?_Ŗ’’’’¬/š‘lp»ģ€č Ė‚ …‚’‚….ƒ1‚ ‡ `@ “& MathType0‡ „ūĄž‚Š"Helv…-…!(‡s˜!Long a callable bond‡ …!)‡ „! ‡?…!=‡D…!(‡® ’!Long an option‡ …!-‡š ! free bond‡ļ…!)‡ä „! ‡äE†!Sh‡äź†!or‡ä …!t‡ä_ ‡! a ‡äŖ …!c‡äd –!all option on the ‡äk…!b‡ä/‡!ond‡äŁ„ūĄž‚ŽSymbol…-…š…!+‡äŃ„ūĄž‚Š"Helv…-…š…!(‡äh…!)‡äĢ…!.…ū ‚¼"System…-…šŖųųųųųųĒĒųųųųųųųųųųųųų/&;)z4„Ŗ ~%ß÷’’---’’’’|CONTEXTUj|CTXOMAPĘX|FONTūV|KWBTREE÷Y|KWDATAŃX|KWMAPęY|SYSTEM)~|TOPICÜ~|TTLBTREE&b|bm0„r|bm1‡†|bm10‡ |bm112 |bm12°|bm13Å|bm14ū$|bm15I+|bm16ż/|bm173|bm18c7|bm19|=|bm2£š|bm20l@|bm21yC|bm22©G|bm23÷K|bm24’P|bm25=R|bm26ąS|bm27uV|bm28 Y|bm29ć\|bm3+°|bm30†d|bm31Jl|bm32Jo|bm33Hr|bm34ęu|bm35Pz|bm4ĶÅ|bm5pą|bm6¼ś|bm7>|bm8|bm9Ł’’’’’’m6n6n6A‡A—A‡A‡A~A–AN’9ʔ&J(Óź“ŽAG:š#G; æ(æ0æ8æ_;_;_;WBŗO@ĀO@ŹO@ŅO@ŚO@āO@źO@ĒĒĒ Ē(Ē0Ē8Ēæææ’’ņ’9Ā&®„lpÉ€ S‚ …‚’‚….ƒ1‚ ‡ @€ “& MathType ‡ „ūĄž‚Œ ArialV…-…!(‡ œ!Value of a callable bond‡ ģ …!)‡ „! ‡1…!=‡4…!(‡°–!Value of an option‡…!-‡· ! free bond‡…!)‡ą „! ‡ąļ…!-‡ą…!(‡ą¦!"Value of a call option on the bond‡ą¹…!)‡ą…!.…ū ‚¼"System…-…š…«‹Epson EPL-60„HPP«¢lpcģ€  R‚ …‚’‚….ƒ1‚ ‡ `Ą “& MathType0‡ „ūĄž‚Š"Helv…-…!(‡ƒ–!Value of an option‡ø …!-‡@ ! free bond‡•…!)‡ „! ‡?…!=‡D…!(‡¾œ!Value of a callable bond‡(…!)‡į „! ‡įō…!+‡įę…!(‡įi¦!"Value of a call option on the bond‡į…!)‡įb…!.…ū ‚¼"System…-…šƒ     ~ulpegĄ °  Ł‚ …‚’‚….ƒ1‚ ‡  € “& MathTypeĄ…ś…-‡  ‡ ż‡ H‡ X ‡ŗą‡ŗ= ‡)o„ūĄž‚Œ Arial#3…-…!$‡)Ö…!.‡1…!(‡…!.‡…!)‡)±…!$‡) …!.‡Y…!(‡B…!.‡) …!)‡Cń…!$‡Cķ…!$‡CT …!.‡1Æ…!(‡1˜…!.‡1P …!)‡ œ …!$‡ b…!.‡ …!.‡)"…!5‡)E†!25‡•…!1‡ ‰!03500‡)d …!5‡)‡ †!25‡½…!1‡© ‰!04010‡C¤‡!100‡C …!5‡CĆ †!25‡1…!1‡1’ ‰!04531‡ O ‡!102‡ ɇ!075‡‰b„ū’‚Œ Arial2…-…š…!1‡‰ …!2‡£¾ …!3‡óK„ūĄž‚ŽSymbol…-…š…!+‡C’…!+‡  …!=‡Į „ūĄž‚Œ Arial#3…-…š „! ‡ ą…!+…ū ‚¼"System…-…š ’m‚g€ lpŖĄ@Ž B‚ …‚’‚….ƒ1‚ ‡  ` “& MathType`‡+„ūĄž‚Œ Arial?5…- ‰!Year ‡:…!1‡Ļ…!:‡†! ‡` ‰!Year ‡`7…!2‡`ä…!:‡`Z†! ‡„! ‰!Year ‡„0…!3‡„ä…!:‡„Z†! ‡?„ūĄžƒŒ ArialC,…-…š…!z‡±…!f‡` …!z‡`. …!f‡`„ …!f‡„ …!z‡„. …!f‡„„ …!f‡„ …!f‡kׄū’‚Œ Arial?5…-…š…!1‡kó…!1‡°·…!2‡°p …!1‡°Ł …!2‡ŅF…!1‡Ņ…!2‡õ±…!3‡õp …!1‡õŁ …!2‡õ\…!3‡Ļ…!1‡Ÿ…!3‡`z„ūĄž‚Œ ArialC,…-…š…!1‡`Š …!1‡`ā…!1‡„z…!1‡„Š …!1‡„Y…!1‡„k…!1‡²„ūĄž‚ŽSymbol…-…š…!=‡`²…!=‡`@ …!+‡`– …!+‡`…!-‡„²…!=‡„@ …!+‡„– …!+‡„…!+‡„Ž…!-‡{„ūĄž‚Œ Arial?5…-…š…!,‡`±…![‡`…!(‡`ų …!)‡`l …!(‡`…!)‡`ż…!]‡`g…!,‡„±…![‡„…!(‡„ų …!)‡„l …!(‡„…!)‡„õ…!(‡„ …!)‡„†…!]‡„š…!.‡ŅĮ„ū’‚Œ ArialC,…-…š…!/‡J…!/…ū ‚¼"System…-…šœB;#2'˜(ž(Ü( )tL#T(2(v(Ŗ'Ģ'6 - lpŹ(@  ƒ‚ …‚’‚….ƒ1‚ ‡ `% “& MathTypeP‡„b„ūĄž‚Œ Arial…- ‰!Year ‡„—…!1‡„…!:‡„A„!"" ‡„(…![‡„…!(‡„ń…!1‡„v…!.‡„Ż ‰!03500‡„]…!)‡„Ł…!]‡Ą_ ‰!Year ‡Ąv…!2‡Ą#…!:‡ĄZ„! ‡üm ‰!Year ‡ü|…!3‡ü0…!:‡ü¦†! ‡ö"„ū’‚Œ Arialƒ°…-…š…!1‡ö…!/‡öå…!1‡„®„ūĄž‚ŽSymbol…-…š…!-‡„c…!=‡„P…!=‡Ą®…!-‡Ąc…!=‡ĄP…!=‡ü®…!-‡üc…!=‡ü!…!=‡„‹„ūĄž‚Œ Arialƒ°…-…š…!1‡„b…!0‡„} ‰!03500‡„O …!3‡„j!‡!500‡Ąz …!1‡Ąf ‰!03500‡Ą¾…!1‡ĄŖ ‰!04523‡Ą‹…!1‡Ąb…!0‡Ą} ‰!04010‡Ą^ …!4‡Ąy!‡!010‡ü6…!1‡ü" ‰!03500‡üz …!1‡üf ‰!04523‡ü¾…!1‡üŖ ‰!05580‡ü‹…!1‡üb…!0‡ü} ‰!04531‡ü/ …!4‡üJ!‡!531‡2ļ„ū’‚ˆ Arial…-…š…!1‡2Å…!2‡nļ…!1‡næ…!3‡„„ūĄž‚Œ Arialƒ°…-…š…!.‡„!…!.‡„ƒ#…!%‡„—$…!,‡Ą± …![‡Ą …!(‡Ą’ …!.‡Ąę…!)‡ĄZ…!(‡ĄC…!.‡Ą*…!)‡Ą¦…!]‡Ą…!.‡Ą!…!.‡Ą’#…!%‡Ą¦$…!,‡üm…![‡üŅ…!(‡ü»…!.‡ü¢ …!)‡ü …!(‡ü’ …!.‡üę…!)‡üZ…!(‡üC…!.‡ü*…!)‡ü¦…!]‡ü…!.‡üć …!.‡üc#…!%‡üw$…!.‡2j„ū’‚ˆ Arial…-…š…!/‡nj…!/…ū ‚¼"System…-…ššV,¤’W,É’Y,‘’r,É’v,¶’w,Ū’y,NElpēžĄ P  X‚ …‚’‚….ƒ1‚ ‡ @  “& MathType …ś…-‡µ ‡µ€‡µĖ‡µo‡”3‡”‡>1„ūĄž‚Œ Arial…-…!$‡>˜…!.‡ž1…!(‡ž…!.‡ž…!)‡>~ …!$‡>å …!.‡žÜ…!(‡žÅ…!.‡ž¬ …!)‡ž …!(‡ž …!.‡žš…!)‡Š…!$‡† …!$‡ķ …!.‡ŻD…!(‡Ż-…!.‡Ż …!)‡Żˆ …!(‡Żq …!.‡ŻX…!)‡ŻĢ…!(‡Żµ…!.‡Żœ…!)‡ēz…!$‡ē@…!.‡ēĮ…!.‡>ä…!5‡>†!25‡ž•…!1‡ž ‰!03500‡>1 …!5‡>T †!25‡ž@…!1‡ž, ‰!03500‡ž„ …!1‡žp ‰!04523‡= ‡!100‡9 …!5‡\†!25‡ŻØ…!1‡Ż” ‰!03500‡Żģ …!1‡ŻŲ ‰!04523‡Ż0…!1‡Ż ‰!05580‡ē-‡!102‡ē§‡!075‡Ī„ūĄž‚ŽSymbol…-…š…!+‡ē6…!+‡˜ …!+‡ē{…!=‡ē „ūĄž‚Œ Arial…-…š „! …ū ‚¼"System…-…š5“«lpi'3Ą   [‚ …‚’‚….ƒ1‚ ‡  Ą# “& MathTypeP‡ „ūĄž“Symbol…-…!s‡ @„ūĄž‚ŽSymbol…-…š…!=‡ G„ūĄž‚ˆ Arial…-…š©!%assumed volatility of the forward one‡ Æ…!-‡ #š!year rate at all times‡ ^#…!,‡8…!=‡C‡!one‡Š…!-‡ųÆ!+year rate one year forward if rates decline‡…!,‡…ˆ! and‡ 8…!=‡ C‡!one‡ Š…!-‡ ų©!%year rate one year forward if rates r‡ £‡!ise‡ =…!.‡„ūĄžƒŒ Arial÷O…-…š…!r‡ …!r‡P„ū’ƒˆ Arial…-…š…!D‡Y …!U‡Pm„ū’‚Œ Arial÷O…-…š…!1‡Ys…!1‡PÕ…!,‡YŪ…!,…ū ‚¼"System…-…šŖ$ lpDW€č ‚ …‚’‚….ƒ1‚ ‡  € “& MathTypeP‡€„ūĄžƒŒ ArialT…-…!r‡€Q…!r‡Š„ū’ƒŒ ArialkN…-…š…!U‡ŠT…!D‡Šq„ū’‚ˆ Arial…-…š…!1‡Š±…!1‡ņŻ…!2‡ŠŁ…!,‡Š…!,‡€„ūĄž‚Œ ArialkN…-…š…!,‡€B„ūĄž‚ŽSymbol…-…š…!=‡€!„ūĄž‚Œ ArialkN…-…š…!e‡Žu„ū’“Symbol…-…š…!s…ū ‚¼"System…-…š‚ą‚ą‚Ą‚€LClpūW L .‚ …‚’‚….ƒ1‚ ‡    “& MathTypeP‡€„ūĄžƒŒ Arial…-…!r‡Š÷„ū’ƒŒ ArialI…-…š…!U‡Ša„ū’‚Œ Arial…-…š…!1‡ņņ…!2‡ņ …!0‡ņÜ †!10‡€@„ūĄž‚Œ ArialI…-…š…!4‡€[‡!074‡€z …!4‡€•‡!976‡ŠÉ„ū’‚Œ Arial…-…š…!,‡ņ® …!.‡€ō„ūĄž‚Œ ArialI…-…š…!.‡€t…!%‡€.…!.‡€®…!%‡€Ā…!.‡€2„ūĄž‚ŽSymbol…-…š…!=‡€¶…!ׁ‡€l …!=‡ņ‰ „ū’‚ŽSymbol…-…š…!“‡€6„ūĄž‚ˆ Arial…-…š…!e…ū ‚¼"System…-…š¤  !!!lpž%¬ ę  )'‚ …‚’‚….ƒ1‚ ‡   " “& MathType ‡„ūĄžƒŒ Arial…-…!r‡…!r‡ī…!r‡l7„ū’ƒŒ Arial/M…-…š†!DD‡U*†!UU‡> *†!UD‡lt„ū’‚Œ Arial…-…š…!2‡Ut…!2‡> t…!2‡lü…!,‡Uü…!,‡> ü…!,‡#„ūĄž‚ŽSymbol…-…š…!=‡*„ūĄž‚Œ Arial…-…š‡!one‡…!-‡"Ÿ!year rate two years forward‡7…!,‡% %„!= ±assuming rates fall in the first and second years‡%Į…!;‡…!=‡‡!one‡s…!-‡Ÿ!year rate two years forward‡)…!,‡ %„!= ±assuming rates rise in the first and second years‡(…!;‡žˆ! and‡ī …!=‡ī‡!one‡īl…!-‡ī Ÿ!year rate two years forward‡ī"…!,‡÷ '„!A µassuming rates either rise in the first year and fall‡× '„!A µin the second year or fall in the first year and rise‡· „! ’in the second year‡·™…!.…ū ‚¼"System…-…š®cotcovcscdegdetdimexpgcdglbhominfišēlpČ W” l‚ …‚’‚….ƒ1‚ ‡  ą “& MathTypeP‡€„ūĄžƒŒ Arial…-…!r‡€…!r‡Š*„ū’ƒŒ Arial#/…-…š†!UD‡Š;†!DD‡Št„ū’‚Œ Arial…-…š…!2‡Šx…!2‡ņ|…!2‡Šü…!,‡Š…!,‡€„ūĄž‚ŽSymbol…-…š…!=‡€Ą„ūĄž‚Œ Arial…-…š…!e‡×„ū’“Symbol…-…š…!s…ū ‚¼"System…-…š؄š¢g Ąt’6„’6‚’6ˆ’6†’6ŠšĄ?hPč lp2 W€Ī ‚ …‚’‚….ƒ1‚ ‡  @ “& MathTypeP‡€„ūĄžƒŒ Arial…-…!r‡€…!r‡Š*„ū’ƒŒ Arial…-…š†!UU‡ŠB†!DD‡Št„ū’‚Œ Arial…-…š…!2‡Š…!2‡ņ‰…!4‡Šü…!,‡Š…!,‡€į„ūĄž‚Œ Arial…-…š…!.‡€„ūĄž‚ŽSymbol…-…š…!=‡€Ē„ūĄž‚Œ Arial…-…š…!e‡×#„ū’“Symbol…-…š…!s…ū ‚¼"System…-…š!„ ˆ0'lpŲWĄ !‚ …‚’‚….ƒ1‚ ‡   “& MathTypeP‡€„ūĄžƒŒ Arial…-…!r‡Š*„ū’ƒŒ Arial…-…š†!UD‡Št„ū’‚Œ Arial…-…š…!2‡ņ# …!2‡ņQ …!0‡ņ †!10‡€$„ūĄž‚Œ Arial…-…š…!4‡€?†!53‡€œ …!5‡€·‡!532‡Šü„ū’‚Œ Arial…-…š…!,‡ņß …!.‡€Ų„ūĄž‚Œ Arial…-…š…!.‡€„…!%‡€P…!.‡€Š…!%‡€„ūĄž‚ŽSymbol…-…š…!=‡€ē…!ׁ‡€ …!=‡ņŗ „ū’‚ŽSymbol…-…š…!“‡€g„ūĄž‚ˆ Arial…-…š…!e…ū ‚¼"System…-…šžõõĆÄÅĘæĒĻŹ§ĖĶĪNElpAW P -‚ …‚’‚….