MMQ Problems Solutions with Calculators. Managerial Finance

Transcription

1 MMQ Problems Solutios with Calculators Maagerial Fiace 2008 Adrew Hall. MMQ Solutios With Calculators. Page 1

2 MMQ 1: Suppose Newma s spi lads o the prize of $100 to be collected i exactly 2 years, but that Newma wats to istead have moey to sped today ad ot wait. The Moey Machie ca trade future cash for cash today usig a aual iterest rate of 10% -- but how much cash? Kow Ukow 2 $100 2 PV??? r 10% 0.10 Preset Value of a Lump Sum: PV ( 1 + r) (3.6) APPS ENTER {Set Decimals to 6} [See News Page Chapter 14.] 2 d FINANCE 2 d DISP 6 [Sets the places of decimals displayed to 6] ALL [Clears all of the stored variables] 2 d FORMAT 6 ENTER 2 d QUIT TVM {Set Decimals to 6} [I have t got a Sharp, so look i the maual.} {Clear all variables} [look i the maual.] 100 [ 100 Store As ] [ 2 Store As N ] 10 I/YR [ 10 Store As I/YR ] 10 I/Y 10 i cursor i PV0 cursor i PV0 PV to get CPT PV to get COMP PV to get 2008 Adrew Hall. MMQ Solutios With Calculators. Page 2

3 MMQ 2: MMQ 2: Although the wheel lads o the amout of $100 to be received immediately, Newma would rather wait 2 years. How much will he receive i the mail 2 years from today i exchage for his prize? Kow Ukow PV $ ??? r 10% 0.10 Future Value of a Lump Sum: PV( 1 + r) (3.2) APPS ENTER 2 d FINANCE ALL [Clears all of the stored variables] TVM {Clear all variables} [look i the maual.] 100 PV [ 100 Store As PV ] 100 PV 100 PV [ 2 Store As N ] 10 I/YR [ 10 Store As I/YR ] 10 I/Y 10 i cursor i 0 cursor i 0 to get CPT to get COMP to get 2008 Adrew Hall. MMQ Solutios With Calculators. Page 3

4 MMQ 3: Remember Newma? He spis the wheel, ad receives $10,000 immediately. Alteratively, he ca leave it with the machie for 2 years to ear iterest of 10% compouded semi-aually. How much will he have? Kow Ukow PV $10, ??? r 10% 0.10 m2 We use formula PV PV m 1 + { rm} 1+ { } 4 [ ] 2 PV 5 2*2 Compoudig semi-aually at 10% per aum over two years is the same thig as compoudig periodically at 10% / 2 (where 2 is the # of compoudig periods i a year) or 5% per period over 4 periods: So we could use formula 3.2 but with r 5% ad 4 4 PV PV 10000* ( 1+ r) 4 ( ) ( ) * , All that is left is to crak the calculatio through the calculator APPS {Set Decimals to 2 d DISP 6 2 d FORMAT 6} [See News [Sets the 6 ENTER Page Chapter places of 14.] decimals 2 d QUIT displayed to 6] ENTER 2 d FINANCE ALL [Clears all of the stored variables] TVM {Set Decimals to 6} [I have t got a Sharp, so look i the maual.} {Clear all variables} [look i the maual.] 2008 Adrew Hall. MMQ Solutios With Calculators. Page 4

5 [ 10,000 Store As PV ] [ 2 Store As N ] 10 I/YR [ 10 Store As I/YR ] 10 I/Y 10 i cursor i 0 cursor i d xp/yr [ 2 Store As xp/yr ] to get {Set Coupos Per Period to 2} [see below] CPT to get {Set Coupos Per Period to 2} [look i the maual.] COMP to get Settig Coupos per period with the TI-BAII Plus. 2 d P/Y takes you ito a subroutie. The up ad dow keys toggle betwee settig P/Y ad settig C/Y. C/Y is coupos per period or the umber of times iterest is compouded i the period for which the iterest rate is give. Toggle to C/Y eter the value you wat (i our problem mc/y2). Hit the ENTER key. Leave the sub-routie by 2 d QUIT 2008 Adrew Hall. MMQ Solutios With Calculators. Page 5

