Question

Ch 2 HW Compute the present values of the following annuities frst assuming that payments are made o Saved Help Save & Exit Submit Check my work Problem 2-27 (LG 2-9) Compute the present values of the following annuities first assuming that payments are made on the last day of the period and then assuming payments are made on the first day of the period: (Do not round intermediate calculations. Round your answers to 2 decimal places. (e.g., 32.16) Present Value Present Value Payment Years Interest Rate s 778.09 Years (Annuall Payment made on (Payment made on first day of period) last day of period) 81 141% 14 8,968 26 21.32.93 24 70,712 54 32 Next > Prev5 of 15

Answer 1

In general, Present Value formula: PV = A/(1+r) + A/(1+r)^2 + ...+ A/(1+r)^n ; A =annuity; r = yearly interest rate, n = number of years.

Assuming payments made at the end of the period (EOP)

• Annuity, A = 778.09; for years = 8 at rate = 14%
  • PV =778.09/(1+0.14) + 778.09/(1+0.14)^2 + .... + 778.09/(1+0.14)^8 = $3,609.45
• Annuity, A = 8,968.26; for years = 14 at rate = 7%
  • PV = 8,968.26/(1+0.07) + 8,968.26/(1+0.07)^2 + .... + 8,968.26/(1+0.07)^14 = $78,431.63
• Annuity, A = 21,322.93; for years = 24 at rate = 5%
  • PV = 21,322.93/(1+0.05) + 21,322.93/(1+0.05)^2 + .... + 21,322.93/(1+0.05)^24 = $294,227.47
• Annuity, A = 70,712.54; for years = 5 at rate = 32%
  • PV = 70,712.54/(1+0.32) + 70,712.54/(1+0.32)^2 + .... + 70,712.54/(1+0.32)^5 = $165,835.41

Assuming payments made at the beginning of the period (BOP)

• Annuity, A = 778.09; for years = 8 at rate = 14%
  • PV =778.09 + 778.09/(1+0.14) + 778.09/(1+0.14)^2 + .... + 778.09/(1+0.14)^7 = $4,114.78
• Annuity, A = 8,968.26; for years = 14 at rate = 7%
  • PV = 8,968.26 + 8,968.26/(1+0.07) + 8,968.26/(1+0.07)^2 + .... + 8,968.26/(1+0.07)^13 = $83,921.84
• Annuity, A = 21,322.93; for years = 24 at rate = 5%
  • PV =21,322.93 + 21,322.93/(1+0.05) + 21,322.93/(1+0.05)^2 + .... + 21,322.93/(1+0.05)^23 = $308,938.85
• Annuity, A = 70,712.54; for years = 5 at rate = 32%
  • PV = 70,712.54 + 70,712.54/(1+0.32) + 70,712.54/(1+0.32)^2 + .... + 70,712.54/(1+0.32)^4 = $218,902.74

Summary:

Annuity	Years	rate	PV calc at EOP	PV calc at BOP
778.09	8	14%	$3,609.45	$4,114.78
8968.26	14	7%	$78,431.63	$83,921.84
21,322.93	24	5%	$294,227.47	$308,938.85
70,712.54	5	32%	$165,835.41	$218,902.74

Future Value:

In general, Future Value formula = A(1+r) + A(1+r)^2 + ... + A(1+r)^n

[EOP = End of period, BOP = Beginning of period]

a) A=5000 invested at the EOP, n=7 years, rate = 7%, compounded annually

FV = 5000(1+0.07)^6 + 5000(1+0.07)^5 + ...+ 5000(1+0.07)^1 + 5000 = $43,270.11

b) A=5000 invested at the EOP, n=7 years, rate = 7%, compounded quarterly

number of compounding periods = 7 * 4 = 28; compounding rate = 7%/4 =  1.75%

FV = 5000(1+0.0175)^27 + 5000(1+0.0175)^26 + ...+ 5000(1+0.0175)^1 + 5000 = $178,688.40

c) A=5000 invested at the BOP, n=7 years, rate = 7%, compounded annually

FV = 5000(1+0.07)^7 + 5000(1+0.07)^6 + ...+ 5000(1+0.07)^1 = $46,299.01

d) A=5000 invested at the BOP, n=7 years, rate = 7%, compounded quarterly

number of compounding periods = 7 * 4 = 28; compounding rate = 7%/4 =  1.75%

FV = 5000(1+0.0175)^28 + 5000(1+0.0175)^27 + ...+ 5000(1+0.0175)^1 = $178,691.02

Summary:

Annuity	investment period	annual rate	Compounding	Invested at	Future Value
5000	7	7%	annual	EOP	$43,270.11
5000	7	7%	quarterly	EOP	$178,689.40
5000	7	7%	annual	BOP	$46,299.01
5000	7	7%	quarterly	BOP	$178,691.02