Annuity Due 55o5e

ANNUITY DUE Some annuities are ordinary annuities and others are referred to as annuities due. The main difference between the two types is when the payment is made. In the case of an ordinary annuity, the payments are made at the end of each period. With an annuity due, the payment is made at the beginning of each period.

Present Value of Annuity Due

The formula for the present value of an annuity due, sometimes referred to as an immediate annuity, is used to calculate a series of periodic payments, or cash flows, that start immediately.

Alternative Formula for the Present Value of an Annuity Due The present value of an annuity due formula can also be stated as

which is (1+r) times the present value of an ordinary annuity. This can be shown by looking again at the extended version of the present value of an annuity due formula of

This formula shows that if the present value of an annuity due is divided by (1+r), the result would be the extended version of the present value of an ordinary annuity of

If dividing an annuity due by (1+r) equals the present value of an ordinary annuity, then multiplying the present value of an ordinary annuity by (1+r) will result in the alternative formula shown for the present value of an annuity due.

Future Value of Annuity Due

The future value of annuity due formula is used to calculate the ending value of a series of payments or cash flows where the first payment is received immediately. The first cash flow received immediately is what distinguishes an annuity due from an ordinary annuity. An annuity due is sometimes referred to as an immediate annuity. The future value of annuity due formula calculates the value at a future date. The use of the future value of annuity due formula in real situations is different than that of the present value for an annuity due. For example, suppose that an individual or company wants to buy an annuity from someone and the first payment is received today. To calculate the price to pay for this particular situation would require use of the present value of annuity due formula. However, if an individual is wanting to calculate what their balance would be after saving for 5 years in an interest bearing and they choose to put the first cash flow into the today, the future value of annuity due would be used.

Example of Future Value of Annuity Due Formula To elaborate on the prior example of the future value of an annuity due, suppose that an individual would like to calculate their future balance after 5 years with today being the first deposit. The amount deposited per year is $1,000 and the has an effective rate of 3% per year. It is important to note that the last cash flow is received one year prior to the end of the 5th year. For this example, we would use the future value of annuity due formula to come to the following equation:

After solving, the balance after 5 years would be $5468.41.

Formula Although the present value (PV) of an annuity can be calculated by discounting each periodic payment separately to the starting point and then adding up all the discounted figures, however, it is more convenient to use the 'one step' formulas given below.

1 − (1 + i)-n PV of an Ordinary Annuity = R × i

1 − (1 + i)-n PV of an Annuity Due = R ×

× (1 + i) i

Where, i is the interest rate per compounding period; n are the number of compounding periods; and R is the fixed periodic payment.

Example 1: Calculate the present value on Jan 1, 2011 of an annuity of $500 paid at the end of each month of the calendar year 2011. The annual interest rate is 12%. Solution

We have, Periodic Payment

R

= $500

Number of Periods

n

= 12

Interest Rate

i

= 12%/12 = 1%

Present Value

PV

= $500 × (1-(1+1%)^(-12))/1% = $500 × (1-1.01^-12)/1% ≈ $500 × (1-0.88745)/1% ≈ $500 × 0.11255/1% ≈ $500 × 11.255 ≈ $5,627.54

Example 2: A certain amount was invested on Jan 1, 2010 such that it generated a periodic payment of $1,000 at the beginning of each month of the calendar year 2010. The interest rate on the investment was 13.2%. Calculate the original investment and the interest earned. Solution

Periodic Payment

R

= $1,000

Number of Periods

n

= 12

Interest Rate

i

= 13.2%/12 = 1.1%

Original Investment

= PV of annuity due on Jan 1, 2010 = $1,000 × (1-(1+1.1%)^(-12))/1.1% ×

(1+1.1%) = $1,000 × (1-1.011^-12)/0.011 × 1.011 ≈ $1,000 × (1-0.876973)/0.011 × 1.011 ≈ $1,000 × 0.123027/0.011 × 1.011 ≈ $1,000 × 11.184289 × 1.011 ≈ $11,307.32 Interest Earned

≈ $1,000 × 12 − $11,307.32 ≈ $692.68

THE END