ƒ1‚ ‡  ` “& MathTypeP‡€„ūĄžƒŒ Arial…-…!r‡Š*„ū’ƒŒ ArialG/…-…š†!UU‡Št„ū’‚Œ Arial…-…š…!2‡ņ0 …!4‡ņd …!0‡ņ †!10‡€+„ūĄž‚Œ ArialG/…-…š…!4‡€F†!53‡€· …!6‡€Ņ‡!757‡Šü„ū’‚Œ Arial…-…š…!,‡ņņ …!.‡€ß„ūĄž‚Œ ArialG/…-…š…!.‡€¬…!%‡€k…!.‡€ė…!%‡€’…!.‡€„ūĄž‚ŽSymbol…-…š…!=‡€ī…!ׁ‡€° …!=‡ņĶ „ū’‚ŽSymbol…-…š…!“‡€n„ūĄž‚ˆ Arial…-…š…!e…ū ‚¼"System…-…š—ī’Ō’’’“ĶĶŒżó’ą’›’lpF'ž€ ź S‚ …‚’‚….ƒ1‚ ‡ @ # “& MathType ‡-œ„ūĄžƒŒ Arial…-…!V‡ ž’…!V‡öų’…!V‡ß„…!C‡]±„ū’ƒŒ Arialć<…-…š…!U‡Fø…!D‡-ģ„ūĄž‚ŽSymbol…-…š…!=‡ ģ…!=‡öģ…!=‡ßģ…!=‡-ś„ūĄž‚Œ Arialć<…-…š Œ!the bond‡-ą…!'‡-?£!s value at the node in question‡-4…!,‡ ś Œ!the bond‡ ą…!'‡ ?¢!s value in one year if the one‡ q…!-‡ Ó“!year rate rises‡ ¾ …!,‡öś Œ!the bond‡öą…!'‡ö?¢!s value in one year if the one‡öz…!-‡öĖ“!year rate falls‡öO …!,‡öÅ ˆ! and‡ßś–!the coupon payment‡ßb…!,…ū ‚¼"System…-…šš9j&I<ƒ öQƒ I0öA0I"«¢lp"Ź@ ‡ あ …‚’‚….ƒ1‚ ‡  Ą “& MathType@‡ ų’„ūĄžƒŒ Arial…-…!V‡ ²…!C‡p«„ū’ƒŒ Arial…-…š…!U‡ Ō„ūĄž‚ŽSymbol…-…š…!+…ū ‚¼"System…-…š˜--LZ6ZZQZZ-Z£šlp"Ź@ś‡ あ …‚’‚….ƒ1‚ ‡  Ą “& MathType@‡ ų’„ūĄžƒŒ Arial×K…-…!V‡ ø…!C‡pø„ū’ƒŒ Arial 8…-…š…!D‡ Ś„ūĄž‚ŽSymbol…-…š…!+…ū ‚¼"System…-…šŽ¼C½R¾ •Œlphø@Ž :‚ …‚’‚….ƒ1‚ ‡ ` “& MathType°…ś…-‡­ ‡­Ł‡6„ūĄžƒŒ Arial…-…!V‡6Ņ…!C‡$?…!r‡†Ė„ū’ƒŒ ArialcG…-…š…!U‡6ō„ūĄž‚ŽSymbol…-…š…!+‡$Q…!+‡$‹„ūĄž‚Œ ArialcG…-…š…!1‡–Ū„ū’‚ˆ Arial…-…š…!*…ū ‚¼"System…-…š¬~~~~~_~uuuullcZZZZZZQ•Œlphø@Ž :‚ …‚’‚….ƒ1‚ ‡ ` “& MathType°…ś…-‡­ ‡­ß‡6„ūĄžƒŒ Arial+6…-…!V‡6Ų…!C‡$B…!r‡†Ų„ū’ƒŒ Arial6…-…š…!D‡6ś„ūĄž‚ŽSymbol…-…š…!+‡$T…!+‡$Ž„ūĄž‚Œ Arial+6…-…š…!1‡–Ž„ū’‚Œ Arial6…-…š…!*…ū ‚¼"System…-…š¬~~~~~_~uuuullcZZZZZZQŁŠlpµøf ģ‚ …‚’‚….ƒ1‚ ‡ `@ “& MathType°…ś…-‡­V‡­/‡­n‡­± ‡ų’„ūĄžƒŒ Arial…-…!V‡6Ś…!V‡6”…!C‡6~ …!V‡6> …!C‡$Ņ…!r‡†„ū’ƒŒ Arial+6…-…š…!U‡†> …!D‡H„ūĄž‚ŽSymbol…-…š…!=‡6¶…!+‡65…!+‡6` …!+‡$ä…!+‡6w„ūĄž‚Œ Arial+6…-…š…!1‡öo…!2‡$…!1‡6…!(‡6ˆ…!)‡6# …!(‡62 …!)‡Ł …!.‡–n „ū’‚ˆ Arial…-…š…!*…ū ‚¼"System…-…š£ l…`eø-” Ä Ę!Ž,¶.(‚E°£šlp&] @ś ż‚ …‚’‚….ƒ1‚ ‡   “& MathTypeą…ś…-‡–V‡–/‡–n‡–± ‡IV‡I/‡In‡IĪ‡ĪV‡ĪS ‡éų’„ūĄžƒŒ Arial…-…!V‡Ś…!V‡”…!C‡~ …!V‡> …!C‡ Ņ…!r‡o„ū’ƒŒ Arial…-…š…!U‡o> …!D‡éH„ūĄž‚ŽSymbol…-…š…!=‡¶…!+‡5…!+‡` …!+‡ ä…!+‡œH…!=‡ŅĮ…!+‡ŅĆ …!+‡Ņś…!+‡! H…!=‡W3…!+‡Ü H…!=‡w„ūĄž‚Œ Arial…-…š…!1‡ßo…!2‡ …!1‡Ņw…!1‡’o…!2‡Ņ¦†!98‡Ņm‡!562‡Ņb …!4‡ŅŲ †!99‡Ņ¦‡!522‡Ņ›…!4‡’’ …!1‡’~ ‡!035‡W†!99‡WŲ‡!113‡WŌ‡!100‡WU ‡!021‡ …!2‡Ü ś†!99‡Ü ȇ!567‡…!(‡ˆ…!)‡# …!(‡2 …!)‡Ņ…!(‡Ņó…!$‡Ņ…!.‡ŅÆ …!$‡Ņ …!)‡Ņ± …!(‡Ņ% …!$‡Ņ?…!.‡Ņč…!$‡ŅO…!)‡’ …!.‡Wg…!$‡W…!.‡W!…!$‡Wī …!.‡Ü G…!$‡Ü a…!.‡Ü ā…!.‡n „ū’‚ˆ Arial…-…š…!*…ū ‚¼"System…-…š Ž@@˜¢¢¤…¢¢Ä»lp¤ €< ‚ …‚’‚….ƒ1‚ ‡ ` ą “& MathTypeš…ś…-‡„V‡„/‡„n‡„± ‡zV‡z/‡zn‡z“‡’V‡’” ‡ųų’„ūĄžƒŒ Arial6…-…!V‡.Ś…!V‡.”…!C‡.~ …!V‡.> …!C‡īŪ…!r‡~„ū’ƒŒ Arialļ5…-…š…!U‡~> …!D‡ųH„ūĄž‚ŽSymbol…-…š…!=‡.¶…!+‡.5…!+‡.` …!+‡īķ…!+‡ĶH…!=‡ …!+‡¢ …!+‡ą…!+‡R H…!=‡ˆC…!+‡ H…!=‡.w„ūĄž‚Œ Arial6…-…š…!1‡īo…!2‡ī'…!1‡w…!1‡Ćo…!2‡¦†!99‡t‡!071‡A …!4‡· †!99‡…‡!929‡…!4‡Ć… …!1‡Ćq ‡!035‡ˆ†!99‡ˆč‡!586‡ˆä‡!100‡ˆt ‡!414‡H Ø…!2‡ ś‡!100‡ {„!0‡>8 „ū’‚Œ Arialļ5…-…š…!0‡.„ūĄž‚Œ Arial6…-…š…!(‡.ˆ…!)‡.# …!(‡.2 …!)‡…!(‡ó…!$‡ …!.‡Ž …!$‡õ …!)‡ …!(‡ …!$‡…!.‡Ī…!$‡5…!)‡Ć …!.‡ˆg…!$‡ˆ…!.‡ˆ1…!$‡ˆž …!.‡ G…!$‡ …!.‡ •…!.…ū ‚¼"System…-…š„"’’  R`x’Ę÷lpĪÉ@“ ƒ‚ …‚’‚….ƒ1‚ ‡ @ą “& MathType ‡ „ūĄž‚Œ Arial…-…!(‡ ˜!Value of call option‡  …!)‡ „! ‡ø“!Value of option‡5 …!-‡Ź ˆ!free‡Ś ‰! bond‡ł…!)‡ą „! ‡ąQš!Value of callable bond‡=„ūĄž‚ŽSymbol…-…š…!=‡ąč…!-‡<„ūĄž‚Œ Arial…-…š…!(‡ąÕ…!(‡ą¤…!)‡ą…!.…ū ‚¼"System…-…š‰ŖhKŠ-‚ “žõlpĪÉ@° ƒ‚ …‚’‚….ƒ1‚ ‡ @ą “& MathType ‡ „ūĄž‚Œ ArialA…-…!(‡ —!Value of put option‡  …!)‡ „! ‡ø“!Value of option‡5 …!-‡Ź ˆ!free‡Ś ‰! bond‡ł…!)‡ą „! ‡ąQ™!Value of putable bond‡=„ūĄž‚ŽSymbol…-…š…!=‡ąč…!-‡<„ūĄž‚Œ ArialA…-…š…!(‡ąÕ…!(‡ą„…!)‡ąš…!.…ū ‚¼"System…-…š‰Ŗh·KŠ-‚ “ž•lpĪø€š Ņ‚ …‚’‚….ƒ1‚ ‡ `ą “& MathType°…ś…-‡­F ‡­¶‡ „ūĄž‚Œ Arial…-–!Effective Duration‡8 „ūĄž‚ŽSymbol…-…š…!=‡6Ć …!-‡6 …!-‡6> „ūĄžƒŒ Arial…-…š…!V‡6į…!V‡ö …!V‡6C „ūĄž‚Œ Arial…-…š…!(‡6`…!)‡6ę…!(‡67…!)‡ö•…!(‡ö½…!)‡6‘ „ūĄž‚ŽSymbol…-…š…!D‡6h…!D‡ö?…!D‡öō „ūĄž‚Œ Arial…-…š…!2‡ö …!0…ū ‚¼"System…-…šœl6¢zzzz#!618Z¢jalp’@ ˆ ?‚ …‚’‚….ƒ1‚ ‡  € “& MathTypeŠ…ś…-‡­ ‡­č‡ „ūĄž‚Œ Arialׇ…-—!Effective Convexity‡ž „ūĄž‚ŽSymbol…-…š…!=‡6‰ …!-‡6Ó…!-‡6>…!+‡6 „ūĄžƒŒ Arialׇ…-…š…!V‡6d…!V‡6…!V‡$ź…!V‡6 „ūĄž‚Œ Arial…-…š…!(‡6&…!)‡6i…!(‡6‘…!)‡6…!(‡6i…!)‡$ļ…!(‡$…!)‡…!.‡6W„ūĄž‚ŽSymbol…-…š…!D‡6š…!D‡$™…!D‡6Č„ūĄž‚Œ Arialׇ…-…š…!2‡6Ż…!0‡$c…!0‡–H„ū’‚Œ Arial…-…š…!2…ū ‚¼"System…-…š¢¢6¢Q6™zzl-6ZZYZ*ZŁŠlpBø@f ų‚ …‚’‚….ƒ1‚ ‡ `Ą “& MathType…ś…-‡ķ0‡ķŠ‡ķī‡ķ” ‡w„ūĄžƒŒ Arial‡*…-…!r‡wŖ…!e‡wć…!r‡w’…!r‡w½ …!r‡wl …!r‡@…!r‡ég„ū’‚Œ Arial_*…-…š…!2‡5 „ūĄž‚Œ Arial‡*…-…š…!2‡w…!2‡5t…!2‡éų„ū’“Symbol…-…š…!s‡wŗ„ūĄž“Symbol…-…š…!s‡@ų …!s‡w÷„ūĄž‚ŽSymbol…-…š…!-‡@ę…!»‡wÓ…!+‡w€ …!-‡@ņ „ūĄž‚Œ Arial[*…-…š…!=…ū ‚¼"System…-…š   !³ŖlŪ-/A Model for Valuing Bonds and Embedded OptionsFCreateButton("QuickHelp", "&?", "PopupId(`faj.hlp', `HC_QUICKHELP')")BrowseButtons()ŲŲ’’’’ 9’’’’E1Ŗ’’’’Ģ’’’’E¶ContentsX1 ' €b€ø˜B˜€‚’A Model for Valuing Bonds and Embedded Optionsa0Ež1 2€`€ø˜ā(LĄ€€‰€‚’Financial Analysts Journal, May/June 1993|Dz8 @€ˆ€ø˜ā$Šū€‰ā]GŠū‰āœĢ‰‚’Constructing a Binomial Interest Rate Tree Using the Binomial Tree .¼“@r ²€y€ZŒRųRć*5°€‰€€‚敧‰€‚ć Ķo €‰€€‚ćt®K‰€‚ćäĖĶр‰€€‚ćMĪ -‰€‚’Valuing a Callable Bond Determining the Call Option Value Transaction Costs Other Embedded Options OAS, Effective Duration and Convexity After-tax Valuation M - *€@€ZŒRČRć„A-'€‰‚’The Challenge of Valuation )@¶& €€ø˜€‚’7ķ1{€’’’’ķ1Credit0 ¶' €€ø˜B˜€‚’Creditė¼ķ/ ,€y€ø˜€€€€‚’Used, with permission, from Financial Analysts Journal, May/June 1993. Copyright 1991, Association for Investment Management and Research, Charlottesville, Va. All rights reserved.)1& €€ø˜€‚’Ct1ō’’’’Š’’’’t% Andrew J. Kalotay<1°' €*€ø˜B˜€‚’Andrew J. Kalotay0 tą& €€ø˜€‚’Andrew J. Kalotay is associate professor of finance in the Graduate School of Business of Fordham University and an authority in the field of debt management, having structured transactions for both the issuance and retirement sides amounting to over $60 billion.E°% ( €;€ø˜€‚‚’Prior to establishing his own firm, Dr. Kalotay was Director-research in Salomon Brothers' Bond Portfolio Analysis Group. Before coming to Wall Street, he was with the treasury department of AT&T and with Bell Laboratories. He is a trustee of the Financial Management Association.Cąh 1 ”%’’’’h 0 George O. Williams<% ¤ ' €*€ø˜B˜€‚’George O. Williamsc=h  & €{€ø˜€‚’George O. Williams is a Principal of Andrew Kalotay Associates. He has been a frequent consultant to major financial institutions on computer-related issues ranging from advanced workstation technology and computerized trading systems to simulation techniques. An expert on the theory of random walks, he is widely published in computer science and applied mathematics. His past academic positions include a research faculty appointment at New York University's Courant Institute. Dr. Williams is Dean of Quantitative Analysis for the Fixed Income Analysts Society.)