Question

Eileen R Edwards wanted to borrow $27,951 from the bank. The bank responded by saying that she could borrow the money if she repaid $28,951 14 months later. a. What rate would the bank be charging expressed on a per year basis compounded quarterly basis? b. Eileen would like to repay the loan in two instalments. Her first payment would be $20,000 paid after 11 months after borrowing the money, and the second payment would be $89,051. If the bank were to charge her the same interest rate as in the original offer, when would her second repayment have to occur?

Answer 1

Given

Loan amount = $ 27951

Outstanding Balance of loan =$ 28951

Time period = 14 Months

Weknow that 1 Quarter = 3 Months

No.of Quarter in 14 Months = 14/3 =4.66667 Quarters

We know that

Future value = Present value (1+i)^n

Here I = Interest rate per period

n = No.of Payments

$ 28951= $ 27951( 1+i)^4.66667

$ 28951/$ 27951 = (1+i)^4.66667

1.035777= ( 1+i)^4.66667

1.035777^1/4.66667 = 1+i

1.035777^0.21428571= 1+i

1.007561= 1+i

i =0.007561 or 7.561%

Rate of interest per Year = 0.7561*4 = 3.0244% Compounded Quarterly

Computation of timing of Second payment

We know that Present value of the Future Cash flows is equal to the loan amount.

We also know that Present vaue =Future Value / ( 1+i)^n

Here I = Rate of interest per Period

n = No.of Compounding periods

Loan amount = PV of $ 20000+ PV of $ 89051

$ 27951 = $ 20000/ [( 1+0.007561)^11/4] + $ 89051/ [ (1+0.007561)^n

$ 27951= $ 20000/ [ ( 1.007561)^2.75] + $ 89051/ ( 1.007561)^n

$ 27951= $ 20000/ 1.02093057+ $ 89051/ 1.007561^n

$ 27951= $ 19589.9707+ $ 89051/ 1.007561^n

$ 27951-$19589.9707= $ 89051/ 1.007561^n

$ 8361.0293= $ 89051/ 1.007561^n

1.007561^n = $ 89051/$ 8361.0293

1.007561^n = 10.65072

Applying log to base 10 on both sides

log 1.007561^n = log 10.65072

n log 1.007561= log 10.65072

n * 0.003271= 1.027379

n = 1.027379/0.003271

n = 314.0536

No.of Quarters is 314.0536 or 314.0536/4 = 78.51341 Years = 78.51341*12= 942.1609 Months

Hence second payments occurs afrer 942.1609 months from the date of borrowing.

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