6 MMQ 4: Retur to MMQ 3. Same problem, except that 10% iterest compouds daily. How much will he have? Kow Ukow PV $10, ??? r 10% 0.10 M365 (days i a regular year) We use formula PV PV PV m 1 + { rm} 365*2 1 + { } [ ] * , All that is left is to crak the calculatio through the calculator APPS {Set Decimals to 2 d DISP 6 2 d FORMAT 6} [See News [Sets the 6 ENTER Page Chapter places of 14.] decimals 2 d QUIT displayed to 6] ENTER 2 d FINANCE ALL [Clears all of the stored variables] TVM {Set Decimals to 6} [I have t got a Sharp, so look i the maual.} {Clear all variables} [look i the maual.] [ 10,000 Store As PV ] d P/YR [ 365 Store As P/YR ] 2 2 d xp/yr [ 2 * P/YR Store As 10 I/Y 10 i N ] {Set Coupos {Set Coupos 10 I/YR Per Period Per Period [ 10 Store As I/YR ] to 365} [see to 365} [look above] i the 2008 Adrew Hall. MMQ Solutios With Calculators. Page 6

7 maual.] cursor i PV0 cursor i PV0 PV to get CPT PV to get COMP PV to get MMQ 5: Retur to MMQ 3. Same problem, except that 10% iterest compouds cotiuously. How much will he have? Kow Ukow PV $10, ??? r 10% 0.10 m (ifiity) Actually there is aother formula Cotiuous Compoudig: * PV r e (3.4) Calculator Check! Do you have the e key? If yes, the lear how to use it. If ot, the you ll eed to thik of Formula (3.4) as: PV { r } So all we eed to do is crak the umbers: PV Usig e ad the formula o a TI-Calculator e r* 0.10* * * * [ ] , , Adrew Hall. MMQ Solutios With Calculators. Page 7

8 All that is left is to crak the calculatio through the calculator APPS {Set Decimals to 2 d DISP 6 2 d FORMAT 6} [See News [Sets the 6 ENTER Page Chapter places of 14.] decimals 2 d QUIT displayed to 6] ENTER 2 d FINANCE ALL [Clears all of the stored variables] TVM {Set Decimals to 6} [I have t got a Sharp, so look i the maual.} {Clear all variables} [look i the maual.] [ 10,000 Store As PV ] cursor i PV0 cursor i PV d P/YR [ Store As P/YR ] 2 2 d xp/yr [ 2 * P/YR Store As N ] 10 I/YR [ 10 Store As I/YR ] PV to get 10 I/Y {Set Coupos Per Period to } [see above MMQ12] CPT PV to get 10 i COMP PV to get 2008 Adrew Hall. MMQ Solutios With Calculators. Page 8

9 MMQ 6: Let s go back to MMQ 1, where we leared that $82.64 is equivalet to receivig $100 i two years at a particular iterest rate. Solve for that rate. Kow Ukow PV $ r??? 2 $ Solvig for the iterest rate: r PV 1 1 (3.10) So all we eed to do is crak the umbers: Usig the formula o a TI-Calculator r r r r 0.10 or 10 percet All that is left is to crak the calculatio through the calculator APPS {Set Decimals to 2 d DISP 6 2 d FORMAT {Set Decimals 6} [See News [Sets the 6 ENTER to 6} [I Page Chapter places of 2 d QUIT have t got a 14.] decimals Sharp, so look displayed to 6] i the maual.} 2008 Adrew Hall. MMQ Solutios With Calculators. Page 9

10 ENTER 2 d FINANCE ALL [Clears all of the stored variables] TVM {Clear all variables} [look i the maual.] PV [ Store As PV ] PV PV [ Store As ] [ 2 Store As N ] cursor i I%0 cursor i I%0 I/YR to get CPT I/Y to get COMP i to get 2008 Adrew Hall. MMQ Solutios With Calculators. Page 10

11 MMQ 7: The wheel lads o $100,000 to be received immediately. But what Newma really wats i life is to ope a Fat-Free Yogurt Store, ad he kows that to purchase such a store will take $165,000. If Newma takes the optio of leavig the moey with the Moey Machie, ad if the moey ears 10% aually, how log will he have to wait? Kow Ukow PV $100, r 0.10 or 10%??? 2 $165, Rearrage formula (3.2) PV ( 1+ r) to get (3.11) ( 1+ r) { PV } ad take the atural logarithm of both sides of the equatio to get: *l ( 1+ r) l PV Divide both sides by l(1+r) to get l PV l ( 1+ r) So all we eed to do is crak the umbers: l l ( ) ( ) l 1.65 Usig the formula o a TI-Calculator l( 1.10) Ufortuately his dream will take over 5 years to realize. All that is left is to crak the calculatio through the calculator 2008 Adrew Hall. MMQ Solutios With Calculators. Page 11

