{"id":288,"date":"2021-03-25T03:03:10","date_gmt":"2021-03-25T03:03:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&p=288"},"modified":"2021-11-17T23:42:10","modified_gmt":"2021-11-17T23:42:10","slug":"putting-it-together-power-series","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/putting-it-together-power-series\/","title":{"raw":"Putting It Together: Power Series","rendered":"Putting It Together: Power Series"},"content":{"raw":"
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Recall the questions posed in the chapter opener about which is the better way of receiving payouts from lottery winnings. We now revisit those questions and show how to use series to compare values of payments over time with a lump sum payment today. We will compute how much future payments are worth in terms of today\u2019s dollars, assuming we have the ability to invest winnings and earn interest. The value of future payments in terms of today\u2019s dollars is known as the present value<\/em> of those payments.<\/p>\r\n\r\n

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ExAmple: Present Value of Future Winnings<\/h3>\r\n
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<\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]\"This (credit: modification of work by Robert Huffstutter, Flickr)[\/caption]<\/figure>\r\n

Suppose you win the lottery and are given the following three options: (1) Receive 20 million dollars today; (2) receive 1.5 million dollars per year over the next 20 years; or (3) receive 1 million dollars per year indefinitely (being passed on to your heirs). Which is the best deal, assuming that the annual interest rate is 5%? We answer this by working through the following sequence of questions.<\/p>\r\n\r\n

    \r\n \t
  1. How much is the 1.5 million dollars received annually over the course of 20 years worth in terms of today\u2019s dollars, assuming an annual interest rate of 5%?<\/li>\r\n \t
  2. Use the answer to part a. to find a general formula for the present value<\/span> of payments of C<\/em> dollars received each year over the next n<\/em> years, assuming an average annual interest rate r<\/em>.<\/li>\r\n \t
  3. Find a formula for the present value if annual payments of C<\/em> dollars continue indefinitely, assuming an average annual interest rate r<\/em>.<\/li>\r\n \t
  4. Use the answer to part c. to determine the present value of 1 million dollars paid annually indefinitely.<\/li>\r\n \t
  5. Use your answers to parts a. and d. to determine which of the three options is best.<\/li>\r\n<\/ol>\r\n<\/div>\r\n[reveal-answer q=\"44558899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558899\"]\r\n
    \r\n
      \r\n \t
    1. Consider the payment of 1.5 million dollars made at the end of the first year. If you were able to receive that payment today instead of one year from now, you could invest that money and earn 5% interest. Therefore, the present value of that money P<\/em>1<\/sub> satisfies [latex]{P}_{1}\\left(1+0.05\\right)=1.5\\text{million dollars}[\/latex]. We conclude that\r\n<\/span>\r\n
      [latex]{P}_{1}= \\frac{1.5}{1.05} =\\$1.429\\text{ million dollars}[\/latex].<\/div>\r\n\r\n<\/span>\r\nSimilarly, consider the payment of 1.5 million dollars made at the end of the second year. If you were able to receive that payment today, you could invest that money for two years, earning 5% interest, compounded annually. Therefore, the present value of that money P<\/em>2<\/sub> satisfies [latex]{P}_{2}{\\left(1+0.05\\right)}^{2}=1.5\\text{million dollars}[\/latex]. We conclude that\r\n<\/span>\r\n
      [latex]{P}_{2}= \\frac{1.5}{\\left(1.05\\right)}^{2}=\\$1.361\\text{ million dollars}[\/latex].<\/div>\r\n\r\n<\/span>\r\nThe value of the future payments today is the sum of the present values [latex]{P}_{1},{P}_{2},\\dots,{P}_{20}[\/latex] of each of those annual payments. The present value Pk<\/sub><\/em> satisfies\r\n<\/span>\r\n
      [latex]{P}_{k}=\\frac{1.5}{{\\left(1.05\\right)}^{k}}[\/latex].<\/div>\r\n\r\n<\/span>\r\nTherefore,\r\n<\/span>\r\n
      [latex]\\begin{array}{cc}\\hfill P& =\\frac{1.5}{1.05} + \\frac{1.5}{\\left(1.05\\right)^2}+ \\cdots +\\frac{1.5}{\\left(1.05\\right)^20} \\hfill \\\\ & =\\$18.693\\text{ million dollars}\\hfill \\end{array}[\/latex]<\/div>\r\n <\/li>\r\n \t
    2. Using the result from part a. we see that the present value P<\/em> of C<\/em> dollars paid annually over the course of n<\/em> years, assuming an annual interest rate r<\/em>, is given by\r\n<\/span>\r\n
      [latex]P=\\frac{C}{1+r}+\\frac{C}{{\\left(1+r\\right)}^{2}}+\\cdots +\\frac{C}{{\\left(1+r\\right)}^{n}}\\text{dollars}\\text{.}[\/latex]<\/div>\r\n <\/li>\r\n \t
    3. Using the result from part b. we see that the present value of an annuity that continues indefinitely is given by the infinite series\r\n<\/span>\r\n
      [latex]P=\\displaystyle\\sum _{n=0}^{\\infty }\\frac{C}{{\\left(1+r\\right)}^{n+1}}[\/latex].<\/div>\r\n\r\n<\/span>\r\nWe can view the present value as a power series in r<\/em>, which converges as long as [latex]|\\frac{1}{1+r}|<1[\/latex]. Since [latex]r>0[\/latex], this series converges. Rewriting the series as\r\n<\/span>\r\n
      [latex]P=\\frac{C}{\\left(1+r\\right)}\\displaystyle\\sum _{n=0}^{\\infty }{\\left(\\frac{1}{1+r}\\right)}^{n}[\/latex],<\/div>\r\n\r\n<\/span>\r\nwe recognize this series as the power series for\r\n<\/span>\r\n
      [latex]f\\left(r\\right)=\\frac{1}{1-\\left(\\frac{1}{1+r}\\right)}=\\frac{1}{\\left(\\frac{r}{1+r}\\right)}=\\frac{1+r}{r}[\/latex].<\/div>\r\n\r\n<\/span>\r\nWe conclude that the present value of this annuity is\r\n<\/span>\r\n
      [latex]P=\\frac{C}{1+r}\\cdot \\frac{1+r}{r}=\\frac{C}{r}[\/latex].<\/div>\r\n <\/li>\r\n \t
    4. From the result to part c. we conclude that the present value P<\/em> of [latex]C=1\\text{ million dollars}[\/latex] paid out every year indefinitely, assuming an annual interest rate [latex]r=0.05[\/latex], is given by\r\n<\/span>\r\n
      [latex]P=\\frac{1}{0.05}=20\\text{ million dollars}\\text{.}[\/latex]<\/div>\r\n <\/li>\r\n \t
    5. From part a. we see that receiving $1.5 million dollars over the course of 20 years is worth $18.693 million dollars in today\u2019s dollars. From part d. we see that receiving $1 million dollars per year indefinitely is worth $20 million dollars in today\u2019s dollars. Therefore, either receiving a lump-sum payment of $20 million dollars today or receiving $1 million dollars indefinitely have the same present value.<\/li>\r\n<\/ol>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section>","rendered":"
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      \n

