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T 0 = time (age) to failure random variable for a new entity, where the space of T 0 is {t | 0 < t < } and = is possible F 0 (t) = Pr(T 0 t) is the CDF (cumulative distribution function) for T 0 Note that F 0 (t) = 1 S 0 (t) and S 0 (t) = 1 F 0 (t). Sections 3.1, 3.2, 3.3 S 0 (t) = Pr(T 0 > t) is the SDF (survival distribution function) for T 0 f 0 (t) = Pr(T 0 t) is the pdf (probability density function) for T 0 Note that F 0 (t) = f 0 (t) and S 0 (t) = f 0 (t). d — dt d — dt 0 (t) = t = is the HRF (hazard rate function) for T 0 f 0 (t) —— S 0 (t) also called force of mortality for T 0

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1. Chapter 5 Class Exercises Let 0 (t) = t = abt b 1 for 0 t, where a and b are constants. (a) Suppose the values of a and b are such that 0 (t) = t is the hazard rate for a survival distribution; find the possible values for a and b, the survival function S 0 (t), the p.d.f. f 0 (t), and the mode of the distribution. S 0 (t) = exp( at b )

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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|

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1. Chapter 3 Class Exercises Suppose we want to find the present value of an annuity which pays $200 at the end of each quarter-year for 12 years if the rate of interest is 6% convertible quarterly. (a)Write the complete actuarial notation for this present value. a –– 48 | 0.015 200 (b)Calculate this present value using the formula. a –– 48 | 0.015 200= 1 – v n 200 —— = 0.015 With v = 1 / 1.015,$ 6808.51 (Note: On the TI-83 calculator, the | 2nd | | FINANCE | keys should be used in place of the | APPS | key and Finance option.) (c)Calculate this present value on the TI-84 calculator by doing the following:

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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. Select the tvm_PV option, and after pressing the | ENTER | key, the desired result should be displayed. $ 6808.51

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(d)Calculate this present value in Excel by entering the following formula in any cell: $ 6808.51 =PV(0.015,48,-200,0,0) This is the balance remaining (generally 0) A 0 implies payments at the end of each period. A 1 implies payments at the beginning of each period. 2.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? We must have 5000 = R a –– 40 | 0.03 R = (a)Let R be the amount withdrawn (i.e., the payments) at each half- year, and write a formula for R using complete actuarial notation. (b)Calculate the value of R using the appropriate formula. a –– 40 | 0.03 5000 ———

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a –– 40 | 0.03 = 1 – v n —— = 0.03 With v = 1 / 1.03,23.1148 R = 5000 ——— = 23.1148 $ 216.31 (Note: On the TI-83 calculator, the | 2nd | | FINANCE | keys should be used in place of the | APPS | key and Finance option.) (c)Calculate the value of R on the TI-84 calculator by doing the following: We must have 5000 = R a –– 40 | 0.03 R = (a)Let R be the amount withdrawn (i.e., the payments) at each half- year, and write a formula for R using complete actuarial notation. (b)Calculate the value of R using the appropriate formula. a –– 40 | 0.03 5000 ———

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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. Select the tvm_Pmt option, and after pressing the | ENTER | key, the desired result should be displayed. $ 216.31

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(d)Calculate the value of R in Excel by entering the following formula in any cell: $ 216.31 =5000/PV(0.03,40,-1,0,0) This is the balance remaining (generally 0) A 0 implies payments at the end of each period. A 1 implies payments at the beginning of each period.

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3. Chapter 3 Class Exercises Find 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)The entire loan plus accumulated interest is paid in one lump sum at the end of 6 years. 3000(1.075) 6 =$4629.90Total Interest Paid =$1629.90 (b)Interest is paid each year as accrued, and the principal is repaid at the end of 6 years. Each year, the interest on the loan is3000(0.075) =$225 Total Interest Paid =$1350 (c)The loan is repaid with level payments at the end of each year over the 6-year period. 3000 = R a –– 6 | 0.075 R =$639.13 Total Interest Paid =6(639.13) – 3000 =$834.78 Let R be the level payments.

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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.

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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|

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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

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4. Chapter 3 Class Exercises 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? (a)Let R be the payments each year, and write a formula for R using complete actuarial notation. We must have (b)Calculate the value of R using the appropriate formula. 3000 = R.. s –– 14 | 0.08 R = 3000 ———— =.. s –– 14 | 0.08 3000 —————— s –– 15 | 0.08 – 1

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(a)Let R be the payments each year, and write a formula for R using complete actuarial notation. We must have (b)Calculate the value of R using the appropriate formula. 3000 = R.. s –– 14 | 0.08 R = 3000 ———— =.. s –– 14 | 0.08 3000 —————— s –– 15 | 0.08 – 1 (1 + 0.08) 14 – 1 —————— = 0.08/1.08 Using either.. s –– = 14 | 0.08 26.1521 or s –– = 15 | 0.08 (1 + 0.08) 14 – 1 —————— = 0.08 we have R = 27.1521 $ 114.71 (Note: On the TI-83 calculator, the | 2nd | | FINANCE | keys should be used in place of the | APPS | key and Finance option.) (c)Calculate the value of R on the TI-84 calculator by doing the following:

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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. Select the tvm_Pmt option, and after pressing the | ENTER | key, the desired result should be displayed. $ 114.71

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(d)Calculate the value of R in Excel by entering the following formula in any cell: $ 114.71 =3000/FV(0.08,14,-1,0,1) This is the balance remaining (generally 0) A 0 implies payments at the end of each period. A 1 implies payments at the beginning of each period.

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