¤ 0 & €€ø˜€‚’A q 1¤Š’’’’’’’’q ą@Frank J. Fabozzi:0 « ' €&€ø˜B˜€‚’Frank J. FabozziÅq ·@; D€‹€ø˜€€€€€€€€‚’Frank J. Fabozzi is editor of the Journal of Portfolio Management and a consultant to numerous financial institutions and money management firms. He is on the boards of directors of the complex of closed-end funds of BlackRock Financial Management and the family of open-end funds sponsored by The Guardian. Dr. Fabozzi has edited and authored numerous books on investment management. He recently co-authored two books with Professor Franco Modigliani, Capital Markets: Institutions and Instruments (Prentice Hall) and Mortgage a« ·@0 nd Mortgage-backed Securities Markets (Harvard Business School Press). From 1986 to 1992, he was a full-time professor of finance at MIT's Sloan School of Management.)« ą@& €€ø˜€‚’9·@A1źĢł„’’’’AŹFAbstractX1ą@qA' €b€ø˜B˜€‚’A Model for Valuing Bonds and Embedded Optionsa0AŅA1 2€`€ø˜ā(LĄ€€‰€‚’Financial Analysts Journal, May/June 1993¬qqA~E; D€ć€ø˜ā$Šū€‰ā]GŠū‰ā€ø˜B˜€‚’Spot Rates and Forward Rates(ülÅŚĒ, &€ł€ø˜€ā×^e‰‚’The relationship among the yields to maturity on securities selling at par with the same credit quality but different maturities is called the on-the-run yield curve. The yield curve is typically constructed using the maturities and observed yields of Treasury securities. As market participants do not perceive these securities to have any default risk, this government yield curve reflects the effect of maturity alone on yield. However, a theoretical yield curve can be constructed for any issuer.)²ÅČ& €€ø˜€‚’; ŚĒ>Č1‡‚ö‡ ’’’’>ȊĢSpot Rates4 ČrČ' €€ø˜B˜€‚’Spot Ratesj0>ČÜĖ: B€a€ø˜€āT”;T‰āzŗ8@‰‚āģƏ¼‰‚’The yield curve shows the relationship between the yield on full-coupon securities and maturity. What we are interested in is the relationship between the yield on zero coupon (i.e., spot rates) and maturity. Pure expectations theory and arbitrage arguments can be used to determine this relationship, called the spot rate curve. (Footnote 5)Table I gives, for example, the yield curve (based on current on-the-run yields) for annual-pay, full-coupon par bonds as well as the spot rate for each year. Now consider an option-free bond with three years remaining to maturity and a coupon rate of 5.25%. The value of this bond is the present value of the cash flows with each cash flow discounted at the corresponding spot rate. So the value of our hypothetical bond, assuming a par value of $100, is: ®|rȊĢ2 4€ś€ø˜€ƒ†"€ ‚‚‚’This procedure will ascribe a value of 100 to each of the three securities used to define the on-the-run yield curve.> ÜĖČĢ1č <„¼ ’’’’ČĢ_Forward Rates7ŠĢ’Ģ' € €ø˜B˜€‚’Forward RatespCČĢoĻ- (€‡€ø˜€ā½č%¬‰‚’A forward rate is an interest rate on a security that begins to pay interest at some time in the future. For example, a one-year rate one year forward is an interest rate on a one-year investment beginning one year from today; a one-year rate two years forward is an interest rate on a one-year investment beginning two years from today. Forward rates, like spot rates, can be derived from the yield curve, using arbitrage arguments. Consider once again the hypothetical yield curve used to derive the spot rates. Table II gives the arbitrage-free forward one-year rates.ą‘’Ģ[O l€#€ø˜€€€‚€€€€€€€€€€€‚’Note that the forward one-year rate for Year 1 is the rate on a ooĻ[ŠĢne-year security issued today, the forward one-year rate for Year 2 is the rate on a one-year security issued one year from today, and so forth. In saying that these rates are arbitrage-free, we mean that an investor would be indifferent between, say, a sequence of three one-year investments at these rates and a three-year investment at the on-the-run three-year rate, all else being equal.Spot rates and the arbitrage-free forward rates are related. Specifically, letting ft denote the forward one-year rate in year t and zt the spot t-year rate today, one can show that:¹oĻ_K d€y€ø˜€ƒ†"€ ‚‚ƒ†"€ ‚‚ƒ†"€‚‚‚’Substituting in the forward one-year rates, we can verify this relationship:Consequently, discounting cash flows at the arbitrage-free forward one-year rates is equivalent to discounting at the current spot rates. This can be verified using the 5.25% coupon bond with three years to maturity. The value of the bond found by discounting at the arbitrage-free forward one-year rates is:This is the same value as found earlier.I[Ø1dö‡^ ’’’’ØĆInterest Rate VolatilityB_ź' €6€ø˜B˜€‚’Interest Rate Volatility°„Øš, &€ €ø˜€€€‚’Once we allow for embedded options, consideration must be given to interest rate volatility. This can be done by introducing a binomial interest rate tree. This tree is nothing more than a discrete representation of the possible evolution over time of the one-period rate based on some assumption about interest rate volatility. The construction of this tree is illustrated below.)źĆ& €€ø˜€‚’Lš1× ¼ ’’’’”CBinomial Interest Rate TreeEĆT' €<€ø˜B˜€‚’Binomial Interest Rate Treeč›< M h€7€ø˜āB‹É€‰€€€€€€€€€ā{ŗ8@‰‚’Figure A shows an example of a binomial interest rate tree. In this tree, the nodes are spaced one year apart in time (left to right). Each node N is labeled with one or more subscripts indicating the path the one-year rate followed in reaching that node. The one-year rate may move to one of two values in the following year: U represents the higher of these two values and D represents the lower. It is important to note that these rates are not necessarily higher or lower than the one-year rate in the preceding year. Thus node NUU simply means that the one-year rate followed the upper path in both the first and second years. (Footnote 6)? T{ 6 :€€ø˜€€€€€€‚’Look first at the point denoted N. This is the root of the tree and is nothing more than the current one-year rate, which we denote by r0. What we have assumed in creating this tree is that the one-year rate can take on two possible values one year from today. In what follows, we also assume that these two future rate environments are equally likely. For convenience, we will refer to the higher of the two rates in the next period as resulting from a "rise" in rates and the lower as resulting from a "fall" in rates (although we stress that the rate has not necessarily fallen or risen relative to its last value). The tree we construct will be a discrete representation of a lognormal distribution over time of future one-year rates with a certain volatility.