EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR

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1 EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR 8S CHAPTER 8 EXAMPLES EXAMPLE 8.4A THE INVESTMENT NEEDED TO REACH A PARTICULAR FUTURE VALUE What amount must you nvest now at 4% compoune monthly to accumulate $10,000 after Gven: j 4%, m 12, FV $10,000, Term 3.5 years Then j m 4% % an n m Term Enter the known varables an then compute the present value. 42 n FV 0 Answer: 8, years? Note that we entere the $10,000 as a postve value because t s the cash nflow you wll receve 3.5 years from now. The answer s negatve because t represents the nvestment (cash outflow) that must be mae toay. Roune to the cent, the ntal nvestment requre s $ EXAMPLE 8.4B CALCULATING AN EQUIVALENT VALUE OF TWO PAYMENTS Two payments of $10,000 each must be mae one year an four years from now. If money can earn 9% compoune monthly, what sngle payment two years from now woul be equvalent to the two scheule payments? Gven: j 9% compoune monthly makng m 12 an j m 9% % Other ata an the soluton strategy are shown on the tmelne below. FV 1 represents the future value of the frst scheule payment an 2 represents the present value of the secon payment Years $10,000 $10,000 = 0.75%, n = 24 = 0.75%, n = 12 2 FV 1

2 2 Example Problems Solve Usng the Sharp EL-733A Calculator The sngle equvalent payment s FV 1 2. Before we start crunchng numbers, let s exercse your ntuton. Do you thnk the equvalent payment wll be greater or smaller than $20,000? It s clear that FV 1 s greater than $10,000 an that 2 s less than $10,000. When the two amounts are ae, wll the sum be more than or less than $20,000? We can answer ths queston by comparng the tme ntervals through whch we shft each of the $10,000 payments. The frst payment wll have one year s growth ae but the secon payment wll be scounte by two years growth. Therefore, 2 s farther below $10,000 than FV 1 s above $10,000. Hence, the equvalent payment wll be less than $20,000. If your calculate equvalent payment turne out to be more than $20,000, you woul know that your soluton ha an error. Returnng to the calculatons, FV 1 : n 0 FV Answer: 10, : Do not clear the values an settngs currently n memory. Then you nee enter only those values an settngs that change FV 24 n Answer: 8, The equvalent payment two years from now s $10, $ $19, EXAMPLE 8.4C CALCULATING TWO UNKNOWN LOAN PAYMENTS Kramer borrowe $4000 from George at an nterest rate of 7% compoune semannually. The loan s to be repa n three nstalments. The frst payment, $1000, s ue two years after the ate of the loan. The secon an thr payments are ue three an fve years, respectvely, after the ntal loan. Calculate the amounts of the secon an thr payments f the secon payment s to be twce the sze of the thr payment. Gven: j 7% compoune semannually makng m 2 an j m 7% 2 3.5% Let x represent the thr payment. Then the secon payment must be 2x. As ncate n the followng agram, 1, 2, an 3 represent the present values of the frst, secon, an thr payments Years $4000 n = 4, = 3.5% $1000 n = 6, = 3.5% n = 10, = 3.5% 2x x Snce the sum of the present values of all payments equals the orgnal loan, then $ : 1000 FV 4 n Answer: At frst, we may be stumpe as to how to procee for 2 an 3. Let s thnk about the thr payment of x ollars. We can compute the present value of just $1 from the x ollars. 1 FV 10 n Answer: The present value of $1 pa fve years from now s $ (almost $0.71). Conser the followng questons (Q) an ther answers (A). Q: What s the present value of $2? A: It s about 2 $0.71 $1.42. Q: What s the present value of $5? A: It s about 5 $0.71 $3.55. Q: What s the present value of $x? A: Extenng the preceng pattern, the present value of $x s about x $0.71 $0.71x. Precsely, t s 3 $ x. 1

