3. Present value of Annuity Problems


 Sherilyn Dickerson
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1 Mathematcs of Fnance The formulae 1. A = P(1 +.n) smple nterest 2. A = P(1 + ) n compound nterest formula 3. A = P(1.n) deprecaton straght lne 4. A = P(1 ) n compound decrease dmshng balance 5. P =  present value of annuty 6. A =  future value of annuty 7. A =  future value of annuty one nstalment does not earn nterest Use for snkng fund. 8. B n =  m s remanng perod or payments all payments equal. 9. P =  d s deferred perod A. Calculaton Problems 1. Use the formula A = P(1 +.n) to fnd 1.1 A f P = R5000 ; = 10% pa; n= 6 years 1.2 P f A = R10000 ; = 9% pa; n = 5 years 1.3 f A = R20000; P = R10000 ; n = 10years 1.4 n f A = R15000; P = R10000; = 7 years 2. Use the formula A = P(1 + ) n to fnd 2.1 A f P = R5000 ; = 10% pa compounded monthly; n= 6 years 2.2 P f A = R10000 ; = 9% pa compounded quarterly; n = 5 years 2.3 f A = R20000; P = R10000 ; n = 10years; nterest compounded monthly 2.4 n f A = R15000; P = R10000; = 7 years; nterest compounded quarterly 2.5 A f P = R60000 ; = 12%pa compounded daly; n = 5 years 3. Use the formula A = to fnd 3.1 A f x = R3500; = 10% pa compounded monthly; n = 10 years. 3.2 x f A = R500000; = 13% pa compounded monthly; n = 8 years 3.3 n f A = R600000; = 12%; x = R10000 ; nterest compounded monthly. 4. Use the formula A = to fnd 4.1 A f x = R3500; = 10% pa compounded monthly; n = 10 years. 4.2 x f A = R500000; = 13% pa compounded monthly; n = 8 years 4.3 n f A = R600000; = 12%; x = R10000 ; nterest compounded monthly. 5. Use the formula P = 5.1 P f x = R2000; = 8% pa compounded monthly and n = 10 years 5.2 P f x = R50000; = 10% pa compounded annually; n = 6 years 5.3 x f P = R ; = 9% pa compounded monthly; n = 20 years 5.4 n f P = R ; = 8,3% pa compounded monthly; n = 18 years.
2 6. Determne the effectve rate of nterest for the year f the nomnal rate of nterest s % pa compounded monthly % pa compounded monthly % pa compounded daly % pa compounded daly % pa compounded quarterly 7. Determne the nomnal rate of nterest pa f the effectve rate of nterest s % pa; compounded monthly ,4 % pa ; compounded monthly ,8% pa compounded daly ,2% pa compounded quarterly 8. Determne the effectve rate of nterest for 6 months f 8.1 the nomnal rate of nterest s 13,6% pa compounded monthly 8.2 the nomnal rate of nterest s 11,4% pa compounded monthly. B. Problems on Mathematcs of Fnance 1. Mrs Khumalo wanted to nvest a certan amount of money to ensure that her two chldren, ages 3 years and 5 years and whose brthday fall on the same day, are guaranteed a sum of R each when they each reach 19. She was gven an nterest rate of 9,6% compounded monthly for the gven perod. Determne the amount she must nvest now n order to acheve her objectve. 2. Mr Zee saved a certan amount so that on hs 40 th brthday he would receve R and on hs 50 th brthday he would receve R He s 24 years old when he dd hs nvestment at a rate of nterest of 15% pa compounded monthly. What must he nvest now? 3. Present value of Annuty Problems 3.1 Mr Y took out a loan for a certan amount. He pad ths loan off n 10 equal monthly nstalments of R1000 each at an nterest rate of 12% pa compounded annually. Determne the amount of the loan taken f the frst payment starts at the end of the frst year.
