(1) Find the present value (one period before the first payment) of an annuity- immediate that lasts five years and pays $3,000 at the end of each month, using a nominal interest rate of 3% convertible monthly. Then repeat the problem using an annual effective discount rate of 3%. Which is higher

(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.

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