&Ć< ”c ”€€ø˜€‚ƒ†"€‚‚ƒ†"€‚€€€€€ €‚ƒ†"€‚€€‚’We will use the following notation to describe the tree in the first year. LetGiven our assumption that the two one-year rates one year forward are equally likely, the relationship between the two is simply:where e is the base of the natural logarithm 2.71828... Suppose, for example, that r1,D is 4.074% and s is 10% per year. Then:In the second year, the one-year rate has three possible values, which we denote as follows:Æ{ \B” '€ø˜€ƒ†"€‚€€€€‚ƒ†"€‚‚ƒ†"€‚€€€€”\BĆ€ €‚ƒ†"€‚‚ƒ†"€‚āB‹É‰€€ €€€€€āC‹É‰‚’The relationship between r2,DD and the other two forward rates is as follows:andFor example, if r2,DD is 4.53% and, once again, s is 10%, thenandFigure A shows the notation for the binomial interest rate tree over three years. We can simplify the notation by letting rt denote the one-year rate t years forward if interest rates always decline since all the other rates t years forward will depend on that rate. Figure B shows the interest rate tree using this simplified notation.8””C4 6€ €ø˜€€ €€ €‚‚’Before we go on to show how to use this binomial interest rate tree to value bonds, let's focus on two issues. First, what does the volatility parameter s in the expression exp(2s) represent? Second, how do we find the value of the bond at each node?R!\BęC1\^tƒ ’’’’ęCšFVolatility and Standard DeviationK$”C1D' €H€ø˜B˜€‚’Volatility and Standard Deviation–IęCĒFM h€“€ø˜€€€€ €ā|ŗ8@‰€ €€€€€ €‚’The standard deviation of the one-year rate one year forward is equal to r0s. (Footnote 7) The standard deviation is a statistical measure of volatility. This means that volatility is measured relative to the current level of rates. For example, if s is 10% and the one-year rate (r0) is 4%, then the standard deviation of the one-year rate one year forward is 4% times 10%, which equals 0.4% or 40 basis points. However, if the current one-year rate is 12%, the standard deviation of the one-year rate one year forward would be 12% times 10%, or 120 basis points.)1DšF& €€ø˜€‚’PĒF@G1Ž ’’’’@G Determining the Value at a NodeI"šF‰G' €D€ø˜B˜€‚’Determining the Value at a Node×@G§IG \€Æ€ø˜€āC‹É‰€€€€€€€€€‚’To find the value of the bond at a node, first calculate the bond's value at the two nodes to the right of the node of interest. For example, in Figure B, to determine the bond's value at node NU, values at nodes NUU and NUD must be determined. (Hold aside for now how we get these two values; as we will see, the process involves starting from the last year in the tree and working backward to get the final solution, so these two values will be known.)§_‰GNMH ^€æ€ø˜€€€€€€€€€€€€‚’What we are saying, in effect, is that the value of a bond at a given node will depend on future cash flows. We can separate the contributions from the future cash flows into (1) the coupon payment one year from now and (2) the bond's value one year from now. The former is known. The latter depends on whether the one-year rate rises or falls in the coming year. The two nodes to the right of any given node show the bond's value when rates rise or fall. Suppose we intend to sell the bond at the next node date. The cash flow we will receive on that future date will be either (1) the bond's value if rates rise plus the coupon payment or (2) the bond's value if rates fall plus the coupon payment. The bond's value at NU, then, will be either its value at NUU plus the coupon payment or its value at NUD plus the coupon payment, discounted.Ī§ITO8 >€Ÿ€ø˜€€€‚ƒ†"€‚’In calculating the present value of expected future cash flows, the appropriate discount rate is the one-year rate at the node where we seek the value. Now, there are two future values to discount--the value if the one-year rate rises over the coming year and the value if it falls. We assume both outcomes are equally likely, so we can simply average the two future values. Assuming that the one-year rate is r* at the node being valued, and letting:«BNM i  €€ø˜€‚ƒ†"€‚‚ƒ†"€‚€€‚ƒ†"€‚‚ƒ†"€‚‚ƒ†"€‚‚’the cash flow at a node is either:if the one-year rate rises oTO šFrif the one-year rate falls. The present values of these two cash flows, using the one-year rate r* at the node, areif the one-year rate rises andif the one-year rate falls. The value of the bond at the node is then found as follows:[*TOf1tƒd‡’’’’fgÉConstructing a Binomial Interest Rate TreeT- ŗ' €Z€ø˜B˜€‚’Constructing a Binomial Interest Rate Tree"šf܂2 2€į€ø˜€āģƏ¼‰€ €‚’To see how to construct a binomial interest rate tree, assume on-the-run yields are as given in Table I and that the volatility, s, is 10%. We construct a two-year model that correctly values a two-year bond with a 4% coupon at 100.Xŗk†7 <€±€ø˜āD‹É€‰€€€‚‚’Figure C shows a binomial interest rate tree that gives the cash flow at each node. The root rate for the tree, r0, is simply the current one-year rate, 3.5%. This rate was chosen because it will value a one-year bond with a 3.5% coupon at 100.There are two possible one-year rates one year forward---one if rates fall and one if rates rise. What we want to find are the two forward rates consistent with the volatility assumption that result in a value of 100 for a 4% full-coupon, two-year bond. While one can construct a simple algebraic expression for these two rates, the formulation becomes increasingly complex beyond the first year. Furthermore, because one may want to implement this procedure on a computer, the natural approach is to find these rates by an iterative process (i.e., trial-and-error). The steps are described below:nź܂Łˆ„ րÕ€ø˜€ƒ€€€€€€€€€‚ƒ€€€€€€ €€ €€€€€ €€ €āD‹É‰€€€‚’Step 1: Select a value for r1. Recall that r1 is the one-year rate one year forward if rates fall. In this first trial we arbitrarily selected a value of 4.5%.Step 2: Determine the corresponding value for the one-year rate one year forward if rates rise. As explained earlier, this rate is given by r1×exp(2s). Since r1 is 4.5%, the forward one-year rate if rates rise is 5.496% (= 4.5%“exp(2“0.10)). This value is reported in Figure C at node NU.