Question

The Altoona Company issued a 25-year bond 5 years ago with a face value of $1,000. The bond pays interest semiannually at a 9.4% annual rate.

a. What is the bond's price today if the interest rate on comparable new issues is 12%? Do not round intermediate calculations. Round PVFA and PVF values in intermediate calculations to four decimal places. Round the answer to the nearest cent. $

b. What is the price today if the interest rate is 8%? Do not round intermediate calculations. Round PVFA and PVF values in intermediate calculations to four decimal places. Round the answer to the nearest cent. $

c. Explain the results of parts (a) and (b) in terms of opportunities available to investors.

d. What is the price today if the interest rate is 9.4%? Round the answer to the nearest cent.

Comment on the answer to part (d).

Answer 1

Part A)

Coupon rate x par value Price = a Coupon rate x par value par value +...+- (1 + 5)2x0.5 (1 + 5)2x20 (1 + 5)2x20

0.094 x 1000 0.091 x 1000 1000 Price = (1 + 0.12 2x0,5 + ... + + 0.12 2x20 + (1 + 0.12 2x 20

• As the bond pays semi annual coupons, the coupon rate and the discount rate are divided by 2 and time is multiplied by 2
• As 20 years are remaining the maturity =20 years and number of periods = 20 x 2 = 40
• Price comes to  $804.40

Period CF Discount Factor 11 $ 47.00 1/(1+0.12/2)^1= 2 $ 47.00 1/(1+0.12/2)^2= 3 $ 47.00 1/(1+0.12/2)^3= 41 $ 47.00 1/(1+0.12/2)^4= 5 $ 47.00 1/(1+0.12/2)^5= 6 $ 47.00 1/(1+0.12/2)^6= 71 $ 47.00 1/(1+0.12/2)^7= 8 $ 47.00 1/(1+0.12/2)^8= 9 $ 47.00 1/(1+0.12/2)^9= 10 $ 47.00 1/(1+0.12/2)^10= 111 $ 47.00 1/(1+0.12/2)^11= 121 $ 47.00 1/(1+0.12/2)^12= 13 $ 47.00 1/(1+0.12/2)^13= 14 $ 47.00 1/(1+0.12/2)^14= 151 $ 47.00 1/(1+0.12/2)^15= 16 $ 47.00 1/(1+0.12/2)^16= 17 $ 47.00 1/(1+0.12/2)^17= 18 $ 47.00 1/(1+0.12/2)^18= 19 $ 47.00 1/(1+0.12/2)^19= 20 $ 47.00 1/(1+0.12/2)^20= 21 $ 47.00 1/(1+0.12/2)^21= 22 $ 47.00 (1/(1+0.12/2)^22= 231 $ 47.00 1/(1+0.12/2)^23= 24 $ 47.00 