Question

A 20-year annuity pays $2,100 per month, and payments are made at the end of each month. If the interest rate is 11 percent compounded monthly for the first ten years, and 7 percent compounded monthly thereafter, what is the present value of the annuity? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Answer 1

- Periodic annuity per month = $2100

Interest rate compounded monthly for first 10 years = 11%

Interest rate compounded monthly for last 10 years = 7%

- Calculating the Present Value today of the first 10 years:-

Where, C= Periodic Payments = $2100

r = Periodic Interest rate = 11%/12 = 0.916666%

n= no of periods = 10 years*12 = 120

Present Value = $152,450.08

So, Present Value today of first 10 years = $152,450.08

Now, Calculating the Present value at year end 10 of the periodic payment of the last 10 years :-

Where, C= Periodic Payments = $2100

r = Periodic Interest rate = 7%/12 = 0.5833333%

n= no of periods = 10 years*12 = 120

Present Value = $180,865.34

Present Value at year end 10 of the periodic payment of the last 10 years is $180,865.34

We will now discount present value at year end 10 to present value today using Interest rate of 11% as it is the interest rate in first 10 years:-

Present Value today = Present Value at year end 10/(1+r)^n

r = Periodic Interest rate = 11%/12 = 0.916666%

n= no of periods = 10 years*12 = 120

Present Value today = $180,865.34/(1+0.00916666)^120

Present Value today = $180,865.34/2.98914960064

Present Value today of the last 10 years = $60,507.29

- Total present value of the annuity = $152,450.08 + $60,507.29

Total present value of the annuity = $212,957.37

If you need any clarification, you can ask in comments.    

If you like my answer, then please up-vote as it will be motivating