Question

A 20-year annuity pays $2,250 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?

Answer 1

First we will calculate PV of annuity of first 10 years

Here

PMT= $2250

rate = 11/12= 0.917% as it is compounded monthly

N= 10*12= 120 as it is compounded monthly

PV= PMT*((1-(1+r)^-n)/r)

= 2250*((1-1.00917^-120)*0.00917)

=2250*((1-0.3344)/0.00917)

=2250*(0.6656/0.00917)

=$163311.75

Now we will calculate the value of annuity in 10th year for last 10 years

Here rate will be 7/12=0.583%

PV in 10th year = PMT*((1-(1+r)^-n)/r)

= 2250*((1-1.00583^-120)*0.00583)

=2250*((1-0.4978)/0.00583)

=2250*(0.5022/0.00583)

=$193818.71

Now we need to discount it back to present day

PV =FV/(1+r)^n

=193818.71/(1.00917^1200

193818.71/2.9903

=$64815.056

Present value will be $163311.75+$64815.056= $228126.81