1/(1+0.12/2)^24= 25 $ 47.00 1/(1+0.12/2)^25= 26 $ 47.00 1/(1+0.12/2)^26= 271 $ 47.00 1/(1+0.12/2)^27= 28 $ 47.00 1/(1+0.12/2)^28= 29 $ 47.00 1/(1+0.12/2)^29= 30 $ 47.00 1/(1+0.12/2)^30= 31$ 47.00 1/(1+0.12/2)^31= 32 $ 47.00 (1/(1+0.12/2)^32= 33 $ 47.00 1/(1+0.12/2)^33= 34 $ 47.00 1/(1+0.12/2)^34= 35 $ 47.00 (1/(1+0.12/2)^35= 36 $ 47.00 1/(1+0.12/2)^36= 371 $ 47.00 1/(1+0.12/2)^37= 38 $ 47.00 1/(1+0.12/2)^38= 39 $ 47.00 1/(1+0.12/2)^39= 40 $ 1,047.00 1/(1+0.12/2)^40= Discounted CF 0.9434 0.943396226415094*47= $ 44.34 0.8900 0.88999644001424*47= $ 41.83 0.8396 0.839619283032302*47= $ 39.46 0.7921 0.79209366323802*47= $ 37.23 0.7473|0.747258172866057*47= $ 35.12 0.7050 0.704960540439676*47= $ 33.13 0.6651 0.665057113622336*47= $ 31.26 0.6274 0.627412371341826*47= $ 29.49 0.5919 0.591898463530025*47= $ 27.82 0.5584 0.558394776915118*47= $ 26.24 0.5268|0.526787525391621*47= $ 24.76 0.4970 0.496969363577001*47= $ 23.36 0.4688 0.468839022242453*47= $ 22.04 0.4423 0.442300964379673*47= $ 20.79 0.4173 0.41726506073554*47= $ 19.61 0.39360.393646283712774*47= $ 18.50 0.3714 0.371364418596957*47= $ 17.45 0.3503 0.350343791129204*47= $ 16.47 0.3305 0.330513010499249*47= $ 15.53 0.3118 0.311804726886084*47= $ 14.65 0.2942 0.294155402722721*47= $ 13.83 0.2775 0.277505096908227*47= $ 13.04 0.2618|0.261797261234177*47= $ 12.30 0.2470 0.246978548334129*47= $ 11.61 0.2330|0.232998630503895*47= $ 10.95 0.21980.21981002877726*47= $ 10.33 0.2074|0.20736795167666*47= $ 9.75 0.1956 0.195630143091189*47= $ 9.19 0.1846 0.184556738765272*47= $ 8.67 0.1741 0.174110130910634*47= $ 8.18 0.1643|0.16425484048173*47= $ 7.72 0.1550 0.154957396680878*47= $ 7.28 0.1462 0.146186223283847*47= $ 6.87 0.1379 0.137911531399856*47= $ 6.48 0.1301|0.130105218301751*47= $ 6.11 0.1227 0.122740771982784*47= $ 5.77 0.1158 0.115793181115834*47= $ 5.44 0.1092 0.109238850109277*47= $ 5.13 0.1031 0.103055518971016*47= $ 4.84 0.0972 0.0972221877085056*1047= $ 101.79 NPV = Sum of all Discounted CF | $ 804.40