®~k†‡‰0 0€ü€ø˜€ƒ€€€‚’Step 3: Compute the bond's value in each of the two interest rate states one year from now, using the following steps.æŁˆ‹I `€€ø‘€€ƒ€€‚ƒ€€€€€€€€‚’a: Determine the bond's value two years from now. Since we are using a two-year bond, the bond's value is its maturity value of $100 plus its final coupon payment of $4, or $104.b: Calculate the bond's present value at node NU. The appropriate discount rate is the forward one-year rate assuming rates rise, or 5.496% in our example. The present value is $98.582 (= $104/1.05496). This is the value of VU referred to earlier."į‡‰±ŒA P€Ć€ø‘€€ƒ€€€€€€ €€‚’c: Calculate the bond's present value at node NU. The appropriate discount rate is the forward one-year rate assuming rates fall, or 4.500%. The present value is $99.522 (= $104/1.04500) and is the value of VD.Ģ=‹‰Ą ģ€}€ø˜€ƒ€€€€€€€€€€€€€€€€€€€€‚ƒ†"€‚€ƒ€€€€€€€€€‚’Step 4: Add the coupon to the cash flows VU and VD computed in Step 3 to get the total cash flow at NU and ND, respectively. Then calculate the average of the present values of these two cash flows using the assumed value of r0, 3.5%, for r*.Step 5: Compare the value in obtained Step 4 with the target value of $100. If the two values are the same, then the r1 used in this trial is the one we seek. It is the forward one-year rate to be used in the binomial interest rate tree for a decline in rates as well as the rate to be used if rates rise. If, however, the value found in Step 4 is not equal to the target value of $100, our assumed value is not consistent±Œ‰Ą  with the volatility assumption and the yield curve. In this case, repeat the five steps with a different value for r1.Į±Œ¦Ā\ †€ƒ€ø˜€€€€€€€€€€āE‹É‰‚ƒ€€€€€€‚’If we use 4.5% for r1, a value of $99.567 results in Step 4. This is smaller than the target value of $100. Therefore, 4.5% is too large. The five steps must be repeated, with a smaller value for r1. It turns out that the correct value for r1 is 4.074%. The corresponding binomial interest rate tree is shown in Figure D. Steps 1 through 5, using the correct rate, are as follows.Step 1: Select a value of 4.074% for r1. ”‰Ą³Äl ¦€C€ø˜€ƒ€€€€ €€ €‚ƒ€€€€€€€€€€€€€€€‚’Step 2: The corresponding value for the forward one-year rate if rates rise is 4.976% (= 4.074%“exp(2“0.10))Step 3: The bond's value one year from now is determined from the bond's value two years from now---$104, just as in the first trial; the bond's present value at node NU---VU, or $99.071 (= $104/1.04976); and the bond's present value at node ND---VD, or $99.929 (= $104/1.04074).‹ž¦Ā>ɍ 耒€ø˜€ƒ€€€€€€€€€‚ƒ†"€‚€ƒ€€€€€‚€€€€€€€€€€ €€āF‹É‰‚’Step 4: Add the coupon to both VU and VD and average the present values of the resulting cash flows.Step 5: Since the average present value is equal to the target value of $100, r1 is 0.04074 = 4.074%.We're not done. Suppose that we want to "grow" this tree for one more year---that is, we must determine r2. We will use a three-year on-the-run 4.5% coupon bond to get r2. The same five steps are used in an iterative process to find the one-year rates two years forward. Our objective now is to find the value of r2 that will produce a value of $100 for the 4.5% on-the-run bond and will be consistent with (1) a volatility assumption of 10%, (2) a current one-year forward rate of 3.5%, and (3) the two forward rates one year from now of 4.074% and 4.976%. The desired value of r2 is 4.530%. Figure E shows the completed binomial interest rate tree. Using this binomial interest rate tree, we can value a one-year, two-year, or three-year bond, whether option-free or not.)³ÄgÉ& €€ø˜€‚’H>ÉÆÉ1Į|Š’’’’ÆÉ(ĶUsing the Binomial TreeAgÉšÉ' €4€ø˜B˜€‚’Using the Binomial Tree8žÆÉ(Ķ: B€ż€ø˜€āģƏ¼‰āF‹É‰āG‹É‰‚‚’To illustrate how to use the binomial interest rate tree, consider an option-free 5.25% bond with two years remaining to maturity. Assume that the issuer's on-the-run yield curve is the one given in Table I so that the appropriate binomial interest rate tree is the one in Figure E. Figure F shows the various values obtained in the discounting process. If produces a bond value of $102.075.It is important to note that this is value is identical to the bond value found earlier by discounting at either the zero-coupon rates or the forward one-year rates. We should expect to find this result, since our bond is option-free. This clearly demonstrates that, for an option-free bond, the valuation model is consistent with the standard valuation model.HšÉpĶ1 d‡ž’’’’pĶsValuing a Callable BondA(Ķ±Ķ' €4€ø˜B˜€‚’Valuing a Callable BondąŗpĶ‘Ļ& €u€ø˜€‚’The binomial interest rate tree can also be applied to callable bonds. The valuation process proceeds in the same fashion as in the case of an option-free bond, with one exception: When the call option can be exercised by the issuer, the bond's value at a node must be changed to reflect the lesser of its value if it is not called (i.e., the value obtained by applying the recursive valuation formula described above) or the call price.(±ĶW |€Q€ø˜€āH‹É‰āG‹É‰€€€€€€€ €€€€€‚’Consider a 5.25% bond w‘Ļ(Ķith three years remaining to maturity that has an embedded call option exercisable in years one and two at $100. Figure G shows the value at each node of the binomial interest rate tree. The discounting process is identical to that shown in Figure F, except that at two nodes, ND and NDD, the values from the recursive valuation formula ($101.002 at ND and $100.689 at NDD) exceed the call price ($100) and therefore have been struck out and replaced with $100. The value for this callable bond is $101.432..ū‘ĻJ3 4€÷€ø˜€ā}ŗ8@‰ā~ŗ8@‰‚’Note that the technique which we have just illustrated allows the direct valuation of a bond with an embedded option. While others have implemented interest rate processes and valuation models on discrete tree structures, they have generally valued options directly, not as an integral part of the underlying security. (Footnote 8) The distinction becomes even more important when confronted with a package of interrelated embedded options, such as those found in sinking fund bonds. (Footnote 9))s& €€ø˜€‚’R!JÅ1ō|Š!’’’’