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.

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:

${\displaystyle a_{{\overline {n}}|i}={\frac {1-(1+i)^{-n}}{i}},}$

where ${\displaystyle n}$ is the number of terms and ${\displaystyle i}$ is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, or rent ${\displaystyle R}$ is:

${\displaystyle {\text{PV}}(i,n,R)=R\times a_{{\overline {n}}|i}.}$

In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest ${\displaystyle I}$ is stated as a nominal interest rate, and ${\textstyle i=I/12}$.

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 n-th payment. The future value is given by:

${\displaystyle s_{{\overline {n}}|i}={\frac {(1+i)^{n}-1}{i}},}$

where ${\displaystyle n}$ is the number of terms and ${\displaystyle i}$ is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, or rent ${\displaystyle R}$ is:

${\displaystyle {\text{FV}}(i,n,R)=R\times s_{{\overline {n}}|i}}$

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

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

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:

${\displaystyle s_{{\overline {n}}|i}=(1+i)^{n}\times a_{{\overline {n}}|i}}$

and

${\displaystyle {\frac {1}{a_{{\overline {n}}|i}}}-{\frac {1}{s_{{\overline {n}}|i}}}=i}$

Annuity-due

An annuity-due is an annuity whose payments are made at the beginning of each period.^[3] Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities due.

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

${\displaystyle {\ddot {a}}_{{\overline {n|}}i}=(1+i)\times a_{{\overline {n|}}i}={\frac {1-(1+i)^{-n}}{d}},}$

${\displaystyle {\ddot {s}}_{{\overline {n|}}i}=(1+i)\times s_{{\overline {n|}}i}={\frac {(1+i)^{n}-1}{d}},}$

where ${\displaystyle n}$ is the number of terms, ${\displaystyle i}$ is the per-term interest rate, and ${\displaystyle d}$ is the effective rate of discount given by ${\displaystyle d={\frac {i}{i+1}}}$.

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

${\displaystyle {\ddot {s}}_{{\overline {n}}|i}=(1+i)^{n}\times {\ddot {a}}_{{\overline {n}}|i},}$

${\displaystyle {\frac {1}{{\ddot {a}}_{{\overline {n}}|i}}}-{\frac {1}{{\ddot {s}}_{{\overline {n}}|i}}}=d.}$

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:

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

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

An annuity-due with n 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:

${\displaystyle {\ddot {a}}_{{\overline {n|}}i}=a_{{\overline {n}}|i}(1+i)=a_{{\overline {n-1|}}i}+1}$. The value at the time of the first of n payments of 1.

${\displaystyle {\ddot {s}}_{{\overline {n|}}i}=s_{{\overline {n}}|i}(1+i)=s_{{\overline {n+1|}}i}-1}$. The value one period after the time of the last of n payments of 1.

Perpetuity

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

${\displaystyle \lim _{n\,\rightarrow \,\infty }{\text{PV}}(i,n,R)=\lim _{n\,\rightarrow \,\infty }R\times a_{{\overline {n}}|i}=\lim _{n\,\rightarrow \,\infty }R\times {\frac {1-\left(1+i\right)^{-n}}{i}}=\,{\frac {R}{i}}.}$

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

${\displaystyle a_{{\overline {\infty }}|i}={\frac {1}{i}}{\text{ and }}{\ddot {a}}_{{\overline {\infty }}|i}={\frac {1}{d}},}$

where ${\displaystyle i}$ is the interest rate and ${\displaystyle d={\frac {i}{1+i}}}$ is the effective discount rate.