ࡱ> BDAf5@ bjbj22 pXX~j.8.t|.**@@###j.l.l.l.l.l.l.$/R1F.]("#@](](.@V.w*w*w*](@:Rzj.w*](j.w*zw**R+<J"z+z `Zz3(j+,.0.r+92y)92z+92z+<#$w*J%%###..DRI*.R There are many ways of borrowing and paying back a sum of money. There may be administration fees, regular instalments, insurance payments or other fees which a borrower is required to pay. The Annual Percentage Rate (APR) provides a way of comparing the costs of different schemes. It is, by law, given in all advertisements for borrowing money. This activity will show you how to calculate the APR in the simplest case where a sum of money is borrowed at a particular time and paid back, with interest, in a single payment at a later date. APR formula for a single repayment You may have met the idea of the present and future value of a sum of money before. The future value after n years is given by FV = PV (1 + r)n where PV is the present value and r is the annual interest rate (written as a decimal). This formula can be rearranged to give: PV =  EMBED Equation.3  Replacing PV by the size of a loan, C, FV by the amount repaid, A, and r by the annual rate, i, gives the following formula.  To find the APR, i, you need to rearrange this formula as shown in the following example: Example A loan of 2000 is repaid by a payment of 2500 made 3 years later.  2000 =  EMBED Equation.3  (  EMBED Equation.3  (  EMBED Equation.3  (  EMBED Equation.3  (  EMBED Equation.3  ( i = 0.07721 The APR is 7.7% (to 1 dp). The arithmetic is more complex when the period of the loan is not an exact number of years. Example A borrower is lent 8750 on 10th December 2007 and agrees to repay this loan by a single repayment of 10 000 on 13th May 2009. The time period from 10th December 2007 to 13th May 2009 is: 1 year and 154 days (22 weeks) = 1 + 154 ( 365 = 1.4219178 years. When working with such difficult numbers, use your calculator's memory or work to 6 decimal places. At the end of the calculation, round the APR to 1 decimal place. 8750 =  EMBED Equation.3  (  EMBED Equation.3  (  EMBED Equation.3  (  EMBED Equation.3  (  EMBED Equation.3  ( i = 0.098 460 The APR is 9.8% (to 1 dp). Some to try: 1 Find the APR for each of the following: a) Loan 1000 repaid by single repayment of 1200 made 2 years later. b) Loan 4000 repaid by single repayment of 6000 made 4 years later. c) Loan 5000 repaid by single repayment of 7200 made 5 years later. d) Loan 12 000 repaid by single repayment of 15 000 made 1 years later. e) Loan 6400 repaid by single repayment of 8000 made 2 years later. f) Loan 25 000 repaid by single repayment of 26 000 made 6 months later. 2 Find the APR for each of the loans described in the table below 3 a) Show that the formula C =  EMBED Equation.3  can be rearranged to give  EMBED Equation.3  b) Use this formula to check your answers to question 1.  Unit Advanced Level, Mathematical Principles for Personal Finance Notes on Activity This activity shows learners how to find the APR of a loan in the simplest case where the loan is repaid in a single instalment. If your students require a calendar for use in question 2, many are available on the internet. The suggestion to work to 6 decimal places and round final answers to 1 decimal place is consistent with the advice given in the document 'Credit Charges and APR' which is available from the Office of Fair Trading ( HYPERLINK "http://www.oft.gov.uk" www.oft.gov.uk). This document also gives examples of how to calculate APR in many more complex cases. Calculating the APR in cases which involve a series of instalments will be covered in another activity. Answers 1 a) 9.5% b) 10.7% c) 7.6% d) 16.0% e) 9.3% f) 8.2% 2 a) 14.4% b) 18.1% c) 12.2% d) 24.5% e) 9.6% f) 10.5% A Resource for Free-standing Mathematics Qualifications APR - Annual Percentage Rate  The Nuffield Foundation  PAGE 3 APR formula for a single repayment C =  EMBED Equation.3  gives the APR, i, as a decimal when a loan of C is paid back by a single repayment A after n years. Note  EMBED Equation.3  =  EMBED Equation.3  Amount borrowedDate of LoanAmount repaidDate of repaymenta)10003rd December 2007115017th December 2008 b)350022nd December 2007420026th January 2009c)560011th January 2008640011th March 2009d)75003rd March 200810 00026th June 2009e)20 0001st April 200824 8002nd August 2010f)25 00016th April 200832 50030th November 2010 Teacher Notes Photo-copiable  EMBED GALLERYClipart    ! & ' ) 2 S b c   ' ( ) . 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