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 years from 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

Answer: We have 3 timelines available in this scenario:

Timelines:

1. Humpty has 15 Yrs to retire.

2. 20 Yrs to his death

2. after death 4 years for Harvard study for his son

Lets first try to calculate the funds needed to be put aside for the Harvard education after his death

PV(at start of education of Jack) = Annuity * PVAF(10%,3)

we have taken 3 years reason being for the first year directly 50000 will be paid and for the rest 3 10% compounding will happen

PVAF (10%,3 ) = 2.4869 from the PCAF table

Annuity amount at the starting of each year = 50000

PV( at start of education) = 50000*2.4869

PV = 124345

This is for the next 3 years but in the starting of first year 50000 needs to be paid as well.

So, total amount Humpty needs to have on his death for jacks education= 124345 + 50000 = $174345

Now, We will have to equate the Present value of the total amount Humpty would get in his 20 yrs of retired life to the future value of the investments done by him in the 15 years of his work life.

Equate timeline has been taken his retirement year

FV (15yrs work life) = (1000 shares * $20) *(1+ interest rate ) ^ (no of years) + Amount of IRA after 15 yrs at 10%

FV ( 15 Yrs work life) = 20,000 * (1+ 0.2) ^(15) + Annuity amount of IRA * CVAF (10%,15)

FV( 15 yrs work life ) = 20000 * 15.4 + annuity amount * 31.7725

FV( 15 Yrs of work life ) = $308140.4 + annuity amount * 31.7725

PV( of retirement yrs annuity payments) = annuity amount * PVAF( 10%, 19) + first year annuity payment of Humpty + College education discounted amount

Again here Humpty will get 100000 at the starting only and will put the rest amount he will get from his retirement in to account at 10%

Both the equations should be equal as they both are of the same time

308140 + annuity amount *31.7725 = 100,000 * 8.3649 + 100,000 + 174,345 /(1+r ) ^ ( no of years)

308140 + annuity amount *31.7725 = 836490 + 100,000 + 174345 /(1.1)^20

On Solving this you will get

308140 + annuity amount * 31.7725 = 8,36,490 + 100,000 + 25915.27

Annuity amount * 31.7725 = 654264.8

Annuity amount of payment in IRA Humpty needs to do is = $20,592.17