# Annuities Certain. 1 Introduction. 2 Annuities-immediate. 3 Annuities-due

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1 Annuities Certain 1 Introduction 2 Annuities-immediate 3 Annuities-due

2 Annuities Certain 1 Introduction 2 Annuities-immediate 3 Annuities-due

3 General terminology An annuity is a series of payments made * at specified intervals (e.g., yearly - whence the name) called the payment periods * for a certain (defined in advance) length of time If the length of time is fixed (deterministic), then the annuity is called annuity certain If not, then it is called a contingent annuity; an important example is the life annuity If the length of time is deterministic and infinite, the annuity is called a perpetuity

4 General terminology An annuity is a series of payments made * at specified intervals (e.g., yearly - whence the name) called the payment periods * for a certain (defined in advance) length of time If the length of time is fixed (deterministic), then the annuity is called annuity certain If not, then it is called a contingent annuity; an important example is the life annuity If the length of time is deterministic and infinite, the annuity is called a perpetuity

5 General terminology An annuity is a series of payments made * at specified intervals (e.g., yearly - whence the name) called the payment periods * for a certain (defined in advance) length of time If the length of time is fixed (deterministic), then the annuity is called annuity certain If not, then it is called a contingent annuity; an important example is the life annuity If the length of time is deterministic and infinite, the annuity is called a perpetuity

6 General terminology An annuity is a series of payments made * at specified intervals (e.g., yearly - whence the name) called the payment periods * for a certain (defined in advance) length of time If the length of time is fixed (deterministic), then the annuity is called annuity certain If not, then it is called a contingent annuity; an important example is the life annuity If the length of time is deterministic and infinite, the annuity is called a perpetuity

7 General terminology An annuity is a series of payments made * at specified intervals (e.g., yearly - whence the name) called the payment periods * for a certain (defined in advance) length of time If the length of time is fixed (deterministic), then the annuity is called annuity certain If not, then it is called a contingent annuity; an important example is the life annuity If the length of time is deterministic and infinite, the annuity is called a perpetuity

8 General terminology An annuity is a series of payments made * at specified intervals (e.g., yearly - whence the name) called the payment periods * for a certain (defined in advance) length of time If the length of time is fixed (deterministic), then the annuity is called annuity certain If not, then it is called a contingent annuity; an important example is the life annuity If the length of time is deterministic and infinite, the annuity is called a perpetuity

9 Annuities Certain 1 Introduction 2 Annuities-immediate 3 Annuities-due

10 The definition If an annuity is such that the payments are made at the end of the payment periods, then it is called an annuity-immediate Caveat: The terminology may seem counter-intuitive... If the amount paid at the end of each period is equal to \$1, we call this annuity the basic annuity-immediate In general, if the payments are equal for all the periods of an annuity, we say that the annuity is level; otherwise, we say it is a nonlevel annuity

11 The definition If an annuity is such that the payments are made at the end of the payment periods, then it is called an annuity-immediate Caveat: The terminology may seem counter-intuitive... If the amount paid at the end of each period is equal to \$1, we call this annuity the basic annuity-immediate In general, if the payments are equal for all the periods of an annuity, we say that the annuity is level; otherwise, we say it is a nonlevel annuity

12 The definition If an annuity is such that the payments are made at the end of the payment periods, then it is called an annuity-immediate Caveat: The terminology may seem counter-intuitive... If the amount paid at the end of each period is equal to \$1, we call this annuity the basic annuity-immediate In general, if the payments are equal for all the periods of an annuity, we say that the annuity is level; otherwise, we say it is a nonlevel annuity

13 The definition If an annuity is such that the payments are made at the end of the payment periods, then it is called an annuity-immediate Caveat: The terminology may seem counter-intuitive... If the amount paid at the end of each period is equal to \$1, we call this annuity the basic annuity-immediate In general, if the payments are equal for all the periods of an annuity, we say that the annuity is level; otherwise, we say it is a nonlevel annuity

14 Notation: Present value a n... denotes the present value of all the payments made by a basic annuity-immediate over the length of time equal to n periods Assume that the rate of interest is constant and equal to i in each payment period Writing the time 0 equation of value (with the help of a time-line), we get a n = v + v v n = v 1 v n 1 v = 1 v n i If we want to emphasize the exact value of the interest rate used, we write a n i

