Understanding Financial Management: A Practical Guide Problems and Answers

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Aubrey Richard
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1 Udestadig Fiacial Maagemet: A Pactical Guide Poblems ad Aswes Chapte 4 Time Value of Moey Note: You ca use a fiacial calculato to check the aswes to each poblem. 4.2 Futue Value of a Peset Amout. If a ivesto deposits $00,000 today, how much will she have 0 yeas fom today if she eas a aual iteest ate of (A) 6%, (B) 8%, ad (C) 0%? 4.3 Peset Value of a Futue Amout 2. What is the peset value of a $750,000 paymet that is expected to be eceived 20 yeas fom today assumig a aual discout ate of (a) 6%, (b) 8%, ad (c) 0%? 3. To settle a legal dispute, Lemo Ic. has ageed to pay damages to a competito of $.25 millio oe yea fom today, $.5 millio two yeas fom today, ad $.75 millio thee yeas fom today. At a discout ate of 6%, what is the peset value of the settlemet paymets? 4.4 Futue Value of a Auity 4. Jaze Cop plas to deposit $2 millio at the ed of each of the ext 5 yeas ito a sikig fud accout i ode to have sufficiet fuds to etie a lage bod issue. If Jaze ca ea 8% pe yea o these fuds, how much will the fim have i the sikig fud accout 5 yeas fom today? 5. To fud a futue lage capital expeditue, Lexo Ic. deposited $ millio today ad plas to also deposit $500,000 at the ed of each yea fo the ext eight yeas. If Lexo ca ea 9% aual etu o these deposits, how much will the fim accumulate eight yeas fom today? 4.5 Peset Value of a Auity 6. Compute the peset value of 20-yea auity with aual paymets of $20,000 usig a discout ate of 8%. A. Assume paymets occu at the ed of each yea. B. Assume paymets occu at the begiig of each yea.

2 7. Xyla Compay pomises to pay a etiig employee $0,000 at the ed of each yea fo the ext fou yeas followed by $5,000 at the ed of each yea fo six moe yeas. What is the peset value of these futue cash flows assumig a discout ate of 9%? 4.6 Peset Value of a Pepetuity 8. Ifiity Compute Ic. has just issued a pefeed stock with o matuity that pomises to pay divideds of $80 a yea. If ivestos equie a 0% ate of etu, what is the peset value of these futue divided paymets? 4.7 Compoudig Fequecies 9. A small fim has ecetly ivested $50,000. If the fim expects to eceive a etu of 0% pe yea, compouded semi-aually, o this ivestmet, what will be the futue value of the ivestmet five yeas fom today? What if iteest is compouded quately? 0. A ivesto cuetly has $25,000 i his ivestmet accout. He plas to deposit a additioal $,000 at the ed of each moth fo the ext fou yeas. If he ca ea 9% pe yea, compouded mothly, how much will the accout be woth fou yeas fom ow?. Fist Motgage Ic. loaed a compay $300,000 at a aual iteest ate (compouded mothly) of 2%. What is the ed-of-the-moth paymet ove the 30-yea life of the motgage loa? 2. A ivesto has deposited $20,000 ito a accout that pays 9.5% pe yea, compouded cotiuously. How much will the accout be woth six yeas fom today? 4.8 Nomial ad Effective Iteest Rates 3. AMV Bak is offeig a motgage ate of 6.25% a yea. If the bak equies ed-of-the moth paymets, what is the effective aual iteest ate o the motgage? 4.9 Solvig fo a Ukow Iteest Rate 4. A fim just epaid a $ millio loa by makig five equal, ed-of-yea paymets of $300,000 i yeas though 5. What is the implied iteest ate o this loa? 5. A ivestmet offes to pay ivestos $25 at the ed of each yea fo 0 yeas ad to make a additioal lump-sum paymet of $,000 at the ed of yea 0. The ivestmet is cuetly sellig at a pice of $925. What is the implied iteest ate o this ivestmet? 6. A auto deale is offeig the followig deal o a $2,350 ca: o dow paymet; $245 pe moth paymet fo the fist thee yeas; $295 pe moth paymet fo the followig two yeas. What iteest ate is implied by this deal? 2

3 4.0 Othe Time Value Applicatios 7. Geeous Motos is offeig its customes two fiacig choices o its popula lie of Vetua automobiles. Ude Optio A, customes eceive a $,000 ebate. Ude Optio B, customes eceive a special fiacig ate of 2.4% pe yea, compouded mothly. Assume a custome who chooses Optio A ca fiace the full puchase pice of the ca ove a fou-yea peiod at a iteest ate of 9% pe yea, compouded mothly. Assume the custome will puchase the ca ad keep it fo fou yeas ad ca fiace the etie puchase pice. A. Assume the puchase pice of the ew ca is $20,000 ad the custome plas to keep the ca fo the etie fou yeas. Which optio should you choose? B. What ebate amout would make the custome idiffeet betwee Optio A ad Optio B? 3

