Time Value of Money and Present Value

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Date: 14/11/2012

52. Annuities:

You are saving for the college education of your two children. They are two years apart in age; one will begin college 15 years from today and the other will begin 17 years from today. You estimate your children’s college expenses to be $23,000 per year per child, payable at the beginning of each school year. The annual interest rate is 5.5 percent. How much money must you deposit in account each year to fund your children’s education? Your deposits begin one year from today. You will make your last deposit when your oldest child enters college. Assume four years of college

Solution:

Cost of 1 year at university = 23,000

N=4

I=5.5%

PMT=23,000

CPT PV = 80,618.45

For the first child the PV = 80,618.45/ (1.055) ^14 = $38,097.81

For the second child the PV = 80,618.45/ (1.055) ^16 = $34,229.07

Therefore the total cost today of your children’s college expense will be the addition of the 2

= $72,326.88

This is the present value of my annual savings, which are an annuity, so to get the amount I am supposed to save each year would be:

PV=72,326.88

N=15

I=5.5

CPT PMT = 7,205.6

57. Calculating Annuity Values:

Bilbo Baggins wants to save money to meet three objectives. First, he would like to be able to retire 30 years from now with retirement income of $25,000 per month for 20 years, with the first payment received 30 years and 1 month from now. Second, he would like to purchase a cabin in Rivendell in 10 years at an estimated cost of $350,000. Third, after he passes on at the end of the 20 years of withdrawals, he would like to leave an inheritance of $750,000 to his nephew Frodo. He can afford to save $2,100 per month for the next 10 years. If he can earn an 11 percent EAR before he retires and an 8 percent EAR after he retires, how much will he have to save each month in years 11 through 30?

Solution:

First we get the FV of the 2,100 savings over 10 years

Bilbo Baggins can afford to save $2,100 dollars per month for the next 10 years therefore at 10 years he would have saved:

PMT = 2,100

I = 10.48 / 12 = 0.873

N = 10 x 12 = 120

CPT FV = $442,201.15

So after 10 years he would be able to purchase his yacht at the price of $350,000, and he would be left with a balance of $92,201.15

This $92,201.15 will be our current PV at year 10.

At year 30, the year when Bilbo retires, the $92,201.15 would become 92,201.15*(1.11) ^20 = $620,283.23

Second we have to find out how much the inheritance of 750,000 would be at year 30: 750,000/1.08^20= $160,911.16

Third In order for him to be able to withdraw a sum of 25,000 per month for the next 20 years after his retirement, we should now calculate this annuity’s present value:

N= 20 x 12 = 240

I= 7.72 / 12 = 0.643

PMT= 25,000

CPT PV = $3,052,135.26

Adding up the PV’s of the $750,000 and the annuity, we will get $3,213,046.32

We will subtract the future value at year 30 of the $92,201.15 ($620,283.23) which we saved at year 10 from $3,213,046.32 to get $2,592,763.09

We are now left with an annuity that pays $2,592,763.09 at year 30, and a time period of 20 years (yr11-30)

To calculate the yearly PMT, we have

FV= $2,592,763.09

I= 10.48 / 12 = 0.873

N= 20 x 12 = 240

CPT PMT = 3,207.33

Therefore the monthly PMT Bilbo would have to save each month through years 11-30 would be

= $3,207.33

34. Valuing bonds:

Mallory Corporation has two different bonds, currently outstanding. Bond M has a face value of $20,000 and matures in twenty years. The bond makes no payments for the first six years, then pays $1,200 every 6 months over the subsequent eight years, and finally pays $1,500 every 6 months over the last years. Bond N also has a face value of $20,000 and a maturity of 20 years; it makes n coupon payments over the life of the bond. If the...
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