15 Notation: Present value a n... denotes the present value of all the payments made by a basic annuity-immediate over the length of time equal to n periods Assume that the rate of interest is constant and equal to i in each payment period Writing the time 0 equation of value (with the help of a time-line), we get a n = v + v v n = v 1 v n 1 v = 1 v n i If we want to emphasize the exact value of the interest rate used, we write a n i

16 Notation: Present value a n... denotes the present value of all the payments made by a basic annuity-immediate over the length of time equal to n periods Assume that the rate of interest is constant and equal to i in each payment period Writing the time 0 equation of value (with the help of a time-line), we get a n = v + v v n = v 1 v n 1 v = 1 v n i If we want to emphasize the exact value of the interest rate used, we write a n i

17 Notation: Present value a n... denotes the present value of all the payments made by a basic annuity-immediate over the length of time equal to n periods Assume that the rate of interest is constant and equal to i in each payment period Writing the time 0 equation of value (with the help of a time-line), we get a n = v + v v n = v 1 v n 1 v = 1 v n i If we want to emphasize the exact value of the interest rate used, we write a n i

18 Notation: Accumulated value s n... denotes the accumulated value at the time of the last payment of all the payments made by a basic annuity-immediate over the length of time equal to n periods, i.e., or, equivalently, s n = a(n)a n a n = v(n)s n In the case of compound interest with the fixed interest rate of i per period, we have that or, equivalently, s n i = (1 + i) n a n i = (1 + i)n 1 i a n i = v n s n i

19 Notation: Accumulated value s n... denotes the accumulated value at the time of the last payment of all the payments made by a basic annuity-immediate over the length of time equal to n periods, i.e., or, equivalently, s n = a(n)a n a n = v(n)s n In the case of compound interest with the fixed interest rate of i per period, we have that or, equivalently, s n i = (1 + i) n a n i = (1 + i)n 1 i a n i = v n s n i

20 Examples I. Consider an annuity which pays \$500 at the end of each half-year for 20 years with the interest rate of 9% convertible semiannually. Find the present value of this annuity. 500 a = = II. Roger invests \$1000 at 8% per annum convertible quarterly. How much will he be able to withdraw from this fund at the end of every quarter to use up the fund exactly at the end of 10 years? We look at Roger s withdrawals as an annuity-immediate lasting for 40 payment periods. If Roger is to deplete the \$1000 that are presently on his account in exactly 10 years - this precisely means that the present value of the annuity immediate equals \$1000. So, if we denote the level amounts of each withdrawal by R, we have the following equation We get that R = R a = 1000

21 Examples I. Consider an annuity which pays \$500 at the end of each half-year for 20 years with the interest rate of 9% convertible semiannually. Find the present value of this annuity. 500 a = = II. Roger invests \$1000 at 8% per annum convertible quarterly. How much will he be able to withdraw from this fund at the end of every quarter to use up the fund exactly at the end of 10 years? We look at Roger s withdrawals as an annuity-immediate lasting for 40 payment periods. If Roger is to deplete the \$1000 that are presently on his account in exactly 10 years - this precisely means that the present value of the annuity immediate equals \$1000. So, if we denote the level amounts of each withdrawal by R, we have the following equation We get that R = R a = 1000

22 Examples I. Consider an annuity which pays \$500 at the end of each half-year for 20 years with the interest rate of 9% convertible semiannually. Find the present value of this annuity. 500 a = = II. Roger invests \$1000 at 8% per annum convertible quarterly. How much will he be able to withdraw from this fund at the end of every quarter to use up the fund exactly at the end of 10 years? We look at Roger s withdrawals as an annuity-immediate lasting for 40 payment periods. If Roger is to deplete the \$1000 that are presently on his account in exactly 10 years - this precisely means that the present value of the annuity immediate equals \$1000. So, if we denote the level amounts of each withdrawal by R, we have the following equation We get that R = R a = 1000

23 Examples I. Consider an annuity which pays \$500 at the end of each half-year for 20 years with the interest rate of 9% convertible semiannually. Find the present value of this annuity. 500 a = = II. Roger invests \$1000 at 8% per annum convertible quarterly. How much will he be able to withdraw from this fund at the end of every quarter to use up the fund exactly at the end of 10 years? We look at Roger s withdrawals as an annuity-immediate lasting for 40 payment periods. If Roger is to deplete the \$1000 that are presently on his account in exactly 10 years - this precisely means that the present value of the annuity immediate equals \$1000. So, if we denote the level amounts of each withdrawal by R, we have the following equation We get that R = R a = 1000

