{"id":560,"date":"2020-04-30T20:00:13","date_gmt":"2020-05-01T00:00:13","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&p=560"},"modified":"2024-10-29T14:19:40","modified_gmt":"2024-10-29T18:19:40","slug":"fractional-periods","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/fractional-periods\/","title":{"raw":"4.9 Fractional Periods","rendered":"4.9 Fractional Periods"},"content":{"raw":" \r\n\r\nOccasionally it may be necessary to deal with compound interest for a fraction of a period, for example taking money out of a savings account after two weeks.\u00a0 In these cases, it is important to understand what the policy of the bank or lender is.\r\n\r\nFor a savings account that pays interest monthly you may still receive interest for the two weeks (at the end of the month) if they pay interest on the daily balance but pay it monthly.\u00a0 In other words, some banks calculate interest based on the equivalent daily rate (j<\/em>365<\/sub>) , but pay monthly. \u00a0The interest rate paid depends on the total daily closing balance. Interest rate is applied to the entire balance, calculated daily, and paid monthly.\r\n\r\nHowever, for GICs if you cash them in early you may get no interest or if it is a redeemable GIC you will get interest for the time that you had it.\u00a0 So if you have a 3-year cashable GIC that pays interest annually and you cash it in after 10 months it is unclear whether they will use simple interest or fractional exponents.\u00a0 We reached out to a bank to ask -\u00a0 the head office of TD Bank has stated that they had no idea, they said the \u201ccomputer calculates it\u201d.\r\n\r\nFor such cases, unless otherwise stated, use the compound-interest formula:\r\n

[latex]FV = PV(1+i)^n[\/latex]<\/p>\r\nwith n having a fractional part. The following example justifies this procedure.\r\n

Example<\/h2>\r\nSuppose that $1,000 was borrowed at an interest rate of 10% compounded annually, and originally was scheduled to be repaid after two years. Instead, it was decided to repay the loan after 1.5 years.\r\n
    \r\n \t
  1. Both parties have agreed to have interest calculated for the partial period<\/strong>.\u00a0 At that time, by the compound-interest formula, the amount to be repaid would be:\r\n[latex]FV=$1000(1.10)^{1.5}=$1,000(1.15369\u2026)=$1,153.69[\/latex].\u00a0 Now suppose this amount were to be reinvested at the same rate for the remaining half year. In this case, the money accumulated would be:[latex]FV=$1,153.69(1.10)^{0.5}=$1,210.00[\/latex], which is exactly what would have resulted from the original two \u00adyear loan:[latex]FV =$1,000(1.10)^2=$1,210.00[\/latex]<\/li>\r\n \t
  2. Both parties have not agreed to have interest calculated for the partial period.<\/strong>\u00a0 In this case, we would round down, so $n = 1$, and we receive no interest for the final year.[latex]FV =$1,000(1.10)^1=$1,100.00[\/latex].\u00a0 In this case, we get a lot less interest!<\/li>\r\n<\/ol>\r\n \r\n\r\nIn fact, part 1 gives us a general property of compound interest: If the balance is found at any time and reinvested <\/em>at the same rate, nothing changes. You may find that some financial institutions prefer to deal with the fractional period by assuming that simple interest is paid for that portion. At one time this method was popular because of the difficulty of performing the calculations for the formula without calculators or computers. It also meant that the institutions would receive more money since, for a partial period, simple interest is slightly higher than compound interest.\r\n\r\n \r\n
    \r\n

    Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

    General property of compound interest: If the balance is found at any time and reinvested at the same rate, nothing changes.<\/div>\r\n<\/div>\r\n \r\n
    \r\n

    Knowledge Check 4.4<\/p>\r\n\r\n<\/header>\r\n

    \r\n\r\nA loan for $6,000 will be taken out for four years at 14% compounded semi-annually. However, it is decided that the money should be repaid after three years and two months.\r\n
      \r\n \t
    1. Find the accumulated amount to be repaid.<\/li>\r\n \t
    2. Check to see that reinvesting this amount for the remaining 10 months would produce the same amount as the original four-year loan.<\/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