Level Annuities with Payments Less Frequent than Each Interest Period

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1 Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due

2 Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due

3 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

4 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

5 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

6 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

7 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

8 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

9 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

10 An Example Fnd an expresson n terms of symbols of the type a n and s n, for the present value of an annuty n whch there are a total of r payments of 1. The frst payment s to be made 7 years from today, and the remanng payments happen at three year ntervals. The present value of ths annuty can be expressed n terms of the annual dscount factor as v 7 + v 10 + v v 3r+4 Calculatng the partal sum of the geometrc seres above, we get v 7 v 3r+7 1 v 3 = (1 v 7 ) + (1 v 3r+7 ) 1 v 3 = 1 v v 3 1 v 3r+7 Caveat: The expresson we obtaned above s not unque! = a 3r+7 a 7 a 3

11 An Example Fnd an expresson n terms of symbols of the type a n and s n, for the present value of an annuty n whch there are a total of r payments of 1. The frst payment s to be made 7 years from today, and the remanng payments happen at three year ntervals. The present value of ths annuty can be expressed n terms of the annual dscount factor as v 7 + v 10 + v v 3r+4 Calculatng the partal sum of the geometrc seres above, we get v 7 v 3r+7 1 v 3 = (1 v 7 ) + (1 v 3r+7 ) 1 v 3 = 1 v v 3 1 v 3r+7 Caveat: The expresson we obtaned above s not unque! = a 3r+7 a 7 a 3

12 An Example Fnd an expresson n terms of symbols of the type a n and s n, for the present value of an annuty n whch there are a total of r payments of 1. The frst payment s to be made 7 years from today, and the remanng payments happen at three year ntervals. The present value of ths annuty can be expressed n terms of the annual dscount factor as v 7 + v 10 + v v 3r+4 Calculatng the partal sum of the geometrc seres above, we get v 7 v 3r+7 1 v 3 = (1 v 7 ) + (1 v 3r+7 ) 1 v 3 = 1 v v 3 1 v 3r+7 Caveat: The expresson we obtaned above s not unque! = a 3r+7 a 7 a 3

13 An Example Fnd an expresson n terms of symbols of the type a n and s n, for the present value of an annuty n whch there are a total of r payments of 1. The frst payment s to be made 7 years from today, and the remanng payments happen at three year ntervals. The present value of ths annuty can be expressed n terms of the annual dscount factor as v 7 + v 10 + v v 3r+4 Calculatng the partal sum of the geometrc seres above, we get v 7 v 3r+7 1 v 3 = (1 v 7 ) + (1 v 3r+7 ) 1 v 3 = 1 v v 3 1 v 3r+7 Caveat: The expresson we obtaned above s not unque! = a 3r+7 a 7 a 3

14 An Example: Unknown fnal payment An nvestment of $1000 s used to make payments of $100 at the end of each year for as long as possble wth a smaller fnal payment to be made at the tme of the last regular payment. If nterest s 7% convertble semannually, fnd the number of payments and the amount of the total fnal payment.

15 An Example: Unknown fnal payment (cont d) Usng the expresson for the present value of ths annuty, we get the equaton of value at tme an s = 1000 where n denotes the unknown number of regular nterest perods that the annuty lasts. The equaton of value yelds a n = 10 s = We get that n = 36 and that 18 regular payments and an addtonal smaller payment must be made. Let R denote the amount of the smaller fnal payment. Then, the tme n equaton of value reads as Thus, R = $10.09 R s s = 1000 (1.035) 36

16 An Example: Unknown fnal payment (cont d) Usng the expresson for the present value of ths annuty, we get the equaton of value at tme an s = 1000 where n denotes the unknown number of regular nterest perods that the annuty lasts. The equaton of value yelds a n = 10 s = We get that n = 36 and that 18 regular payments and an addtonal smaller payment must be made. Let R denote the amount of the smaller fnal payment. Then, the tme n equaton of value reads as Thus, R = $10.09 R s s = 1000 (1.035) 36

17 An Example: Unknown fnal payment (cont d) Usng the expresson for the present value of ths annuty, we get the equaton of value at tme an s = 1000 where n denotes the unknown number of regular nterest perods that the annuty lasts. The equaton of value yelds a n = 10 s = We get that n = 36 and that 18 regular payments and an addtonal smaller payment must be made. Let R denote the amount of the smaller fnal payment. Then, the tme n equaton of value reads as Thus, R = $10.09 R s s = 1000 (1.035) 36