24 Examples I. Consider an annuity which pays \$500 at the end of each half-year for 20 years with the interest rate of 9% convertible semiannually. Find the present value of this annuity. 500 a = = II. Roger invests \$1000 at 8% per annum convertible quarterly. How much will he be able to withdraw from this fund at the end of every quarter to use up the fund exactly at the end of 10 years? We look at Roger s withdrawals as an annuity-immediate lasting for 40 payment periods. If Roger is to deplete the \$1000 that are presently on his account in exactly 10 years - this precisely means that the present value of the annuity immediate equals \$1000. So, if we denote the level amounts of each withdrawal by R, we have the following equation We get that R = R a = 1000

25 Examples I. Consider an annuity which pays \$500 at the end of each half-year for 20 years with the interest rate of 9% convertible semiannually. Find the present value of this annuity. 500 a = = II. Roger invests \$1000 at 8% per annum convertible quarterly. How much will he be able to withdraw from this fund at the end of every quarter to use up the fund exactly at the end of 10 years? We look at Roger s withdrawals as an annuity-immediate lasting for 40 payment periods. If Roger is to deplete the \$1000 that are presently on his account in exactly 10 years - this precisely means that the present value of the annuity immediate equals \$1000. So, if we denote the level amounts of each withdrawal by R, we have the following equation We get that R = R a = 1000

26 An Example: Loan repayment Assume that a \$1000 loan is to be repaid over a 10-year period by level payments at the end of each year. Let the effective rate of interest be 9% per annum. What is the total amount of interest repaid? Let R denote the level payment amount. Then the equation of value at the inception of the loan reads as Hence, R = R a 10 = 1000 The total amount of interest paid is equal to 10 R 1000 = = Caveat: In this example we have ignored the possible rounding off errors that usually result in the last payment being irregular ; see Algorithm in the textbook for a more accurate approach

27 An Example: Loan repayment Assume that a \$1000 loan is to be repaid over a 10-year period by level payments at the end of each year. Let the effective rate of interest be 9% per annum. What is the total amount of interest repaid? Let R denote the level payment amount. Then the equation of value at the inception of the loan reads as Hence, R = R a 10 = 1000 The total amount of interest paid is equal to 10 R 1000 = = Caveat: In this example we have ignored the possible rounding off errors that usually result in the last payment being irregular ; see Algorithm in the textbook for a more accurate approach

28 An Example: Loan repayment Assume that a \$1000 loan is to be repaid over a 10-year period by level payments at the end of each year. Let the effective rate of interest be 9% per annum. What is the total amount of interest repaid? Let R denote the level payment amount. Then the equation of value at the inception of the loan reads as Hence, R = R a 10 = 1000 The total amount of interest paid is equal to 10 R 1000 = = Caveat: In this example we have ignored the possible rounding off errors that usually result in the last payment being irregular ; see Algorithm in the textbook for a more accurate approach

29 An Example: Loan repayment Assume that a \$1000 loan is to be repaid over a 10-year period by level payments at the end of each year. Let the effective rate of interest be 9% per annum. What is the total amount of interest repaid? Let R denote the level payment amount. Then the equation of value at the inception of the loan reads as Hence, R = R a 10 = 1000 The total amount of interest paid is equal to 10 R 1000 = = Caveat: In this example we have ignored the possible rounding off errors that usually result in the last payment being irregular ; see Algorithm in the textbook for a more accurate approach

30 An Example: Loan repayment Assume that a \$1000 loan is to be repaid over a 10-year period by level payments at the end of each year. Let the effective rate of interest be 9% per annum. What is the total amount of interest repaid? Let R denote the level payment amount. Then the equation of value at the inception of the loan reads as Hence, R = R a 10 = 1000 The total amount of interest paid is equal to 10 R 1000 = = Caveat: In this example we have ignored the possible rounding off errors that usually result in the last payment being irregular ; see Algorithm in the textbook for a more accurate approach

31 An Example: An implementation of Algorithm A fund of \$25,000 is to be accumulated by means of deposits of \$1,000 made at the end of every year as long as necessary. Assume that the fund earns an effective interest rate of 8% per annum. Find the number of regular (level) deposits that will be necessary, and the size of the final deposit.