12 APPS {Set Decimals to 2 d DISP 6 2 d FORMAT 6} [See News [Sets the 6 ENTER Page Chapter places of 14.] decimals 2 d QUIT displayed to 6] ENTER 2 d FINANCE ALL [Clears all of the stored variables] TVM {Set Decimals to 6} [I have t got a Sharp, so look i the maual.} {Clear all variables} [look i the maual.] PV [ 100,000 Store As PV ] [ 165,000 Store As ] I/YR [ 10 Store As I/YR ] 10 I/Y 10 i cursor i N0 cursor i N0 N to get CPT N to get COMP N to get 2008 Adrew Hall. MMQ Solutios With Calculators. Page 12

13 MMQ 8: Suppose Newma s spi lads o the prize of $100 to be received three differet times, the first comig i oe year, the secod i 2 years, ad the third i 3 years. Of course, Newma has the optio of receivig the equivalet value of the prize today usig a aual iterest rate of 10% -- but how much cash? Kow Ukow 3 Future Paymets at yearly itervals Its a auity. A$100 PMT(o Calculators) 3 PVA??? r 10% 0.10 No cash right away so END mode Preset Value of a Auity: APPS ENTER PVA 1 1 A r r(1 + r) (3.8) 2 d FINANCE ALL TVM {Clear all variables} 100 PMT [ 100 Store As PMT ] 100 PMT 100 PMT cursor i PV0 cursor i PV0 3 N [ 3 Store As N ] 10 I/YR [ 10 Store As I/YR ] Toggle the MODE such that BGN does ot appear i display usig: 2 d BEG/END PV to get 3 N 10 I/Y 2 d BGN If i END mode 2 d QUIT If i BGN mode 2 d SET the 2 d QUIT CPT PV to get 3 N 10 i Look i Maual to establish what to do with Begi/Ed modes. COMP PV to get 2008 Adrew Hall. MMQ Solutios With Calculators. Page 13

14 MMQ 9: Suppose Newma s spi lads o the prize of $100 to be received three times, but this time the first paymet does ot begi util 2 years from today. This meas that the secod paymet comes i 3 years, ad the third paymet comes i 4 years. What amout today is equivalet to this prize (use a aual iterest rate r 10%)? Time 0 Time 1 Time 2 Time 3 Time 4 Period 1 Period 2 Period 3 Period 4 Receipts Time 0 is today. Period 1 is the first year. Time 1 is the ed of the first year. Period 2 is the secod year ad Time 2 is the ed of the secod year the time we get the first $100. Period 3 is the third year ad Time 3 is the ed of the third year the time we get the secod $100. Period 4 is the fourth year ad Time 4 is the ed of the fourth year the time we get the third $100. IF this were a auity over 4 time periods payable at the ed of the period we could process it o the calculator like MMQ2. Let us preted that today is actually Time 1, ot Time 0 Time 1 Time 2 Time 3 Time 4 Period 2 Period 3 Period 4 Receipts As we stad at Time 1, this ow looks like a auity payable at the ed of each of three periods. So, if we were i Time 1 we could use the calculators as i MMQ2 to calculate the value of the auity, as at Time 1 ad we would get $ as at Time 1. Time 0 Time 1 Time 2 Time 3 Time 4 Period 1 Period 2 Period 3 Period 4 Receipts PVA BUT, what we wat to kow is ot how much at Time 1, rather how much at Time 0. For this we ca use formula 3.6 ad the process we used i MMQ1 o our calculators Adrew Hall. MMQ Solutios With Calculators. Page 14

