1. A series of payments made at equal intervals of time is called an annuity. An annuity where payments are guaranteed to occur for a fixed period of time is called an annuity-certain. Unless otherwise indicated, this is the type of annuity we will assume, and the “certain” will be dropped from the name. An annuity where payments occur only under certain conditions is called an annuity-contingent. The interval between annuity payments is called an payment period, often just called a period. Sections 3.1, 3.2, 3.3

2. An annuity under which payments of 1 are made at the end of each of period for n periods is called an annuity-immediate. 012…n – 1n Payments Periods 1111 The present value of the annuity at time 0 is denoted, where the interest rate i is generally included only if not clear from the context. The accumulated value of the annuity at time n is denoted, where the interest rate i is generally included only if not clear from the context. a – n|i a – n| s – n|i s – n| = v + v 2 + … + v n = 1 – v n v —— = 1 – v 1 – v n —— i = (1 + i) n–1 + (1 + i) n–2 + … + 1 = (1 + i) n – 1 ———— = (1 + i) – 1 (1 + i) n – 1 ———— i Values for and are available from certain calculators & Excel. a – n| s – n|

3. a – n| s – n| 1 – v n = ——  i (1 + i) n – 1 = ————  i a – n| 1 = i + v n The right hand side can be interpreted as the sum of the “present value of the interest payments” and the “present value of 1 (the original investment)” 1 = (1 + i) n  i The right hand side can be interpreted as the “accumulated value of 1 (the original investment)” minus the “accumulated value of the interest payments” s – n| Observe that s – n| a – n| =(1 + i) n Also, 1 —— + i = s – n| i ———— + i = (1 + i) n – 1 i(1 + i) n ———— = (1 + i) n – 1 i —— = 1 – v n 1 —— a – n| This identity will be important in a future chapter.

4. Find the present value of an annuity which pays $200 at the end of each quarter-year for 12 years if the rate of interest is 6% convertible quarterly. a –– 48 | 0.015 200= An investment of $5000 is made at 6% convertible semiannually. How much can be withdrawn each half-year to use up the fund exactly at the end of 20 years?

5. To calculate 200on the TI-84 calculator, do the following: a –– 48 | 0.015 (Note: On the TI-83 calculator, the | 2nd | | FINANCE | keys should be used in place of the | APPS | key and Finance option.) Press the | APPS | key, select the Finance option, and select the TVM_Solver option. Enter the following values for the variables displayed: N = 48 I% = 1.5 PV = 0 PMT = –200 FV = 0 P/Y = 1 C/Y = 1 Select the END option for PMT, press the | APPS | key, and select the Finance option.

6. Select the tvm_PV option, and after pressing the | ENTER | key, the desired result should be displayed. To calculate 200in Excel, enter the following formula: a –– 48 | 0.015 =PV(0.015,48,-200,0,0) This is the balance remaining (generally 0) This implies payments at the end of each period. A 1 implies payments at the beginning of each period.

7. Find the present value of an annuity which pays $200 at the end of each quarter-year for 12 years if the rate of interest is 6% convertible quarterly. a –– 48 | 0.015 200=$6808.51 An investment of $5000 is made at 6% convertible semiannually. How much can be withdrawn each half-year to use up the fund exactly at the end of 20 years? Let R be the amount withdrawn (i.e., the payments) at each half-year. The present value at the time the investment begins is $5000, so the equation of value is 5000 = R a –– 40 | 0.03 R =

8. To calculate R =on the TI-84 calculator, do the following: a –– 40 | 0.03 5000 ——— (Note: On the TI-83 calculator, the | 2nd | | FINANCE | keys should be used in place of the | APPS | key and Finance option.) Press the | APPS | key, select the Finance option, and select the TVM_Solver option. Enter the following values for the variables displayed: N = 40 I% = 3 PV = –5000 PMT = 0 FV = 0 P/Y = 1 C/Y = 1 Select the END option for PMT, press the | APPS | key, and select the Finance option.

