**Present Value of an Annuity**

The present value an annuity is the sum of the periodic payments each discounted at the given rate of interest to reflect the time value of money.

PV of an Ordinary Annuity = R (1 − (1 + i)^{-n})/i

PV of an Annuity Due = R (1 − (1 + i)^{-n})/i × (1 + 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, 2015 of an annuity of 5,000 paid at the end of each month of the calendar year 2015. The annual interest rate is 12%.

**Solution**

We have,

Periodic Payment R = 5,000

Number of Periods n = 12

Interest Rate i = 12%/12 = 1%

Present Value

PV = 5000 × (1-(1+1%)^{(-12)})/1%

= 5000 × (1-1.01^{-12})/1%

= 5000 × (1-0.88745)/1%

= 5000 × 0.11255/1%

= 5000 × 11.255

= 56,275.40

2. A certain amount was invested on Jan 1, 2015 such that it generated a periodic payment of 10,000 at the beginning of each month of the calendar year 2015. The interest rate on the investment was 13.2%. Calculate the original investment and the interest earned.

**Solution**

Periodic Payment R = 10,000

Number of Periods n = 12

Interest Rate i = 13.2%/12 = 1.1%

Original Investment = PV of annuity due on Jan 1, 2015

= 10,000 × (1-(1+1.1%)^{(-12)}/1.1% × (1+1.1%)

= 10,000 × (1-1.011^{(-12)})/0.011 × 1.011

= 10,000 × (1-0.876973)/0.011 × 1.011

= 10,000 × 0.123027/0.011 × 1.011

= 10,000 × 11.184289 × 1.011

= 1,13,073.20

Interest Earned = 10,000 × 12 − 1,13,073.20

= 1,20,000 – 1,13,073.20

= 6926.80

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