Question

You are evaluating two annuities. They are identical in every way, except that one is an ordinary annuity and the other is an annuity due. Which of the following is FALSE?

A. The ordinary annuity must have a higher future value than the annuity due.

B. The ordinary annuity must have a lower present value than the annuity due.

C. The two annuities will differ in present value by the factor of (1+r).

D. The annuity due and the ordinary annuity will make the same number of total payments over time.

Answer 1

Ans:- (A)

Description:- In Ordinary annuity, payment is made at the end of the duration while in Annuity due payment made at the beginning of the period so as per the definition the difference is actually of one compounding period (1+r) in both annuities.

Annuity due = Annuity ordinary * (1+r)

and this implies that the persent value and future value of annuity due will be higher than ordinary Annuity. we will see this with below example:-

Periodic Payment (P)= 100, rate of interest (r)=5%, Term (t)=3 years

FV (Annuity Due)

3rd year payment:- P*(1+r)¹ = 100*(1+0.05)¹  =100*1.05 =105

2nd year payment:- P*(1+r)² = 100*(1+0.05)² =100*(1.05)² =110.25

1st year payment :-P*(1+r)³ =100*(1.05)³ = 115.76

Total FV(Annuity Due) =105+110.25+115.76 =331.01

FV (Ordinary Annuity)

3rd year payment:- P =100 (as payment being made end of the period)

2nd year payment:- P*(1+r)¹ = 100*(1+0.05)¹  =100*1.05 =105

1st year payment:- P*(1+r)² = 100*(1+0.05)² =100*(1.05)² =110.25

Total FV(Annuity Ordinary) =100+105+110.25 =315.25

so the Future value of annuity due is always higher because all the payments are being made to start peiod so interest will be calculated for 1year payment also.

PV (Ordinary Annuity)

1st year:- P/(1+r)¹ = 100/(1+0.05)¹  =100/1.05 =95.24

2nd year:- P/(1+r)² = 100/(1+0.05)² =100/(1.05)² =90.70

3rd year :-P/(1+r)³ =100/(1.05)³ = 86.21

Total PV(Oridinary Annuity) =95.24+90.70+86.21 =272.15

PV (Annuity Due)

1st year:- P =100 (as payment being made start of the period)

2nd year:- P/(1+r) = 100/(1+0.05) =100/(1.05) =95.24

3rd year :-P/(1+r)² =100/(1.05)²= 90.70

Total PV(Annuity Due) =100+95.24+90.70 =285.94

so the present value of annuity due is always higher because all the payments are being made closer to start peiod.

Here also we can see in both the annuity the number of total payment is always same i.e 3