Compound Interest and Rate Essay

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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
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Note that the monthly inflation rate is 2.4%/12, or 0.2% per month. We need to compound the first tuition cash flow, which occurs in 16 years, by 16 X 12 = 192 months, the second tuition cash flow by 17 X 12 = 204 months, etc.

First tuition cash flow = 38,500 * (1.002)192 = 56,501.93

Second tuition cash flow = 38,500 * (1.002)204 = 57,872.99

Third tuition cash flow = 38,500 * (1.002)216 = 59,277.32

Fourth tuition cash flow = 38,500 * (1.002)228 = 60,715.73

If we wish to discount these annual cash flows, we need an effective annual rate. Currently, the interest rate is given to us as 6.0% per year, compounded monthly. This implies an effective monthly rate of 0.50%, or an effective annual rate of:

(1.0050)12 – 1 = 6.1678%

The value at t=16 of these cash flows, with r=6.1678%:

V =56,501.93 + 57,872.99/(1.061678) + 59,277.32/(1.061678)2+ 60,715.73/(1.061678)3

V = $214,339.59

f) Calculate the value, at t=16, of four years’ worth of college tuition if tuition grows at the recent education inflation rate of 6.4% per year, compounded monthly.

Changing the inflation rate in the problem above to 6.4% per year, compounded monthly (or 0.533% per month), we have tuition cash flows of

t Cash flow

16 106,904.64
17 113,950.85
18 121,461.47
19 129,467.13

And the value at t=16, with our opportunity cost of

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