3 Example Problems Solve Usng the Sharp EL-733A Calculator 3 Smlarly, calculate the present value of $1 from the secon payment of 2x ollars. The only varable that changes from the prevous calculaton s n. 6 n Answer: Hence, the present value of $2x s 2 2x 1$ $ x Now, substtute the values for 1, 2, an 3 nto equaton 1 an solve for x. $ x x $ x $ x $ Kramer s secon payment wll be 21$ $ an the thr payment wll be $ EXAMPLE 8.5B ARING GICS HAVING DIFFERENT NOMINAL RATES Suppose a bank quotes nomnal annual nterest rates of 6.6% compoune annually, 6.5% compoune semannually, an 6.4% compoune monthly on fve-year compoun-nterest GICs. Whch rate shoul an nvestor choose? An nvestor shoul choose the rate that results n the hghest maturty value. The gven nformaton may be arrange n a table. j m j m n 6.6% 1 6.6% Choose an amount, say $1000, to nvest. Calculate the maturty values for the three alternatves. FV (1 ) n $1000(1.066) 5 $ for m 1 $1000(1.0325) 10 $ for m 2 $1000( ) 60 $ for m 12 Hereafter, we wll usually present the fnancal calculator keystrokes n a vertcal format. j 6.6% j 6.5% j 6.4% compoune compoune compoune annually semannually monthly / 0 n FV Same, 10 n 3.25 FV Ans: 1, Same, 60 n 0.53 FV Ans: 1, Ans: 1, In the secon an thr columns, we have shown only those values that change from the preceng step. The prevous values for an are automatcally retane f you o not clear the TVM memores. The nvestor shoul choose the GIC earnng 6.5% compoune semannually snce t prouces the hghest maturty value.

4 4 Example Problems Solve Usng the Sharp EL-733A Calculator 9S CHAPTER 9 EXAMPLES EXAMPLE 9.1A CALCULATING THE PERIODIC AND NOMINAL RATES OF INTEREST The maturty value of a three-year, $5000 compoun-nterest GIC s $ To three-fgure accuracy, calculate the nomnal rate of nterest pa on the GIC f nterest s compoune a. annually. b. quarterly. Gven: $5000 an FV $ In Part (a), m 1, n m(term) 1(3) 3 compounng peros. In Part (b), m 4, n m(term) 4(3) 12 compounng peros. Formula (9-1) enables us to calculate the nterest rate for one compounng pero. a. b. a FV b 1/n 1 a $ $ b 1/ % The nomnal rate of nterest on the GIC s j m 1(5.000%) 5.00% compoune annually. a $ $ b 1/ % The nomnal rate of nterest on the GIC s j m 4(1.227%) 4.91% compoune quarterly. 3 n / FV Ans: Same,, FV 12 n Ans: EXAMPLE 9.2A CALCULATING THE NUMBER OF OUNDING PERIODS What s the term of a compoun-nterest GIC f $4000 nveste at 5.5% compoune annually earns nterest totallng $ ? Gven: $4000 j Total nterest $ m 5.5% 1 5.5% The maturty value of the GIC s FV Total nterest $4000 $ $

5 Example Problems Solve Usng the Sharp EL-733A Calculator 5 Metho 1: Metho 2: Use the basc formula FV (1 ) n to calculate the number of compounng peros requre for $4000 to grow to $ Substtute the known values for, FV, an gvng $ $4000(1.055) n Therefore, n $ $ Now take logarthms of both ses. On the left se, use the rule that ln(a n ) n(ln a) Therefore, n(ln 1.055) ln ln an n ln Snce each compounng pero equals one year, the term of the GIC s fve years. Substtute the known values nto the erve formula (9-2). The number of compounng peros requre for $4000 to grow to $ s n ln a FV b ln 11 2 ln a $ $ b ln ln ln / FV n Ans: Snce each compounng pero equals one year, the term of the GIC s fve years. EXAMPLE 9.3A CONVERTING A NOMINAL INTEREST RATE TO AN EFFECTIVE INTEREST RATE What s the effectve rate of nterest corresponng to 10.5% compoune monthly? Gven: j 10.5% an m 12 Then j m 10.5% % per month an f (1 ) m % The effectve nterest rate s 11.02% (compoune annually) n / 0 FV Ans: f

