Solutions to TVM Practice Set II

Transcription

1 Solutions to TVM Pactice Set II. In this poblem you ae solving fo the monthly ate of etun and then the yealy ate of etun. Thee ae 228 months in 9 yeas (9 * 2 = 228). $3, a) To detemine the monthly ate of etun: ( ) =.04% $,00 N = 228, PV = -$,00, FV = $3,438; compute I $3,438 9 b) To detemine the yealy ate of etun : ( ) = 4% $,00 N = 9, PV = -$,00, FV = $3,438; compute I 2. You should disagee with Jim. You do not owe $,900 fo the entie yea since you ae making monthly payments that would educe the outstanding loan balance. If you total inteest payments ae $380, the tue ate of inteest is much highe than 20%. You would need to set up the poblem and solve fo (a monthly ate) in the PVIFA equation. You could convet the monthly ate to an A.P.R. by multiplying it by 2. The tue A.P.R. is 35.07%, not 20% as Jim is suggesting. 2 ( ) => $,900 = $90 * [ ] Now, use a financial calculato to solve fo the monthly ate of inteest (). On a financial calculato: N = 2, PV =,900, PMT = -90; compute I => = 2.923% pe month => 2.923% * 2 = 35.07% APR => The E.A.R on the loan would be found as follows: (.02923) 2 = 4.30% 3. Hee you must solve fo the E.A.R of an annuity due. The easiest way to solve this poblem is to set you calculato to BEGIN mode, diectly solve fo the monthly ate, and then convet the monthly ate to an effective ate. ON BEGIN MODE: N = 24, PV = $50, PMT = -$0; compute I I = 2.032% pe month Convet to an E.A.R.: (.02032) 2 = 27.77% Financial Management TVM pactice set II solutions

2 As an altenative, if you ecognize that the fist payment is at time zeo, you can solve the poblem as follows: 23 ( ) => $50 - $0 = $0 * [ ], which is equivalent to solving the poblem as if the pesent value wee $090 and thee ae 23 months of $0 payments. 23 ( ) => $090 = $0 * [ ]; solve fo on a financial calculato as follows: N = 23, PV = $090, PMT = -$0; compute I; I = 2.032% pe month, Convet to an E.A.R.: (.02032) 2 = 27.77% 4. Thee ae seveal ways to solve this poblem. Fist, you can use bute foce and discount all the cash flows back individually (YUCK!). Altenatively, you can see that thee ae effectively two annuity steams hee; a five peiod odinay annuity of $253 with the fist cash flow one peiod away and then a fou peiod odinay annuity of $253 with the fist cash flow stating 7 peiods fom now. Recalling that the annuity pesent value fomula assumes that payments stat at the end of the peiod (i.e., one peiod away), you can solve the fo pesent value of these cash flows as follows: $ 253* PVIFA (7.9%,4) => PV = $253 * PVIFA(7.9%,5) + () 5 4 () () => = $253 * [ ] + $253 * [ ] => = $, $532.9 = $,545.0 * [ ] () On a financial calculato: Step : Detemine the pesent value of the fist 5 payments: PMT = $253, N = 5, I = 7.9; compute PV = $,02.82 Step 2: Detemine value of the last 4 payments as of t = : PMT = $253, N = 4, I = 7.9; compute PV = $ This is the value at the end of th yea, so you must discount it moe yeas. FV = $839.84, N =, I = 7.9; compute PV = $532.9 Step 3: Add the two values togethe $, $532.9 = $,545.0 Financial Management TVM pactice set II solutions 2

3 Now, a simple appoach. Calculate the value of a 0-yea, $253 annuity and then subtact off the pesent value of the missing payment at t =. => PV = $253 * PVIFA(7.9%,0) - $253 * PVIF(7.9%,) 0 () $253 => $253 * [ ] - () => $, $0.32 = $,545.0 On a financial calculato: Step : Detemine the pesent value of a 0 yea annuity: PMT = $253, N = 0, I = 7.9; compute PV = $, Step 2: Detemine the pesent value of the th payment (single CF) that was missed: FV = $253, N =, I = 7.9; compute PV = $0.32 Step 3: Take the diffeence between these two values: $, $0.32 = $, Fifteen-pecent compounded monthly implies a monthly inteest ate of.5 / 2 =.025. Because the pepetuity fomula assumes that the fist cash flow is one peiod away, you can diectly find the pepetuity s value at t = 0 as follows: => PV = C/ = $257/.025 = $20,50. Fist, find the value as of t = 4 (ecall that the pepetuity fomula assumes that the fist cash flow occus one peiod away), then discount 4 peiods (months) to get a value as of t = 0. Note: 2% compounded monthly implies a monthly ate of % (.2/2). $845 => Value at t = 4: = $84,500.0 $84,500 => Value at t = 0: = $8, (.0) 7. a) Solve fo the monthly payment in each case and then take the diffeence: 30 (.00) 30 yea loan: $32,000 = $C * [ ]; $C = $ On a financial calculato: N = 30, I =., PV = 32,000; compute PMT 80 (.00) 5 yea loan: $32,000 = $C * [ ]; $C = $, Financial Management TVM pactice set II solutions 3

