Question

You are scheduled to receive annual payments of $8,600 for each of the next 27 years. The discount rate is 7.0 percent. What is the difference in the present value if you receive these payments at the beginning of each year rather than at the end of each year?

Answer 1

When the payment is received at the end of the year it is called ordinary annuity, and when the payment is received at the beginning of the year it is called annuity due.

So first we will calculate PV of annuity due and then PC of ordinary annuity.

Present Value of an annuity due is used to determine the present value of a stream of equal payments where the payment occurs at the beginning of each period.

If,

Periodic payment (P) = 8600

Interest rate (i) = 0.07

Time period (n) = 27

Then PV of annuity due = P * (1 + i) [1 - {(1+ i)^-n}/i]

Lets put all the values in the formula to find PV of annuity due,

PV of annuity due = 8600* (1 + 0.07) [{1- (1 + 0.07)^- 27}/ 0.07]

                                      = 8600* (1.07) [{1- (1.07)^- 27}/ 0.07]

                                      = 9202[{1- 0.1609303673}/ 0.07]

                                      = 9202[0.8390696327/ 0.07]

                                      = 9202* 11.98671

                                      = 110301.70542

So PV of annuity due is $110301.71

PV of ordinary annuity

Present value of annuity is the present worth of cash flows that is to be received in the future, if future value is known, rate of interest in r and time is n then PV of annuity is

PV of annuity = P[1- (1+ r)^-n]/ r

= 8600[1- (1+ 0.07)^-27]/ 0.07

= 8600[1- (1.07)^-27]/ 0.07

= 8600[1- 0.16093036730041]/ 0.07

= 8600[0.83906963269959/ 0.07]

= 8600[11.9867090385656]

= 103085.7

Difference = 110301.71 - 103085.7 =$ 7216.01

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