Question

Explain what is annuity & the diff between ordinary annuity and annuity

due. What are the relationship of the present value and future value calculation of these

two types of annuities?

Imagine you buy a car using a bank loan that requires an annual payment of $500 for 10years at the end of each year. If the annual interest rate is 3%, what are the present value of this car? What is the Excel function used for this calculation?

What would the present value be if the loan requires a semi-annual payment of $100 at the

beginning of each period? What is the Excel function for this annuity due?

Answer 1

An annuity is a series of payments that are made at equal intervals. For instance when you make monthly home mortgage payments this is an annuity. Or if you make regular deposits to your savings account then it will again be an annuity. The difference between ordinary annuity and annuity due is that in case of ordinary annuity the payments occur at the end of each period while in case of annuity due the payments occur at the beginning of each period. Present value of an ordinary annuity = P*[(1-(1+r)^(-n))/r]. For computing the present value of an annuity due we simply add the immediate cash flow to the present value of ordinary annuity. Thus present value of annuity due = P + P*[(1-(1+r)^(-n))/r]. The formula for future value of ordinary annuity = P[(1+r)^n – 1]/r. While computing the future value of annuity due we simply multiply the formula of ordinary annuity with (1+r). Hence future value of annuity due = (1+r)* P[(1+r)^n – 1]/r. Thus present value of annuity due is derived from the formula of present value of ordinary annuity and future value of annuity due is derived from the formula of future value of ordinary annuity.

Here PMT = 500, nper = 10 and r = 3%. Thus we have to find present value and hence we will use the “PV” function in excel. The syntax will be: PV (3%, 10, -500). This will give a result of $4,265.10. Thus the present value of car is $4,265.10

In the last case nper will change to 10*2 = 20 and r = 3%/2 (as the payments are semi-annual in nature). Also payments are being done at the beginning of each period so this becomes annuity due and so in the excel function we will select the “type” as 1 (as this represents annuity due). The present value now will be computed again using the PV function and the syntax will be: PV (3%/2, 20, -100,, 1). This will give a result of $1,742.62