Suppose Jennifer entered into a loan with her bank requiring her to make payment

Question

Suppose Jennifer entered into a loan with her bank requiring her to make payments to extinguish the loan as detailed below.

Time of Repayment Amount of Payment
End of Year 1 $100
End of Year 2 $200
End of Year 3 $300
End of Year 10 $1000

Later Jennifer renegotiated the loan such that she would make a single payment, equal to the sum of four original payments in the table above, at some time T.
(a) Determine an exact solution for the time of the single payment using an annual effective rate of interest of 6% and assuming that this new payment has the same present value as the initial series of payments.
(b) Determine an approximation based on the Method of Equated Time for the time of the single payment.

Explanation / Answer

A.) $1,316.32 * (1.06^t) where t is the number of years later that she would pay. Let's find the Present Value of the payments first: 100 * (1.06^1) = 94.3396226415094 200 * (1.06^2) = 177.999288002848 300 * (1.06^3) = 251.88578490969 1000 * (1.06^4) = 792.09366323802 Sum is $1316.32 So the equivalent single payment is equal to: $1,316.32 * (1.06^t) where t is the number of years later that she would pay. B.) Answer is 3.375 years Method of Equated Time is just the summation of the payments times the year divided by the summation of the payments. 5400/1600

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