Question

With total yearly payments of $10,000 for 10 years, compare the compound amount accumulated at the end of the 10 years if the payments are (1) at the end of the year, (2) weekly, and (3) continuous. The effective (annual) interest rate is 8 percent, and the payments are uniform. Also determine the present worth at time zero for each of the three types of payments

Answer 1

yearly payments R = $10,000

Number of years n = 10 years

Interest rate i = 8%

Amount accumulated S =?

Part a

At the end of year

S = R [(1+i)^n - 1]/(i)

= $10000[(1+0.08)^10 - 1]/(0.08)

= $144865.62

Present worth

P = R [(1+i)^n - 1]/[i*(1+i)^n]

= $10000[(1+0.08)^10 - 1]/[0.08*(1+0.08)^10]

= $67100.81

Part b

Weekly

Number of compoundings per year m = 365/7 = 52

Nominal interest rate = r

ieff = ( 1 + r/m)^m - 1

0.08 = ( 1 + r/52)^52 - 1

(1.08)^(1/52) = 1 + r/52

r = 0.077 = 7.70%

S = (R/m) [(1 + r/m)^mn - 1]/(r/m)

S = ($10000/52) [(1 + 0.077/52)^(52*10) - 1]/(0.077/52)

= $(192.307) [1.158] / (0.0014807)

= $150389.07

Present worth

P = S/(1 + r/m)^mn

= ($150389.07) / (1 + 0.077/52)^(52*10)

= $69671.77

Part c

Continuous

ieff = exp(r) - 1

0.08 = exp(r) - 1

exp(r) = 1.08

r = 0.077 = 7.70%

S = R (e^rn - 1)/r

S = $10000 (e^0.077*10 - 1)/0.077

S = $150618.99

Present worth

P = S/e^rn

= ($150618.99)/(e^0.077*10 )

= $69738.56