Question

Background Information: Ansel and Harriet W. were a young, highly educated professional couple both employed by one of the leading resort hotels in the area. They were planning on saving for a new house, which they expected to purchase in 7 years. In addition to that financial requirement, they felt that Harriet would quit working at the time to care for their expected family, and that the loss of her income would make them unable to keep up payments on the house without an annuity to supplement his income. The couple felt that they needed $1,500 a year in supplemental income beginning at the end of the 8th year to assist with the house payments, and that they needed this for each of the next 30 years. They also wanted to have $50,000 with which to make the down payment in 7 years when they planned to buy the house. As both were working, they had plenty of funds for savings and were wondering how much they should put away at the end of each of the next 7 years to be able to make the down payment and buy the annuity. Larson felt that an 11 per cent interest rate applied to their situation. PLEASE SHOW WORK & LABEL EACH QUESTION YOU ARE ANSWERING

Question 1: $50,000 FV for down payment on house in 7 years; how much do they need to pay each year for 7 years to have $50,000: 11% rate. Annuity payment x FV annuity factor = FV of annuity. Annuity payment x FV annuity factor = $50,000. Annuity payment = ?

Question 2 - Part A: $1,500 a year income for them (annuity payment) for 30 years; need to first figure out the present value of this future payment stream (annuity); 11% rate. Annuity payment x PV annuity factor = PV of annuity. $1,500 x PV annuity factor = PV of Annuity.

Question 2 - Part B: Once we know the “PV of Annuity” which is what they will need, we must find out what kind of payment they will need to make over the next 7 years so they will have this amount available. Annuity payment x FV annuity factor = FV of annuity. Therefore, how much do they need to pay each year for 7 years to have the “PV of Annuity” amount available at this “future” time?

Question 3: Now, what is the total of these two payments in 1 & 2 above?

Question 4: How much must the couple save each year and deposit at the end of the year at 11% interest to meet their goals?

Answer 1

Need $1,500 from 9th year starting onwards for next 30 years.

Present value at 8th year will be Cash Flows starting from year 9 to year 38

PV(8) = 1500/(1+0.11) + 1500/(1+0.11)^2 + 1500/(1+0.11)^3 + 1500/(1+0.11)^4 + 1500/(1+0.11)^5 + 1500/(1+0.11)^6 + 1500/(1+0.11)^7 + 1500/(1+0.11)^8 + 1500/(1+0.11)^9

Present value of the annuity payments of $1500 at 8th year, PV(8) = $13,040.69

Present value of this cash flow is $13,040.69/(1+0.11)^8 = $5,658.70

Another cash flow requirement after 7 years for down payment of amount = $50,000

PV(down payment) = $50,000/(1+0.11)^7 = $24,082.92

Total amount required to meet both the payments = $24,082.92 + $5,658.70 = $29,741.62

Yearly amount they have to save per annum to meet the above needs as below:

$29,741.62 = CF/(1+0.11) + CF/(1+0.11)^2 + CF/(1+0.11)^3 + CF/(1+0.11)^4 + CF/(1+0.11)^5 + CF/(1+0.11)^6 + CF/(1+0.11)^7

CF = $6,311.63