Math 373 Fall 2013 Homework Chapter 4

Transcription

1 Math 373 Fall 2013 Hoework Chapter 4 Chapter 4 Secto 5 1. (S09Q3)A 30 year auty edate pays 50 each quarter of the frst year. It pays 100 each quarter of the secod year. The payets cotue to crease aually so that the payets each quarter are 50 hgher tha the prevous year. a. Calculate the preset value of ths auty at a aual effectve terest rate of 10%. Soluto: Use the forula that does t follow the rules (4) 4 FrstPayet (1.1) (1.1) a v ( ) a v 0.1 ( Ia) , b. Calculate the preset value of ths auty at a oal rate of 10% copouded quarterly. Soluto: Now we eed to redefe our terest rates: (4) (4) (1 ) (1.025) The use the forula as we dd part a.

2 30 FrstPayet (1.1038) a v ( ) a v ( Ia) , (S09Q3)A auty due akes othly payets for 15 years. The frst payet s 10. Each subsequet payet s 10 larger tha the prevous payet. I other words, the payet at the start of the secod oth s 20 ad the payet at the start of the thrd oth s 30, etc. Calculate the preset value of ths auty usg a oal terest rate of 12% copouded othly. Soluto: Now we eed to use the P ad Q forula fro our otes. Be careful to use the correct terest rate! P 10; Q 10; 12* PV ( IP. Qa ) 1 ( IP, Qa ) Q Pa ( a v ) ,

3 3. (S13HW) Alca s akg deposts to her retreet accout at the ed of every oth. Durg the frst year, she deposts 100 each oth. Durg the secod year, Alca deposts 200 each oth. Durg the thrd year, the payets are 300. The payets cotue the sae patter utl she deposts 3000 per oth durg the 30 th year. If Alca ears a aual effectve terest rate of 7%, how uch wll she have accuulated after 30 years. Soluto: Use the forula that does t follow the rules. 30; 12 FrstPayet 100 1/ ; (1.07) (1.07) a v ( ) a v 0.07 ( Ia) , We eed to fd the accuulated value, so we ultply our aswer by (1 ). 165, , 257,

4 4. (S12HW) A 25 year auty wth othly payets akes payets of 1000 at the ed of each oth durg the frst year. The payets durg the secod year are 1200 per oth. The payets durg the thrd year are 1400 per oth. The payets cotue to crease by 200 each year wth 5800 beg pad each oth durg the 25 th year. Calculate the preset value of ths auty at a aual effectve terest rate of 8%. Soluto: For ths proble, we wat to separate ths to two dfferet autes. 1. A 25 year auty wth othly payets of $800 at the ed of each oth. 2. A 25 year auty wth payets of $200 for the frst year, $400 for the secod year, etc. We ca calculate the value of each auty ad the add the up. The terest rates that we wll 0.08 use are: 1/12 (1.08) Noral Auty: PV 800a , Forula the does t follow the rules: 25 FrstPayet (1.08) a v 0.08 ( Ia) , Ths gves our fal aswer: 106, , ,

5 5. (S13HW) Matt s recevg a creasg perpetuty. The payets are ade at the ed of each quarter. The payets the frst year are 25. The quarterly payets the secod year are 50. The payets cotue to crease each year by 25, but are level throughout the year. Calculate the preset value of ths perpetuty at a oal rate of 9% copouded othly. Soluto: We ow swtch to the perpetuty forula that breaks the sae rules. 1 ( Ia). * Frst, let s fd the approprate terest rates. (1 ) (1.0075) (4) 1/ Now we ca fd the preset value: (25) 12, ( * ) Chapter 4, Secto 6 6. (S13HW) A cotuous auty pays at a rate of 100 per year for the ext 25 years. Calculate the preset value of ths auty at a force of terest of 8%. Soluto: 25*0.08 1e 1e 100a ,

6 7. (S13HW) A creasg auty akes payets cotuously for the ext 10 years. The payet rate s 1000t at te t. Calculate the preset value of ths auty at aual effectve terest rate of 6%. Soluto: Frst, let s fd the force of terest: (1.06) e l(1.06) Now use the forula: ( Ia) a v (1.06) 10 10(1.06) l(1.06) Aswer: l(1.06) 8. (S13HW) Tao vests oey a fud cotuously over the ext 22 years. The aout vested s at a rate of 4000 per year. Calculate the aout that Tao wll have after 22 years f Soluto: We eed to fd 22*0.12 e 1 e s , (S13HW) Shj s recevg cotuous payets uder a perpetuty that pays 100t at te t. Calculate the preset value of ths perpetuty at aual effectve rate of terest of 20%. Soluto: 1 Ths s a creasg cotuous perpetuty. Our forula s : PV Our aswer s: PV 100 3, l(1.2) 2

7 10. (S09Q3)A 10 year cotuous auty pays at a rate of t 3 at te t. The dscout fucto for ths auty s t 2. Calculate the preset value of ths auty. Soluto: For ths proble, we eed to tegrate. Reeber that PV f ( t) v( t) dt We have, t 0.006t *10 PV t ( t ) dt t 0.006t dt 1,

8 11. (S11HW) Calculate the accuulated value at a costat force of terest of 5% of a 20 year cotuous auty whch pays at the rate of (t+1)/2 per perod at exact oet t. Soluto: Frst, set up the equato for PV t t 0.05 PV e dt 0.5 ( t 1) e dt We are gog to eed to tegrate by parts. u t1 du 1 dv e 0.05t v20e uv vdu 0.05t t 0.05t PV 0.5 t 120 e 20 e dt t 20 t e e (20) Now that we have the preset value, we ca calculate the future value. 20 FV 0.05(20) e Or we ca do ths the easy way whch s t t 0.05t t t 0.05t t PV e dt 0.5 ( t 1) e dt 0.5 tv dt 0.5 v dt sce e v t t 0.5( Ia) 0.5 a sce ( Ia) tv dt ad a v dt Now that we have the preset value, we ca calculate the future value. FV 0.05(20) e