# On some special nonlevel annuities and yield rates for annuities

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1 On some specal nonlevel annutes and yeld rates for annutes 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson

2 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson

3 An Example Assume that an annuty-mmedate provdes 20 annual payments: the frst payment beng \$1,000. The payments ncrease n such a way that each payment s 4% greater than the precedng payment. Fnd the present value of ths annuty at an annual effectve rate of nterest of 7%. Let us look at a more general stuaton. Let n denote the numer of perods. For smplcty, assume that the frst payment s equal to 1, and let all subsequent payments ncrease n geometrc progresson wth the common rato equal to 1 + g Then, the present value of ths annuty equals PV = v + v 2 (1 + g) + v 3 (1 + g) v n (1 + g) n 1 = v [1 + v (1 + g) + (v (1 + g)) (v (1 + g)) n 1] [ ] 1 (v (1 + g)) n = v 1 v(1 + g) = 1 ( 1+g 1+ )n g

4 An Example Assume that an annuty-mmedate provdes 20 annual payments: the frst payment beng \$1,000. The payments ncrease n such a way that each payment s 4% greater than the precedng payment. Fnd the present value of ths annuty at an annual effectve rate of nterest of 7%. Let us look at a more general stuaton. Let n denote the numer of perods. For smplcty, assume that the frst payment s equal to 1, and let all subsequent payments ncrease n geometrc progresson wth the common rato equal to 1 + g Then, the present value of ths annuty equals PV = v + v 2 (1 + g) + v 3 (1 + g) v n (1 + g) n 1 = v [1 + v (1 + g) + (v (1 + g)) (v (1 + g)) n 1] [ ] 1 (v (1 + g)) n = v 1 v(1 + g) = 1 ( 1+g 1+ )n g

5 An Example Assume that an annuty-mmedate provdes 20 annual payments: the frst payment beng \$1,000. The payments ncrease n such a way that each payment s 4% greater than the precedng payment. Fnd the present value of ths annuty at an annual effectve rate of nterest of 7%. Let us look at a more general stuaton. Let n denote the numer of perods. For smplcty, assume that the frst payment s equal to 1, and let all subsequent payments ncrease n geometrc progresson wth the common rato equal to 1 + g Then, the present value of ths annuty equals PV = v + v 2 (1 + g) + v 3 (1 + g) v n (1 + g) n 1 = v [1 + v (1 + g) + (v (1 + g)) (v (1 + g)) n 1] [ ] 1 (v (1 + g)) n = v 1 v(1 + g) = 1 ( 1+g 1+ )n g

6 An Example Assume that an annuty-mmedate provdes 20 annual payments: the frst payment beng \$1,000. The payments ncrease n such a way that each payment s 4% greater than the precedng payment. Fnd the present value of ths annuty at an annual effectve rate of nterest of 7%. Let us look at a more general stuaton. Let n denote the numer of perods. For smplcty, assume that the frst payment s equal to 1, and let all subsequent payments ncrease n geometrc progresson wth the common rato equal to 1 + g Then, the present value of ths annuty equals PV = v + v 2 (1 + g) + v 3 (1 + g) v n (1 + g) n 1 = v [1 + v (1 + g) + (v (1 + g)) (v (1 + g)) n 1] [ ] 1 (v (1 + g)) n = v 1 v(1 + g) = 1 ( 1+g 1+ )n g

7 An Example (cont d) In our case, the frst payment s equal to \$1, 000, so we have to multply the result by that sum. We get that the present value of the annuty from the problem equals ( )20 = 14,

8 An Example (cont d) In our case, the frst payment s equal to \$1, 000, so we have to multply the result by that sum. We get that the present value of the annuty from the problem equals ( )20 = 14,

9 A smplfyng formula Let us revst the formula for the present value: PV = 1 ( 1+g 1+ )n g Defne j = ( g)/(1 + g). Then, we can express the above present value as (1 + ) 1 ä n j In partcular, f = g (.e., f the ncrease n the payment exactly offsets the nterest rate), then the present value becomes n 1 + Assgnment: Example n the textbook; Problems 3.8.1,2,3,4,5 from the textbook Now, let us solve Sample FM Problems #11 and #16 together...

