{"id":506,"date":"2020-04-28T13:17:40","date_gmt":"2020-04-28T17:17:40","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&p=506"},"modified":"2024-10-29T14:18:59","modified_gmt":"2024-10-29T18:18:59","slug":"compound-interest-with-the-baii-plus","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/compound-interest-with-the-baii-plus\/","title":{"raw":"4.7 Compound Interest with the BAII Plus","rendered":"4.7 Compound Interest with the BAII Plus"},"content":{"raw":"

Using Financial Calculator Functions<\/h2>\r\n\"DEcorative\r\n\r\nThe financial calculator recommended for this course is the BAII Plus.\u00a0 Both this and other financial calculators have built-in compound-interest functions. It is possible to do almost all of the course calculations to the same accuracy without these functions, but the process is much faster if they are available.\r\n\r\nThe functions you will use in this chapter are controlled by the following keys:\r\n\r\n \r\n\r\n\r\n\r\n\r\n\r\n
P\/Y and C\/Y<\/strong><\/td>\r\nN<\/strong><\/td>\r\nI\/Y<\/strong><\/td>\r\nPV<\/strong><\/td>\r\nPMT<\/strong><\/td>\r\nFV<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
How many times do we compound per year?(m)<\/em><\/td>\r\nNumber of periods<\/em><\/td>\r\nNominal Interest Rate, jm<\/sub><\/em><\/td>\r\nPresent Value<\/em><\/td>\r\n0<\/em>\r\n\r\n(for now)<\/em><\/td>\r\nFuture Value<\/em>\r\n\r\n(One of PV and FV is negative!)<\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n
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Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

Financial Calculators should have built-in compound-interest functions.<\/div>\r\n<\/div>\r\n\u00a0<\/em>\r\n\r\n<\/div>\r\nIn the same row is the PMT key which you will use in the next chapter. For this chapter, the PMT value should be set at 0.\u00a0 It\u2019s always best practice to set it to 0 each and every time!\r\n

Example 4.7.1<\/h2>\r\nInvest $100 at j2<\/sub><\/em> =6%\u00a0 for 4 years. N = 2\u00d7 4 = 8 periods.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Step<\/strong><\/td>\r\nTo<\/strong><\/td>\r\nPress<\/strong><\/td>\r\nDisplay<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
1<\/td>\r\nClear previous saved values\r\n\r\n(except P\/Y<\/em> and C\/Y<\/em>)<\/td>\r\n[2ND] [CLR TVM]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
2<\/td>\r\nEnter\u00a0 N=8 <\/em>periods<\/td>\r\n[N] [8]<\/td>\r\n \r\n\r\nN = 8<\/td>\r\n<\/tr>\r\n
3<\/td>\r\nEnter nominal interest rate, I\/Y <\/em>= 6%. (Annual interest rate in percentage)<\/em><\/td>\r\n[I\/Y][6]<\/td>\r\nI\/Y = 6<\/td>\r\n<\/tr>\r\n
4<\/td>\r\nSelect P\/Y<\/em> and C\/Y<\/em> worksheet<\/td>\r\n[2ND] [P\/Y]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
5<\/td>\r\nSet number of payments per year, P\/Y<\/em> = 2\r\n\r\n\u00a0<\/em><\/td>\r\n[ENTER] [2]<\/td>\r\nP\/Y = 2<\/td>\r\n<\/tr>\r\n
6<\/td>\r\nSet Number of compounding periods per year, C\/Y<\/em>=2\r\n\r\n(By default, C\/Y is set as the same as P\/Y)<\/em><\/td>\r\n[\u2193] [2] [ENTER]<\/td>\r\nC\/Y = 2<\/td>\r\n<\/tr>\r\n
7<\/td>\r\nReturn to standard calculator mode<\/td>\r\n[2ND] [QUIT]<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
8<\/td>\r\nEnter present value, PV<\/em> =100\r\n\r\n\u00a0<\/em><\/td>\r\n[1][0][0][\u00b1][PV]<\/td>\r\nPV = 100<\/td>\r\n<\/tr>\r\n
9<\/td>\r\nEnter periodic payment, PMT<\/em> =0<\/td>\r\n[0][PMT]<\/td>\r\nPMT = 0<\/td>\r\n<\/tr>\r\n
10<\/td>\r\nCompute future value, FV<\/em>\r\n\r\n(positive value for inflow)<\/em><\/td>\r\n[CPT][FV]<\/td>\r\nFV = -126.6770081<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nWe write this as:\r\n\r\n\r\n\r\n\r\n\r\n
<\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
<\/td>\r\n4<\/td>\r\n<\/td>\r\n4\u00d72=8<\/td>\r\n6<\/td>\r\n+100<\/td>\r\n0<\/td>\r\nCPT: -125.6770<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nLeaving some spaces for Annuities, in Chapter 5.\r\n\r\n \r\n