4 Aswes A. The futue value of a peset amout: FV PV( + ) 0 FV PV( + ) $00,000(.06) $79, B. FV $00,000(.08) 0 $25, C. FV $00,000(.0) 0 $259, A. The peset value of a futue amout: PV PV FV $750, ( + ) (.06) FV ( + ) $233, B. 2C. PV $750,000 $60, (.08) PV $750,000 $, (.0) 3. Calculate the peset value of each futue amout ad the sum these amouts: PV $,250,000 + $,500,000 $,750, (.06) (.06) (.06) $3,983, ( ) + 4. The futue value of the auity: FV PMT FV $2,000,000 (.08) 5 $54,304, Calculate the FV value, the FV of a odiay auity ad the sum these amouts: $7,506, ( ) FV $,000, $500,000 (.09) 8 $,992, $5,54,

5 6A. The peset value of a odiay auity: PV PMT $20, ( + ) (.08) 6B. The peset value of a auity due: PV PMT ( + ) ( + k) 20 $20, $96, (.08) 20 (.08) $22, Compute the PV of both auities ad the add them togethe: 4 6 (.09) (.09) PV $0,000 + $5, (.09) $80, The peset value of the pepetuity: PMT $80 PV $ $32, $47, The FV of the ivestmet assumig semi-aual aual compoudig: FV PV 0 ( + ) $50,000(.05) $8, With quately compoudig: FV $50, ( ) 20 $8, The futue value of this ivestmet accout is: $93, ( ) FV $25, $,000 (.0075) 48 $35, $57, The loa amout is a PV. We use the peset value of a auity equatio to solve fo the paymet: 5

6 PV PMT ( + ) $300,000 PMT 0.0 (.0) 360 Reaagig ad solvig fo PMT, the mothly paymet is $ The futue value amout usig cotiuous compoudig: FV PVe $20,000e (.095 )( 6) (.57) $20,000e $35, Solve the effective aual iteest ate usig m 2 peiods pe yea: m om eff + m o 6.43% 4. Solve the peset value of a auity fomula fo the ukow iteest ate: $,000,000 $300,000 ( + ) Usig tial ad eo: guess 5.24% $,000,000 $300, (.524) Theefoe, the implied iteest ate is 5.24%. 5 Usig the BA II Plus calculato, the coect iteest ate is foud as follows:. 2d CLR TVM 2. 5 N 3.,000,000 PV ,000 +/- 5. CPT I/Y 6

7 The implied iteest ate is 5.24%. 5. Solve fo the ukow iteest ate: $ 925 $25 ( + ) + $, ( + ) Solve fo usig tial ad eo. Guess 3.93% $ 925 $ (.393) $, (.393) Theefoe, the implied iteest ate is 3.93%. Usig the BA II Plus calculato, the coect iteest ate is foud as follows:. 2d CLR TVM 2. 0 N PV /- PMT 5.,000 +/- FV 6. CPT I/Y The implied iteest ate is 3.93%. 6. Solve fo the ukow implied iteest ate: 2, ( + ) ( + ) ( + ) Solvig fo the ukow iteest ate algebaically ivolves a iteative pocedue. Usig tial ad eo, the esult is 0.829% (.00829) (.00829) , (.00829) 7

8 Thus, 0.829% pe moth. 9.95% pe yea ( ) Usig the BA II Plus calculato, the coect iteest ate is foud as follows:. [CF] [2 d ] [CLR WORK] 2. 2,350 [+/-] [ENTER] 3. [ ] 245 [ENTER] 4. [ ] 36 [ENTER] 5. [ ] 295 [ENTER] 6. [ ] 24 [ENTER] 7. [IRR] [CPT] Thus, 0.829% pe moth. 9.95% pe yea ( ). 7A. Compute the mothly paymet fo each optio usig PV of auity fomula: PV PMT PMT ( + ) PV ( + ) Optio A : PMT $9, (.0075) 48 $ Optio B : PMT $20,000 (.002) $ The custome should choose Optio B with the lowe mothly paymet of $ vesus $ ove the 48 moth peiod. 8

9 7B. Solve fo the PV amout ude Optio A that would give a mothly paymet of $437.40: Optio A : $20,000 Rebate PMT $ (.0075) Discoutig $ fo 48 peiods at 0.75% a moth gives a peset value of $7, Thus, $20, $7, $2,423.8, which is the amout of the ebate. 9

Financial Math Lesson #2

Financial Math Lesson #2 Compoud Iteest Fiacial Math Lesso #2 Compoud iteest is a method of calculatig iteest i which the iteest is added to the picipal each peiod so that the picipal cotiues to gow thoughout the life of the loa