6 6 Example Problems Solve Usng the Sharp EL-733A Calculator 10S CHAPTER 10 EXAMPLES EXAMPLE 10.2A THE FUTURE VALUE OF REGULAR INVESTMENTS Henz has been contrbutng $300 at the en of each month for the past 15 months to a savngs plan that earns 6% compoune monthly. What amount wll he have one year from now f he contnues wth the plan? The total amount wll be the future value of n contrbutons of $300 each. Payments an compounng both occur at one-month ntervals. Therefore, the payments form an ornary smple annuty havng 6% % per month. FV c 11 2n 1 $300 c $300 a b $ One year from now, Henz wll have $ n the plan n / FV Ans: 8, EXAMPLE 10.2B CALCULATING THE FUTURE VALUE WHEN THE RATE OF RETURN CHANGES DURING THE TERM OF THE ANNUITY Calculate the future value of an ornary annuty wth payments of $600 every 6 months for 16 years. The rate of return wll be 8% compoune semannually for the frst years an 9% compoune semannually for the subsequent years Because the compounng nterval an the payment nterval are both sx months, we have an ornary smple annuty wth 1 1 j for the frst 5 years, an 9% m 8% 2 4% 2 4.5% for the subsequent 10 years n m(term) 2(5.5) 11 for the frst 52 years, an n 2(10.5) 21 for the subsequent 102 years Snce the rate of return changes urng the term of the annuty, we must conser the frst years separately from the subsequent years. The algebrac soluton has three steps, as ncate n the followng tme agram Years $600 every 6 months n = 11 Step 1 $600 every 6 months FV 1 n = 21 Step 2 n = 21 Step 3 FV 3 FV 2 Sum

7 Example Problems Solve Usng the Sharp EL-733A Calculator 7 Step 1: Step 2: Step 3: Calculate the future value, FV 1, of the frst 11 payments. FV 1 c 11 2n 1 $600 c $600 c $ Determne the future value, FV 2, of the Step 1 result 1 after an atonal 102 years. FV 2 (1 ) n $ (1.045) 21 $20, Calculate the future value, FV 3, of the last 21 annuty payments. Then a FV 2 an FV 3. FV 3 $600 c $20, FV 2 FV 3 $40, The future value of the annuty s $40, $600 c n / FV Ans: 8, Same 21 n / FV Ans: 40, EXAMPLE 10.3A THE PRESENT VALUE OF AN ORDINARY SIMPLE ANNUITY Determne the present value of $500 pa at the en of each calenar quarter for 6% compoune quarterly years. Use a scount rate of Gven: $500, Term years, j 6% compoune quarterly Therefore, 6% 4 1.5% an n 4(6.5) n c $500 c $500 a b $10, The present value of the annuty s $10, Assume s are nflows. 26 n FV Ans: 10,699.32

8 8 Example Problems Solve Usng the Sharp EL-733A Calculator EXAMPLE 10.4A CALCULATING THE PRESENT VALUE OF A DEFERRED ANNUITY Mr. an Ms. Templeton are settng up a fun to help fnance ther granaughter s college eucaton. They want her to be able to wthraw $3000 every three months for three years after she starts college. Her frst wthrawal wll be years from now. If the fun can earn 7.2% compoune quarterly, what sngle amount contrbute toay wll prove for the wthrawals? The money the Templetons nvest now wll have years to grow before wthrawals start. Thereafter, further earnngs of money stll n the fun wll help support the peroc wthrawals. The one-tme up front contrbuton s the present value of the wthrawals. The tme agram s presente below. Vewe from toay, the wthrawals form a eferre annuty. In orer to have an ornary annuty followng the pero of eferral, the pero of eferral must en three months before the frst payment. Ths makes the pero of eferral only years Years Payments = Twelve $3000 payments n = 12 Snce payments an compounng both occur quarterly, we have a eferre smple annuty wth $3000 n an 7.2% 4 1.8% The present value of the payments 5 years from now s n 1 c $3000 a b $32, n FV Ans: 32, The present value of the payments toay s 2 FV(1 ) n $32,119.23(1.018) 21 $22, Same 21 n FV Ans: 22, The Templetons can prove the esre fnancal support for ther granaughter by puttng $22, nto the fun toay.