10 A smplfyng formula Let us revst the formula for the present value: PV = 1 ( 1+g 1+ )n g Defne j = ( g)/(1 + g). Then, we can express the above present value as (1 + ) 1 ä n j In partcular, f = g (.e., f the ncrease n the payment exactly offsets the nterest rate), then the present value becomes n 1 + Assgnment: Example n the textbook; Problems 3.8.1,2,3,4,5 from the textbook Now, let us solve Sample FM Problems #11 and #16 together...

11 A smplfyng formula Let us revst the formula for the present value: PV = 1 ( 1+g 1+ )n g Defne j = ( g)/(1 + g). Then, we can express the above present value as (1 + ) 1 ä n j In partcular, f = g (.e., f the ncrease n the payment exactly offsets the nterest rate), then the present value becomes n 1 + Assgnment: Example n the textbook; Problems 3.8.1,2,3,4,5 from the textbook Now, let us solve Sample FM Problems #11 and #16 together...

12 A smplfyng formula Let us revst the formula for the present value: PV = 1 ( 1+g 1+ )n g Defne j = ( g)/(1 + g). Then, we can express the above present value as (1 + ) 1 ä n j In partcular, f = g (.e., f the ncrease n the payment exactly offsets the nterest rate), then the present value becomes n 1 + Assgnment: Example n the textbook; Problems 3.8.1,2,3,4,5 from the textbook Now, let us solve Sample FM Problems #11 and #16 together...

13 A smplfyng formula Let us revst the formula for the present value: PV = 1 ( 1+g 1+ )n g Defne j = ( g)/(1 + g). Then, we can express the above present value as (1 + ) 1 ä n j In partcular, f = g (.e., f the ncrease n the payment exactly offsets the nterest rate), then the present value becomes n 1 + Assgnment: Example n the textbook; Problems 3.8.1,2,3,4,5 from the textbook Now, let us solve Sample FM Problems #11 and #16 together...

14 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson

15 The Set-up n... the number of tme perods for the annuty-mmedate P... the value of the frst payment Q... the amount by whch the payment per perod ncreases So, the payment at the end of the j th perods s P + Q(j 1) (I P,Q a) n... the present value of the annuty descrbed above (I P,Q s) n... the accumulated value at the tme of the last payment of the annuty descrbed above

16 The Set-up n... the number of tme perods for the annuty-mmedate P... the value of the frst payment Q... the amount by whch the payment per perod ncreases So, the payment at the end of the j th perods s P + Q(j 1) (I P,Q a) n... the present value of the annuty descrbed above (I P,Q s) n... the accumulated value at the tme of the last payment of the annuty descrbed above

17 The Set-up n... the number of tme perods for the annuty-mmedate P... the value of the frst payment Q... the amount by whch the payment per perod ncreases So, the payment at the end of the j th perods s P + Q(j 1) (I P,Q a) n... the present value of the annuty descrbed above (I P,Q s) n... the accumulated value at the tme of the last payment of the annuty descrbed above

18 The Set-up n... the number of tme perods for the annuty-mmedate P... the value of the frst payment Q... the amount by whch the payment per perod ncreases So, the payment at the end of the j th perods s P + Q(j 1) (I P,Q a) n... the present value of the annuty descrbed above (I P,Q s) n... the accumulated value at the tme of the last payment of the annuty descrbed above

19 The Set-up n... the number of tme perods for the annuty-mmedate P... the value of the frst payment Q... the amount by whch the payment per perod ncreases So, the payment at the end of the j th perods s P + Q(j 1) (I P,Q a) n... the present value of the annuty descrbed above (I P,Q s) n... the accumulated value at the tme of the last payment of the annuty descrbed above

20 The Set-up n... the number of tme perods for the annuty-mmedate P... the value of the frst payment Q... the amount by whch the payment per perod ncreases So, the payment at the end of the j th perods s P + Q(j 1) (I P,Q a) n... the present value of the annuty descrbed above (I P,Q s) n... the accumulated value at the tme of the last payment of the annuty descrbed above

