11 Annuities Payable more than Once per Time Unit

11.1 Annuity-Immediate Payable m-thly

For annuities payable more frequently than interest is convertible, let:

$m$ denote the number of payments per one interest conversion period
$n$ be the total number of conversion periods
$mn$ is then the total number of payments for the term of the annuity
$i$ be the interest rate per conversion period.

We will assume that the number of payments per conversion period is an integral number.

Payments of $1$ are being made per interest conversion period with $1/m$ being made at the end of each m-th of an interest conversion period. The present value of such an annuity will be denoted by $a^{(m)}_{\overline{n|}}$.

The formula for the present value can be derived from:

\[a^{(m)}_{\overline{n|}} = \frac{1}{m} a_{\overline{mn|}\frac{i^{(m)}}{m}} = \frac{1}{m}\bigg[v^{\frac{1}{m}}+v^{\frac{2}{m}} + ... + v^{\frac{nm-1}{m}} + v^{n}\bigg] = \frac{v^{\frac{1}{m}}}{m} \times \frac{1-(v^{\frac{1}{m}})^{mn}}{1-v^{\frac{1}{m}}} = \frac{1-v^n}{i^{(m)}}\]

Definition: The present value and accumulated value at time $0$ of an annuity-immediate that makes payments of $1$ per time unit, payable m-thly (i.e. pays the amount of $1/m$ for $m$ times per time unit), for $n$ units of time is: \[a^{(m)}_{\overline{n|}} = a_{\overline{mn|}\frac{i^{(m)}}{m}} = \frac{1-v^n}{i^{(m)}} \hskip7em s^{(m)}_{\overline{n|}} = s_{\overline{mn|}\frac{i^{(m)}}{m}} =\frac{(1+i)^n -1 }{i^{(m)}}\]

Example: The nominal annual interest rate compounded monthly is $12\%$. Find the present value of an annuity-immediate that makes monthly payments at a rate of $\$36$ per year for 10 years.

Solution: You can use the BA II Calculators for this task. You can compute directly or set the payment per year variable:

120 [N] 1 [I/Y] 3 [PMT] [CPT] [PV] -> PV = -209.10
[2nd] [P/Y] 12 [ENTER] [2nd] [QUIT] (setting monthly payments) 10 [2nd] [xP/Y] [N] 12 [I/Y] 3 [PMT] [CPT] [PV] -> PV = -209.10

11.2 Annuity-Due Payable m-thly

Definition: The present value and accumulated value at time $0$ of an annuity-due that makes payments of $1$ per time unit, payable m-thly (i.e. pays the amount of $1/m$ for $m$ times per time unit), for $n$ units of time is: \[\ddot a^{(m)}_{\overline{n|}} = \ddot a_{\overline{mn|}\frac{i^{(m)}}{m}} = \frac{1-v^n}{d^{(m)}} \hskip7em \ddot s^{(m)}_{\overline{n|}} = \ddot s_{\overline{mn|}\frac{i^{(m)}}{m}} =\frac{(1+i)^n -1 }{d^{(m)}}\]

Example: The nominal annual interest rate compounded monthly is $12\%$. Find the accumulated value at the end of 10 years of an annuity-due that makes monthly payments at a rate of $\$36$ per year for 10 years.

Solution: We can calculate the equivalent nominal rate of discount: \[d^{(12)} = 12\times \bigg[1-\bigg(1+\frac{i^{(12)}}{12}\bigg)^{-1}\bigg] = 12\times[1-(1.01)^{-1}] = 0.1188\] Since the annual payment is $36$ per year, then \[FV = 36\times \ddot s^{(12)}_{\overline{10|}} = 36\times \frac{(1.01)^{120} - 1}{d^{(12)}} = 697.02\]

You can also use your calculator for this task by:

120 [N] 1[I/Y] 3[PMT] [CPT] [FV] -> FV = -690.12 [x] 1.01 [=] -697.02

11.3 Level Annuities Payable Continuously

We found the present value of an annuity-immediate that pays 1 per time unit, in increments of $1/m$ at the end of each period: $a_{\overline{n|}} = \frac{1-v^n}{i^{(m)}}$. As $m$ becomes larger, the payments are made more frequently, but the rate of payment remains 1 per time unit. When we let $m$ tends to infinity, recall that $\lim_{m\to\infty} i^{(m)} = \delta$, and the annuity becomes continuously payable.

Definition: The present value and accumulated value at time $0$ of an annuity payable continuously that makes payments of $1$ per time unit for $n$ units of time is: \[\bar{a}_{\overline{n|}} = \frac{1-v^n}{\delta} \hskip8em \bar{s}_{\overline{n|}} = \frac{(1+i)^n-1}{\delta}\] where $\delta$ is the force of interest.

11.3.1 Perpetuity Payable Continuously

The present value of a continuously payable perpetuity of 1 per time unit is found by taking the limit as $n$ goes to infinity: \[\lim_{n\to\infty} \bar{a}_{\overline{n|}} = \bar{a}_{\overline{\infty|}} = \frac{1}{\delta}\] Example: The nominal annual interest rate compounded monthly is $12\%$. Find the present value of an annuity that makes continuous payments at a rate of $\$36$ per year for 10 years.

Solution: The equivalent force of interest is: $\delta = 12\ln(1+i^{(12)}/12) = 0.1194$ We have: \[PV = 36\times \bar a_{\overline{10|}} = 36\times \frac{1-(1.01)^{-12\times 10}}{0.1194} = 210.15\]