Example 4.7.2<\/h2>\r\nTo illustrate the use of the financial calculator, suppose you want to obtain the future value of a $5,000 loan at 8% compounded semi-annually for two years.\r\n\r\n\r\n\r\n\r\n\r\n
<\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
<\/td>\r\n2<\/td>\r\n<\/td>\r\n2\u00d72=8<\/td>\r\n8<\/td>\r\n5,000<\/td>\r\n0<\/td>\r\nCPT<\/strong>: -5,849.29<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nYou will see the answer, $5,849.29, which was obtained earlier in the chapter by an account and by the formula. Note that the answer appears as a negative <\/em>value on the calculator. This is because the calculator performs an equation of value <\/em>in the form of:\r\n

[latex]\\text{Value of Inflows}+\\text{Value of Outflows}=0[\/latex]<\/p>\r\nHence it must make either inflows or outflows negative. (Since PV was made positive, it must make FV negative.)\r\n\r\nFrom now on, you will normally indicate the procedure for solving problems - especially if they are likely to be done with computer functions - by listing the available values of the variables and what is required.\r\n\r\nThe answer would be negative on the calculator, but this will be mentioned only if confusion may arise from the answer.\r\n\r\nWith the calculator functions, any one of the functions N, I\/Y, PV, or FV can be found from the others. How this is done is illustrated in the next example, which uses some previous problems.\r\n\r\nThe calculator assumes each problem has a cash outflow (entered as a negative) and a cash inflow (entered as a positive).\u00a0 For simplicity, we will always show PV as positive, and FV as negative.\r\n

Example 4.7.3<\/h2>\r\nYou borrow $1,000 and agree to repay the loan with a single payment in 2 years. How much should you pay if interest is charged at 8% compounded quarterly?\r\n\r\n\r\n\r\n\r\n\r\n
<\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
<\/td>\r\n4<\/td>\r\n<\/td>\r\n4\u00d72=8<\/td>\r\n8<\/td>\r\n1,000<\/td>\r\n0<\/td>\r\nCPT<\/strong>: -1,171.66<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nTo look at values entered in your calculator, just press [RCL] and then the value you want to check, e.g., [RCL] [N] should show 8.\r\n

Example 4.7.4<\/h2>\r\nIf an invested $8,000 results in a future value of $8,998.91 in nine months, what is the interest rate compounded quarterly?\r\n\r\nYou have:\r\n\r\n\r\n\r\n\r\n\r\n
<\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
<\/td>\r\n4<\/td>\r\n<\/td>\r\n4\u00d7 9\/12 =3<\/td>\r\nCPT<\/strong><\/td>\r\n8,000<\/td>\r\n0<\/td>\r\n-8,998.91<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAnswer: 16% compounded quarterly.\r\n\r\n \r\n\r\nAlternatively, you could solve the algebra problem:\r\n

[latex]$8,000(1+\\frac{j_m}{4})^3=$8,998.91[\/latex]<\/p>\r\nWhich simplifies to:\r\n

[latex]j_m=4\\left(\\sqrt[3]{(\\frac{FV}{PV})-1)}\\right)=4\\left(( \\frac{FV}{PV})^{1\/3}-1\\right)[\/latex]<\/p>\r\nBut this is a much tougher problem!\r\n\r\n \r\n

Example 4.7.5<\/h2>\r\n \r\n\r\nIf $150,000 is invested at 12% compounded monthly and results in a future value of $169,023.75, for how long must it have been invested?\r\n\r\n\r\n\r\n\r\n\r\n
<\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
<\/td>\r\n12<\/td>\r\n<\/td>\r\nCPT<\/strong><\/td>\r\n12<\/td>\r\n150,000<\/td>\r\n0<\/td>\r\n-169,023.75<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nAnswer:\u00a0 11.9999973 or 12 months.\r\n\r\n \r\n\r\nAlternatively, we could solve the algebra problem:\r\n

[latex]$150,000\\left(1+\\frac{0.12}{12}\\right)^n=$169,023.75[\/latex]<\/p>\r\nWhich simplifies, using logarithms to:\r\n

[latex]n=\\log_{1.01} \\left(\\frac{$169,023.75}{$150,000}\\right)[\/latex]<\/p>\r\nIn general, the calculator is a very good option \u2013 you do not need to use logarithms, and can solve much faster.\r\n\r\n \r\n

\r\n

Knowledge Check 4.5<\/p>\r\n\r\n<\/header>\r\n

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    \r\n \t
  1. Find the future value of a loan of $12,000 for 16 months at 15% compounded monthly. In doing this, you should write down the values entered into the TVM:<\/li>\r\n<\/ol>\r\n
    <\/div>\r\n\r\n\r\n\r\n\r\n\r\n
    <\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    <\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
      \r\n \t
    1. How much must be invested at 11% quarterly to get $9,500 in two years?\r\n\r\n\r\n\r\n\r\n\r\n
      <\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      <\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t
    2. If a bank deposit of $80,000 amounts to $84,934.22 after gaining interest compounded monthly for one year, what was the nominal rate per month?\r\n\r\n\r\n\r\n\r\n\r\n
      <\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      <\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\nSolutions at the end of the chapter<\/a>\r\n\r\n<\/div>\r\n<\/div>\r\n \r\n

      Your Own Notes<\/h2>\r\n