Answer:
1. Present value on December 31, 2018 = $18,023.88
2. Present value on December 31, 2018 = $20,186.75
3. Present value on December 31, 2018 = $17,780.59
Explanation:
1. The first payment is received on December 31, 2019, and interest is compounded annually.
This is an example of ordinary annuity. Therefore, the present value on December 31, 2018 can be calculated using the formula for calculating the present value of an ordinary annuity as follows:
PV = P * ((1 - (1 / (1 + r))^n) / r) …………………………………. (1)
Where;
PV = present value on December 31, 2018 = ?
P = Annual annuity = $5,000
r = Annual interest rate = 12%, or 0.12
n = number of years = 5
Substitute the values into equation (1), we have:
PV = $5,000 * ((1 - (1 / (1 + 0.12))^5) / 0.12)
PV = $5,000 * 3.60477620234501
PV = $18,023.88
2. The first payment is received on December 31, 2018, and interest is compounded annually.
This is an example of annuity due. Therefore, the present value on December 31, 2018 can be calculated using the formula for calculating the present value of an annuity due as follows:
PV = P * ((1 - [1 / (1+r))^n) / r) * (1+r) .................................. (2)
Where;
Where;
PV = present value on December 31, 2018 = ?
P = Annual annuity = $5,000
r = Annual interest rate = 12%, or 0.12
n = number of years = 5
Substitute the values into equation (1), we have:
PV = $5,000 * ((1 - [1 / (1+0.12))^5) / 0.12) * (1+0.12)
PV = $5,000 * 3.60477620234501 * 1.12
PV = $5,000 * 4.03734934662641
PV = $20,186.75
3. The first payment is received on December 31, 2019, and interest is compounded quarterly.
Note: See the calculation of the present value on December 31, 2018 in the attached excel file.
This is also an example of ordinary annuity.
In the attached excel file, the following formula is used:
Discounting factor = 1 / (1 + r)^n .............. (1)
Where;
r = Quarterly interest rate = Annual interest rate / Number of quarters in a year = 12% / 4 = 0.12 / 4 = 0.03
n = number of quarters = number of years * Number of quarters in a year
From the attached excel file, we have:
Present value on December 31, 2018 = Total present value = $17,780.59
