Part 1: Foster owns a small business and has received an interesting offer from
ID: 1164678 • Letter: P
Question
Part 1:
Foster owns a small business and has received an interesting offer from Sapphire Bank. The bank has offered to pay him a new customer incentive of $151 at the end of each year, with the amount increasing by 7% each subsequent year, as long as he continues to use the bank for all of his business needs. The bank will make the payments to him for 6 years. In addition, the bank will pay him a bonus of $283 at the end of year 2.
His current bank is trying to retain his business by offering to immediately give him a toaster and season tickets to the local sports team if he commits to staying with the bank for 5 more years.
Foster is trying to compare the two offers in terms of monetary value. Calculate the present value of the payments and the bonus from Sapphire Bank using an interest rate of 7% compounded annually. (In other words, how much are all of the payments worth today?)
Note: Foster would receive the first payment from Sapphire Bank at the end of year 1.
Part 2:
Foster deposited $420728 into an account paying 5% interest compounded annually with the goal of letting the money accrue interest until the end of year 32. Unfortunately, Foster had a small health crisis at the end of year 10 and had to use the money in the account for several years to help with expenses. Foster made an annual withdrawal of $41167 at the end of each year for 5 years. Foster's health improved and he was able to let the remainder of the money accrue interest as he had originally planned. After the emergency withdrawals, how much was left in the account at the end of the 32 years?
Note: Foster makes the first withdrawal at the end of year 10. He made a total of 5 withdrawals.
Explanation / Answer
Part 1: The offer from Sapphire bank is:
From above, we can easily calculate total sum of payments.
What we need to do is: Customer incentive + 7% of customer incentive = Total payment for that year. Then, Total payment for previous year + 7% of total payment of previous year and so on. Therefore:
Therefore, Total payment = $597.54
Part 2: Total amount Foster deposited into account = $420728 at 5% interest compounded per year.
Using the same method as above the amount he had in bank at the end of 10th year = $692,943
Total amount withdrawed during 5 years = $41167 x 5 = $205,835
Amount left in his account after 15 years = $692,943 - $205,835 = $487,108.
Now we need to calculate interest of $487,108 from 16th to 32nd year, in other words we need to calculate interest for 32-15= 17 years for the sum of $487,108.
Using the same method the total amount left in his account after 32 years = $1,137,649
Note: In order to solve part 2 faster, the compound interest formula can also be used which is A = P (1 + r/n) ^ nt.
For this formula, P is the principal amount, r is the rate of interest per annum, n denotes the number of times in a year the interest gets compounded, and t denotes the number of years.