21 Formulas for the accumulated and present values Smple algebra yelds: (I P,Q s) n = P s n + Q (s n n) (I P,Q a) n = P a n + Q (a n n v n ) In partcular, f P = Q = 1, the notaton and the equatons can be smplfed to (Is) n = s n n (Ia) n = än n v n

22 Formulas for the accumulated and present values Smple algebra yelds: (I P,Q s) n = P s n + Q (s n n) (I P,Q a) n = P a n + Q (a n n v n ) In partcular, f P = Q = 1, the notaton and the equatons can be smplfed to (Is) n = s n n (Ia) n = än n v n

23 Formulas for the accumulated and present values Smple algebra yelds: (I P,Q s) n = P s n + Q (s n n) (I P,Q a) n = P a n + Q (a n n v n ) In partcular, f P = Q = 1, the notaton and the equatons can be smplfed to (Is) n = s n n (Ia) n = än n v n

24 Formulas for the accumulated and present values Decreasng payments Smple algebra yelds: In partcular, f P = n and Q = 1, then we modfy the notaton to stress that the annuty s decreasng and get (Ds) n = n(1 + )n s n (Da) n = n a n Assgnment: Examples 3.9.8, Problems 3.9.1,2,3,4

25 Formulas for the accumulated and present values Decreasng payments Smple algebra yelds: In partcular, f P = n and Q = 1, then we modfy the notaton to stress that the annuty s decreasng and get (Ds) n = n(1 + )n s n (Da) n = n a n Assgnment: Examples 3.9.8, Problems 3.9.1,2,3,4

26 An Example Fnd the expresson for the present value of an annuty-mmedate such that payments start at the amount of 1 dollar, ncrease by annual amounts of 1 to a payment of n, and then decrease by annual amounts of 1 to a fnal payment of 1. You are allowed to use the standard notaton for present values of basc annutes-mmedate. The present value s (Ia) n + v n (Da) n 1 = än n v n + v n (n 1) a n 1 = 1 [ an n v n + n v n v n v n ] a n 1 = 1 [ an 1 (1 v n ) + (1 v n ) ] = 1 (1 v n )(1 + a n 1 ) = a n ä n

27 An Example Fnd the expresson for the present value of an annuty-mmedate such that payments start at the amount of 1 dollar, ncrease by annual amounts of 1 to a payment of n, and then decrease by annual amounts of 1 to a fnal payment of 1. You are allowed to use the standard notaton for present values of basc annutes-mmedate. The present value s (Ia) n + v n (Da) n 1 = än n v n + v n (n 1) a n 1 = 1 [ an n v n + n v n v n v n ] a n 1 = 1 [ an 1 (1 v n ) + (1 v n ) ] = 1 (1 v n )(1 + a n 1 ) = a n ä n

28 An Example: A Perpetuty-Immedate Fnd the present value of a perpetuty-mmedate whose successve payments are 1, 2, 3,... at an effectve per perod nterest rate of If we take a lmt as n n the formula we get (Ia) n = än n v n (Ia) = 1 = (1 + a n 1 ) n v n, = = 420 Let us look at Sample FM Problems #18 and #6...

29 An Example: A Perpetuty-Immedate Fnd the present value of a perpetuty-mmedate whose successve payments are 1, 2, 3,... at an effectve per perod nterest rate of If we take a lmt as n n the formula we get (Ia) n = än n v n (Ia) = 1 = (1 + a n 1 ) n v n, = = 420 Let us look at Sample FM Problems #18 and #6...

30 An Example: A Perpetuty-Immedate Fnd the present value of a perpetuty-mmedate whose successve payments are 1, 2, 3,... at an effectve per perod nterest rate of If we take a lmt as n n the formula we get (Ia) n = än n v n (Ia) = 1 = (1 + a n 1 ) n v n, = = 420 Let us look at Sample FM Problems #18 and #6...

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