Compound Interest and Rate

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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 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 interest portion can be calculated as the monthly rate of 0.39583% times the beginning mortgage balance. The principal portion is simply the total payment less the interest portion. For the following month, the beginning principal will decline by the principal amount which was paid in the previous month.

Month| Beginning Principal| Interest| Principal Paid| Ending Principal| 1| 575,000.00| 2,276.04| 723.43| 574,276.57|
2| 574,276.57| 2,273.18| 726.29| 573,550.28|
3| 573,550.28| 2,270.30| 729.17| 572,821.11|
4| 572,821.11| 2,267.42| 732.06| 572,089.05|
5| 572,089.05| 2,264.52| 734.95| 571,354.10|
6| 571,354.10| 2,261.61| 737.86| 570,616.24|

Note that you can check your work, since in any month, the ending principal can also be calculated as the discounted value of all future payments.

For example, as of the end of month 6, the ending principal should be:

V = $2,999.47 * Annuity discount factor (0.39583%,354)

V = $2,999.47 * 190.24 = $570,615.82 (the difference is rounding.)

c) Immediately after the sixth payment, what is the balance remaining on the mortgage?

As we can read from either the table in part b, or from the calculation of the value today of the remaining payments, the balance is $570,616.24

d) If you design the mortgage so that the payments will grow at 0.20% per month, what will be the first payment on the mortgage?

Now we’ll use the annuity discount factor formula that encompasses a growth factor, of 0.20% per month.

V0 = 575,000 = C1 * [1-((1.002)/(1.0039583))360]/(0.0039583-0.0020)

775,000 = C1 * 257.80

C1 = $2,230.41

3. You are saving for your child’s college education. Your child will start college in 16 years, and college tuition is due at the beginning of the year (i.e., the first tuition payment will occur at t=16). Average college tuition at a private school this year is $38,500 per year. You may assume that your opportunity cost...
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