{"id":585,"date":"2020-05-01T12:01:37","date_gmt":"2020-05-01T16:01:37","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&p=585"},"modified":"2021-07-12T16:12:52","modified_gmt":"2021-07-12T20:12:52","slug":"average-rates","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/average-rates\/","title":{"raw":"4.10 Average Rates","rendered":"4.10 Average Rates"},"content":{"raw":"Often, interest rates change over the life of an investment.\u00a0 In this case, we often try to find the average rate.\u00a0 To illustrate, here is an example:\r\n

Example 4.10.1<\/h2>\r\n7 years ago, Warren\u2019s grandmother gave each of her grandchildren different amounts of money. \u00a0Warren invested $4,000 of his money with CJC investments.\u00a0 For the first year, CJC put his money in a GIC which paid an effective rate of 5.9%.\u00a0 For the next 2 years, CJC moved the money into Canada Savings Bonds, which paid out a semi-annual rate of 7.8%.\u00a0 To capitalize on the markets, CJC moved the money into a mutual fund, where it earned an average monthly interest rate of 4.8% for three years.\u00a0 For the final year, the money was put it into stocks which paid out 5.6% compounded quarterly.\u00a0 How much money does Warren have today?\r\n\r\nTo solve this, we will take the Present value, and find its value at each point:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
YEAR<\/strong><\/td>\r\nP\/Y\r\n<\/strong><\/td>\r\nC\/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
1<\/td>\r\n1<\/td>\r\n<\/td>\r\n1<\/td>\r\n5.9<\/td>\r\n4,000<\/td>\r\n0<\/td>\r\nCPT: -4236<\/td>\r\n<\/tr>\r\n
2-3<\/td>\r\n2<\/td>\r\n<\/td>\r\n2 \u00d72 = 4<\/td>\r\n7.8<\/td>\r\n4236<\/td>\r\n0<\/td>\r\nCPT: -4936.488637...<\/td>\r\n<\/tr>\r\n
4-6<\/td>\r\n12<\/td>\r\n<\/td>\r\n12\u00d7 3=36<\/td>\r\n4.8<\/td>\r\n4936.488637...<\/td>\r\n0<\/td>\r\nCPT: -5699.43497...<\/td>\r\n<\/tr>\r\n
7<\/td>\r\n4<\/td>\r\n<\/td>\r\n4<\/td>\r\n5.6<\/td>\r\n-5699.43497...<\/td>\r\n0<\/td>\r\nCPT: 6025.36864...<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can also look at this algebraically:\r\n

[latex]FV = $4,000(1.059)^1(1.039)^4(1.004)^{36}(1.014)^4= $6025.36864...[\/latex]<\/p>\r\nSo Warren has $6,025.37.\r\n\r\n \r\n\r\nNow we can work out what Warren's average effective rate is.\u00a0 We take the PV and FV, and use the calculator to find out what effective rate will turn $4,000 to $6025.37 in 7 years.\u00a0 Make sure to use your memory buttons to store all of the decimal points.\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\n1<\/td>\r\n<\/td>\r\n7<\/td>\r\nCPT<\/td>\r\n4,000<\/td>\r\n0<\/td>\r\n-6025.36864..<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe get an effective rate of j1<\/sub><\/em> = 6.027290102%.\u00a0 This is Warren's average rate.\r\n\r\nOnce we have the effective average rate, we can use it to solve other problems:\r\n

Example 4.10.2<\/h2>\r\n
Amy also invested with CJC investments, who invested her money in the same manner.\u00a0 If Amy now has $8150, how much did she invest 7 years ago with CJC?<\/div>\r\n
To solve this, we can use the average rate already computed:<\/div>\r\n
<\/div>\r\n
\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\n1<\/td>\r\n<\/td>\r\n7<\/td>\r\n6.027290102<\/td>\r\nCPT<\/td>\r\n0<\/td>\r\n-8,150<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
\r\n\r\nWhich tells us that Amy invested $5,410.46, 7 years ago.\r\n\r\n<\/div>\r\n

Your Own Notes<\/h2>\r\n
    \r\n \t
  • Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t
  • These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t
  • Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"1\"]","rendered":"

    Often, interest rates change over the life of an investment.\u00a0 In this case, we often try to find the average rate.\u00a0 To illustrate, here is an example:<\/p>\n

