Annuities

In investment, an annuity is a series of payments made at equal intervals. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. Annuities may be calculated by mathematical functions known as "annuity functions".

Timing of payments

Payments of an annuity-immediate are made at the end of payment periods, so that interest accrues between the issue of the annuity and the first payment. Payments of an annuity-due are made at the beginning of payment periods, so a payment is made immediately on issue.

Valuation

Valuation of an annuity entails calculation of the present value of the future annuity payments. The valuation of an annuity entails concepts such as time value of money, interest rate, and future value.

Annuity-certain

If the number of payments is known in advance, the annuity is an annuity certain or guaranteed annuity. Valuation of annuities certain may be calculated using formulas depending on the timing of payments.

Annuity-immediate

If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an annuity-immediate, or ordinary annuity. Mortgage payments are annuity-immediate, interest is earned before being paid. What is Annuity Due? Annuity due refers to a series of equal payments made at the same interval at the beginning of each period. Periods can be monthly, quarterly, semi-annually, annually, or any other defined period. Examples of annuity due payments include rentals, leases, and insurance payments, which are made to cover services provided in the period following the payment.

	↓	↓	...	↓	payments
———	———	———	———	—
0	1	2	...	[math]n[/math]	periods

The present value of an annuity is the value of a stream of payments, discounted by the interest rate to account for the fact that payments are being made at various moments in the future. The present value is given in actuarial notation by:

[[math]]a_{\overline{n}|i} = \frac{1-(1+i)^{-n}}{i},[[/math]]

where [math]n[/math] is the number of terms and [math]i[/math] is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, or rent [math]R[/math] is:

[[math]]\text{PV}(i,n,R) = R \times a_{\overline{n}|i}.[[/math]]

In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest [math]I[/math] is stated as a nominal interest rate, and [math]i = I/12[/math].

The future value of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account. For an annuity-immediate, it is the value immediately after the [math]n[/math]-th payment. The future value is given by:

[[math]]s_{\overline{n}|i} = \frac{(1+i)^n-1}{i},[[/math]]

where [math]n[/math] is the number of terms and [math]i[/math] is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, or rent [math]R[/math] is:

[[math]]\text{FV}(i,n,R) = R \times s_{\overline{n}|i}[[/math]]

Example: The present value of a 5-year annuity with a nominal annual interest rate of 12% and monthly payments of $100 is:

[[math]]\text{PV}\left( \frac{0.12}{12},5\times 12,\$100\right) = \$100 \times a_{\overline{60}|0.01} = \$4,495.50[[/math]]

The rent is understood as either the amount paid at the end of each period in return for an amount PV borrowed at time zero, the principal of the loan, or the amount paid out by an interest-bearing account at the end of each period when the amount PV is invested at time zero, and the account becomes zero with the n-th withdrawal.

Future and present values are related since:

[[math]]s_{\overline{n}|i} = (1+i)^n \times a_{\overline{n}|i}[[/math]]

and

[[math]]\frac{1}{a_{\overline{n}|i}} - \frac{1}{s_{\overline{n}|i}} = i[[/math]]

Proof of annuity-immediate formula

To calculate present value, the [math]k[/math]-th payment must be discounted to the present by dividing by the interest, compounded by [math]k[/math] terms. Hence the contribution of the [math]k[/math]-th payment [math]R[/math] would be [math] \frac{R}{(1+i)^k} [/math]. Just considering [math]R[/math] to be 1, then:

[[math]]\begin{align} a_{\overline n|i} &= \sum_{k=1}^n \frac{1}{(1+i)^k} = \frac{1}{1+i}\sum_{k=0}^{n-1}\left(\frac{1}{1+i}\right)^k \\[5pt] &= \frac{1}{1+i}\left(\frac{1-(1+i)^{-n}}{1-(1+i)^{-1}}\right)\quad\quad\text{by using the equation for the sum of a geometric series}\\[5pt] &= \frac{1-(1+i)^{-n}}{1+i-1}\\[5pt] &= \frac{1-\left(\frac{1}{1+i}\right)^n}{i}, \end{align} [[/math]]

which gives us the result as required.

Similarly, we can prove the formula for the future value. The payment made at the end of the last year would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of [math]n-1[/math] years. Therefore,

[[math]] s_{\overline n|i} = 1 + (1+i) + (1+i)^2 + \cdots + (1+i)^{n-1} = (1+i)^n a_{\overline n|i} = \frac{(1+i)^n-1}{i}.[[/math]]

Annuity-due

An annuity-due is an annuity whose payments are made at the beginning of each period. Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities due.

↓	↓	...	↓		payments
———	———	———	———	—
0	1	...	[math]n[/math] − 1	[math]n[/math]	periods

Each annuity payment is allowed to compound for one extra period. Thus, the present and future values of an annuity-due can be calculated.

[[math]] \ddot{a}_{\overline{n|}i} = (1+i) \times a_{\overline{n|}i} = \frac{1-(1+i)^{-n}}{d}, [[/math]]

[[math]]\ddot{s}_{\overline{n|}i} = (1+i) \times s_{\overline{n|}i} = \frac{(1+i)^n-1}{d}, [[/math]]

where [math]n[/math] is the number of terms, [math]i[/math] is the per-term interest rate, and [math]d[/math] is the e given by [math]d=\frac{i}{i+1}[/math].

