Question

9. A 15-year annuity pays $1,500 per month, and payments are made at the end of each month. If the interest rate is 13% compounded monthly for the first seven years, and 10% compounded monthly thereafter, what is the present value of the annuity? (16 Marks)

Answer 1

Amwan calculation of Present value of annuiety Formula Px -(ltrakel nale First 7 years. x oasi } 1-(1+0:12) - 7x127 0.13 12 $1.500 x 1- (1.010833)-BY 0.010633 251 (SDO Y I 84 (1.010833) 0.010833 #1500 x | - 0.404510 0.010833 #1500*_0.59549 #82454.9986 70 0.010633 For next 8 years. + Xoasi +(+0:00)-8812 18 1.010833) D.10 12 - 96 1500 x - 1500 X 1-(1.008333) O. Moyo 0.008833 1- 0.450835 xo.Moysio 0008333 $ 1500 x 0,519161 x O, MOMS O 0:008333 $ 39987.2916 So, Prevent value of annuity $82464.9986, 4.29,987.2916

Answer 2

SOLUTION :

r = 13/12 %  = 13/1200 /per month for first 7 years (n = 84 months).

=> (1 +r) = 1 + 13/1200 = 1213/1200 

A = monthly payment = 1500 ($) 

PV of annuity of first 7 years 

= A((1+r)^n - 1) / (r*(1+r)^n) 

= 1500((1213/1200)^84 - 1) / ((13/1200)*(1213/1200)^84))

= 82453.99 ($)

After 7 years :

r = 10/12 % = 1/120 per month

=> (1 + r) = 121/120

n = (15 - 7) = 8 years = 96 months.

A = 1500 ($)

PV (at 7 years end) 

= A((1+r)^n - 1) / (r*(1+r)^n) 

= 1500((121/120)^86 - 1) / ((1/120) * (121/120)^96)

= 84517.42  ($)

So, PV at the starting point

= 84517.42 / (121/120)^84

= 42092.02 ($)

So, total PV at the beginning

= 82453.99 + 42092.02

= 124546.01 ($) (ANSWER).