Part B)

Coupon rate x par value Price = a Coupon rate x par value par value +...+- (1 + 5)2x0.5 (1 + 5)2x20 (1 + 5)2x20

0.094 x 1000 0.091 x 1000 1000 Price = (1 0.08 )2x0,5 + ... + + 0.08 )2x20 + (1 + 0.08 )2x 20

• Price comes to  $1138.55

Period CF Discount Factor Discounted CF 1 $ 47.00 |1/(1+0.08/20*1= | 0.9615|0.961538461538461*47= $ 45.19 2] $ 47.00 |1/(1+0.08/2)^2= | 0.9246|0.924556213017751*47= $ 43.45 3| $ 47.00 |1/(1+0.08/20…3= | 0.88900.888996358670915*47= | $ 41.78) 4 $ 47.00 |1/(1+0.08/2545 0.8548|0.854804191029726*47= $ 40.18) 5| $ 47.00 |1/(1+0.08/25*5= | 0.8219|o.82192710675935247= $ 38.63| $ 47.00 |1/(1+0.08/25*6= | 0.7903|0.790314525730146*47= $ 37.14) 7| $ 47.00 |1/(1+0.08/25*7= | 0.7599|o.759917813202063*47= $ 35.72 | 8} $ 47.00 |1/(1+0.08/25^8= | 0.7307|0.730690205001984*47= $ 34.34) 9 $ 47.00 |1/(1+0.08/25*5= | 0.7026|o.70258673557883*47= $ 33.02 | 10| $ 47.00 |1/(1+0.08/2)10= | 0.6756|0.675564168825799*47= | 31.75| 11 $ 47.00 |1/(1+0.08/2)-11= | 0.6496|0.649580931563268%47= $ 30.53| 12| $ 47.00 |1/(1+0.08/25*12= | 0.6246|0.624597049580065*47= 3 29.36] 13| $ 47.00 |1/(1+0.08/25*13 | 0.6006|0.600574086134678%47= $ 28.23) 14| $ 47.00 |1/(1+0.08/20*145 | 0.5775|0.577475082821806*47= $ 27.14) 15[ $ 47.00 |1/(1+0.08/25*15= | 0.5553|0.555264502713275*47= $ 26.10 | 16| $ 47.00 |1/(1+0.08/25*16= | 0.5339|o.533908175685841 *47= $ 25.09| 17| $ 47.00 |1/(1+0.08/25*17= | 0.5134|0.51337324585177*47= $ 24.13] 18| $ 47.00 |1/(1+0.08/20^18= | 0.49360.493628121011317*47= $ 23.20 | | 19| $ 47.00 |1/(1+0.08/2)-19= | 0.4746|0.474642424049344*47= $ 22.31 20 $ 47.00 |1/(10.08/2)-20= | 0.4564|0.456386946201292*47= | $ 21.45) 21| $ 47.00 |1/(1+0.08/25*21s | 0.4388|0.43883360211662747= $ 20.63| 22| $ 47.00 |V(1+0.08/2)^22= | 0.4220|0.421955386650603*47= $ 19.83) 23 $ 47.00|1/10.08/25*23= | 0.4057|0.405726333317887*67= $ 19.07 | [24] $ 47.00 |1/(1+0.08/25*24 | 0.3901|0.390121474344122*47= $ 18.34) 25| $ 47.00 |1/(1+0.08/25*25= | 0.3751|0.375116802253964*47= $ 17.63| 26| $ 47.00 |1/(1+0.08/2)^26= | 0.3607|0.360689232936504*47= $ 16.95| 27| $ 47.00 |1/(1+0.08/2)-27= | 0.3468|0.346816570131253*47= | $ 16.30 | 28| $ 47.00 |1/(1+0.08/2)^28= | 0.3335|0.333477471280051*47= $ 15.67| 29| $ 47.00 |1/(1+0.08/2)^29= | 0.32070.320651414692357*47= $ 15.07| 30| $ 47.00 ||(1+0.08/20^30= | 0.3083|0.30831866797342*47= $ 14.49) | 31| $ 47.00 |1/(1+0.08/25*31= | 0.2965|0.29646025766675*47= | 3 13.93| 32] $ 47.00 ||(10.08/2532= | 0.2851|0.285057940064183*47= $ 13.40 | 33| $ 47.00 |1/(1+0.08/25*33= | 0.2741|0.274094173138637*47= $ 12.88) 34 $ 47.00 |1/(1+0.08/2)^345 | 0.2635|0.263552089556382*147= $ 12.39] 35| $ 47.00 | (1+0.08/2)-35= | 0.2534[o.25341547072729*47= | $ 11.91 36| $ 47.00 |1/(1+0.08/25-36= | 0.243710.243668721853164*47= $ 11.45| 371 $ 47.00|1/10.08/25-37= | 0.2343|0.234296847935735*67= $ 11.01) 38 47.00 |1/(1+0.08/20…38= | 0.2253|0.225285430707437*47= 39| $ 47.00|1/10.08/25-395 | 0.2166|o.216620606449459*47= | $ 10.18 40[ $ 1,047.00 |/(1+0.08/2540= | 0.20830.208289044662941*1047= 1 $ 218.08 NPV = Sum of all Discounted CF $ 1,138.55 TAT IT ITT IIIIIIIIIIIIIIIIIIIII

Part c)

• If current interest rate > the coupon rate that means that the bond that investors hold is worth less than the one available in the market and investors can get better results by investing in the one available in the market.
• If current interest rate < the coupon rate that means that the bond that investors hold is worth more than the one available in the market and investors are better off by staying invested in the one they already have

Part d) If the coupon rate = the current interest rate then the price pf the bond will equal the par value of the bond. So the price of the bond = 1000, we dont even need to solve it however, just for verification we can do it.