      Recall the questions posed in the chapter opener about which is the better way of receiving payouts from lottery winnings. We now revisit those questions and show how to use series to compare values of payments over time with a lump sum payment today. We will compute how much future payments are worth in terms of today\u2019s dollars, assuming we have the ability to invest winnings and earn interest. The value of future payments in terms of today\u2019s dollars is known as the present value<\/em> of those payments.<\/p>\n

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      ExAmple: Present Value of Future Winnings<\/h3>\n
      \n
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      <\/figcaption>
      \"This<\/p>\n

      (credit: modification of work by Robert Huffstutter, Flickr)<\/p>\n<\/div>\n<\/figure>\n

      Suppose you win the lottery and are given the following three options: (1) Receive 20 million dollars today; (2) receive 1.5 million dollars per year over the next 20 years; or (3) receive 1 million dollars per year indefinitely (being passed on to your heirs). Which is the best deal, assuming that the annual interest rate is 5%? We answer this by working through the following sequence of questions.<\/p>\n

        \n
      1. How much is the 1.5 million dollars received annually over the course of 20 years worth in terms of today\u2019s dollars, assuming an annual interest rate of 5%?<\/li>\n
      2. Use the answer to part a. to find a general formula for the present value<\/span> of payments of C<\/em> dollars received each year over the next n<\/em> years, assuming an average annual interest rate r<\/em>.<\/li>\n
      3. Find a formula for the present value if annual payments of C<\/em> dollars continue indefinitely, assuming an average annual interest rate r<\/em>.<\/li>\n
      4. Use the answer to part c. to determine the present value of 1 million dollars paid annually indefinitely.<\/li>\n
      5. Use your answers to parts a. and d. to determine which of the three options is best.<\/li>\n<\/ol>\n<\/div>\n
        Show Solution<\/span><\/p>\n
        \n
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          \n
        1. Consider the payment of 1.5 million dollars made at the end of the first year. If you were able to receive that payment today instead of one year from now, you could invest that money and earn 5% interest. Therefore, the present value of that money P<\/em>1<\/sub> satisfies [latex]{P}_{1}\\left(1+0.05\\right)=1.5\\text{million dollars}[\/latex]. We conclude that
          \n<\/span><\/p>\n
          [latex]{P}_{1}= \\frac{1.5}{1.05} =\\$1.429\\text{ million dollars}[\/latex].<\/div>\n