ÅgDetermining the Call Option ValueK$s' €H€ø˜B˜€‚’Determining the Call Option ValueŠŖÅą& €U€ø˜€‚’From our earlier discussion of the relationships between the value of a callable bond, the value of a noncallable bond, and the value of the call option, we know that:‡Tg3 4€«€ø˜€ƒ†"€‚‚‚’But we have just seen how to determine the values of both a noncallable bond and a callable bond. The difference between the two values is just the value of the call option. In our example, the value of the noncallable bond is $102.075 and the value of the callable bond is $101.432, so that the value of the call option is $0.643.Bą©1=ž ’’’’©¤ Transaction Costs;gä' €(€ø˜B˜€‚’Transaction Costsą³©Ä - (€g€ø˜€€€‚’When an issuer calls a bond, the debt is generally refinanced. The issuer will, in general, incur transaction costs in the refinancing. These costs may be injected into the valuation model by scaling up the call price appropriately. It would be imprudent for an issuer to exercise a call option unless the net-present-value savings realized by calling the bond represented a substantial fraction of the forfeited option value (i.e., unless the intrinsic value of the call option dominates its incremental time value). Indeed, the option-exercise rule embodied in the valuation model presented here assumes that an option will be exercised only if its incremental time value is zero.ąøä¤ ( €q€ø˜€‚‚’By including transaction costs in every option-exercise decision, one obtains an adjusted option value that does not penalize refunding. One must consider, however, what will happen at the final maturity. Debt is rarely extinguished at maturity; it is generally refinanced, again with transaction expenses. Failure to include these costs at maturity will penalize the decision to exercise the call option on a bond close to maturity.GÄ ė 1!l‚’’’’ė gCOther Embedded Options@¤ + ' €2€ø˜B˜€‚’Other Embedded Optionse!ė œ@D V€C€ø˜€āG‹É‰āI‹É‰€ €€€ €€‚’The bond valuation framework presented here can be used to analyze securities such as putable bonds, options on interest rate swaps, caps and floors on floating-rate notes, and the optional accelerated redemption granted to an issuer in fulfilling sinking fund requirements. Let's consider a putable bond. Suppose that a 5.25% bond with three years remaining to maturity has a put option exercisable annually in years one and two at $100. Assume that the appropriate binomial interest rate tree for this issuer is the one in Figure F. Figure H shows the binomial interest rate tree with the bond values altered at two nodes NUU and NUD) becau+ œ@¤ se the bond values at these two nodes are less than $100, the value at which the bond can be put. The value of this putable bond is $102.523.¢h+ >C: B€Ó€ø˜€‚ƒ†"€ ‚€ €‚‚’As the value of a nonputable bond can be expressed as the value of a putable bond minus the value of a put option on that bond:In our example, the value of the putable bond is $102.523 and the value of the corresponding nonputable bond is $102.075, so the value of the put option is -$0.448. The negative sign indicates the issuer has sold the option, or equivalently, the investor has purchased the option.The same general framework can be used to value a bond with multiple or interrelated embedded options. The bond values at each node are simply altered to reflect the exercise of any of the options.)œ@gC& €€ø˜€‚’a0>CČC1Ė  ĀŒ’’’’ČC2OOAS, Effective Duration, and Effective ConvexityZ3gC"D' €f€ø˜B˜€‚’OAS, Effective Duration, and Effective ConvexityŃ«ČCóE& €W€ø˜€‚’Our model determines the theoretical value of a bond. For example, if the observed market price of the three-year, 5.25% callable bond is $101 and the theoretical value is $101.432, the bond is cheap by $0.432. Bond market participants, however, prefer to think not in dollar terms but rather in terms of a yield spread: A "cheap" bond trades at a higher spread and a "rich" bond at a lower spread relative to some basis.śĶ"DķI- (€›€ø˜€€€‚’The market convention has been to think of a yield spread as the difference between the yield to maturity on a particular bond and the yield on a Treasury bond or other benchmark security of comparable maturity. This is inappropriate, however, because yield to maturity is just a conversion of a dollar price to a yield, given the contractual cash flows, and may have little to do with the underlying yield curve that determines the price. A step in the right direction is to determine a discounting spread over the issuer's spot or forward rate curve. In terms of our binomial interest rate tree, one seek the constant spread that, when added to all the forward rates on the tree, makes the theoretical value equal to the market price. This quantity is called the option-adjusted spread (OAS). It is "option-adjusted" in that a bond fairly priced relative to an issuer's yield curve will have an OAS of zero whether the bond is option-free or has embedded options.õĀóEāM3 4€…€ø˜€€€€€‚’Investors also want to know the sensitivity of a bond's price to changes in interest rates. Unwarranted emphasis has been placed on modified duration, which is a measure of the sensitivity of a bond's price, not to changes in interest rates, but to changes in the bond's yield. It has the further shortcoming of ignoring any dependence of cash flows on interest rate levels, hence is unsuited for bonds with embedded options. The correct duration measure---called effective duration---quantifies the price sensitivity to small changes in interest rates while simultaneously allowing for changing cash flows. In terms of our binomial interest rate tree, price response to changing interest rates is found by shifting the tree up and down by a few basis points. Any measure of convexity should be derived from changes in the yield curve, rather than the yield to maturity. Effective convexity recognizes the dependence of cash flows on interest rates.PēķI2Oi  €Ó€ø˜€€€€€€€€ €€ €€€€ €‚ƒ†"€!