9 Example Problems Solve Usng the Sharp EL-733A Calculator 9 11S CHAPTER 11 EXAMPLES EXAMPLE 11.1A CALCULATING THE PERIODIC INVESTMENT NEEDED TO REACH A SAVINGS TARGET Markham Auto Boy wshes to accumulate a fun of $300,000 urng the next 18 months n orer to open at a secon locaton. At the en of each month, a fxe amount wll be nveste n a money market savngs account wth an nvestment ealer. What shoul the monthly nvestment be n orer to reach the savngs objectve? The plannng assumpton s that the account wll earn 3.6% compoune monthly. The savngs target of $300,000 represents the future value of the fxe monthly nvestments. Snce earnngs are compoune monthly, the en-of-month nvestments form an ornary smple annuty. We are gven Step 1: FV $300,000 n 18 an 3.6% % per month Step 2: Substtute the gven values nto formula (10-1). Step 3: $300,000 ( ) Step 4: FV c 11 2n 1 $300,000 c $300, $16, n ,000 FV Ans: 16, Markham Auto Boy shoul make monthly nvestments of $16, n orer to accumulate $300,000 after 18 months. EXAMPLE 11.1B CALCULATING THE PERIODIC LOAN PAYMENTS THAT FORM AN ORDINARY GENERAL ANNUITY A $5000 loan requres payments at the en of each quarter for four years. If the nterest rate on the loan s 9% compoune monthly, what s the sze of each payment? The orgnal loan equals the present value of all payments scounte at the loan s nterest rate. Snce nterest s compoune monthly an payments are mae at the en of each quarter, we have an ornary general annuty wth $5000 n an 9% % per month

10 10 Example Problems Solve Usng the Sharp EL-733A Calculator Step 1: Then, an 12 compounngs per year c 3 4 payments per year c per quarter Step 2: Substtute the preceng values nto formula (10-2) n c $5000 c Step 3: $ Step 4: $ $ The sze of each quarterly payment s $ n FV Ans: EXAMPLE 11.2A CALCULATING n GIVEN THE FUTURE VALUE OF AN ORDINARY GENERAL ANNUITY One month from now, Maurce wll make hs frst monthly contrbuton of $250 to an RRSP. Over the long run, he expects to earn 8% compoune annually. How long wll t take for the contrbutons an accrue nterest to reach $100,000? (Roun n to the next larger nteger.) Snce compounng occurs annually but the contrbutons are mae monthly, the payments form a general annuty havng FV $100,000 $250 an 8% 1 8% To obtan the peroc rate matchng the monthly payment nterval, frst calculate Then 1 compounng per year c payments per year c per month Substtute these values nto formula (10-1n). n ln a 1 FV b ln11 2 ln c $100,0002 $ ln / FV n Ans:

11 Example Problems Solve Usng the Sharp EL-733A Calculator 11 The annuty has 199 payments takng 199 months. We nee to express the tme requre n years an months. 199 months years years 16 years months2 16 years, 7 months It wll take 16 years an 7 months for Maurce to accumulate $100,000. EXAMPLE 11.3A FINDING THE RATE OF RETURN ON FUNDS USED TO PURCHASE AN ANNUITY A lfe nsurance company avertses that $50,000 wll purchase a 20-year annuty payng $ at the en of each month. What nomnal rate of return an effectve rate of return oes the annuty nvestment earn? The purchase prce of an annuty equals the present value of all payments. Hence, the rate of return on the $50,000 purchase prce s the scount rate that makes the present value of the payments equal to $50,000. The payments form an ornary annuty wth $50,000 $ m 12 an n 12(20) 240 Enter these values n your calculator as ncate n the box at rght. The peroc rate of return we obtan s 0.45% (per month). Then j m 12(0.45%) 5.40% compoune monthly an the corresponng effectve nterest rate s f (1 ) m % / n FV Ans: S CHAPTER 12 EXAMPLES EXAMPLE 12.1A CALCULATING THE FUTURE VALUE OF A SIMPLE ANNUITY DUE To the nearest ollar, how much wll Stan accumulate n hs RRSP by age 60 f he makes semannual contrbutons of $2000 startng on hs twenty-seventh brthay? Assume that the RRSP earns 8% compoune semannually an that no contrbuton s mae on hs sxteth brthay. The accumulate amount wll be the future value of the contrbutons on Stan s sxteth brthay. Vewe from the future value s focal ate at hs sxteth brthay, the RRSP contrbutons form an annuty ue. Snce the payment nterval equals the compounng nterval, we have a smple annuty ue wth $2000 8% 2 4% an n payments Substtute the preceng values nto formula (12-1) FV1ue2 c 11 2n $2000 a b $2000 a b $640,156 Stan wll have $640,156 n hs RRSP at age BGN moe 66 n / FV Ans: 640,156

