Answer:
(a) The present value using a nominal interest rate of 3% convertible monthly is $166,785.44.
(b) the present value using an annual effective discount rate of 3% is $166,957.07.
(c) The present value of $166,957.07 obtained using an annual effective discount rate of 3% is higher than the present value of $166,957.07 obtained using a nominal interest rate of 3% convertible monthly is $166,785.44.
Explanation:
(a) Find the present value using a nominal interest rate of 3% convertible monthly.
To find this, we first convert the nominal interest to effective annual discount rate using the following formula:
EAR = ((1 + (i / n))^n - 1 .............................(1)
Where;
EAR = Effective annual discount rate = ?
i = Stated nominal interest rate = 3%, or 0.03
n = Number of compounding periods or months = 12
Substituting the values into equation (1), we have:
EAR = ((1 + (0.03 / 12))^12) - 1 = 0.0304159569135067
Since an annuity-immediate is also known as an ordinary annuity, the present value can now 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 = ?
P = Monthly payment = $3,000
r = Monthly EAR = 0.0304159569135067 / 12 = 0.00253466307612556
n = number of months = 5 years * 12 months = 60
Substitute the values into equation (1) to have:
PV = $3,000 * ((1 - (1 / (1 + 0.00253466307612556))^60) / 0.00253466307612556)
PV = $3,000 * 55.595148258086
PV = $166,785.444774258
Approximating to 2 decimal places, we have:
PV = $166,785.44
Therefore, the present value using a nominal interest rate of 3% convertible monthly is $166,785.44.
(b) Then repeat the problem using an annual effective discount rate of 3%.
Since this already an annual effective discount rate, there is no need for any conversion here.
As this is also an annuity-immediate which is also known as an ordinary annuity, the present value can aslso 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 = ?
P = Monthly payment = $3,000
r = Monthly annual effective discount rate = 3% / 12 = 0.03 / 12 = 0.0025
n = number of months = 5 years * 12 months = 60
Substitute the values into equation (1) to have:
PV = $3,000 * ((1 - (1 / (1 + 0.0025))^60) / 0.0025)
PV = $3,000 * 55.6523576868044
PV = 166,957.073060413
Approximating to 2 decimal places, we have:
PV = $166,957.07
Therefore, the present value using an annual effective discount rate of 3% is $166,957.07.
c. Which is higher?
From part a and b above, we have:
Present value using a nominal interest rate of 3% convertible monthly = $166,785.44.
Present value using an annual effective discount rate of 3% = $166,957.07
Based on these, the present value of $166,957.07 obtained using an annual effective discount rate of 3% is higher than the present value of $166,957.07 obtained using a nominal interest rate of 3% convertible monthly is $166,785.44.