9. Select the tvm_Pmt option, and after pressing the | ENTER | key, the desired result should be displayed. =5000/PV(0.03,40,-1,0,0) To calculate R =in Excel, enter the following formula: a –– 40 | 0.03 5000 ———

10. Find the present value of an annuity which pays $200 at the end of each quarter-year for 12 years if the rate of interest is 6% convertible quarterly. a –– 48 | 0.015 200=$6808.51 An investment of $5000 is made at 6% convertible semiannually. How much can be withdrawn each half-year to use up the fund exactly at the end of 20 years? Let R be the amount withdrawn (i.e., the payments) at each half-year. The present value at the time the investment begins is $5000, so the equation of value is 5000 = R a –– 40 | 0.03 R =$216.31

11. Compare the total amount of interest that would be paid on a $3000 loan over a 6-year period with an effective rate of interest of 7.5% per annum, under each of the following repayment plans: (a) (b) (c) The entire loan plus accumulated interest is paid in one lump sum at the end of 6 years. Interest is paid each year as accrued, and the principal is repaid at the end of 6 years. The loan is repaid with level payments at the end of each year over the 6-year period. 3000(1.075) 6 =$4629.90Total Interest Paid =$1629.90 Each year, the interest on the loan is3000(0.075) =$225 Total Interest Paid =$1350 3000 = R a –– 6 | 0.075 R =$639.13 Total Interest Paid =6(639.13) – 3000 =$834.78 Let R be the level payments.

12. An annuity under which payments of 1 are made at the beginning of each period for n periods is called an annuity-due. 012…n – 1n Payments Periods 1111 The present value of the annuity at time 0 is denoted, where the interest rate i is generally included only if not clear from the context. The accumulated value of the annuity at time n is denoted, where the interest rate i is generally included only if not clear from the context. = 1 + v + v 2 + … + v n–1 = 1 – v n —— = 1 – v 1 – v n —— d = (1 + i) n + (1 + i) n–1 + … + (1 + i) = (1 + i) n – 1 (1 + i)———— = (1 + i) – 1 (1 + i) n – 1 ———— d.. a – n|i.. s – n|i.. a – n|.. s – n|

13. Observe that = (1 + i) n Also, 1 —— + d = 1 ——.. a – n|.. s – n| In addition, observe that a – n| =(1 + i).. a – n| =(1 + i).. s – n| s – n| a ––– n–1| =1 +.. a – n| =– 1.. s – n| s ––– n+1| These last four formulas demonstrate that annuity-immediate and annuity-due are really just the same thing at two different points in time, as is illustrated graphically in Figure 3.3 of the textbook... s – n|.. a – n| d ———— + d = (1 + i) n – 1 d(1 + i) n ———— = (1 + i) n – 1 d —— = 1 – v n

14. An investor wishes to accumulate $3000 at the end of 15 years in a fund which earns 8% effective. To accomplish this, the investor plans to make deposits at the end of each year, with the final payment to be made one year prior to the end of the investment period. How large should each deposit be? Let R be the payments each year. The accumulated value of the investment at the end of the investment period is to be $3000, so the equation of value is 3000 = R.. s –– 14 | 0.08 R = 3000 ———— =.. s –– 14 | 0.08 3000 —————— = s –– 15 | 0.08 – 1

15. To calculate R =on the TI-84 calculator, do the following: 3000 ———.. s –– 14 | 0.08 (Note: On the TI-83 calculator, the | 2nd | | FINANCE | keys should be used in place of the | APPS | key and Finance option.) Press the | APPS | key, select the Finance option, and select the TVM_Solver option. Enter the following values for the variables displayed: N = 14 I% = 8 PV = 0 PMT = 0 FV = –3000 P/Y = 1 C/Y = 1 Select the BEGIN option for PMT, press the | APPS | key, and select the Finance option.

16. =3000/FV(0.08,14,-1,0,1) To calculate R =in Excel, enter the following formula: 3000 ———.. s –– 14 | 0.08 Select the tvm_Pmt option, and after pressing the | ENTER | key, the desired result should be displayed.

17. An investor wishes to accumulate $3000 at the end of 15 years in a fund which earns 8% effective. To accomplish this, the investor plans to make deposits at the end of each year, with the final payment to be made one year prior to the end of the investment period. How large should each deposit be? Let R be the payments each year. The accumulated value of the investment at the end of the investment period is to be $3000, so the equation of value is 3000 = R.. s –– 14 | 0.08 R = 3000 ———— =.. s –– 14 | 0.08 3000 —————— = s –– 15 | 0.08 – 1 3000 ——— = 26.1521 $114.71