15 Time 0 Time 1 Time 2 Time 3 Time 4 Period 1 Period 2 Period 3 Period 4 Receipts PVA PV Rouded OR, we ca treat the problem as a auity over four time periods payable at the ed of the period, LESS the preset value of $100 payable at the ed of Period 1 Time 0 Time 1 Time 2 Time 3 Time 4 Period 1 Period 2 Period 3 Period 4 Auity PVA PV Net Rouded For PVA use your calculator as i MMQ2 ad for PV1 as i MMQ1. MMQ 10: Suppose Newma s spi lads o the prize of $100 to be received three times, but this time the first paymet does ot begi util 18 years from today. This meas that the secod paymet comes i 19 years, ad the third paymet comes i 20 years. What amout today is equivalet to this prize ( r 10%)? Period # Time# Receipts Auity Auity A20 - A Period # Time# Receipts Auity Auity A20 - A So we calculate the preset value of a auity over 20 years as i MMQ2. We calculate the preset value of a auity over 17 years also as i MMQ2. Subtract the 2008 Adrew Hall. MMQ Solutios With Calculators. Page 15

16 17 year auity s preset value from the 20 year auity s preset value ad that is the preset value of 3 paymets of $100, oe due at ed of 18; aother at the ed of 19 ad the last at the ed of 20 years. PVA PVA PVA20 PVA Roud to two places of decimals MMQ 11: Suppose Newma s spi lads o the prize of $100 to be received three times: the first i 1 year, the secod i two years, ad the third i four years (there is o year 3 paymet). What amout today is equivalet to this prize ( r 10%)? We ca treat the problem as a auity over four time periods payable at the ed of the period, LESS the preset value of $100 payable at the ed of Period 3. Time 0 Time 1 Time 2 Time 3 Time 4 Period 1 Period 2 Period 3 Period 4 Receipts Auity PVA PV Net Rouded MMQ 12: Newma spis the wheel ad it lads o the prize of $1 millio to be received immediately. Newma is scared however to get his hads o that much cash at oe time, ad cofesses this fear to the Moey Machie. Have o fear says the MM as I ca deliver the prize i whatever package you d prefer. How bout this: I ll give you $50,000 smakeroos today, the followed by a equal amout of moey to be received at the ed of each year for the ext 25 years. How s that soud Mr. Newma? Newma is overjoyed. Help him compute that equal amout usig a iterest rate of 10%. This is askig for a auity comprisig 50,000 NOW ad 25 lots of X. e at the ed of each of the ext 25 years. The 25 lots of X is a simple auity. The preset value of this simple auity together with the first paymet of $50,000 should be worth $1millio. So the PV of the simple auity should be $950, Adrew Hall. MMQ Solutios With Calculators. Page 16

17 Kow Ukow PVA $950, PMT??? r 10% 0.10 per aum 1 1 PVA A r r(1 + r) 1 1 r r(1 r) ad reorgaize to get + PVA $950, 000 A r r(1 + r) ( ) $950, 000 $950, (1.1) 0.1( ) $950, 000 $950, { } $950, 000 $104, Preset Value of a Auity: Divide both sides by { } (3.8) MMQ 13: Leavig Newma aside, suppose you purchase a ew car for $21,500, put $5,000 dow, ad the fiace the balace over 36 moths at a iterest rate of 6%. What will be the size of the mothly car loa paymet? The loa will be for $16,500. You kow the term of the loa is 36 moths, ad sice you are solvig for the mothly paymet you eed to compute the mothly rate of iterest o the loa:.06 /12.005, which is oe half of oe percet. The first loa paymet usually happes at the ed of the moth that the loa was grated i.e. paymets at the ed. Kow Ukow PVA $16, PMT??? r 6% 0.06 per aum or per moth or 0.5% per moth 2008 Adrew Hall. MMQ Solutios With Calculators. Page 17

18 PVA A Use 1 1 r r(1 r) + as we did for MMQ7 Or use the calculator, much as we did i MMQ2 2 d FINANCE APPS ENTER ALL TVM {Clear all variables} PV [ Store As PV ] PV PV cursor i PMT0 cursor i PMT0 36 N [ 3 Store As N ] 0.5 I/YR [ 0.5 Store As I/YR ] Toggle the MODE such that BGN does ot appear i display usig: 2 d BEG/END PMT to get 36 N 0.5 I/Y 2 d BGN If i END mode 2 d QUIT If i BGN mode 2 d SET the 2 d QUIT CPT PMT to get 36 N 0.5 i Look i Maual to establish what to do with Begi/Ed modes. COMP PMT to get 2008 Adrew Hall. MMQ Solutios With Calculators. Page 18