‚‚ƒ†"€"‚‚’If V(0) is the value of a bond including accrued interest, V(D) is its value when the tree is shifted up by D basis points, and V(-D) is its value when the tree is shifted down by a similar amount, then:andDāMvO1¤ l‚Ž’’’’vO„ˆAfter-tax Valuation=2O³O' €,€ø˜B˜€‚’After-tax Valuationŗ”vOy‚& €)€ø˜€‚’Certain market participants, particula³Oy‚2Orly corporations, must value bonds on an after-tax basis. The standard approach is to discount after-tax cash flows at after-tax rates. One begins by multiplying the entire on-the-run yield curve by (1 - Tax Rate). Using the resulting after-tax yield curve, one can then determine either a set of zero-coupon rates or a set of forward one-period rates which are then used to discount the after-tax cash flows. As is the case of pretax valuation, however, this discounting methodology does not take the volatility of interest rates into account, hence is unable to properly capture the value of any embedded options.³Œ³O,…' €€ø˜€‚’Using a binomial tree for after-tax valuation would seem to be simple. There is a significant catch, however. If an after-tax binomial interest rate tree is constructed by simply multiplying each one-period rate on a pretax tree by (1 - Tax Rate), the after-tax expected cost of an option-free bond has a is dependent on volatility. If one constructs an interest rate tree based on an after-tax yield curve---so that the after-tax discounted present value of the after-tax cash flows of a bond is independent of volatility---the implied underlying pretax interest rate process is different from that obtained directly from the pretax yield curve.yRy‚„ˆ' €„€ø˜€‚’There is another interesting implication of the after-tax valuation of option-bearing bonds. Consider for a moment the pretax valuation of a callable bond. The valuation process described in the paper is essentially an optimization procedure designed to produce the lowest possible value for an investor or, equivalently, the lowest possible expected cost of servicing the outstanding debt to the issuer. It follows, then, that any other set of option-exercise decisions can only increase the value of the bond to the investor, (who, we assume, is not a taxpayer). But suppose that the valuation process is carried out which minimizes the issuer's after-tax cost of servicing the debt. In this case what's good for the goose is also good for the gander. The reason is that this is a three-player game, with the third player being the taxman.P,…õˆ1!ĀŒ’’’’’’’’õˆʋThe Challenge of ImplementationI"„ˆ>‰' €D€ø˜B˜€‚’The Challenge of Implementationˆbõˆʋ& €Å€ø˜€‚’To transform the basic interest rate tree into a practical tool requires several refinements. For one thing, the spacing of the node lines in the tree must be much finer, particularly if American options are to be valued. However, the fine spacing required to value short-dated securities becomes computationally inefficient if one seeks to value, say, 30-year bonds. While one can introduce a time-dependent node spacing, caution is required; it is easy to distort the term structure of volatility. Other practical difficulties include the management of cash flow dates that fall between two node lines.; >‰Œ1ō’’’’¢ ’’’’ŒŗFootnote 14 ʋ5Œ' €€ø˜B˜€‚’Footnote 1\0Œ‘, &€a€ø˜€€€‚’For an illustration of coupon stripping and why the theoretical price of a Treasury bond will be equal to the present value of the cash flows discounted at the theoretical spot rates, see Chapter 10 in F. J. Fabozzi, Bond Markets, Analysis and Strategies (Englewood Cliffs, NJ: Prentice Hall, 1993).)5Œŗ& €€ø˜€‚’; ‘õ1¼b¤ ’’’’õvFootnote 24 ŗ)Ž' €€ø˜B˜€‚’Footnote 2$ņõM2 2€å€ø˜€€€€€‚’See, for example, M. L. Dunetz and J. Mahoney, "Using Duration and Convexity in the Analysis of Callable Bonds," Financial Analysts Journal, May/June 1988; and F. J. Fabozzi, Fixed Income Mathematics (Chicago: Probus Publishing, 1988).))Žv& €€ø˜€‚’; M±1¢  ƒ’’’’±›ĆFootnote 34 v Ą' €€ø˜B˜€‚’Footnote 3± Ąvf ±rĆ\ †€€ø˜€€€€€€€€€€€€€€€€€€€‚’See F. Black, E. Derman, and W. Toy, "A One-factor Model of Interest Rates and its Application to Treasury Bond Options," Financial Analysts Journal, January/February 1990; J. Hull and A. White, "Valuing Derivative Securities Using the Explicit Finite Difference Method," Journal of Financial and Quantitative Analysis 25 (1990), pp. 87--100; J. Hull and A. White, "Pricing Interest-rate Derivative Securities," Review of Financial Studies 3 (1990), pp. 573--592; T. S. Y. Ho and S. B. Lee, "Term Structure Movements and Pricing Interest Rate Contingent Claims," Journal of Finance 41 (1986), pp. 1011--1029; and Ravi Dattatreya and Frank J. Fabozzi, "A Simplified Model for Valuing Debt Options,'' Journal of Portfolio Management 15 (1989), pp. 64--73.) Ą›Ć& €€ø˜€‚’; rĆÖĆ1Ť „’’’’ÖĆ`ÅFootnote 44 ›Ć Ä' €€ø˜B˜€‚’Footnote 4-ÖĆ7Å, &€€ø˜€€€‚’For an explanation of these options and their valuation using the model presented in this article, see A. J. Kalotay and G. O. Williams, "The Valuation and Management of Bonds with Sinking Fund Provisions," Financial Analysts Journal, March/April 1992.) Ä`Å& €€ø˜€‚’; 7Å›Å1p ƒŌ„’’’’›ÅŠĘFootnote 54 `ÅĻÅ' €€ø˜B˜€‚’Footnote 5Ų¦›Å§Ę2 2€M€ø˜€€€€€‚’The method used is called "bootstrapping." For an illustration of how this is done, see Chapter 10 in Fabozzi, Bond Markets, Analysis and Strategies, op. cit.)ĻÅŠĘ& €€ø˜€‚’; §Ę Ē1„ †’’’’ ĒēČFootnote 64 ŠĘ?Ē' €€ø˜B˜€‚’Footnote 6& Ē¾ČY €€M€ø˜€€€€€€€€€€€€€€€€€€‚’Note that this tree is said to be recombinant, so that NUD is equivalent to NDU in the second year, and in the third year NUUD is equivalent to both NUDU and NDUU. We have simply selected one label for a node rather than clutter up the exhibit with unnecessary information.)?ĒēČ& €€ø˜€‚’; ¾Č"É1QŌ„’†’’’’"É8ŹFootnote 74 ēČVÉ' €€ø˜B˜€‚’Footnote 7«t"ÉŹ7 >€č€ø˜€€ €€ €€ €‚’This can be seen by noting that exp(2s) » 1+2s. Then the standard deviation of forward one-period rates is7VÉ8Ź1 2€€ø˜€ƒ†"€#‚‚’; ŹsŹ1‡ †eˆ’’’’sŹæĢFootnote 84 8Ź§Ź' €€ø˜B˜€‚’Footnote 8ļĆsŹ–Ģ, &€‡€ø˜€€€‚’See, for example, Black, Derman, and Toy, "A One-factor Model of Interest Rates and its Application to Treasury Bond Options," op. cit. Like ours, their model is a one-factor (the short-term interest rate) process represented on a binomial tree. However, theirs takes as inputs a zero-coupon yield curve and its volatility term structure. The stochastic differential equations which underlie the two interest rate processes are also different.)§ŹæĢ& €€ø˜€‚’; –ĢśĢ10’†’’’’ ’’’’śĢļĶFootnote 94 æĢ.