3 Frst Prncples:  P = (1,12) (1,12) (1,12) (1,12) 10 P = (1,12) 1 [ 1 + (1,12) 1 + (1,12) (1,12) 10 ] 1 + (1,12) 1 + (1,12) (1,12)  9 = a[r n 1] = [(1,12) 1 ] 10 1 r 1 (1,12) 1 1 P = 1000 (1,12) 1 [(1,12) 10 1] (1,12) 1 1 = 1000 [(1,12) 10 1] 1,12[(1,12) 1 1] = 1000 [(1,12) 10 1] 1 1,12 = 1000 [1 (1,12) 10 ] 0,12 = R5 650,22 P = x [1 (1 + ) n ] 3.2 A homeowner takes a loan of R The nterest s 9,8% p.a. compounded monthly. The loan s to be repad monthly n 240 nstalments wth the frst payment starts at the end of the frst month Determne the monthly nstalment Determne total amount pad Calculate the actual nterest If nflaton s calculated at a rate of 9% p.a. compounded annually determne the new value of the house after 10 years Dscusson Bascally all loans are deferred payments. No one takes out a loan and pay an amount mmedately. Suppose the borrower wants to pay R5 000 mmedately then that person should take out a loan for R The formula s for one deferred payment. P= x [1 (1 + ) n ] = x [1 (1 + 9,8 1200) 240 ] 9, x = R3807,24 (nstalment s actually R3808, not less) A = 3808 x 240 = R = x. n P = 3808 x = R New Value = ( ) 10 = R
4 3.3 A homeowner takes a loan of R The nterest s 8,9% p.a. compounded monthly. The loan s to be repad monthly n 240 nstalments wth the frst payment starts at the end of the frst month Determne the monthly nstalment Determne total amount pad Calculate the actual nterest If nflaton s calculated at a rate of 8% p.a. compounded annually determne the new value of the house n 20 years tme. 3.4 A small busness enterprse decded to take a loan for R at a low nterest of 6% p.a. compounded monthly for 5 years Determne the monthly nstalment Determne the total nterest pad on ths loan Determne the new monthly nstalment f the frst payment s deferred for 6 months and the total nterest pad. Dscusson Let us consder a smpler problem: A loan of R was to be repad n yearly nstalments at a rate of 10% p.a compounded annually. Two cases: (a) Payment to commence 1 year after the loan was awarded. (b) Payment to commence 3 years after the loan s awarded. Determne the nstalment n each case. (a) Standard formula: P= x [1 (1 + ) n ] = x [1 ( ) 10 ] x = R (b) Let us go back to frst prncples P = x (1,10) 3 + x (1,10) 4 + x (1,10) x (1,10) 12 = x(1,10) 2. (1,10) 1 [1 +(1,12) 1 + (1,12) (1,12) 10 ] The only dfference s the addtonal factor of (1,10) 2 P = (1,1) 2.x [1(1+) 10 ] x = R19693 (c) New formula: P = (1+) d.x [1(1+) n ] d s deferred perod = deferred per A homeowner takes a home loan of R Ths amount s to be repad n monthly nstalments over 20 years at a rate of 9.6% p.a. compounded monthly. Calculate the monthly nstalments f the frst payment s made at the end of the 6 months.
5 3.6 A shopkeeper takes a loan and pays t off n 12 equal yearly nstalments of R startng at the end of the 4 th year at an nterest rate of 10% p.a. compounded annually. How much can she borrow? Hnt: Remember one postponed payment (standard) s (1 + ) 1. In ths case we start wth (1 + ) 4. P = (1 + ) (1 + ) (1 + ) A company takes out a loan for R at an nterest rate of 7,5% pa compounded monthly for 10 years Determne the monthly nstalment How long wll t take to pay of the loan f they double the nstalment? 3.8 Jojo nvests R wth a bank that agreed to gve hm a return of 18% pa compounded monthly. How much wll Jojo receve every month for 10 years? 3.9 Patence nvested R wth a bank that agreed to gve her 15% pa compounded monthly on ths nvestment. What wll her balance be after 5 years f she s gven a monthly annuty of R6000? 4. Future Value of Annuty 4.1 Shrley nvested R5000 per month startng on 1 May 2000 earnng nterest at a rate of 17% pa compounded monthly. What can Shrley expect on 30 Aprl 2012? 4.2 Trevor nvested R per annum startng on 1 July 2003 earnng nterest at a rate of 15,8%pa compounded monthly. What s the total amount that Trevor receved on 30 June 2011? 4.3 Dudu wanted to become a mllonare n 8 years tme. The bank was very happy wth her commtment to save and offered her a whoppng 20% pa return compounded monthly. She nvested her money at the end of each month startng on 31 August At what date can she expect to be a mllonare? What must her monthly nvestment be to acheve ths objectve? 4.4 Randy took out an nsurance polcy whch was fxed at R500 per month. The guaranteed nterest rate was 12% pa compounded annually. What wll Randy s polcy be worth n 30 years tme?