4 On a financial calculato: N = 80, I =., PV = 32,000; compute PMT The payments ae $305.2 ($, $89.00) highe pe month fo the 5 yea loan. b) ($89.00 * 30) - ($,20.2 * 80) = $0, c and d) The E.A.R.s ae both the same. Both loans have the same A.P.R. and both.072 ae compounded on a monthly basis: E.A.R. = [ ] 2 = 7.442% 2 8. Note that you will be financing 80% of the $40,000 puchase pice, o a total of $2,000. Also, the monthly ate is.0/2 = 05. => $C = $2,000 ; $C = (5) [ ] 5 On a financial calculato: N = 240, I =./2 =.55, PV = $2,000; compute PMT = $84.5 You will need to constuct an amotization schedule fo the fist 2 months of the loan in ode to answe these questions. A sample is povided below Amotization Schedule. Peiod Loan Balance Payment Inteest Reduction of Pincipal End Balance $2, $84.5 $.00 $225.5 $, $, $84.5 $4.7 $22.89 $, $,547.4 $84.5 $3.5 $228.4 $, $,39.32 $84.5 $2.2 $ $, $, $84.5 $0.99 $230. $0, $0, $84.5 $09.73 $23.92 $0, $0,27.35 $84.5 $08.45 $ $0, $0,394.5 $84.5 $07.7 $ $0, $0,59.7 $84.5 $05.88 $ $09, $09, $84.5 $04.58 $ $09,8.82 $09,8.82 $84.5 $03.28 $ $09, $09, $84.5 $0.97 $239.8 $09, $7, a) As you can see fom the amotization schedule, the total amount of inteest that will be paid ove the fist 2 months of the loan is $ b) The loan balance afte one yea of monthly payments would be $09, Financial Management TVM pactice set II solutions 4

5 9. MAKE A TIME LINE! => In 8 yeas (9 months), Betsy will have $00,000() 9 = $,44.27 => In 8 yeas (9 months), Betsy will need $32,000 +$558, = $90, () ($4,000 * [ ] = $558,323.09) => Betsy will have to make up the diffeence ($528,908.82) with monthly payments ove the 8 yea (9 month) peiod. () 9 => $528, = $C * [ ] $C = $4,30.07 Betsy must save $4,30.07 pe month fo the next 8 yeas (9 months). Wish Betsy luck! 0. Lance s contact does not include a signing bonus and can be valued as follows: 2 () The value of the fist yea s cash flows (as of t = 0) is $80,000 * [ ] $2,09, The value of the 2 nd yea s cash flows (as of t = ) is $2,09, *.03 = $2,54,50.0. The value of the 3 d yea s cash flows (as of t = 2) = $2,09, * (.03) 2 = $2,28, = The pesent value of Lance s contact is theefoe: $2,54,50.0 $2,28,774.5 => $ 2,09, = $,088, () () The value of Geoge s contact (befoe the signing bonus) can be found as follows: 2 () The value of the fist yea s cash flows (as of t = 0) is $20,000 * [ ] = $,394, The value of the 2 nd yea s cash flows (as of t = ) is $,394,27.85 *.05 = $,43, The value of the 3 d yea s cash flows (as of t = 2) = $,394,27.85 * (.05) 2 = $,537,84.7. The pesent value of Geoge s contact befoe the signing bonus is theefoe: $,43, $,537,84.7 => $,394,27.85 = $4,3, () () In ode fo the two deals to have the same pesent value, Geoge s signing bonus must have been: $,088, $4,3, = $,95, Financial Management TVM pactice set II solutions 5