Question

What is the value today of an investment that pays $2,000 every two years forever starting one year from today and $4,000 every two years forever starting two years from today if the APR is 8.00% compounded quarterly? That is, a $2,000 payment occurs 1 year from today, a $4,000 payment 2 years from today, a $2,000 payment 3 years from today, and so on.

	$35,016
	$35,913
	$36,811
	$37,709
	$38,607

Answer 1

Sum of an infinite gp = a/(1-r)

where a is the first term and r is the common ratio

Annual equivalent of 8% compounded quarterly = (1+0.08/4)^4 -1 = 0.08243216

Investment of $2000 geometric progression (Obtained by discounting the cash-flows to PV)

2000/(1.08243216^1) + 2000/(1.08243216^3) +2000/(1.08243216^5) +2000/(1.08243216^7) ............

a = 2000/(1.08243216^1)

r = 1/(1.08243216^2)

Sum of $2000 investments = (2000/(1.08243216^1))/(1-(1/(1.08243216^2))) = $12611.39

Investment of $4000 geometric progression (Obtained by discounting the cash-flows to PV)

4000/(1.08243216^2) + 4000/(1.08243216^4) +4000/(1.08243216^6) +4000/(1.08243216^8) ............

a = 4000/(1.08243216^2)

r = 1/(1.08243216^2)

Sum of $2000 investments = (4000/(1.08243216^2))/(1-(1/(1.08243216^2))) = $23301.9

Total PV = $12611.39 + $23301.9 = $35913

Hence, $35,913