A 10-year annuity pays $1,700 per month, and payments are made at the end of each month. If the interest rate is 12 percent compounded monthly for the first five years, and 8 percent compounded monthly thereafter, what is the present value of the annuity?

Present value (PV) of the annuity = $156,988.13

Explanation:

Since the payments are made at the end of each month, the formula for calculating the present value  of an ordinary annuity is the relevant to use as follows:

PV = P × [{1 - [1 ÷ (1+r)]^n} ÷ r] …………………………………. (1)

Where for the first 5 years;

PV = Present value of the payments today =?

P = monthly payment = $1,700

r = monthly interest rate = 12%/12 = 1%, or 0.01

n = number of months = 5* 12 = 60

Substitute the values into equation (1) to have:

PV = 1,700 × [{1 - [1 ÷ (1+0.01)]^60} ÷ 0.01] = $76,423.57  

Where for the last 5 years;

PV = Present value of the payments today =?

P = monthly payment = $1,700

r = monthly interest rate = 8%/12 = 0.67% , or 0.0067

n = number of months = 5* 12 = 60

Substitute the values into equation (1) to have:

PV_5 = 1,700 × [{1 - [1 ÷ (1+0.0067)]^60} ÷ 0.0067] = $83,841.34  

PV after five years is:

PV = $83,841.34 ÷ (1 + 0.0067)^6 = $80,564.57  

PV of the annuity = $76,423.57 + $80,564.57 = $156,988.13

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