Answer:
1. The PV of option 1 which is $94,000 is the highest. Therefore, Alex will choose option 1.
2. The fund balance after the last payment is made on December 31, 2030 will be $2,752,446.87.
Explanation:
1. Assuming an interest rate of 7%, determine the present value for the above options.
The option that Alex will choose will be the one that has the highest present value (PV).
We can calculate the present value of each option as follows:
Option 1: $94,000 cash immediately
PV of option 1 = $94,000
Option 2: $38,000 cash immediately and a six-period annuity of $9,700 beginning one year from today
PV of $38,000 cash immediately = $38,000
PV of a six-period annuity of $9,700 beginning one year from today can be determined using the formula for calculating the present value of an ordinary annuity as follows:
PV of $9,700 annuity = P * ((1 - (1 / (1 + r))^n) / r) …………………………………. (1)
Where;
PV = Present value of the $9,700 annual payments today =?
P = Annual payment = $9,700
r = interest rate = 7% = 0.07
n = number of years = 6
Substitute the values into equation (1) to have:
PV of $9,700 annuity = $9,700 * ((1 - (1 / (1 + 0.07))^6) / 0.07)
PV of $9,700 annuity = $9,700 * 4.7665396597641
PV of $9,700 annuity = $46,235.43
Therefore,
PV of option 2 = PV of $38,000 cash immediately + PV of $9,700 annuity = $38,000 + $46,235.43 = $84,235.43
Option 3: a six-period annuity of $19,600 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 $19,600 annual payments today =?
P = Annual payment = $19,600
r = interest rate = 7% = 0.07
n = number of years = 6
Substitute the values into equation (2) to have:
PV of option 3 = $19,600 * ((1 - (1 / (1 + 0.07))^6) / 0.07)
PV of option 3 = $19,600 * 4.7665396597641
PV of option 3 = $93,424.18
Based on the calculations, the PV of option 1 which is $94,000 is the highest. Therefore, Alex will choose option 1.
2. Assuming that the bank account pays 8% interest compounded annually, what will be the fund balance after the last payment is made on December 31, 2030?
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 = $190,000
r = annual interest rate = 8%, or 0.08
n = number of years = 10
Substituting the values into equation (3), we have:
FV = $190,000 * (((1 + 0.08)^10 - 1) / 0.08)
FV = $190,000 * 14.4865624659098
FV = $2,752,446.87
Therefore, the fund balance after the last payment is made on December 31, 2030 will be $2,752,446.87.