The future and present values for annuities due are related since:

[[math]]\ddot{s}_{\overline{n}|i} = (1+i)^n \times \ddot{a}_{\overline{n}|i},[[/math]]

[[math]]\frac{1}{\ddot{a}_{\overline{n}|i}} - \frac{1}{\ddot{s}_{\overline{n}|i}} = d.[[/math]]

Example: The final value of a 7-year annuity-due with a nominal annual interest rate of 9% and monthly payments of $100 can be calculated by:

[[math]] \text{FV}_{\text{due}}\left(\frac{0.09}{12},7\times 12,\$100\right) = \$100 \times \ddot{s}_{\overline{84}|0.0075} = \$11,730.01. [[/math]]

In Excel, the PV and FV functions take on optional fifth argument which selects from annuity-immediate or annuity-due.

An annuity-due with [math]n[/math] payments is the sum of one annuity payment now and an ordinary annuity with one payment less, and also equal, with a time shift, to an ordinary annuity. Thus we have:

[[math]] \ddot{a}_{\overline{n|}i}=a_{\overline{n}|i}(1 + i)=a_{\overline{n-1|}i}+1[[/math]]

. The value at the time of the first of [math]n[/math] payments of 1.

[[math]] \ddot{s}_{\overline{n|}i}=s_{\overline{n}|i}(1 + i)=s_{\overline{n+1|}i}-1[[/math]]

. The value one period after the time of the last of [math]n[/math] payments of 1.

Perpetuity

A perpetuity is an annuity for which the payments continue forever. Observe that

[[math]] \lim_{n\,\rightarrow\,\infty} \text{PV}(i,n,R) = \lim_{n\,\rightarrow\,\infty} R \times a_{\overline{n}|i} = \lim_{n\,\rightarrow\,\infty} R \times \frac{1-\left(1+i\right)^{-n}}{i} = \,\frac{R}{i}. [[/math]]

Therefore a perpetuity has a finite present value when there is a non-zero discount rate. The formulae for a perpetuity are

[[math]] a_{\overline{\infty}|i} = \frac{1}{i} \text{ and } \ddot{a}_{\overline{\infty}|i} = \frac{1}{d},[[/math]]

where [math]i[/math] is the interest rate and [math]d=\frac{i}{1+i}[/math] is the effective discount rate.

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.

Theorem (Present value of arithmetic varying annuities)

Suppose payments in an [math]n[/math]-period annuity-immediate begin at [math]P[/math] and increase by [math]D[/math] per period thereafter . Then, the present value of payments is [math]\left(P+\frac{D}{i}\right)a_{\overline n|i}-\frac{Dn}{i}\cdot v^n[/math] with effective interest rate [math]i[/math] during each of [math]n[/math] periods.

Show Proof

Consider the following time diagram:

                                 Row
                           D     1st
                     D     D     2nd
                     .     .
                     .     .
                     .     .
         D           D     D    n-1 th
   P     P           P     P    nth
---*-----*----...----*-----*
0  1     2    ...   n-1    n    t

• For payments in the [math]n[/math]th row, the present value is [math]Pa_{\overline n|i}[/math];
• for payments in the [math]n-1[/math]th row, the present value is [math]Da_{\overline {n-1}|i}v=Dv\cdot\frac{1-v^{n-1}}{i}[/math];
• ...
• for payments in the [math]n-j[/math]th row, the present value is [math]Da_{\overline {n-j}|i}v^j=Dv^j\cdot\frac{1-v^{n-j}}{i}[/math];
• for payments in the 2nd row, the present value is [math]Da_{\overline 2|i}v^{n-2}=Dv^{n-2}\cdot\frac{1-v^2}{i}[/math];
• for payments in the 1st row, the present value is [math]Da_{\overline 1|i}v^{n-1}=Dv^{n-1}\cdot\frac{1-v}{i}[/math].
• So, the present value of all payments is

[[math]] \begin{align} Pa_{\overline n|i}+\sum_{k=1}^{n-1}Dv^k\cdot\frac{1-v^{n-k}}{i} &=Pa_{\overline n|i}+\frac{D}{i}\sum_{k=1}^{n-1}(v^k-v^n)\\ &=Pa_{\overline n|i}+\frac{D}{i}\left(\sum_{k=1}^{n-1}(v^k)-(n-1)v^n\right)\\ &=Pa_{\overline n|i}+\frac{D}{i}\bigg(\underbrace{\sum_{k=1}^{n-1}(v^k)+v^n}_{=\sum_{k=1}^{{\color{blue}n}}v^k\overset{\text{ def }}=a_{\overline n|i}}-nv^n\bigg)\\ &=Pa_{\overline n|i}+\frac{D}{i}(a_{\overline n|i}-nv^n)\\ &=\left(P+\frac{D}{i}\right)a_{\overline n|i}-\frac{Dn}{i}\cdot v^n.\\ \end{align} [[/math]]

■

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.

Example

• An annuity has payments at the beginning of each year without stopping, which begin at 100 at the beginning of first year, and then increase by 10% per year.
• Suppose the interest rate is 20% payable quarterly.
• Calculate the present value of the annuity.

Solution:

The nominal interest rate of 20% implies the quarterly effective interest rate is 5%.So, the annual effective interest rate is [math]1.05^4-1\approx 21.551\%[/math]. Consider the time diagram:

100     100(1.1)  100(1.1)^2   100(1.1)^3
*---------*---------*------------*------
0         1         2            3

It follows that the present value of the annuity is

[[math]] 100+100(1.1)(1.21551)^{-1}+100(1.1)^2(1.21551)^{-2}+\dotsb =100+100\cdot\frac{1.1}{1.21551}+100\cdot\left(\frac{1.1}{1.21551}\right)^2+\dotsb =\frac{100}{1-\frac{1.1}{1.21551}} \approx 1052.33. [[/math]]

Wikipedia References

• Wikipedia contributors. "Annuity". Wikipedia. Wikipedia. Retrieved 5 November 2023.

General References

• Wikibooks contributors. "Financial Math FM/Annuities,". Wikibooks. Wikibooks. Retrieved 5 November 2023.