Question

A loan is negotiated with the lender agreeing to accept $8,000 after one year, $9,000 after two years, and $20,000 after four years in full repayment Of the loan. The loan is negotiated so that the borrower makes a single payment Of $37,000 at time T and this results in the same total present value Of payments when calculated using an annual effective rate of 5%. Estimate T using the method of equated time. Also find T exactly.

Answer 1

Taking the present time as t= 0, the scheduled loan repayments is,

0 ---->

1 -----> $8,000

2 -----> $9,000

3 ----->

4 -----> $20,000

According to the method of equated time,Tis a dollar-weighted average of thepayment times; that is, with time measured in years,

T = ( $8,000 /$37,000)1 + ( $9,000 /$37,000)2 + ( $20,000 /$37,000)4

= ( 106,000 /37,000 ) ≈ 2.864864865 ≈ 2.86487.

The equation of value at time t= 0 is,

`(37,000v)^(t)`=
`8,000v + 9,000v^(2) + 20,000v^(4)`

wherev= (1.05)
`^(-1)`, Solving for T gives:

T =
`ln(((8)/(37)v + (9)/(37)v^(2) + (20)/(37)v^(4))/(ln(v))`

=2.82480766.