Compound Interest and Rate

Solution to Problem Set 1

1. You are considering various retirement plans. Your goal is to have a lump sum of $3,000,000 available (‘in the bank’) when you retire at age 67. The various plans, with their payment schedules, are listed below. In each case, calculate the payment(s) that must be made into the plan to ensure that you have the $3,000,000 available. For each plan, you may assume that your opportunity cost of funds is 6% per year; for each plan, you may assume that the phrase “at age XX” means the same thing as “on your XX’th birthday”.

Plan 1: Single lump sum at age 25

Plan 2: Single lump sum at age 50

Plan 3: Equal annual payments, commencing at age 31 and ending at age 67

Plan 4: Equal annual payments, commencing at age 51 and ending at age 67

Plan 5: Equal annual panicky payments, commencing at age 60 and ending at age 67

Plan 1:

V0 * (1.06)42 = 3,000,000

V0 = 3,000,000/(1.06)42

V0 = $259,582.20

Plan 2:

V0 * (1.06)17 = 3,000,000

V0 = 3,000,000/(1.06)17

V0 = $1,114,093.26

Plan 3:

C * Annuity Compound Factor (6%, 37) = 3,000,000

C * [((1.06)37 – 1)/0.06] = 3,000,000

C *127.27 = 3,000,000

C = $23,572.28

Plan 4:

C * Annuity Compounding Factor (6%,17) = 3,000,000

C * 28.21 = 3,000,000

C = $106,334.41

Plan 5:

C * Annuity Compounding Factor (6%,8) = 3,000,000

C * 9.90= 3,000,000

C = $303,107.83

2. You have just taken out a mortgage for $575,000, at a fixed rate of 4.75% per year, compounded monthly, and a term of 30 years.

a) Calculate the monthly payments

The payments must discount to a value that is equivalent to $575,000 today, assuming a monthly rate of (4.75%/12), or 0.39583% per month, for 360 months.

C * Annuity discount factor (0.39583%,360) = 575,000

C * 191.70 = 575,000

C = $2,999.47

b) For the first six months’ payments, calculate the portion that is interest and the portion that is principal

Each month, the