Present Value Annuities
Present Value Annuities
For present value annuities, regular equal payments/installments are made to pay back a loan or bond over a given time period. The reducing balance of the loan is usually charged compound interest at a certain rate. In this section, we learn how to determine the present value of a series of payments.
Consider the following example:
Kate needs to withdraw \(\text{₦}\,\text{1,000}\) from her bank account every year for the next three years. How much must she deposit into her account, which earns \(\text{10}\%\) per annum, to be able to make these withdrawals in the future? We will assume that these are the only withdrawals and that there are no bank charges on her account.
To calculate Kate's deposit, we make \(P\) the subject of the compound interest formula:
\begin{align*} A &= P ( 1 + i )^{n} \\ \cfrac{A}{( 1 + i )^{n}} &= P \\ \therefore P &= A ( 1 + i )^{-n} \end{align*}
We determine how much Kate must deposit for the first withdrawal:
\begin{align*} P &= \text{1 000} ( 1 + \text{0.1} )^{-1} \\ &= \text{909.09} \end{align*}
We repeat this calculation to determine how much must be deposited for the second and third withdrawals:
\begin{align*} \text{Second withdrawal: } P &= \text{1 000} ( 1 + \text{0.1} )^{-2} \\ &= \text{826.45} \\ \text{Third withdrawal: } P &= \text{1 000} ( 1 + \text{0.1} )^{-3} \\ &= \text{751.31} \end{align*}
Notice that for each year's withdrawal, the deposit required gets smaller and smaller because it will be in the bank account for longer and therefore earn more interest. Therefore, the total amount is:
\[\text{₦}\,\text{909.09} + \text{₦}\,\text{826.45} + \text{₦}\,\text{751.31} = \text{₦}\,\text{2,486.85}\]
We can check these calculations by determining the accumulated amount in Kate's bank account after each withdrawal:
|
Calculation |
Accumulated amount |
|
|
Initial deposit |
\(\text{₦}\,\text{2,486.85}\) |
|
|
Amount after one year |
= \(\text{2 486.85}(1+\text{0.1})\) |
= \(\text{₦}\,\text{2,735.54}\) |
|
Amount after first withdrawal |
= \(\text{₦}\,\text{2,735.54} - \text{₦}\,\text{1,000}\) |
= \(\text{₦}\,\text{1,735.54}\) |
|
Amount after two years |
= \(\text{1 735.54} (1+\text{0.1})\) |
= \(\text{₦}\,\text{1,909.09}\) |
|
Amount after second withdrawal |
= \(\text{₦}\,\text{1,909.09}- \text{₦}\,\text{1,000}\) |
= \(\text{₦}\,\text{909.09}\) |
|
Amount after three years |
= \(\text{909.09}(1+\text{0.1})\) |
= \(\text{₦}\,\text{1,000}\) |
|
Amount after third withdrawal |
= \(\text{₦}\,\text{1,000} - \text{₦}\,\text{1,000}\) |
= \(\text{₦}\,\text{0}\) |
Completing this table for a three year period does not take too long. However, if Kate needed to make annual payments for \(\text{20}\) years, then the calculation becomes very repetitive and time-consuming. Therefore, we need a more efficient method for performing these calculations.
This lesson is part of:
Finance and Growth