10.2 Future Value and Present Value of an Ordinary Simple Annuity


 Neil Manning
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1 348 Chapter 10 Annutes 10.2 Future Value and Present Value of an Ordnary Smple Annuty In compound nterest, 'n' s the number of compoundng perods durng the term. In an ordnary smple annuty, payments are made at the end of each payment perod and the compoundng perod s equal to the payment perod. In ths secton, you wll learn how to calculate the future value and present value of an ordnary smple annuty. Future Value of an Ordnary Smple Annuty Consder an example where Abrella decdes to nvest $1000 at the end of every year for fve years n a savngs account that earns an nterest rate of 10% compounded annually. She wants to fnd out how much she would have at the end of the fveyear perod. In other words, she wants to fnd out the future value of her nvestments at the end of fve years. To calculate the future value of her nvestments at the end of fve years, we could calculate the future value of each of her nvestments usng the compound nterest formula and then add all the future values. From the compound nterest Formula 9.1(a), we know that the future value of each payment s FV = PV(1 + ) n where, FV = future value, PV = present value, = nterest rate for the compoundng perod (perodc nterest rate), and n = total number of compoundng perods for each payment. In ths example: 'PV' for each payment s $1000 j 010. '' for each payment s = = = 0.10 per annum m 1 'n' for each payment s not the same. 'n' for each payment startng from the 1 st payment s 4, 3, 2, 1, and 0. The future value of an annuty s the sum of the accumulated value of each perodc payment. Indvdual future values calculated are n geometrc seres wth 'n' terms, where the 1 st term s PMT and the common rato s (1+). By applyng the formula for the sum of a geometrc seres you wll get the smplfed 'FV' Formula 10.2 (a). Exhbt 10.2(a): Future Value of Ordnary Smple Annuty Payments Sum of the future values of her nvestment = = $ Therefore, f she nvests $1000 every year for fve years at 10% compounded annually n a savngs account, she would have a total of $ at the end of fve years.
2 Chapter 10 Annutes 349 Now, f there were many payments for an annuty (e.g., she nvested the same amount for 15 years, compounded annually), the above method would become too tmeconsumng. A smplfed formula to calculate the future value of an ordnary smple annuty s gven by: Formula 10.2(a) Future Value of an Ordnary Smple Annuty FV = PMT In an annuty, 'n' s the number of payments durng the term. Example 10.2(a) where 'n' s the number of payments durng the term, 'PMT' s the amount of the perodc payment, and '' s the perodc nterest rate. Calculatng the future value of her nvestment at the end of fve years usng ths smplfed formula, n = 1 payment/year # 5 years = 5 annual payments j 010. = = = 0.10 m 1 PMT = $1000 n ^1+ h  1 FV = PMT ; E 5 ^ h  1 = 1000; E 010. = 1000 [6.1051] = $ Therefore, the future value of the nvestment s $ In the calculaton of the future value of annutes, the amount of nterest s calculated as follows: Amount of Interest Earned = Future Value of the Investments  Amount Invested Over the Term I = FV  n(pmt) In ths example, I = ( ) = = $ Therefore, she would earn $ from ths nvestment. Calculatng the Future Value, Total Investment, and Interest Earned n an Ordnary Smple Annuty Rta nvested $200 at the end of every month for 20 years nto an RRSP. Assume that the nterest rate was constant at 6% compounded monthly over the entre term. () What was the accumulated value of the nvestment at the end of the term? () What was the total nvestment over the term? () What was the amount of nterest earned? Ths s an ordnary smple annuty as: Payments are made at the end of each payment perod (monthly) Compoundng perod (monthly) = payment perod (monthly)
3 350 Chapter 10 Annutes contnued () Usng Formula 10.2(a), ^1+ h n  1 FV = PMT ; E 240 ^ h  1 = 200 ; E = 200 [ ] = $92, Therefore, the accumulated value of the nvestment at the end of the term was $92, () Total Investment = $200 per month # 240 payments = $48, Therefore, the total nvestment over the term was $48, () Interest Earned = $92, $48, = $44, Therefore, the amount of nterest earned was $44, Usng the Fnancal Calculator to Solve Problems Before you start solvng problems, check and set the followng n the Texas Instruments BA II Plus fnancal calculator: 1. Set the number of decmals to 9. (Set ths only when you start usng the calculator.) Press 2ND then press FORMAT (ths s the secondary functon of the decmal key). Enter 9 then press ENTER to set the number of decmals to 9. Press 2ND then QUIT (ths s the secondary key above CPT). 2. Check the settngs for endofperod (ordnary annuty) or begnnngofperod (annuty due) calculatons. (You need to check ths before you work out problems.) Pressng 2ND then BGN (secondary functon above PMT) wll dsplay the current settng  ether END or BGN. (Select END for ordnary annuty problems and BGN for annuty due problems.) END would mean endofperod settng (for an ordnary annuty). To change ths settng to begnnngofperod settng (for annuty due), press 2ND then SET (secondary functon above ENTER). You can swtch between END and BGN by pressng 2ND then SET agan. If BGN settng s selected, a small BGN appears at the toprght corner of your screen and wll reman there throughout your calculatons. However, f END s selected, nothng wll appear. Once you select your settng, press 2ND then QUIT (secondary functon above CPT) and return to your calculaton.
