Present Value of Annuity

The Present Value (PV) of an annuity can be found by calculating the PV of each individual payment and then summing them up . As in the case of finding the Future Value (FV) of an annuity, it is important to note when each payment occurs. Annuities-due have payments at the beginning of each period, and ordinary annuities have them at the end.

Recall that the first payment of an annuity-due occurs at the start of the annuity, and the final payment occurs one period before the end. The PV of an annuity-due can be calculated as follows:

$\displaystyle{ P_0 = \frac{P_n}{(1+i)^n} = P \frac{1-(1+i)^{-n}}{i} \cdot (1+i) }$

where $P$ is the size of the payment (sometimes $A$ or $pmt$), $i$ is the interest rate, and $n$ is the number of periods.

An ordinary annuity has annuity payments at the end of each period, so the formula is slightly different than for an annuity-due. An ordinary annuity has one full period before the first payment (so it must be discounted) and the last payment occurs at the termination of the annuity (so it must be discounted for one period more than the last period in an annuity-due). The formula is:

$\displaystyle{ { P }_{ 0 }=\frac { { P }_{ n } }{ { (1+i) }^{ n } } =P\cdot \sum _{ k=1 }^{ n }{ \frac { 1 }{ { (1+i) }^{ n+k-1 } } } =P\cdot \frac { 1-\left[ \frac { 1 }{ { (1+i })^{ n } } \right] }{ i }}$

where, again, $P$, $i$, and $n$ are the size of the payment, the interest rate, and the number of periods, respectively.

Both annuities-due and ordinary annuities have a finite number of payments, so it is possible, though cumbersome, to find the PV for each period. For perpetuities, however, there are an infinite number of periods, so we need a formula to find the PV. The formula for calculating the PV is the size of each payment divided by the interest rate.

Example 1

Suppose you have won a lottery that pays $1,000 per month for the next 20 years. But, you prefer to have the entire amount now. If the interest rate is 8%, how much will you accept?

Consider for argument purposes that two people, Mr. Cash, and Mr. Credit, have won the same lottery of $1,000 per month for the next 20 years. Now, Mr. Credit is happy with his $1,000 monthly payment, but Mr. Cash wants to have the entire amount now. Our job is to determine how much Mr. Cash should get. We reason as follows: If Mr. Cash accepts x dollars, then the x dollars deposited at 8% for 20 years should yield the same amount as the $1,000 monthly payments for 20 years. In other words, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like the future values to be equal.

Since Mr. Cash is receiving a lump sum of x dollars, its future value is given by the lump sum formula:

$\displaystyle{ {x \left( \frac{1+.08}{12} \right)} ^{240} }$

Since Mr. Credit is receiving a sequence of payments, or an annuity, of $1,000 per month, its future value is given by the annuity formula:

$\displaystyle{ \frac{1000 \left[ \left( \frac{1+0.08}{12} \right) ^{240}-1 \right]} {\frac{0.08}{12}} }$

The only way Mr. Cash will agree to the amount he receives is if these two future values are equal. So we set them equal and solve for the unknown:

$\displaystyle{ {x \left( \frac{1+.08}{12} \right)} ^{240} = \frac{1000 \left[ \left( \frac{1+0.08}{12} \right) ^{240}-1 \right]} {\frac{0.08}{12}}}$

$x\cdot (4.9268) = \ 1,000 \cdot (589.02041)$

$x \cdot 4.9268 = \ 589,020.41$

$x = \ 119,554.36$

The reader should also note that if Mr. Cash takes his lump sum of $119,554.36 and invests it at 8% compounded monthly, he will have $589,020.41 in 20 years.

Example 2

Find the monthly payment for a car costing $15,000 if the loan is amortized over five years at an interest rate of 9%.

Again, consider the following scenario: Two people, Mr. Cash and Mr. Credit, go to buy the same car that costs $15,000. Mr. Cash pays cash and drives away, but Mr. Credit wants to make monthly payments for five years. Our job is to determine the amount of the monthly payment.

We reason as follows: If Mr. Credit pays x dollars per month, then the x dollar payment deposited each month at 9% for 5 years should yield the same amount as the $15,000 lump sum deposited for 5 years. Again, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like them to be the same.

Since Mr. Cash is paying a lump sum of $15,000, its future value is given by the lump sum formula:

$\displaystyle{ \$ 15,000\cdot { \left( \frac{1+.09}{12} \right) }^{60}}$

Mr. Credit wishes to make a sequence of payments, or an annuity, of x dollars per month, and its future value is given by the annuity formula:

$\displaystyle{ \frac { x{ \left[ { \left( \frac{1+0.09}{12} \right) }^{ 60 }-1 \right] } }{ \frac{0.09}{12} }}$

We set the two future amounts equal and solve for the unknown:

$\displaystyle{15,000\cdot { \left( \frac{1+.09}{12} \right) }^{60} = \frac { x{ \left[ { \left( \frac{1+0.09}{12} \right) }^{ 60 }-1 \right] } }{ \frac{0.09}{12} }}$

$\ 15,000 \cdot 1.5657 = x \cdot 75.4241$

$\ 311.38 = x$