32 An Example: An implementation of Algorithm (cont d) The final time n equation of value is So, 1000 s n = s n = (1 + i)n 1 = (1.08)n 1 = 25 i 0.08 Using your calculator, you get that n = 15 After the end of the 14 th year, i.e., after the first 14 payments were made, the balance in the fund is 1000s 14 = 24, 215 With interest only, the amount in the fund grows to be 24, = 26, 152 We get a somewhat irregular answer, that the amount in the fund is greater than the needed amount. Hence, the final payment should be negative; namely, \$1152

33 An Example: An implementation of Algorithm (cont d) The final time n equation of value is So, 1000 s n = s n = (1 + i)n 1 = (1.08)n 1 = 25 i 0.08 Using your calculator, you get that n = 15 After the end of the 14 th year, i.e., after the first 14 payments were made, the balance in the fund is 1000s 14 = 24, 215 With interest only, the amount in the fund grows to be 24, = 26, 152 We get a somewhat irregular answer, that the amount in the fund is greater than the needed amount. Hence, the final payment should be negative; namely, \$1152

34 An Example: An implementation of Algorithm (cont d) The final time n equation of value is So, 1000 s n = s n = (1 + i)n 1 = (1.08)n 1 = 25 i 0.08 Using your calculator, you get that n = 15 After the end of the 14 th year, i.e., after the first 14 payments were made, the balance in the fund is 1000s 14 = 24, 215 With interest only, the amount in the fund grows to be 24, = 26, 152 We get a somewhat irregular answer, that the amount in the fund is greater than the needed amount. Hence, the final payment should be negative; namely, \$1152

35 An Example: An implementation of Algorithm (cont d) The final time n equation of value is So, 1000 s n = s n = (1 + i)n 1 = (1.08)n 1 = 25 i 0.08 Using your calculator, you get that n = 15 After the end of the 14 th year, i.e., after the first 14 payments were made, the balance in the fund is 1000s 14 = 24, 215 With interest only, the amount in the fund grows to be 24, = 26, 152 We get a somewhat irregular answer, that the amount in the fund is greater than the needed amount. Hence, the final payment should be negative; namely, \$1152

36 An Example: An implementation of Algorithm (cont d) The final time n equation of value is So, 1000 s n = s n = (1 + i)n 1 = (1.08)n 1 = 25 i 0.08 Using your calculator, you get that n = 15 After the end of the 14 th year, i.e., after the first 14 payments were made, the balance in the fund is 1000s 14 = 24, 215 With interest only, the amount in the fund grows to be 24, = 26, 152 We get a somewhat irregular answer, that the amount in the fund is greater than the needed amount. Hence, the final payment should be negative; namely, \$1152

37 A useful equality i + 1 i = i + s n i (1 + i) n 1 [ ] 1 = i 1 + (1 + i) n 1 (1 + i) n = i (1 + i) n 1 1 = i 1 v n = 1 a n i Question: Can we interpret this formula?

38 A useful equality i + 1 i = i + s n i (1 + i) n 1 [ ] 1 = i 1 + (1 + i) n 1 (1 + i) n = i (1 + i) n 1 1 = i 1 v n = 1 a n i Question: Can we interpret this formula?

39 An Example revisited Let us consider the loan repayment problem once more; as we have seen, we can set up a repayment schedule as a level annuity-immediate This time we propose a different scheme to repay the loan A \$1000 loan is to be repaid over a 10-year period in the following way: * at the end of each year the interest accrued is repaid * the entire principal is repaid at the end of 10 years.

40 An Example revisited Let us consider the loan repayment problem once more; as we have seen, we can set up a repayment schedule as a level annuity-immediate This time we propose a different scheme to repay the loan A \$1000 loan is to be repaid over a 10-year period in the following way: * at the end of each year the interest accrued is repaid * the entire principal is repaid at the end of 10 years.

41 An Example revisited Let us consider the loan repayment problem once more; as we have seen, we can set up a repayment schedule as a level annuity-immediate This time we propose a different scheme to repay the loan A \$1000 loan is to be repaid over a 10-year period in the following way: * at the end of each year the interest accrued is repaid * the entire principal is repaid at the end of 10 years.

