Question

Future Value of an Annuity for Various Compounding Periods Find the future values of the following ordinary annuities. FV of $800 each 6 months for 9 years at a nominal rate of 8%, compounded semiannually. Do not round intermediate calculations. Round your answer to the nearest cent. $ FV of $400 each 3 months for 9 years at a nominal rate of 8%, compounded quarterly. Do not round intermediate calculations. Round your answer to the nearest cent. $ The annuities described in parts a and b have the same amount of money paid into them during the 9-year period, and both earn interest at the same nominal rate, yet the annuity in part b earns more than the one in part a over the 9 years. Why does this occur?

Answer 1

Problem-1
	FV of annuity
	If compounding is done on semi-annual basis
	P = PMT x ((((1 + r) ^ n) - 1) / i)
	Where:
	P = the future value of an annuity stream	To be calculated
	PMT = the dollar amount of each annuity payment	$               1,600.00	800*2
	r = the effective interest rate (also known as the discount rate)	8.16%	((1+8%/2)^2)-1)
	i=nominal Interest rate	8.00%
	n = the number of periods in which payments will be made	9
	FV of annuity=	PMT x ((((1 + r) ^ n) - 1) / i)
	FV of annuity=	1600*((((1 + 8.16%) ^ 9) - 1) / 8%)
	FV of annuity=	$             20,516.33
	If compounding is done on quarterly basis
	P = PMT x ((((1 + r) ^ n) - 1) / i)
	Where:
	P = the future value of an annuity stream	To be calculated
	PMT = the dollar amount of each annuity payment	$               1,600.00	400*4
	r = the effective interest rate (also known as the discount rate)	8.24%	((1+8%/4)^4)-1)
	i=nominal Interest rate	8.00%
	n = the number of periods in which payments will be made	9
	FV of annuity=	PMT x ((((1 + r) ^ n) - 1) / i)
	FV of annuity=	1600*((((1 + 8.24%) ^ 9) - 1) / 8%)
	FV of annuity=	$             20,786.84
	Since the compounding is done quarterly, the effective interest rate goes up from 8.16% to 8.24% and hence
	the future value in quarterly compounding is higher.