    Example 4.10.1<\/h2>\n

    7 years ago, Warren\u2019s grandmother gave each of her grandchildren different amounts of money. \u00a0Warren invested $4,000 of his money with CJC investments.\u00a0 For the first year, CJC put his money in a GIC which paid an effective rate of 5.9%.\u00a0 For the next 2 years, CJC moved the money into Canada Savings Bonds, which paid out a semi-annual rate of 7.8%.\u00a0 To capitalize on the markets, CJC moved the money into a mutual fund, where it earned an average monthly interest rate of 4.8% for three years.\u00a0 For the final year, the money was put it into stocks which paid out 5.6% compounded quarterly.\u00a0 How much money does Warren have today?<\/p>\n

    To solve this, we will take the Present value, and find its value at each point:<\/p>\n\n\n\n\n\n\n\n\n
    YEAR<\/strong><\/td>\nP\/Y
    \n<\/strong><\/td>\n    		C\/Y<\/strong><\/td>\nN<\/strong><\/td>\nI\/Y<\/strong><\/td>\nPV<\/strong><\/td>\nPMT<\/strong><\/td>\nFV<\/strong><\/td>\n<\/tr>\n<\/thead>\n
    1<\/td>\n1<\/td>\n<\/td>\n1<\/td>\n5.9<\/td>\n4,000<\/td>\n0<\/td>\nCPT: -4236<\/td>\n<\/tr>\n
    2-3<\/td>\n2<\/td>\n<\/td>\n2 \u00d72 = 4<\/td>\n7.8<\/td>\n4236<\/td>\n0<\/td>\nCPT: -4936.488637…<\/td>\n<\/tr>\n
    4-6<\/td>\n12<\/td>\n<\/td>\n12\u00d7 3=36<\/td>\n4.8<\/td>\n4936.488637…<\/td>\n0<\/td>\nCPT: -5699.43497…<\/td>\n<\/tr>\n
    7<\/td>\n4<\/td>\n<\/td>\n4<\/td>\n5.6<\/td>\n-5699.43497…<\/td>\n0<\/td>\nCPT: 6025.36864…<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    We can also look at this algebraically:<\/p>\n

    [latex]FV = $4,000(1.059)^1(1.039)^4(1.004)^{36}(1.014)^4= $6025.36864...[\/latex]<\/p>\n

    So Warren has $6,025.37.<\/p>\n

     <\/p>\n

    Now we can work out what Warren’s average effective rate is.\u00a0 We take the PV and FV, and use the calculator to find out what effective rate will turn $4,000 to $6025.37 in 7 years.\u00a0 Make sure to use your memory buttons to store all of the decimal points.<\/p>\n\n\n\n\n\n
    <\/th>\nP\/Y<\/th>\nC\/Y<\/th>\nN<\/th>\nI\/Y<\/th>\nPV<\/th>\nPMT<\/th>\nFV<\/th>\n<\/tr>\n<\/thead>\n
    <\/td>\n1<\/td>\n<\/td>\n7<\/td>\nCPT<\/td>\n4,000<\/td>\n0<\/td>\n-6025.36864..<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    We get an effective rate of j1<\/sub><\/em> = 6.027290102%.\u00a0 This is Warren’s average rate.<\/p>\n

    Once we have the effective average rate, we can use it to solve other problems:<\/p>\n

    Example 4.10.2<\/h2>\n
    Amy also invested with CJC investments, who invested her money in the same manner.\u00a0 If Amy now has $8150, how much did she invest 7 years ago with CJC?<\/div>\n
    To solve this, we can use the average rate already computed:<\/div>\n
    <\/div>\n
    \n\n\n\n\n\n
    <\/th>\nP\/Y<\/th>\nC\/Y<\/th>\nN<\/th>\nI\/Y<\/th>\nPV<\/th>\nPMT<\/th>\nFV<\/th>\n<\/tr>\n<\/thead>\n
    <\/td>\n1<\/td>\n<\/td>\n7<\/td>\n6.027290102<\/td>\nCPT<\/td>\n0<\/td>\n-8,150<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
    \n

    Which tells us that Amy invested $5,410.46, 7 years ago.<\/p>\n<\/div>\n

    Your Own Notes<\/h2>\n
      \n
    • Are there any notes you want to take from this section? Is there anything you’d like to copy and paste below?<\/li>\n
    • These notes are for you only (they will not be stored anywhere)<\/li>\n
    • Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n
      \n