4 Chapter 10 Annutes 351 Solvng Example 10.2(a)() usng the Texas Instruments BA II Plus calculator, to fnd 'FV' Where, P/Y = 12, C/Y = 12, N = 240, I/Y = 6, PMT = $200 CashFlow Sgn Conventon Transacton PV FV Investment Outflow () Inflow (+) Clear past values n the memory of the functon keys. Ths opens the P/Y, C/Y worksheet to set values. Set payments per year (P/Y) equal to compoundng perods per year (C/Y) as 12. You can scroll down usng the down arrow key to vew C/Y, whch wll be automatcally set to 12. Ths closes the P/Y, C/Y worksheet. Number of payments (n). Nomnal nterest rate per year (j). In annuty problems, we wll usually fnd ether FV or PV. To avod errors when you are fndng FV set PV to 0 and vce versa. Perodc payments can be cash nflows or outflows, so set the sgn accordngly. In ths problem, t s a cash outflow (money pad out for the nvestment), therefore, the perodc payment s negatve. Loan Inflow (+) Outflow () Example 10.2(b) When the payment date of the nvestment s not stated, t s assumed to be at the end of the payment perod. Calculatng the Future Value, Total Investment, and Interest Earned n an Ordnary Smple Annuty Combned wth a Compound Interest Perod Jack deposted $1500 nto an account every three months for a perod of four years. He then let the money grow for another sx years wthout nvestng any more money nto the account. The nterest rate on the account was 6% compounded quarterly for the frst four years and 9% compounded quarterly for the next sx years. Calculate () the accumulated amount of money n the account at the end of the 10year perod and () the total nterest earned. Ths s an ordnary smple annuty as: Payments are assumed to be at the end of each payment perod (quarterly) Compoundng perod (quarterly) = payment perod (quarterly)
5 352 Chapter 10 Annutes contnued () Usng Formula 10.2(a), n ^1+ h  1 FV = PMT ; E FV 1 = 1500 = ^ h G = 1500[ ] = $26, Ths 'FV 1 ' amount now becomes the present value for the compoundng perod. We now need to fnd the future value of ths amount usng the compound nterest formula. FV 2 = PV 2 (1 + ) n Here, 'n', the number of compoundng perods = m # t = 24 FV 2 = 26, ( ) 24 = $45, Therefore, the accumulated amount of money n the account at the end of the 10year perod was $45, () Total Invested = n(pmt) = 16 # = $24, Interest Earned = FV  n(pmt) = 45, , = $21, Therefore, the total nterest earned over the perod was $21, Example 10.2(c) Calculatng the Future Value when Payment Changes Grace saved $500 at the end of every month n an RRSP for fve years and thereafter $600 at the end of every month for the next three years. If the nvestment was growng at 3% compounded monthly, calculate the maturty value of her RRSP at the end of eght years. Ths s an ordnary smple annuty as: Payments are made at the end of each payment perod (monthly) Compoundng perod (monthly) = payment perod (monthly)
6 Chapter 10 Annutes 353 contnued Calculatng the future value of PMT () at the end of the fve years Usng Formula 10.2(a), n ^1+ h  1 FV = PMT ; E FV 1 = ^ h 1 ; E = 500[ ] = $32, FV 1 becomes the present value for the compoundng perod. We now need to fnd the future value of ths amount usng the compound nterest formula. FV 2 = PV 2 (1 + ) n Here 'n', the number of compoundng perods = m # t = 36 FV 2 = 32, ( ) 36 = $35, Calculatng the future value of PMT () at the end of the term: Usng Formula 10.2(a), n ^1+ h  1 FV = PMT ; E ^ h 1 FV 3 = 600 ; E = 600[ ] = $22, Maturty value of nvestment at the end of the term: = FV 2 + FV 3 = 35, , =$57, Therefore, the maturty value of her nvestment at the end of eght years s $57, Present Value of an Ordnary Smple Annuty Consder an example where Margaret wshes to wthdraw $1000 at the end of every year for the next fve years from an account that pays nterest at 10% compounded annually. How much money should she depost nto ths account now? To calculate the present value of all the payments at the begnnng of the fveyear perod, we can calculate the present value of each payment and then add all the present values usng the compound nterest formula as shown below: As payments that she would be recevng n the future have to be dscounted, the present value of FV each payment s gven by the compound nterest formula: PV = n = FV(1 + ) n ^1 + h where 'n' s the number of compoundng perods for each payment. 'n' for each payment startng from the 1 st payment s 1,2,3,4, and 5.
