Annuity Formula

An annuity formula is used to find the present and future value of an amount. An annuity is a fixed amount of income that is given annually or at regular intervals. An annuity is an agreement with an insurance company in which you make a lump sum payment (one-time big payment) or series of payments and, in return, receive a regular fixed income, beginning either immediately or after some predefined time in the future. The annuity formula is used to find the present and future value of an amount. The annuity formula is explained below along with solved examples.

What is Annuity Formula?

The annuity formula helps in determining the values for annuity payment and annuity due based on the present value of an annuity due, effective interest rate, and several periods. Hence, the formula is based on an ordinary annuity that is calculated based on the present value of an ordinary annuity, effective interest rate, and several periods.  The annuity formulas are: 

Annuity = r * PVA _Ordinary / [1 – (1 + r)^-n]

Annuity = r * PVA _Due / [{1 – (1 + r)^-n} * (1 + r)]

The annuity formula for the present value of an annuity and the future value of an annuity is very helpful in calculating the value quickly and easily. The Annuity Formulas for future value and present value are:

The future value of an annuity, FV = P×((1+r)ⁿ−1) / r

The present value of an annuity, PV = P×(1−(1+r)^-n) / r

Annuity Formula 

The formula is calculated based on two important aspects - The present Value of the Ordinary Annuity and the Present Value of the Due Annuity. 

Annuity = r * PVA _Ordinary / [1 – (1 + r)^-n]

Where,

• PVA _Ordinary = Present value of an ordinary annuity
• r = Effective interest rate
• n = Number of periods

Annuity = r * PVA _Due / [{1 – (1 + r)^-n} * (1 + r)]

Where,

• PVA _Due = Present value of an annuity due
• r = Effective interest rate
• n = number of periods

The Annuity Formulas for future value and present value is:

The future value of an annuity, FV = P×((1+r)ⁿ−1) / r

The present value of an annuity, PV = P×(1−(1+r)^-n) / r

where,

• P = Value of each payment
• r = Rate of interest per period in decimal
• n = Number of periods

Examples Using Annuity Formula

Example 1: Dan was getting $100 for 5 years every year at an interest rate of 5%. Find the future value of this annuity at the end of 5 years? Calculate it by using the annuity formula.

Solution 

The future value 

Given:  r = 0.05, 5 years  = 5 yearly payments, so n = 5, and P = $100

FV = P×((1+r)ⁿ−1) / r

FV = $100 × ((1+0.05)⁵−1) / 0.05

FV = 100 × 55.256

FV = $552.56

Therefore, the future value of annuity after the end of 5 years is $552.56.

Example 2: If the present value of the annuity is $20,000. Assuming a monthly interest rate of 0.5%, find the value of each payment after every month for 10 years. Calculate it by using the annuity formula.

Solution:

Given:

r = 0.5% = 0.005

n = 10 years x 12 months = 120, and PV = $20,000

Using formula for present value

PV = P×(1−(1+r)^-n) / r

Or, P = PV × ( r / (1−(1+r)⁻ⁿ))

P = $20,000 × (0.005 / (1−(1.005)⁻¹²⁰))

P = $20,000 × (0.005/ (1−0.54963))

P = $20,000 × 0.011...

P = $220

Therefore, the value of each payment is $220.

Example 3: Jane won a lottery worth $20,000,000 and has opted for an annuity payment at the end of each year for the next 10 years as a payout option. Determine the amount that Jane will be paid as annuity payment if the constant rate of interest in the market is 5%.

Solution: 

Given:

PVA (ordinary) = $20,000,000 (since the annuity to be paid at the end of each year)

r = 5%

n = 10 years

Using the Annuity Formula, 

Annuity = r * PVA Ordinary / [1 – (1 + r)^-n]

Annuity = 5% × 20000000 / [1 - (1 + 0.05)^-10

Annuity = $2,564,102.56

Therefore, Jane will pay an annuity amount of $2,564,102.56