Answer:
1. The PV of option 3 which is $90,000 is the highest. Therefore, Alex will choose option 3 because it has the highset PV.
2. The fund balance after the last payment is made on December 31, 2027 will be approximately $1,934,302.71.
Explanation:
1. Assuming an interest rate of 6%, determine the present value for the above options. Which option should Alex choose?
Alex will choose the option with the highest present value (PV). The present value of each option can be determined as follows:
Option 1: $74,000 cash immediately
PV of option 1 = $74,000
Option 2: $26,000 cash immediately and a six-period annuity of $8,300 beginning one year from today
PV of $26,000 cash immediately = $26,000
PV of a six-period annuity of $8,300 beginning one year from today can be determined using the formula for calculating the present value of an ordinary annuity as follows:
PV of $8,300 annuity = P * ((1 - (1 / (1 + r))^n) / r) …………………………………. (1)
Where;
PV = Present value of the $8,300 annual payments today =?
P = Annual payment = $8,300
r = interest rate = 6% = 0.06
n = number of years = 6
Substitute the values into equation (1) to have:
PV of $8,300 annuity = $8,300 * ((1 - (1 / (1 + 0.06))^6) / 0.06)
PV of $8,300 annuity = $8,300 * 4.9173243260054
PV of $8,300 annuity = $40,813.79
Therefore,
PV of option 2 = PV of $26,000 cash immediately + PV of $8,300 annuity = $26,000 + $40,813.79 = $66,813.79
Option 3: a six-period annuity of $15,000 beginning one year from today
The PV of option 2 can be determined using the formula for calculating the present value of an ordinary annuity as follows:
PV of option 3 = P * ((1 - (1 / (1 + r))^n) / r) …………………………………. (2)
Where;
PV of option 3 = Present value of the $15,000 annual payments today =?
P = Annual payment = $15,000
r = interest rate = 6% = 0.06
n = number of years = 6
Substitute the values into equation (2) to have:
PV of option 3 = $15,000 * ((1 - (1 / (1 + 0.06))^6) / 0.06)
PV of option 3 = $15,000 * 4.9173243260054
PV of option 3 = $90,000
Based on the calculations, the PV of option 3 which is $90,000 is the highest. Therefore, Alex will choose option 3.
2. Assuming that the bank account pays 7% interest compounded annually, what will be the fund balance after the last payment is made on December 31, 2027?
This can be determined using the formula for calculating the Future Value (FV) of an Ordinary Annuity is used as follows:
FV = M * (((1 + r)^n - 1) / r) ................................. (3)
Where,
FV = Future value of the deposits after 10 years =?
M = Annual deposits = $140,000
r = annual interest rate = 7%, or 0.07
n = number of years = 10
Substituting the values into equation (3), we have:
FV = $140,000 * (((1 + 0.07)^10 - 1) / 0.07)
FV = $140,000 * 13.8164479612795
FV = $1,934,302.71
Therefore, the fund balance after the last payment is made on December 31, 2027 will be approximately $1,934,302.71.