12 12 Example Problems Solve Usng the Sharp EL-733A Calculator EXAMPLE 12.1B CALCULATING THE FUTURE VALUE OF A GENERAL ANNUITY DUE Repeat Example 12.1A wth the change that the RRSP earns 8% compoune annually nstea of semannually. We now have a general annuty snce the compounng nterval (one year) ffers from the payment nterval (sx months). The value we must use for n the FV formula s the peroc rate for the sx-month payment nterval. 8% (It wll be about 2 4%.) Substtute nto formula (9-4c) gvng Number of compounngs per year 8% 1 8% an c 1 Number of payments per year Use ths value for n formula (12-1) gvng 2 (1 ) c 1 (1.08) per sx months FV1ue2 c 11 2n $2000 a b $2000 a b $618,606 Stan wll have $618,606 n hs RRSP at age 60. BGN moe n / FV Ans: 618,606 EXAMPLE 12.3A CALCULATING THE SIZE OF LEASE PAYMENTS A lease that has years to run s recore on a company s books as a lablty of $27,369. If the company s cost of borrowng was 6% compoune monthly when the lease was sgne, what s the amount of the lease payment at the begnnng of each month? The book value of the lease lablty s the present value of the remanng lease payments. The scount rate employe shoul be the nterest rate the company woul have pa to borrow funs. The lease payments consttute a smple annuty ue wth 1ue2 $27, 369 n an 6% % per month Substtute the gven values nto formula (12-2) an solve for n 1ue2 c 11 2 $27, 369 a b ( )(1.005) ( ) $ The monthly lease payment s $ BGN moe 30 n FV Ans:

13 Example Problems Solve Usng the Sharp EL-733A Calculator 13 EXAMPLE 12.3E CALCULATING n GIVEN THE PRESENT VALUE OF A GENERAL ANNUITY DUE An nvestment fun s worth $210,000 an earns 9% compoune semannually. If $2000 s wthrawn at the begnnng of each month startng toay, when wll the fun become eplete? The ntal amount n the account equals the present value of the future wthrawals. Snce the frst wthrawal occurs toay, an the payment nterval ffers from the compounng nterval, the wthrawals form a general annuty ue havng The value we must use for n formula (12-2n) s the peroc rate for the one-month payment nterval. Substtute nto 2 (1 ) c 1 (1.045) Substtute the known values nto formula (12-2n). n $210, 0002 ln c 1 $ ln ue2 $210,000 $2000 an 9% 2 4.5% c Number of compounngs per year Number of payments per year ln c 1 1ue ln per month BGN moe / 0 FV Ans: The fun wll permt 199 monthly wthrawals. The fnal wthrawal, smaller than $2000, wll occur at the begnnng of the 199th payment nterval. But that wll be 198 months from now. So, the fun wll be eplete at the tme of the 199th payment, whch s 198 months or 16 years an 6 months from now. n EXAMPLE 12.3F CALCULATING THE INTEREST RATE FOR AN ANNUITY DUE Therese ntens to contrbute $3000 at the begnnng of each sx-month pero to an RRSP. What rate of return must her RRSP earn n orer to reach $600,000 after 25 years? The payments form an annuty ue whose future value after 25 years s to be $600,000. That s, FV(ue) $600,000 $3000 an n m(term) 2(25) 50 Enter these values n the calculator memores an compute. Ths gves the peroc nterest rate for one payment nterval (sx months). Then j m 2(4.713%) 9.43% compoune semannually. Therese s RRSP must earn 9.43% compoune semannually. BGN moe 50 0 n / FV Ans: 4.713

14 14 Example Problems Solve Usng the Sharp EL-733A Calculator EXAMPLE 12.3G CALCULATING THE INTEREST RATE BUILT INTO AN INSTALMENT PAYMENT OPTION A $100,000 lfe nsurance polcy requres an annual premum of $420 or a monthly premum of $ In ether case, the premum s payable at the begnnng of the pero of coverage. What s the effectve rate of nterest polcyholers pay when they choose the monthly payment plan? In effect, the nsurance company lens the $420 annual premum to polcyholers choosng the monthly payment opton. These polcyholers then repay the loan wth 12 begnnng-of-month payments of $ Hence, $420 s the present value of the 12 payments that form an annuty ue. We have (ue) $420 $37 an n 12 Enter these values n the calculator memory an compute. Ths gves the peroc nterest rate for one payment nterval (one month). Then f (1 ) m 1 ( ) % The effectve nterest rate on the monthly payment plan s 13.04%. 37 BGN moe 12 n / 0 FV Ans:

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