18 An Example: Unknown fnal payment (cont d) Usng the expresson for the present value of ths annuty, we get the equaton of value at tme an s = 1000 where n denotes the unknown number of regular nterest perods that the annuty lasts. The equaton of value yelds a n = 10 s = We get that n = 36 and that 18 regular payments and an addtonal smaller payment must be made. Let R denote the amount of the smaller fnal payment. Then, the tme n equaton of value reads as Thus, R = $10.09 R s s = 1000 (1.035) 36

19 An Example: Unknown fnal payment (cont d) Usng the expresson for the present value of ths annuty, we get the equaton of value at tme an s = 1000 where n denotes the unknown number of regular nterest perods that the annuty lasts. The equaton of value yelds a n = 10 s = We get that n = 36 and that 18 regular payments and an addtonal smaller payment must be made. Let R denote the amount of the smaller fnal payment. Then, the tme n equaton of value reads as Thus, R = $10.09 R s s = 1000 (1.035) 36

20 Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due

21 Value at ssuance and accumulated value Agan, consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k Ths annuty has a payment at the begnnng of each k nterest perods Then, the value at ssuance of ths annuty-due s ä r I and ä r I = (1 + I ) a r I = a n a k Smlarly, we get that the accumulated value equals s r I = s n a k Caveat: The above accumulated value s k nterest converson perods after the last payment...

22 Value at ssuance and accumulated value Agan, consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k Ths annuty has a payment at the begnnng of each k nterest perods Then, the value at ssuance of ths annuty-due s ä r I and ä r I = (1 + I ) a r I = a n a k Smlarly, we get that the accumulated value equals s r I = s n a k Caveat: The above accumulated value s k nterest converson perods after the last payment...

23 Value at ssuance and accumulated value Agan, consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k Ths annuty has a payment at the begnnng of each k nterest perods Then, the value at ssuance of ths annuty-due s ä r I and ä r I = (1 + I ) a r I = a n a k Smlarly, we get that the accumulated value equals s r I = s n a k Caveat: The above accumulated value s k nterest converson perods after the last payment...

24 Value at ssuance and accumulated value Agan, consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k Ths annuty has a payment at the begnnng of each k nterest perods Then, the value at ssuance of ths annuty-due s ä r I and ä r I = (1 + I ) a r I = a n a k Smlarly, we get that the accumulated value equals s r I = s n a k Caveat: The above accumulated value s k nterest converson perods after the last payment...

25 Value at ssuance and accumulated value Agan, consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k Ths annuty has a payment at the begnnng of each k nterest perods Then, the value at ssuance of ths annuty-due s ä r I and ä r I = (1 + I ) a r I = a n a k Smlarly, we get that the accumulated value equals s r I = s n a k Caveat: The above accumulated value s k nterest converson perods after the last payment...

26 An Example: Accumulated value Fnd the accumulated value at the end of four years of an nvestment fund n whch $100 s deposted at the begnnng of each quarter for the frst two years and $200 s deposted at the begnnng of every quarter for the second two years. Assume that the fund earns 12% convertble monthly. The rate of nterest s 1% per month. In ths annuty-due, there are 48 nterest perods and each payment perod conssts of 3 nterest coverson perods. So, the accumulated value s 100 s s a = = $2999 Assgnment: Examples 4.2.9, 12 Problems 4.2.1,3

27 An Example: Accumulated value Fnd the accumulated value at the end of four years of an nvestment fund n whch $100 s deposted at the begnnng of each quarter for the frst two years and $200 s deposted at the begnnng of every quarter for the second two years. Assume that the fund earns 12% convertble monthly. The rate of nterest s 1% per month. In ths annuty-due, there are 48 nterest perods and each payment perod conssts of 3 nterest coverson perods. So, the accumulated value s 100 s s a = = $2999 Assgnment: Examples 4.2.9, 12 Problems 4.2.1,3

28 An Example: Accumulated value Fnd the accumulated value at the end of four years of an nvestment fund n whch $100 s deposted at the begnnng of each quarter for the frst two years and $200 s deposted at the begnnng of every quarter for the second two years. Assume that the fund earns 12% convertble monthly. The rate of nterest s 1% per month. In ths annuty-due, there are 48 nterest perods and each payment perod conssts of 3 nterest coverson perods. So, the accumulated value s 100 s s a = = $2999 Assgnment: Examples 4.2.9, 12 Problems 4.2.1,3

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