An annuity is a series of equal payments made at regular intervals over a
period of time. The two most common types are ordinary annuities
(payments made at the end of each period) and annuities due (payments
made at the beginning of each period).
Future Value and Present Value of Annuities
1. Future Value (FV): This is the value of a series of cash flows at some
point in the future, considering the effect of interest.
2. Present Value (PV): This is the current worth of a series of future cash
flows, discounted to account for the time value of money.
Formulas for Future and Present Value
1. Future Value of an Ordinary Annuity
In an ordinary annuity, payments are made at the end of each period. The
formula for the future value is:
n
FV =P[((1+r ) −1)/r ]
Where:
FV = Future value of the annuity
P = Payment amount per period
r = Interest rate per period
n = Total number of payments (periods)
2. Future Value of an Annuity Due
In an annuity due, payments are made at the beginning of each period. The
formula is:
n
FV =P[((1+r ) −1)/r ]•(1+ r)
Where:
FV = Future value of the annuity
P = Payment amount per period
r = Interest rate per period
n = Total number of payments (periods)
The difference between the future value of an annuity due and an ordinary
annuity is the extra compounding for each payment, since payments are
made at the beginning rather than the end.
3. Present Value of an Ordinary Annuity
The present value is the sum of all payments, discounted to the present day.
The formula is:
PV =P[(1−(1+ r)(−n) )/r ]
Where:
PV = Present value of the annuity
P= Payment amount per period
r = Interest rate per period
n = Total number of payments (periods)
4. Present Value of an Annuity Due
For an annuity due, the present value formula is:
(−n)
PV =P[( 1−(1+ r) )/r ]•(1+ r)
Where:
PV = Present value of the annuity
P= Payment amount per period
r = Interest rate per period
n = Total number of payments (periods)
Breakdown of Variables
P (Payment): This is the fixed amount paid or received each period. For
annuities, this value is constant for each interval.
r (Interest rate per period): This is the interest rate applied per period. If the
annual interest rate is given, divide it by the number of periods per year to
find the rate per period.
n (Number of periods): This represents the total number of payment
intervals, such as monthly or annually, over the life of the annuity.
FV (Future value): This refers to the total amount of money accumulated in
the future, after all payments and interest have been accounted for.
PV (Present value): This is the amount of money today that would be
equivalent to a series of future payments, discounted back to the present
using the interest rate.
Summary of Key Differences
Ordinary annuity: Payments are made at the end of each period (e.g., at
the end of a year or month). As a result, future value and present value are
slightly lower compared to an annuity due because there’s no extra period of
interest accumulation.
Annuity due: Payments are made at the beginning of each period. This
means each payment has an extra period to accrue interest, leading to a
higher future value and present value compared to an ordinary annuity.
These formulas are used in various financial scenarios, including retirement
savings, loan repayments, and investment planning.
