Question

If you could include work and formulas that would be great, thanks!

A 9-year annuity pays $3,000 per month, and payments are made at the end of each month. The interest rate is 9 percent compounded monthly for the first four 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.) Present valuc References eBook & Resources

Answer 1

Monthly payment = $3,000

For the period after 4 years:

Annual interest rate = 7.00%
Monthly interest rate = 0.5833%

Period = 5 years or 60 months

Value of Payments at the end of Year 4 = $3,000/1.005833 + $3,000/1.005833^2 + … + $3,000/1.005833^59 + $3,000/1.005833^60
Value of Payments at the end of Year 4 = $3,000 * (1 - (1/1.005833)^60) / 0.005833
Value of Payments at the end of Year 4 = $3,000 * 50.502475
Value of Payments at the end of Year 4 = $151,507.425

For the period first 4 years:

Annual interest rate = 9.00%
Monthly interest rate = 0.75%

Period = 4 years or 48 months

Present Value = $3,000/1.0075 + $3,000/1.0075^2 + … + $3,000/1.0075^47 + $3,000/1.0075^48 + $151,507.425/1.0075^48
Present Value = $3,000 * (1 - (1/1.0075)^48) / 0.0075 + $151,507.425/1.0075^48
Present Value = $3,000 * 40.184782 + $151,507.425 * 0.698614
Present Value = $226,399.55

Answer 2

SOLUTION :

r = 9/12 % = 0.75% = 0.0075/per month for first 4 years (n = 48 months).

=> (1 +r) = 1 + 0.0075 = 1.0075 

A = monthly payment = 3000 ($) 

PV of annuity of first 4 years 

= A((1+r)^n - 1) / (r*(1+r)^n) 

= 3000(1.0075^48 - 1) / (0.0075*1.0075^48)

= 120554.35 ($)

After 4 years :

r = 7/12 % = 7/1200 per month

=> (1 + r) = 1207/1200 

n = (9 - 4) = 5 years = 60 months.

A = 3000 ($)

PV (at 4 years end) 

= A((1+r)^n - 1) / (r*(1+r)^n) 

= 3000((1207/1200)^60 - 1) / ((7/1200) * (1207/1200)^60)

= 151505.98  ($)

So, PV at the starting point

= 151505.98 / (1207/1200)^48

= 114598.95 ($)

So, total PV at the beginning

= 120554.35 + 114598.95

= 235153.30 ($) (ANSWER).