Period CF Discount Factor Discounted CF 11 $ 47.00 11/(1+0.094/2)^1= 0.9551 0.955109837631328*47= $ 44.89 2 $ 47.00 (1/(1+0.094/2)^2= 0.9122 0.912234801940141*47= $ 42.88 3 $ 47.00 1/(1+0.094/2)^3= 0.8713 0.871284433562695*47= $ 40.95 41 $ 47.00 1/(1+0.094/2)^4= 0.8322|0.832172333870768*47= $ 39.11 5 $ 47.00 1/(1+0.094/2)^5= 0.7948|0.794815982684593*47= $ 37.36 6 $ 47.00 1/(1+0.094/2)^6= 0.7591 0.759136564168666*47= $ 35.68 71 $ 47.00 1/(1+0.094/2)^7= 0.7251 0.725058800543138*47= $ 34.08 8 $ 47.00 1/(1+0.094/2)^8= 0.6925 0.692510793259922*47= $ 32.55 9 $ 47.00 1/(1+0.094/2)^9= 0.6614 0.661423871308426*47= $ 31.09 10 $ 47.00 1/(1+0.094/2)^10= 0.6317 0.631732446330875*47= $ 29.69 111 $ 47.00 1/(1+0.094/2)^11= 0.6034 0.603373874241523*47= $ 28.36 12$ 47.00 (1/(1+0.094/2)^12= 0.5763|0.576288323057807*47= | $ 27.09 13 $ 47.00 1/(1+0.094/2)^13= 0.5504 0.550418646664572*47= $ 25.87 14 $ 47.00 1/(1+0.094/2)^14= 0.5257 0.525710264245054*47= $ 24.71 15 $ 47.00 1/(1+0.094/2)^15= 0.5021 0.502111045124216*47= $ 23.60 16 $ 47.00 (1/(1+0.094/2)^16= 0.47960.479571198781486*47= $ 22.54 171 $ 47.00 1/(1+0.094/2)^17= 0.45800.458043169800847*47= $ 21.53 18 $ 47.00 1/(1+0.094/2)^18= 0.4375 0.437481537536625*47= $ 20.56 19 $ 47.00 1/(1+0.094/2)^19= 0.4178 0.41784292028331*47= $ 19.64 20 $ 47.00 1/(1+0.094/2)^20= 0.3991 0.399085883747192*47= $ 18.76 21 $ 47.00 1/(1+0.094/2)^21= 0.3812 0.381170853626735*47= $ 17.92 22 $ 47.00 (1/(1+0.094/2)^22= 0.3641|0.364060032117226*47= $ 17.11 23 $ 47.00 1/(1+0.094/2)^23= 0.34770.347717318163539*47= $ 16.34 241 $ 47.00 1/(1+0.094/2)^24= 0.3321 0.332108231292779*47= $ 15.61 25 $ 47.00 1/(1+0.094/2)^25= 0.3172 0.317199838866073*47= $ 14.91 26 $ 47.00 (1/(1+0.094/2)^26= 0.3030 0.302960686596059*47= $ 14.24 271 $ 47.00 1/(1+0.094/2)^27= 0.2894 0.289360732183437*47= $ 13.60 28 $ 47.00 1/(1+0.094/2)^28= 0.2764 0.276371281932605*47= $ 12.99 29 $ 47.00 1/(1+0.094/2)^29= 0.2640 0.263964930212612*47= $ 12.41 30 $ 47.00 1/(1+0.094/2)^30= 0.2521 0.252115501635733*47= $ 11.85 31 $ 47.00 1/(1+0.094/2)^31= 0.2408 0.240797995831645*47= $ 11.32 32 $ 47.00 (1/(1+0.094/2)^32= 0.2300 0.229988534700712*47= $ 10.81 33 $ 47.00 (1/(1+0.094/2)^33= 0.2197 0.219664312035064*47= $ 10.32 34 $ 47.00 (1/(1+0.094/2)^34= 0.2098 0.209803545401207*47= $ 9.86 35 $ 47.00 1/(1+0.094/2)^35= 0.20040.200385430182624*47= 9.42 36 $ 47.00 1/(1+0.094/2)^36= 0.1914 0.19139009568541*47= $ 9.00 371 $ 47.00 (1/(1+0.094/2)^37= 0.18280.182798563214336*47= $ 8.59 38 $ 47.00 (1/(1+0.094/2)^38= 0.1746|0.174592706030884*47= $ 8.21 39 $ 47.00 1/(1+0.094/2)^39= 0.16680.166755211108772*47= $ 7.84 40 $ 1,047.00 1/(1+0.094/2)^40= 0.1593|0.159269542606277*1047= $ 166.76 NPV = Sum of all Discounted CF $ 1,000.00