7 354 Chapter 10 Annutes Indvdual present values calculated are n geometrc seres wth 'n' terms, where the 1 st term s 'PMT' and the common rato s (1+ ) 1. By applyng the formula for the sum of a geometrc seres you wll get the smplfed 'PV' Formula 10.2 (b). The present value of an annuty s the sum of the dscounted values of each perodc payment. Formula 10.2(b) Exhbt 10.2(b): Present Value of Ordnary Smple Annuty Payments Sum of Present Values of her Investment = = $ Now, smlar to the smplfed formula used n FV calculatons, the smplfed PV formula s gven by: Present Value of an Ordnary Smple Annuty PV = PMT where 'n' s the number of payments durng the term and 'PMT' s the perodc payment. Calculatng the present value of her payments usng ths smplfed formula: 1 ^ h 5 PV = 1000 ; E = 1000[ ] 0.10 = $ Therefore, she would have to depost $ at the begnnng of the fveyear perod to be able to wthdraw $1000 at the end of each year for fve years. In calculatng the present value of an annuty, the amount of nterest s calculated as follows: Amount of Interest Earned = Amount Receved over the Term  Present Value I = n(pmt)  PV In ths example, I = 5( )  $ = $ = $ Therefore, she would earn nterest of $ Example 10.2(d) Calculatng the Present Value and Interest Earned n an Ordnary Smple Annuty Zack purchased an annuty that provded hm wth payments of $1000 every month for 25 years at 5.4% compounded monthly. () How much dd he pay for the annuty? () What was the total amount receved from the annuty and how much of ths amount was the nterest earned? Ths s an ordnary smple annuty as: Payments are assumed to be made at the end of each payment perod (monthly) Compoundng perod (monthly) = payment perod (monthly)
8 Chapter 10 Annutes 355 contnued () Usng Formula 10.2(b), 1 ^1+ h  n PV = PMT ; E 1 ^ h 300 = 1000 ; E = 1000 [ ] = $164, Therefore, he pad $164, for the annuty. () Amount Receved = n(pmt) = 300 # $ = $300, Interest Earned = n(pmt)  PV = 300, , = $135, Example 10.2(e) Calculatng the Present Value, Amount Invested, and Total Interest Charged n an Ordnary Smple Annuty Andrew pad $20,000 as a down payment towards the purchase of a machne and receved a loan for the balance amount at an nterest rate of 3% compounded monthly. He settled the loan n ten years by payng $1500 at the end of every month. () What was the purchase prce of the machne? () What was the total amount pad to settle the loan and what was the amount of nterest charged? Ths s an ordnary smple annuty as: Payments are made at the end of each payment perod (monthly) Compoundng perod (monthly) = payment perod (monthly)
9 356 Chapter 10 Annutes contnued () Usng Formula 10.2(b),  n 1 ^1+ h PV = PMT ; E 1. = ^ h ; 120 E = 1500[ ] = 155, = $155, As he pad $20, as down payment for the machne: Purchase Prce = Down Payment + PV of All Payments = 20, , = $175, Therefore, the purchase prce of the machne was $175, () Calculatng the total amount pad and the nterest amount Amount Pad = n(pmt) = 120 # = $180, Interest Charged = n(pmt)  PV = 180, , = $24, Therefore, the total amount pad to settle the loan was $180, and the amount of nterest charged was $24, Example 10.2(f) Calculatng the Present Value when the Interest Rate Changes How much should Halfax Steel Inc. nvest today n a fund to be able to wthdraw $15,000 at the end of every three months for a perod of sx years? The money n the fund s expected to grow at 4.8% compounded quarterly for the frst two years and 5.6% compounded quarterly for the next four years. Ths s an ordnary smple annuty as: Wthdrawals are made at the end of each payment perod (quarterly) Compoundng perod (quarterly) = payment perod (quarterly)