42 An Example revisited Let us consider the loan repayment problem once more; as we have seen, we can set up a repayment schedule as a level annuity-immediate This time we propose a different scheme to repay the loan A \$1000 loan is to be repaid over a 10-year period in the following way: * at the end of each year the interest accrued is repaid * the entire principal is repaid at the end of 10 years.

43 An Example revisited Let us consider the loan repayment problem once more; as we have seen, we can set up a repayment schedule as a level annuity-immediate This time we propose a different scheme to repay the loan A \$1000 loan is to be repaid over a 10-year period in the following way: * at the end of each year the interest accrued is repaid * the entire principal is repaid at the end of 10 years.

44 An Example revisited (still) In order to be able to repay the entire principal at the end of the loan term, we deposit equal amounts of money at the end of every year during the loan term. Thus, we have to deposit exactly 1/s n at the end of every year for every borrowed dollar On the other hand, the accrued interest must also be paid at the end of every year and that amount is i per every dollar borrowed Altogether, we should pay at the end of every year ( 1000 i + 1 ) s n

45 An Example revisited (still) In order to be able to repay the entire principal at the end of the loan term, we deposit equal amounts of money at the end of every year during the loan term. Thus, we have to deposit exactly 1/s n at the end of every year for every borrowed dollar On the other hand, the accrued interest must also be paid at the end of every year and that amount is i per every dollar borrowed Altogether, we should pay at the end of every year ( 1000 i + 1 ) s n

46 An Example revisited (still) In order to be able to repay the entire principal at the end of the loan term, we deposit equal amounts of money at the end of every year during the loan term. Thus, we have to deposit exactly 1/s n at the end of every year for every borrowed dollar On the other hand, the accrued interest must also be paid at the end of every year and that amount is i per every dollar borrowed Altogether, we should pay at the end of every year ( 1000 i + 1 ) s n

47 An Example revisited (still) In order to be able to repay the entire principal at the end of the loan term, we deposit equal amounts of money at the end of every year during the loan term. Thus, we have to deposit exactly 1/s n at the end of every year for every borrowed dollar On the other hand, the accrued interest must also be paid at the end of every year and that amount is i per every dollar borrowed Altogether, we should pay at the end of every year ( 1000 i + 1 ) s n

48 An Example revisited (still... ) If you compare this to the level annual payment amount R which was calculated earlier - you will realize that in that case annual payments were R = 1000 a n These two examples together illustrate formula i + 1 s n i = 1 a n i Assignment: Examples , Problems 3.2.1,2,3,8

49 An Example revisited (still... ) If you compare this to the level annual payment amount R which was calculated earlier - you will realize that in that case annual payments were R = 1000 a n These two examples together illustrate formula i + 1 s n i = 1 a n i Assignment: Examples , Problems 3.2.1,2,3,8

50 An Example revisited (still... ) If you compare this to the level annual payment amount R which was calculated earlier - you will realize that in that case annual payments were R = 1000 a n These two examples together illustrate formula i + 1 s n i = 1 a n i Assignment: Examples , Problems 3.2.1,2,3,8

51 Annuities Certain 1 Introduction 2 Annuities-immediate 3 Annuities-due

52 The definition If an annuity is such that the payments are made at the beginning of the payment periods, then it is called an annuity-due If the amount paid at the end of each period is equal to \$1, we call this annuity the basic annuity-due ä n... denotes the present value of all the payments made by a basic annuity-due over the length of time equal to n periods Again, if we assume that the rate of interest is constant and equal to i in each payment period Writing the time 0 equation of value (with the help of a time-line), we get ä n i = 1 v n d

53 The definition If an annuity is such that the payments are made at the beginning of the payment periods, then it is called an annuity-due If the amount paid at the end of each period is equal to \$1, we call this annuity the basic annuity-due ä n... denotes the present value of all the payments made by a basic annuity-due over the length of time equal to n periods Again, if we assume that the rate of interest is constant and equal to i in each payment period Writing the time 0 equation of value (with the help of a time-line), we get ä n i = 1 v n d

54 The definition If an annuity is such that the payments are made at the beginning of the payment periods, then it is called an annuity-due If the amount paid at the end of each period is equal to \$1, we call this annuity the basic annuity-due ä n... denotes the present value of all the payments made by a basic annuity-due over the length of time equal to n periods Again, if we assume that the rate of interest is constant and equal to i in each payment period Writing the time 0 equation of value (with the help of a time-line), we get ä n i = 1 v n d

