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 annutymmedate 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 annutymmedate 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 annutymmedate 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 annutymmedate 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 Setup n... the number of tme perods for the annutymmedate 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 Setup n... the number of tme perods for the annutymmedate 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 Setup n... the number of tme perods for the annutymmedate 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 Setup n... the number of tme perods for the annutymmedate 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 Setup n... the number of tme perods for the annutymmedate 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 Setup n... the number of tme perods for the annutymmedate 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 annutymmedate 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 annutesmmedate. 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 annutymmedate 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 annutesmmedate. 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 PerpetutyImmedate Fnd the present value of a perpetutymmedate 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 PerpetutyImmedate Fnd the present value of a perpetutymmedate 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 PerpetutyImmedate Fnd the present value of a perpetutymmedate 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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