Question

A 20-year annuity pays $1,450 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

`\large\boxed{\large\boxed{\$ 167,404.57}}}`

Step-by-step explanation:

1. Present value of the payments of the first 10 years at the interest rate of 11% compounded monthly.

Formula to calculate the present value, PV, of an annuity:

`\large\boxed{PV=C[(1)/(r)-(1)/(r(1+r)^t)]}`

Where:

• C = monthly payment = $1,450

• r = monthly intererst rate = 11% / 12 = 0.11/12

• t = number of moths = 10 × 12 = 120

Substitute and compute

`PV=C[(1)/(r)-(1)/(r(1+r)^t)]\\\\PV=\$ 1,450[(1)/((0.11/12))-(1)/(((0.11/12)(1+0.11/12)^(120))]`

`PV=\$ 105,263.15`

2. Present value of the payments of the second first 10 years at the interest rate of 7% compounded montly

To calculate this, you must calculate the present value of an annuality for 20 years and subtract the value of an annuity for 10 years, both with the interest rate of 7%.

a) Annuity for 20 years

`PV=C[(1)/(r)-(1)/(r(1+r)^t)]\\\\PV=\$ 1,450[(1)/((0.07/12))-(1)/(((0.07/12)(1+0.07/12)^(240))]`

`PV=\$ 187,024.63`

b) Annuity for 10 years

`PV=C[(1)/(r)-(1)/(r(1+r)^t)]\\\\PV=\$ 1,450[(1)/((0.07/12))-(1)/(((0.07/12)(1+0.07/12)^(120))]`

`PV=\$ 124,883.21`

c) Difference

`PV=\$ 187,024.63-\$ 124,883.21=\$ 62,141.42`

3. Total present value

• Total present value = $105,263.15 + $62,141.42

• Total present value = $167,404.57 ← answer