10 Chapter 10 Annutes 357 contnued Calculatng PV when nterest rate s 4.8% compounded quarterly Usng Formula 10.2(b),  n 1 ^1+ h PV = PMT ; E PV 1 = 15,000 ; 1  ( ) E = 15,000[ ] = $113, Calculatng PV when nterest rate s 5.6% compounded quarterly Usng Formula 10.2(b),  n 1 ^1+ h PV = PMT ; E PV 2 = 15,000 ; 1  ( ) E = 15,000[ ] = $213, PV 2 becomes the future value for the compounded perod. We now need to fnd the present value of that amount usng the compound nterest formula. PV 3 = FV 3 (1 + ) n Intal value of the nvestment = 213, ( ) 8 = $194, = PV 1 + PV 3 = 113, , = 308, Therefore, n order to be able to wthdraw $15,000 at the end of every three months for sx years, Halfax Steel Inc. should nvest $308, today Exercses Answers to the oddnumbered problems are avalable at the end of the textbook 1. Calculate the future value of each of the followng ordnary smple annutes: Perodc Payment Payment Perod Term of Annuty Interest Rate Compoundng Frequency a. $2500 Every year 10 years 4.50% Annually b. $1750 Every 6 months 7.5 years 5.10% Semannually c. $900 Every 3 months 5 years 4.60% Quarterly d. $475 Every month 4.5 years 6.00% Monthly
11 358 Chapter 10 Annutes 2. Calculate the future value of each of the followng ordnary smple annutes: Perodc Payment Payment Perod Term of Annuty Interest Rate Compoundng Frequency a. $4500 Every year 12 years 4.75% Annually b. $2250 Every 6 months 6 years 5.00% Semannually c. $800 Every 3 months 8.5 years 4.80% Quarterly d. $350 Every month 10 years 3 months 5.76% Monthly 3. Calculate the present value of each of the ordnary smple annutes n Problem Calculate the present value of each of the ordnary smple annutes n Problem Alana saved $50 at the end of every month n her savngs account at 6% compounded monthly for fve years. a. What s the accumulated value of the money at the end of fve years? b. What s the nterest earned? 6. Sharleen contrbuted $400 towards an RRSP at the end of every month for four years at 2.5% compounded monthly. a. What s the accumulated value of the money at the end of four years? b. What s the nterest amount earned? 7. Shanelle saves $600 at the end of every month n an RESP at 4.5% compounded monthly for 15 years for her chld's educaton. a. How much wll she have at the end of 15 years? b. If she leaves the accumulated money n the savngs account for another two years, earnng the same nterest rate, how much wll she have at the end of the perod? 8. Lue makes deposts of $250 at the end of every month for ten years n a savngs account at 3.5% compounded monthly. a. How much wll he have at the end of ten years? b. If he plans to leave the accumulated amount n the account for another fve years at the same nterest rate, how much wll he have at the end of the perod? 9. Carre saved $750 of her salary at the end of every month n an RRSP earnng 4% compounded monthly for 20 years. How much more would she have earned f she had saved ths amount n an RRSP that was earnng an nterest rate of 4.25% compounded monthly? 10. Adran nvests $500 at the end of every three months n a savngs account at 6% compounded quarterly for 7 years and 9 months. How much more would he have earned f he had saved t n a fund that was provdng an nterest rate of 6.5% compounded quarterly? 11. What s the dscounted value of the followng stream of payments: $3000 receved at the end of every 3 months for 10 years and 6 months at 3% compounded quarterly? 12. What s the dscounted value of the followng stream of payments: $1250 receved at the end of every month for 3 years and 2 months? Assume that money s worth 2.75% compounded monthly. 13. How much should Cortland have n a savngs account that s earnng 4% compounded monthly f he plans to wthdraw $1500 from the account at the end of every month for ten years? 14. Calculate the amount of money that Chn Ho should depost n an nvestment account that s growng at 6% compounded monthly to be able to wthdraw $700 at the end of every month for four years.
