Question

An​ amount, P, must be invested now to allow withdrawals of ​$900 per year for the next 13 years and to permit ​$250 to be withdrawn starting at the end of year 6 and continuing over the remainder of the 13​-year period as the $250 increases by 5% per year thereafter. That​ is, the withdrawal at EOY seven will be $262.50 ,​$275.63 at EOY eight​, and so forth for the remaining years. The interest rate is 12​% per year.

What is the Present value?

Please show all work

Answer 1

Uniform withdrawl of 900 from EOY 1 to EOY 13

i= 12% = 0.12

cash flow series starting in EOY 6 at 250 and increasing by 5% is a geometric series uptil EOY 13 (total 8 annual withdrawl)

Present value at EOY 5 of this series is given by = C * [(1+g)^n/(1+i)^n - 1] / (g-i)

here c= 250 ,g= 0.05, n = 8, i = 0.12

Putting values in the formula

P at EOY 5 = 250 * [(1+0.05)^8/(1+0.12)^8 - 1] / (0.05-0.12)

= 250 * [(1.05)^8/(1.12)^8 - 1] / (-0.07)

= 250* 5.761150

= 1440.29

Present value of total series at EOY 0 = 900* (P/A, 12%,13) + 1440.29 *(P/F, 12%,5)

= 900*6.4235484 + 1440.29 *0.5674268

= 6598.45

Cash flow diagram