Ķ' €€ø˜B˜€‚’Footnote 9˜mśĢĘĶ+ &€Ś€ø˜€€€‚’See Kalotay and Williams, "The Valuation and Management of Bonds with Sinking Fund Provisions," op. cit.).ĶļĶ& €€ø˜€‚’8ĘĶ'Ī1N’’’’!!’’’’'Ī+Table I1 ļĶXĪ' €€ø˜B˜€‚’Table I='Ī•Ī% €0€ø˜€‚’Yields and Spot Rates»7XĪPĻ„#Ų€n€HĶ h © "€€ø˜€€‚’"€€ø˜€€‚’"€B€ø˜€€‚’"€T€ø˜€€‚’’’MaturityYield-to-MaturityValueSpot RateŒ•ĪÜĻo#®€:€HĶ h © €€ø˜€‚’€€ø˜‚’€ €ø˜‚’€*€ø˜‚’’’1 year3.50%1003.500%PĻuo#®€<€HĶ hÜĻuļĶ © €€ø˜€‚’€€ø˜‚’€"€ø˜‚’€,€ø˜‚’’’2 years4.00%1004.010%ÜĻo#®€<€HĶ h © €€ø˜€‚’€€ø˜‚’€"€ø˜‚’€,€ø˜‚’’’3 years4.50%1004.531%)u+& €€ø˜€‚’9d13āˆ’’’’"’’’’d^Table II2 +–' €€ø˜B˜€‚’Table II5dĖ% € €ø˜€‚’Forward Ratesq!–<P#p€B\Hń "€€ø˜€€‚’"€€ø˜€€‚’’’YearOne-year Forward RateS ĖG#^€\Hń €€ø˜€‚’€€ø˜‚’’’13.500%S <āG#^€\Hń €€ø˜€‚’€€ø˜‚’’’24.523%S 5G#^€\Hń €€ø˜€‚’€€ø˜‚’’’35.580%)ā^& €€ø˜€‚’95—1ī’’’’¾#’’’’—LFigure A2 ^É' €€ø˜B˜€‚’Figure AN)—% €R€ø˜€‚’Three-Year Binomial Interest Rate Tree5ÉL0 0€ €ø˜€‡"€‚‚’9…1G„G$’’’’…“Figure B2 L·' €€ø˜B˜€‚’Figure BkF…"% €Œ€ø˜€‚’Three-Year Binomial Interest Rate Tree with Forward One-Year Rates*q7·“: D€p€ø˜€‡"€‚€€€‚‚’*rt equals forward one-year rate if rate falls9"Ģ1*¾½%’’’’Ģ½Figure C2 “ž' €€ø˜B˜€‚’Figure CŠeĢˆ% €Ź€ø˜€‚’Finding the Forward One-Year Rates for Year One Using Two-Year, 4%, On-the-Run Issue: First Trial5ž½0 0€ €ø˜€‡"€‚‚’9ˆö1G&’’’’öĶFigure D2 ½(' €€ø˜B˜€‚’Figure DpKö˜% €–€ø˜€‚’Forward One-Year Rates for Year One Using Two-Year, 4%, On-the-Run Issue5(Ķ0 0€ €ø˜€‡"€‚‚’9˜1½y'’’’’įFigure E2 Ķ8' €€ø˜B˜€‚’Figure EtO¬% €ž€ø˜€‚’Forward One-Year Rates for Year Two Using Three-Year, 4.5%, On-the-Run Issue58į0 0€ €ø˜€‡"€‚‚’9¬ 1Ū(’’’’ ÷ Figure F2 įL ' €€ø˜B˜€‚’Figure FvQ Ā % €¢€ø˜€‚’Valuing an Option-Free Bond with Three Years to Maturity and a Coupon of 5.25%5L ÷ 0 0€ €ø˜€‡"€‚‚’9Ā 0 18y_)’’’’0 / Figure G2 ÷ b ' €€ø˜B˜€‚’Figure G˜s0 ś % €ę€ø˜€‚’Valuing a Callable Bond with Three Years to Maturity and a Coupon of 5.25%, Callable in Years One and Two at 1005b / 0 0€ €ø˜€‡"€‚‚’9ś h 16Ū’’’’*’’’’h e Figure H2 / š ' €€ø˜B˜€‚’Figure H–qh 0 % €ā€ø˜€‚’Valuing a Putable Bond with Three Years to Maturity and a Coupon of 5.25%, Putable in Years One and Two at 1005š e 0 0€ €ø˜€‡"€‚‚’A0 ¦ 1O’’’’’’’’+’’’’¦ “ Embedded Options:e ą ' €&€ø˜B˜€‚’Embedded Options«…¦ ‹ & € €ø˜€‚’Options that are part of the structure of a bond, as opposed to bare options, which trade separately from any underlying security.)ą “ & €€ø˜€‚’A‹ õ 1ū’’’’’’’’,’’’’õ ÆOption-free Bond:“ /' €&€ø˜B˜€‚’Option-free BondW2õ †% €d€ø˜€‚’A bond that does not have any embedded options.)/Æ& €€ø˜€‚’G†ö1G’’’’’’’’-’’’’ö @On-the-run Yield Curve@Æ6' €2€ø˜B˜€‚’On-the-run Yield Curve—röĶ% €ä€ø˜€‚’The relationship between the yield-to-maturity and maturity for bonds of similar credit quality trading at par.)6 @& €€ø˜€‚’Ķ @Æ: ĶF@1ó’’’’’’’’.’’’’F@ų@Spot Rate3 @y@' €€ø˜B˜€‚’Spot RateV1F@Ļ@% €b€ø˜€‚’The interest rate on a zero-coupon instrument.)y@ų@& €€ø˜€‚’; Ļ@3A1m’’’’’’’’/’’’’3AeBVolatility4 ų@gA' €€ø˜B˜€‚’VolatilityÕÆ3APrint Topic.älDåG7 <€É€ø˜€‚€€‚‚€€‚‚’Back Button: Displays the topic you viewed prior to the one you are viewing now.History Button: Displays the history of topics you have explored in this session, and allows you to select a particular one for review.1ŹF’’’’1’’’’’’’’2’’’’’’’’’’’’ĖĀ ŠHelv’’’’’’ß’’’’’’’’Times New Roman’’’Symbol’’÷’’’’’’’’’’Arial’’’’’’’’ż’’’’’Tms Rmn’’’’’’’’’’’’MS Sans Serif’’’’’’Courier’’’’’’’’’’’’Helveticaż’’’’’’’’’Arial Narrow’’’’’’’Eras’’’’’’’’’’’’’     €ĀŒ”Š%¼^tƒd‡|Šž|ŠžŽl‚Ģl‚„¾G½yŪ_b¢ ¤  ƒ„Ō„ †’†eˆ‚ö‡uˆŽł„tƒl‚ ”Š%‚<„āˆ!ĀŒ!ˆd‡|Šž! l‚ĀŒtƒ¼^/&;)i2424 FAJ_AJK 0’’’’’’AbstractAfter-tax ValuationAuthorsBinomial TreeCallable Bond4Callable Bonds8Conclusion<Convexity@Credit LineDDurationHFiguresLFootnotes lForward ratesHelp˜HistoryœImplementation Introduction¤Node ValueØOAS¬Other Options°Quality“Spot ratesĄTablesČTaxesŠTitle PageŌTransaction CostsŲValuationÜValue at a NodeüVolatilityITY d:\c\wpp\primer\faj.rtf 8 JCD793F9D FAJ_DEFN_FWDRATE 1 FAJ_DEFN_FWDRATE d:\c\wpp\primer\faj.rtf 8 F5D3641F4 FAJ_SPOTANDFWD 0 FAJ_SPOTANDFWD d:\c\wpp\primer\faj.rtf 9 L655E19D7 FAJ_DEFN_ONTHERUN 1 FAJ_DEFN_ONTHERUN d:\c\wpp\primer\faj.rtf 9 E470F794B FAJ_SPOTRATES 0 FAJ_SPOTRATES d:\c\wpp\primer\faj.rtf 10 M543BA154 FAJ_DEFN_SPOTRATE 1 FAJ_DEFN_SPOTRATE d:\c\wpp\primer\faj.rtf 10 ;4038BA7A FAJ_FN_5 1 FAJ_FN_5 d:\c\wpp\primer\faj.rtf 10 ABC8FC3EC FAJ_TABLE_I 1 FAJ_TABLE_I d:\c\wpp\primer\faj.rtf 10 C18F8BDA6 FAJ_FWDRATES 0 FAJ_FWDRATES d:\c\wpp\primer\faj.rtf 11 CAC25E8BD FAJ_TABLE_II 1 FAJ_TABLE_II d:\c\wpp\primer\faj.rtf 11 GB22B8903 FAJ_VOLATILITY 0 FAJ_VOLATILITY d:\c\wpp\primer\faj.rtf 12 A870EFE7F FAJ_BINTREE 0 FAJ_BINTREE d:\c\wpp\primer\faj.rtf 13 =C9818B42 FAJ_FIG_A 1 FAJ_FIG_A d:\c\wpp\primer\faj.rtf 13 ;4038BA7B FAJ_FN_6 1 FAJ_FN_6 d:\c\wpp\primer\faj.rtf 13 =C9818B42 FAJ_FIG_A 1 FAJ_FIG_A d:\c\wpp\primer\faj.rtf 13 =C9818B43 FAJ_FIG_B 1 FAJ_FIG_B d:\c\wpp\primer\faj.rtf 13 ?6634D2F6 FAJ_VOLSTD 0 FAJ_VOLSTD d:\c\wpp\primer\faj.rtf 14 ;4038BA7C FAJ_FN_7 1 FAJ_FN_7 d:\c\wpp\primer\faj.rtf 14 C9498024C FAJ_DETVALUE 0 FAJ_DETVALUE d:\c\wpp\primer\faj.rtf 15 =C9818B43 FAJ_FIG_B 1 FAJ_FIG_B d:\c\wpp\primer\faj.rtf 15 K6D6D8279 FAJ_CONSTRUCTING 0 FAJ_CONSTRUCTING d:\c\wpp\primer\faj.rtf 16 ABC8FC3EC FAJ_TABLE_I 1 FAJ_TABLE_I d:\c\wpp\primer\faj.rtf 16 =C9818B44 FAJ_FIG_C 1 FAJ_FIG_C d:\c\wpp\primer\faj.rtf 16 =C9818B44 FAJ_FIG_C 1 FAJ_FIG_C d:\c\wpp\primer\faj.rtf 16 /&;)Lz’†_’’ūŪ·’„æ’’’3Ÿ3’’’’ContentsĢCredit”Andrew J. KalotayŠGeorge O. Williams%Frank J. Fabozzi€Abstractł„A Model for the Valuation of Bonds and Embedded OptionsˆA Brief History of Bond Valuation‚Spot Rates and Forward Rates<„Spot Ratesö‡Forward Rates¼Interest Rate Volatility^Binomial Interest Rate TreeVolatility and Standard DeviationtƒDetermining the Value at a NodeConstructing a Binomial Interest Rate Treed‡Using the Binomial Tree|ŠValuing a Callable BondžDetermining the Call Option Value!Transaction Costs Other Embedded Optionsl‚OAS, Effective Duration, and Effective ConvexityĀŒAfter-tax ValuationŽThe Challenge of ImplementationbFootnote 1¢ Footnote 2¤ Footnote 3 ƒFootnote 4„Footnote 5Ō„Footnote 6 †Footnote 7’†Footnote 8eˆFootnote 9āˆTable I!Table II„Figure A¾Figure BGFigure C½Figure DFigure EyFigure FŪFigure G_Figure HįEmbedded Options|Option-free BondÄOn-the-run Yield Curve€Spot Rate@€Volatility’€Forward RateuQuick Help on 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