55 The definition If an annuity is such that the payments are made at the beginning of the payment periods, then it is called an annuity-due If the amount paid at the end of each period is equal to \$1, we call this annuity the basic annuity-due ä n... denotes the present value of all the payments made by a basic annuity-due over the length of time equal to n periods Again, if we assume that the rate of interest is constant and equal to i in each payment period Writing the time 0 equation of value (with the help of a time-line), we get ä n i = 1 v n d

56 The definition If an annuity is such that the payments are made at the beginning of the payment periods, then it is called an annuity-due If the amount paid at the end of each period is equal to \$1, we call this annuity the basic annuity-due ä n... denotes the present value of all the payments made by a basic annuity-due over the length of time equal to n periods Again, if we assume that the rate of interest is constant and equal to i in each payment period Writing the time 0 equation of value (with the help of a time-line), we get ä n i = 1 v n d

57 Notation: Accumulated value s n... denotes the accumulated value at time n of all the payments made by a basic annuity-due over the length of time equal to n periods, i.e., or, equivalently, s n = a(n)ä n ä n = v(n) s n Again, in the particular case of compound interest with the fixed interest rate of i per period, we write or, equivalently, s n i = (1 + i) n ä n i = (1 + i)n 1 d ä n i = v n s n i

58 Notation: Accumulated value s n... denotes the accumulated value at time n of all the payments made by a basic annuity-due over the length of time equal to n periods, i.e., or, equivalently, s n = a(n)ä n ä n = v(n) s n Again, in the particular case of compound interest with the fixed interest rate of i per period, we write or, equivalently, s n i = (1 + i) n ä n i = (1 + i)n 1 d ä n i = v n s n i

59 Connection between annuity-immediate and annuity-due The most basic relationship is obtained through discounting: and, equivalently, ä n i = (1 + i)a n i s n i = (1 + i)s n i The other basic relationship is obtained by shifting the time by one and ä n = 1 + a n 1 s n + 1 = s n+1

60 Connection between annuity-immediate and annuity-due The most basic relationship is obtained through discounting: and, equivalently, ä n i = (1 + i)a n i s n i = (1 + i)s n i The other basic relationship is obtained by shifting the time by one and ä n = 1 + a n 1 s n + 1 = s n+1

61 An Example Roger wants to accumulate (at least) \$1000 in a fund at the end of 12 years. To accomplish this he plans to make deposits at the end of each year, the final payment being made one year prior to the end of the investment period. How large (at least) should each deposit be if the fund earns 7% effective per annum? We are interested in the accumulated value of the investments one year after the last payment, so the equation of value should involve an annuity-due. Denote the unknown level annual payment by R. Then, we can formalize the conditions in the example as... and so R = R s 11 = = 59.21

62 An Example Roger wants to accumulate (at least) \$1000 in a fund at the end of 12 years. To accomplish this he plans to make deposits at the end of each year, the final payment being made one year prior to the end of the investment period. How large (at least) should each deposit be if the fund earns 7% effective per annum? We are interested in the accumulated value of the investments one year after the last payment, so the equation of value should involve an annuity-due. Denote the unknown level annual payment by R. Then, we can formalize the conditions in the example as... and so R = R s 11 = = 59.21

63 An Example Roger wants to accumulate (at least) \$1000 in a fund at the end of 12 years. To accomplish this he plans to make deposits at the end of each year, the final payment being made one year prior to the end of the investment period. How large (at least) should each deposit be if the fund earns 7% effective per annum? We are interested in the accumulated value of the investments one year after the last payment, so the equation of value should involve an annuity-due. Denote the unknown level annual payment by R. Then, we can formalize the conditions in the example as... and so R = R s 11 = = 59.21

64 An Example Roger wants to accumulate (at least) \$1000 in a fund at the end of 12 years. To accomplish this he plans to make deposits at the end of each year, the final payment being made one year prior to the end of the investment period. How large (at least) should each deposit be if the fund earns 7% effective per annum? We are interested in the accumulated value of the investments one year after the last payment, so the equation of value should involve an annuity-due. Denote the unknown level annual payment by R. Then, we can formalize the conditions in the example as... and so R = R s 11 = = 59.21

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