### Calculating Different Types of Annuities

Definition

An annuity is a series of payments required to be made or received over time at regular intervals. The most common payment intervals are yearly (once a year), semi-annually (twice a year), quarterly (four times a year), and monthly (once a month).

Some examples of annuities: Mortgages, Car payments, Rent, Pension fund payments, Insurance premiums.

TYPES OF ANNUITIES

Ordinary Annuity:
An Ordinary Annuity has the following characteristics:

• The payments are always made at the end of each interval
• The interest rate compounds at the same interval as the payment interval

For calculating the sum of a series of regular payments the following formula should be used:

R[(1+i)^ n -1]
S n = —————–
i

Example: Alan decides to set aside \$50 at the end of each month for his child’s college education. If the child were to be born today, how much will be available for its college education when s/he turns 19 years old? Assume an interest rate of 5% compounded monthly.

Solution:
First, we assign all the terms:

R= \$50
i= 0.05/12 or 0.004166
n= 18 x 12, or 216

Now substituting into our formula, we have:

R[(1+i)^n-1]
S n = ——————-
i

\$50[(1+0.05/12)^216 -1]
S n = ——————————–
0.05 / 12

S n = \$50(349.2020206)

S n = \$17,460.10

Formula for calculating present value of a simple annuity:

R[1-(1+i)^-n]
A n = ——————–
i

Example: Alan asks you to help him determine the appropriate price to pay for an annuity offering a retirement income of \$1,000 a month for 10 years. Assume the interest rate is 6% compounded monthly.

Solution:
Substituting into our formula, we have:

R = \$1,000
i = 0.06 /12 or 0.005
n = 12 x 10, or 120

\$1,000[1-(1+0.005)^-120]
A n = ———————————–
0.005

A n = \$90,073.45

Annuity Due:
In an annuity due, the payments occur at the beginning of the payment period.

For calculating the sum of a series of regular payments the following formula should be used:

R(1+i)[(1+i)^ n -1]
S n (due)= ———————–
i

Example: Alan wants to deposit \$300 into a fund at the beginning of each month. If he can earn 10% compounded interest monthly, how much amount will be there in the fund at the end of 6 years?

Solution:

R = \$300
i = 0.10/12 or 0.008333
n = 12 x 6 or 72

Substituting into our formula yields:

\$300(1+0.10/12)[(1+0.10/12)^72-1]
S n (due) = ————————————————-
0.10/12

S n (due) = \$300(98.93)
S n (due) = \$29,679

Formula for calculating present value of an annuity due:

R(1+i)[1-(1+i)^-n]
A n(due) = ————————-
i

Example: The monthly rent on an apartment is \$950 per month payable at the beginning of each month. If the current interest is 12% compounded monthly, what single payment 12 months in advance would be equal to a year’s rent?

Solution:
R= \$950
i= 0.12/12 or 0.01
n= 12

Substituting into the formula gives:

\$950(1+0.03)[1-(1+0.01)^-12]
A n(due) = ———————————————-
0.01

A n(due) = \$950(11.37)

A n(due) = \$10,801.50

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