A 15-year annuity pays $1,900 per month, and payments are made at the end of each month. If the

A 15-year annuity pays $1,900 per month, and payments are made at the end of each month. If the interest rate is 10 percent compounded monthly for the first seven years, and 6 percent compounded monthly thereafter, what is the present value of the annuity? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

 

Present value

$

Explanation:

This question is asking for the present value of an annuity, but the interest rate changes during the life of the annuity. We need to find the present value of the cash flows for the last eight years first. The PV of these cash flows is:

 

PVA2 = $1,900[{1 – 1 / [1 + (0.06/12)]96} / (0.06/12)] = $144,580.91

Note that this is the PV of this annuity exactly seven years from today. Now we can discount this lump sum to today. The value of this cash flow today is:

PV = $144,580.91 / [1 + (0.10/12)]84 = $72,005.31

   

Now we need to find the PV of the annuity for the first seven years. The value of these cash flows today is:

PVA1 = $1,900 [{1 – 1 / [1 + (0.10/12)]84} / (0.10/12)] = $114,449.67

The value of the cash flows today is the sum of these two cash flows, so:

PV = $72,005.31 + 114,449.67 = $186,454.98

   

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

   

Enter

84

10% / 12

$1,900

N

I/Y

PV

PMT

FV

Solve for

$114,449.67

  

Enter

96

6% / 12

$1,900

N

I/Y

PV

PMT

FV

Solve for

$144,580.91

   

Enter

84

10% / 12

$144,580.91

N

I/Y

PV

PMT

FV

Solve for

$72,005.31

  

$114,449.67 + $72,005.31 = $186,454.98