Question

Bilbo Baggins wants to save money to meet three objectives. First, he would like to be able to retire 30 years from now with a retirement income of $30,000 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 $380,000. Third, after he passes on at the end of the 20 years of withdrawals, he would like to leave an inheritance of $950,000 to his nephew Frodo. He can afford to save $3,300 per month for the next 10 years. If he can earn an EAR of 10 percent before he retires and an EAR of 7 percent after he retires, how much will he have to save each month in Years 11 through 30? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.

Answer 1

The correct answer is $ 3,201.53

In order to solve the question, let's first move to the time, 30 years from now, when he retires.

Let's say Bilbo has retired today.

For the next 20 years,

Annuity required = A₁ = $ 30,000 / month

Frequency = monthly

Interest rate per period = interest rate per month in the retirement period, R₁ = (1 + EAR_{post retirement})^(1/12) - 1

= (1 + 7%)^(1/12) - 1 = 0.565415% =  0.005654145

Period, N₁ = nos. of months in 20 years = 12 x 20 = 240

Hence, kitty size required today (i.e t = 30) to provide an annuity of A₁

= $ 3,934,711.33

After he passes on at the end of the 20 years of withdrawals, he would like to leave an inheritance of $ 950,000 to his nephew Frodo.

Kitty size required today (i.e.at t = 30) to provide for this inheritance = PV of $ 950,000 = $ 950,000 / (1 + R₁)^N₁ = $ 950,000 / (1 + 0.005654145)²⁴⁰ = $ 245,498.05

Hence, total kitty size required at t=30 will be = $ 3,934,711.33 + $ 322,436.06 = $ 4,180,209.38

Now come back in time, come to present situation, i.e. at t= 0 i.e. today.

He can afford to save $3,300 per month for the next 10 years.

Annuity, A₂ = $ 3,300

Interest rate per period, R₂ = interest rate per month = (1 + EAR_{pre retirement})^(1/12) -  = (1 + 10%)^(1/12) - 1 = 0.797414%
= 0.00797414

Period = N₂ = nos. of month in 10 years = 12 x 10 = 120

Future Value of this annuity A₂ at the end of t = 10 years from now

Bilbo would like to purchase a cabin in Rivendell in 10 years at an estimated cost of $380,000.

Hence, balance amount in the savings account after purchasing the cabin = $  659,550.73 - $380,000 = $ 279,550.73

So, Bilbo will have this much amount that will grow over 10 years from t = 10 to t = 30

Hence, value of this amount at t = 30 will be = $ 279,550.73 x (1 + EAR_{pre retirement})^{(30 - 10)} = $ 279,550.73 x (1 + 10%)²⁰ = $ 1,880,677.50

Recall, in the first part of the solution, we have calculated the kitty size required at the end of t = 30 as $ 4,180,209.38

Shortfall in kitty size at t = 30 will be = $ 4,180,209.38 - $ 1,880,677.50 = $ 2,299,531.88

This shortfall in kitty has to be met by monthly saving over the period t = 11 to t = 30

Let the monthly saving desired in this period be A

Interest rate per period, R₂ = 0.797414%

Period = N₃ = nos. of month in 20 years = 12 x 20 = 240

Future Value of this annuity A at the end of t = 30 that is 20 years from t = 10 should be equal to bridge the shortfall in kitty size required at t = 30

Hence, 718.26 x A = $ 2,299,531.88

Hence, A = $ 2,299,531.88 / 718.26 = $ 3,201.53

Hence, the correct answer is $ 3,201.53