          \n<\/span>
          \nSimilarly, consider the payment of 1.5 million dollars made at the end of the second year. If you were able to receive that payment today, you could invest that money for two years, earning 5% interest, compounded annually. Therefore, the present value of that money P<\/em>2<\/sub> satisfies [latex]{P}_{2}{\\left(1+0.05\\right)}^{2}=1.5\\text{million dollars}[\/latex]. We conclude that
          \n<\/span><\/p>\n

          [latex]{P}_{2}= \\frac{1.5}{\\left(1.05\\right)}^{2}=\\$1.361\\text{ million dollars}[\/latex].<\/div>\n


          \n<\/span>
          \nThe value of the future payments today is the sum of the present values [latex]{P}_{1},{P}_{2},\\dots,{P}_{20}[\/latex] of each of those annual payments. The present value Pk<\/sub><\/em> satisfies
          \n<\/span><\/p>\n

          [latex]{P}_{k}=\\frac{1.5}{{\\left(1.05\\right)}^{k}}[\/latex].<\/div>\n


          \n<\/span>
          \nTherefore,
          \n<\/span><\/p>\n

          [latex]\\begin{array}{cc}\\hfill P& =\\frac{1.5}{1.05} + \\frac{1.5}{\\left(1.05\\right)^2}+ \\cdots +\\frac{1.5}{\\left(1.05\\right)^20} \\hfill \\\\ & =\\$18.693\\text{ million dollars}\\hfill \\end{array}[\/latex]<\/div>\n

           <\/li>\n

        2. Using the result from part a. we see that the present value P<\/em> of C<\/em> dollars paid annually over the course of n<\/em> years, assuming an annual interest rate r<\/em>, is given by
          \n<\/span><\/p>\n
          [latex]P=\\frac{C}{1+r}+\\frac{C}{{\\left(1+r\\right)}^{2}}+\\cdots +\\frac{C}{{\\left(1+r\\right)}^{n}}\\text{dollars}\\text{.}[\/latex]<\/div>\n

           <\/li>\n

        3. Using the result from part b. we see that the present value of an annuity that continues indefinitely is given by the infinite series
          \n<\/span><\/p>\n
          [latex]P=\\displaystyle\\sum _{n=0}^{\\infty }\\frac{C}{{\\left(1+r\\right)}^{n+1}}[\/latex].<\/div>\n


          \n<\/span>
          \nWe can view the present value as a power series in r<\/em>, which converges as long as [latex]|\\frac{1}{1+r}|<1[\/latex]. Since [latex]r>0[\/latex], this series converges. Rewriting the series as
          \n<\/span><\/p>\n

          [latex]P=\\frac{C}{\\left(1+r\\right)}\\displaystyle\\sum _{n=0}^{\\infty }{\\left(\\frac{1}{1+r}\\right)}^{n}[\/latex],<\/div>\n


          \n<\/span>
          \nwe recognize this series as the power series for
          \n<\/span><\/p>\n

          [latex]f\\left(r\\right)=\\frac{1}{1-\\left(\\frac{1}{1+r}\\right)}=\\frac{1}{\\left(\\frac{r}{1+r}\\right)}=\\frac{1+r}{r}[\/latex].<\/div>\n


          \n<\/span>
          \nWe conclude that the present value of this annuity is
          \n<\/span><\/p>\n

          [latex]P=\\frac{C}{1+r}\\cdot \\frac{1+r}{r}=\\frac{C}{r}[\/latex].<\/div>\n

           <\/li>\n

        4. From the result to part c. we conclude that the present value P<\/em> of [latex]C=1\\text{ million dollars}[\/latex] paid out every year indefinitely, assuming an annual interest rate [latex]r=0.05[\/latex], is given by
          \n<\/span><\/p>\n
          [latex]P=\\frac{1}{0.05}=20\\text{ million dollars}\\text{.}[\/latex]<\/div>\n

           <\/li>\n

        5. From part a. we see that receiving $1.5 million dollars over the course of 20 years is worth $18.693 million dollars in today\u2019s dollars. From part d. we see that receiving $1 million dollars per year indefinitely is worth $20 million dollars in today\u2019s dollars. Therefore, either receiving a lump-sum payment of $20 million dollars today or receiving $1 million dollars indefinitely have the same present value.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n\n\t\t\t
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          Candela Citations<\/h3>\n\t\t\t\t\t
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          CC licensed content, Shared previously<\/div>