12 Chapter 10 Annutes Adler took a loan from a bank at 7% compounded monthly to purchase a car. He was requred to pay the bank $300 at the end of every month for the next three years. What was the cash prce of the car? 16. What would be the purchase prce of an annuty that provdes $500 at the end of every month for fve years and earns an nterest rate of 4% compounded monthly? 17. Calculate the accumulated value of annuty contrbutons of $500 at the end of every month for fve years followed by contrbutons of $750 at the end of every month for the next four years f money s worth 4.2% compounded monthly. 18. Kumar nvested $1000 at the end of every sx months for sx years followed by $1250 at the end of every sx months for the next three years nto a fund that earns 3.25% compounded semannually. Calculate the accumulated amount at the end of nne years. 19. Donovan contrbuted $900 at the end of every three months for seven years nto an RRSP fund that earned nterest at 3.9% compounded quarterly for the frst four years and 3.8% compounded quarterly for the next three years. Calculate the accumulated value of the contrbuton at the end of seven years and the amount of nterest earned. 20. Calculate the accumulated value and the amount of nterest earned on deposts of $125 made at the end of every month nto an RESP fund for 12 years f the fund earned nterest at 3.75% compounded monthly for 7 years and 4.35% compounded monthly for the next 5 years. 21. Jordan nvested $1500 at the end of every sx months for four years and then $750 at the end of every three months for the next two years. The nvestment earned nterest at 5% compounded semannually the frst four years and 4.8% compounded quarterly for the next two years. Calculate the accumulated value at the end of sx years and the amount of nterest earned. 22. Calculate the accumulated value of an annuty wth payments of $1500 at the end of every three months for three years at 4.1% compounded quarterly and $1750 at the end of every three months for the next two years at 4.25% compounded quarterly. 23. Falco Inc. pad $25,000 as a down payment for a machne. The balance amount was fnanced wth a loan at 3.25% compounded semannually, whch requred a payment of $9000 at the end of every sx months for fve years to settle the loan. a. What was the purchase prce of the machne? b. What was the total amount of nterest charged? 24. Brandon purchased a computercontrolled machne for hs machne shop by payng a down payment of $17,500. He fnanced the balance amount wth a loan at 4.75% compounded semannually, whch requred a payment of $10,000 at the end of every sx months for three years. a. What was the purchase prce of the machne? b. What was the total amount of nterest charged? 25. How much would you have to pay now for a retrement annuty that would provde $3000 at the end of every three months at 5% compounded quarterly for the frst ten years and $2500 at the end of each month at 6% compounded monthly for the followng fve years? 26. How much should Drake pay today for a retrement annuty that would provde hm wth $4000 at the end of every month for fve years at 3.5% compounded monthly and $20,000 every sx months for the next ten years at 4% compounded semannually? 27. Kayla wanted to purchase a storage locker for $5000 at her apartment buldng. She could ether pay the entre amount or take a loan from the bank for ths amount at 6.5% compounded monthly. She would have to pay $150 every month to settle the loan n four years. Whch opton should she choose and why? 28. Ronald and Jll were wonderng f they should pay $30,000 for a parkng space that was for sale n ther condomnum buldng or take a loan from the bank at 4% compounded monthly for fve years, payng monthly repayment amounts of $460. Whch opton should they choose and why?
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