6 5. General Problems 5.1 A loan of R was taken by a homeowner. Interest was to be pad at 9.5%p.a. compounded monthly. The repayment perod s 20 years. 5.1 Calculate the monthly nstalment. 5.2 Calculate the balance on the loan after (a) 5 years (b) 10 years 5,2 ABC Incorporated nvested a certan amount n a bank earnng nterest at a rate of 12,6% p.a compounded monthly. They ntend wthdrawng R9 000 per month for 10 years. How much must they nvest now to ensure that ths wthdrawal can be made? 5.3 Mrs Wse nvested a certan amount n a bank earnng nterest at a rate of 15% p.a compounded monthly. She ntend wthdrawng R per month for 10 years. How much must they nvest now to ensure that ths wthdrawal can be made? 5.4 A father decded to buy a house for hs famly for R He agreed to pay monthly nstalments of R10000 on a loan whch ncurred nterest at a rate of 14% pa compounded monthly. The frst payment was made at the end of the frst month Show that the loan would be pad off n 234 months Suppose the father encountered unexpected problems and was unable to pay any nstalments at the end of the 120 th ; 121 st ; 122 nd ; 123 rd months. At the end of the 124 th month he ncreased hs payment so as to stll pay off the loan n 234 months by 111 equal monthly payments. Calculate the value of ths new nstalment. 5.5 A company purchased a buldng for R They took out a loan for ths amount at an nterest rate of 6% per annum compounded monthly. They pad a fxed amount of R40000 per month How many payments must be made? Determne the fnal payment What wll the balance be after 36 months? 5.6 Snkng Fund A company purchased a new vehcle for R The company wshed to replace the vehcle n 5 years tme. The rate of deprecaton s 10% p.a on dmnshng balance. It s envsaged that a new vehcle wll apprecate n value at a rate of 11% p.a. Calculate (a) the resdual value of the vehcle n 5 years tme. (b) the prce of the new vehcle n 5 years tme. (c) the value that the snkng fund must attan n 5 years tme. (d) the monthly payments nto the snkng fund ( payment commenced 1 month after the snkng fund s set up) at a rate of 13,6%p.a compounded monthly.
7 A borehole machne presently cost R It deprecates n value at a rate of 12 % on a reducng balance. A new borehole machne wll be requred n 5 years tme. The cost of the borehole machne s expected to grow at a rate of 10% p.a. compound nterest. Determne (a) scrap value of old borehole machne. (b) cost of new borehole machne (c) (d) and hence the value that the snkng fund must attan. equal monthly nstalments whch wll start n 6 months tme and fnsh n 5 years tme. Interest s at 11,2% p.a. compounded monthly A car presently cost R It deprecates at a rate of 10% p.a. on a reducng balance. A new car wll be needed n 6 years tme. The cost of the new car s expected to grow by 8% p.a. compounded annually. Determne: (a) the scrap value of the car to the nearest R. (b) the cost of the new car to the nearest R. (c) (d) hence the value that the snkng fund must attan. the equal monthly nstalments whch wll start n 6 months tme and fnsh n 5 years tme. Interest s calculated at 14% p.a. compounded monthly 6. Suppose an amount of R3000 s wthdrawn every 6 months startng n one year s tme and fnsh 6 months before the purchase. What wll the new nstalment be for and 5.6.3?
8 HIRE PURCHASE 1. Mr X purchase a car to the value of R He takes out a loan for ths amount. He repays ths loan at a rate of 12% p.a. compounded monthly over 48 months. Calculate hs monthly nstalments. Rather than take out a bank loan the purchaser sgned a H.P agreement because the H.P charges were 9%p.a. smple nterest. Fnd 1.1 the new nstalments. 1.2 nomnal rate of nterest 1.3 effectve rate of nterest 2. Applances at home cost a new homeowner R The owner pays a depost of 20% and pays the balance over 36 months n equal monthly nstalments. 2.1 Fnd the nstalments f the borrower borrows the outstandng amount and agrees to pay nterest at a rate of 11% p.a. compounded monthly. 2.2 Suppose the homeowner borrows the outstandng balance and agrees to pay the H.P at a rate of 10% p.a. smple nterest. Fnd new nstalments nomnal rate of nterest effectve rate of nterest. 3. Student loans are beng offered by a leadng bankng nsttuton to tertary students. It offers loans of up to R The nterest s fxed at 10% of the amount. A further condton s that ths loan be pad over 10 months. Student A takes a loan of R Student B takes a loan of R Determne the monthly repayment for student A and B. 3.2 Calculate the nomnal and effectve rate of nterest p.a. 3.3 Suppose a parent was offered these loans and pad at 12% p.a. compounded monthly over one year. Calculate the monthly nstalment and the actual nterest pad on these loans. Comment on these two optons and how would you advse.
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