Calculating Equal Monthly Repayments

Loans are usually arranged to be repaid at a constant fixed monthly rate. This means that to start with a large fraction of the repayment made each month is used to pay interest. As time progresses the amount used to pay interest decreases and with the last payment the loan is paid off completely.

It is quite tricky to calculate the month repayment. The method is illustrated in this example.

A loan of £500 is taken out at 11% annual interest to be repaid at a fixed monthly rate. Interest is to be charged from the date of the loan and the first repayment is to be made a month after the loan is taken out. Find the monthly repayment.

Call the monthly repayment x. Since the annual rate of interest is 11%, the monthly rate of interest is the solution tocalculating-eqiual-monthly-on-a-loan-html-m237211e2.gif" NAME="Object1" ALIGN=BOTTOM HSPACE=8 WIDTH=291 HEIGHT=32 BORDER=0 >

After a month the amount owing in £ iscalculating-eqiual-monthly-on-a-loan-html-m13ca2f7e.gif" NAME="Object2" ALIGN=BOTTOM HSPACE=8 WIDTH=72 HEIGHT=19 BORDER=0 >and after the first monthly payment the amount owing iscalculating-eqiual-monthly-on-a-loan-html-m6559598e.gif" NAME="Object3" ALIGN=BOTTOM HSPACE=8 WIDTH=97 HEIGHT=19 BORDER=0 >

After two months the amount owing in £ iscalculating-eqiual-monthly-on-a-loan-html-m5174c8df.gif" NAME="Object4" ALIGN=BOTTOM HSPACE=8 WIDTH=289 HEIGHT=21 BORDER=0 >and after the second monthly payment the amount owing iscalculating-eqiual-monthly-on-a-loan-html-ma9d46f9.gif" NAME="Object5" ALIGN=BOTTOM HSPACE=8 WIDTH=174 HEIGHT=21 BORDER=0 >

After three months the amount owing in £ iscalculating-eqiual-monthly-on-a-loan-html-m60a0d429.gif" NAME="Object6" ALIGN=BOTTOM HSPACE=8 WIDTH=429 HEIGHT=21 BORDER=0 >and after the second monthly payment the amount owing iscalculating-eqiual-monthly-on-a-loan-html-m6a1d24b1.gif" NAME="Object7" ALIGN=BOTTOM HSPACE=8 WIDTH=228 HEIGHT=21 BORDER=0 >

We will carry on in the same way, until after two years, twenty four monthly repayments, the whole loan is paid off. After the last monthly payment there is no money left owing so

calculating-eqiual-monthly-on-a-loan-html-69dfdfb.gif" NAME="Object8" ALIGN=BOTTOM HSPACE=8 WIDTH=331 HEIGHT=21 BORDER=0 >calculating-eqiual-monthly-on-a-loan-html-6a22c199.gif" NAME="Object9" ALIGN=BOTTOM HSPACE=8 WIDTH=309 HEIGHT=21 BORDER=0 >

The expression on the right is a geometric series with first termcalculating-eqiual-monthly-on-a-loan-html-32bd34c6.gif" NAME="Object13" ALIGN=BOTTOM HSPACE=8 WIDTH=17 HEIGHT=18 BORDER=0 >and common ratiocalculating-eqiual-monthly-on-a-loan-html-m5a6f836c.gif" NAME="Object14" ALIGN=BOTTOM HSPACE=8 WIDTH=34 HEIGHT=18 BORDER=0 >so the right hand side equalscalculating-eqiual-monthly-on-a-loan-html-210ea9bb.gif" NAME="Object12" ALIGN=BOTTOM HSPACE=8 WIDTH=99 HEIGHT=43 BORDER=0 >and

calculating-eqiual-monthly-on-a-loan-html-m40459d64.gif" NAME="Object10" ALIGN=BOTTOM HSPACE=8 WIDTH=297 HEIGHT=43 BORDER=0 >

Hencecalculating-eqiual-monthly-on-a-loan-html-275c11f4.gif" NAME="Object11" ALIGN=BOTTOM HSPACE=8 WIDTH=221 HEIGHT=45 BORDER=0 >

Substituting the value ofcalculating-eqiual-monthly-on-a-loan-html-13674e44.gif" NAME="Object15" ALIGN=BOTTOM HSPACE=8 WIDTH=98 HEIGHT=32 BORDER=0 >into this expression gives a monthly repayment of £23.18 to the nearest penny. The total amount repaid is £556.41 to the nearest penny.

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