometric-series formula has been used to sum the second expression. Therefore, with x = (1 + i(m)/m)−1 and 1 − x =
and (2.5) has been used in the last step. Another way to reach the same result is to recognize the increasing perpetual annuity-due as 1/m multiplied by the superposition of perpetuities-due ¨a(m)∞e paid at times 0, 1/m, 2/m, . . . , and therefore its present value must be ¨a(m)∞e · ¨a(m)∞e . As an aid in recognizing this equivalence, consider each annuity-due ¨a(m)∞e paid at a time j/m as being equivalent to a stream of payments 1/m at time j/m, 1/m at (j + 1)/m, etc. Putting together all of these payment streams gives a total of (k +1)/m paid at time k/m, of which 1/m comes from the annuity-due starting at time 0, 1/m from the annuity-due starting at time 1/m, up to the payment of 1/m from the annuity-due starting at time k/m.
(iii). The increasing perpetual annuity-immediate (I(m)a)(m)∞e — the same payment stream as in the increasing annuity-due, but deferred by a time 1/m — is related to the perpetual annuity-due in the obvious way
(I(m)a)(m)∞e = v1/m(I(m)¨a)(m)∞e = (I(m)¨a)(m)∞e .
(1 + i(m)/m) = 1 i(m)d(m)
(iv). Now consider the increasing annuity-due of finite duration n years. This is the present value (I(m)¨a)(m)ne of the payment-stream of (k + 1)/m2 at time k/m, for k = 0, . . . , nm − 1. Evidently, this payment-stream is equivalent to (I(m)¨a)(m)∞e minus the sum of n multiplied by an annuity-due ¨a(m)∞e starting at time n together with an increasing annuity-due (I(m)¨a)(m)∞e starting at time n. (To see this clearly, equate the payments 0 = (k + 1)/m2 − n · m1 − (k − nm + 1)/m2 received at times k/m for k ≥ nm.) Thus
(I(m)¨a)(m)ne = (I(m)¨a)(m)∞e
³
1 − (1 + i(m)/m)−nm´
− n¨a(m)∞e (1 + i(m)/m)−nm
= ¨a(m)∞e
³¨a(m)∞e − (1 + i(m)/m)−nmh
¨a(m)∞e + ni ´
= ¨a(m)∞e
³¨a(m)ne − n vn´
where in the last line recall that v = (1 + i)−1 = (1 + i(m)/m)−m and that ¨a(m)ne = ¨a(m)∞e (1 − vn). The latter identity is easy to justify either by the formulas (2.4) and (2.5) or by regarding the annuity-due payment stream as a superposition of the payment-stream up to time n − 1/m and the payment-stream starting at time n. As an exercise, fill in details of a second, intuitive verification, analogous to the second verification in pargraph (ii) above.
(v). The decreasing annuity (D(m)¨a)(m)ne is defined as (the present value of) a stream of payments starting with n/m at time 0 and decreasing by 1/m2 every time-period of 1/m, with no further payments at or after time n. The easiest way to obtain the present value is through the identity
(I(m)¨a)(m)ne + (D(m)¨a)(m)ne = (n + 1 m ) ¨a(m)ne
Again, as usual, the method of proving this is to observe that in the payment-stream whose present value is given on the left-hand side, the payment amount at each of the times j/m, for j = 0, 1, . . . , nm − 1, is
j + 1
m2 + (n
m − j
m2) = 1
m(n + 1 m)