Question

Albert and Allison borrow exactly the same amount from Liberty Financial. Albert will repay his loan with 20 end of year annual payments. Albert’s first payment will be 1000, and each of his successive payments will be 1000 greater than the one before. Allison will make level payments of 50000 at times T,2T,3T,4T. Both Albert and Allison’s loans are subject to the same annual effective rate of 5%. Determine the time of Allison’s first payment.
(a) 2.7-2.9 years (b) 3-3.2 years (c) 3.5-3.7 years (d) 5-5.2 years (e) 5.5-5.7 years

Answer 1

The amount of loan taken A is given by

A = 1000/1.05+ 2000/1.05^2+3000/1.05^3 ... +20000/1.05^20 (Present value of Albert's payments)

=110950.62 (use Excel)

This should be equal to Present value of Allison's payments

=> 50000/1.05^T+ 50000/1.05^2T+ 50000/1.05^3T+ 50000/1.05^4T = 110950.62

=> 50000/1.05^T* (1-1/1.05^4T)/(1-1/1.05^T) = 110950.62

Using Hit and trial method

Let T= 2, The value of LHS of the equation becomes =157639.34

T= 3, The value of LHS of the equation becomes =140574.97

T= 4, The value of LHS of the equation becomes =125724.54

T= 5, The value of LHS of the equation becomes =112767.30

T = 5.2, The value of LHS of the equation becomes =110378.39

So, the correct value of T lies between 5 and 5.2

So, the first payment has to be made between 5 and 5.2 years (d is the correct option)