Question

Little Jack Horner is tired of sitting in the corner and wants to attend college. In fact, he intends to go to Harvard when he is old enough and he wants to major in nursery rhyme writing. He will stay in school for 4 years after which he will have a BA degree. His father, Humpty Dumpty, is sympathetic with Jack’s aspirations, but he believes Jack must take care of him first. Therefore, Humpty will finance Jack’s education with a bequest at his death and, until that time, Jack is to look after his father. Harvard will cost $50,000 per year for the four years (payable at the beginning of each year) and Humpty specifies in his Will, that, at his death, the necessary funds will be deposited in an account paying 10% annual interest. Jack will begin college immediately after Humpty’s death.

Humpty will retire 20 years before his death and during his retirement he wants to have $100,000 per year (beginning-of-year). The funds necessary for Jack’s education as well as Humpty’s retirement pay will come from an amount he will have accumulated during his remaining working years. Upon his retirement, Humpty plans to invest all of his funds into an account which will earn 10% annual interest (this rate will be earned during Humpty’s retirement years and during Jack’s college years).

Currently Humpty does not have any money saved; however, he plans to retire 15 yearsfrom now. Humpty wants to accumulate enough funds over his remaining 15 working years to enable him to fulfill his plans as described above. Humpty plans to accumulate the necessary funds in two ways:

1. He just purchased 1,000 shares of SkyRocket company common stock for $20 per share. He believes that the price will increase at a 20% annual rate and he will sell the stock when he retires. He does not expect the stock to pay a dividend over the next 15 years.

2. He will put aside a fixed amount at the end of each year beginning this year (during his working years) in an IRA which he will withdraw at the end of his working period (ignore taxes). The IRA will pay 10% annual interest.

What annual payment must Humpty make into the IRA account in order to carry out his plans? Please provide a timeline, a description of all of your math, and calculator inputs.

Answer 1

Time till retirement of Humpty- 15 years

Amount required at the time of retirement after 15 years from now comprise of the following:

(a) Present value of College expenses of Jack Horner which is the PV of Annuity Due of $50,000 for 4 years, commencing 20 years after retirement.

(b) Present value of annuity due of $100,000 towards retirement income for 20 years

(a) Amount for college expenses:

Present value of annuity due after 15+20 = 35 years= $ 174,342.60 as follows:

Present value of this amount after at the time of retirement is the discounted value, discounted for 20 years, ascertained at $25,914.92 as follows:

(b) Present value of annuity due comprising retirement income for 20 years, at the time of retirement is $936,492.01 calculated as follows:

Therefore, amount required at the time of retirement= A + b = $25,914.92 + $936,492.01= 962,406.93

Two methods of fund accumulation:

(1) Number of shares of Skyrocket purchased: 1,000

Purchase price per share= $20. Hence total investment= 1,000*$20 = $20,000

Growth rate expected over 15 years till retirement=20% per year

Therefore, sale proceeds expected is the future value of the total investment arrived at $ 308,140.43 as follows

(2) Net amount required to be accumulated in IRA account= $902,406.93-$308,140.43 = $654,266.50

Amount to be paid annually, in order to accumulate the above amount is the ordinary annuity of $20,592.24 as follows: