# Week 3

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Chapter   6   Basic   questions   (1–16)   1.   The   yield   to   maturity   (YTM)   is   the   required   rate   of    return   on   a   bond   expressed   as   a   nominal   annual   interest   rate.   For   non ‐ callable   bonds,   the   yield   to   maturity   and   required   rate   of    return   are   interchangeable   terms.   Unlike   YTM   and   required   return,   the   coupon   rate   is   not   a   return   used   as   the   interest   rate   in   bond   cash   flow   valuation,   but   it   is   a   fixed   percentage   of    face   value   over   the   life   of    the   bond   used   to   set   the   coupon   payment   amount.   For   the   example   given,   the   coupon   rate   on   the   bond   is   still   10%,   and   the   YTM   is   8%.   2.   Price   and   yield   move   in   opposite   directions;   if    interest   rates   rise,   the   price   of    the   bond   will   fall.   This   is   because   the   fixed   coupon   payments   determined   by   the   fixed   coupon   rate   are   not   as   valuable   when   interest   rates   rise.   Hence,   the   price   of    the   bond   decreases.   NOTE:   Most     problems   do   not    explicitly    list    a    par    value    for    bonds.   Even   though   a   bond    can   have   any     par    value,   in   general,   we   have   adopted    a    par    value   of    \$1000.   We   will    use   this    par    value   in   all     problems   unless   a   different     par    value   is   explicitly    stated.   3.   The   price   of    any   bond   is   the   PV   of    the   interest   payment,   plus   the   PV   of    the   par   value.   Notice   this   problem   assumes   an   annual   coupon.   The   price   of    the   bond   will   be:   P   =   \$60({1    –   [1/(1   +   0.08)] 9 }   /   0.08)   +   \$1000[1   /   (1   +   0.08) 9 ]   P   =   \$875.06   We   would   like   to   introduce   shorthand   notation   here.   Rather   than   write   (or   type,   as   the   case   may   be)   the   entire   equation   for   the   PV   of    a   lump   sum,   or   the   PVA   equation,   it   is   common   to   abbreviate   the   equations   as:   PVIF R,t    =   1   /   (1   +   R ) t    which   stands   for   Present   Value   Interest   Factor      PVIFA R,t    =   ({1    –   [1/(1   +   R )] t    }   /   R )   which   stands   for   Present   Value   Interest   Factor   of    an   Annuity   These   abbreviations   are   shorthand   notation   for   the   equations   in   which   the   interest   rate   and   the   number   of    periods   are   substituted   into   the   equation   and   solved.   We   will   use   this   shorthand   notation   in   the   remainder   of    the   solutions   key.   The   bond   price   equation   for   this   problem   would   be:   P   =   \$60(PVIFA 8%,9 )   +   \$1000(PVIF 8%,9 )   P   =   \$875.06   4.   Here,   we   need   to   find   the   YTM   of    a   bond.   The   equation   for   the   bond   price   is:   P   =   \$1038.50   =   \$70(PVIFA R% ,9 )   +   \$1000(PVIF R %,9 )   Notice   the   equation   cannot   be   solved   directly   for   R .   Using   a   spreadsheet,   a   financial   calculator,   or   trial   and   error,   we   find:   R   =   YTM   =   6.42%   If    you   are   using   trial   and   error   to   find   the   YTM   of    the   bond,   you   might   be   wondering   how   to   pick   an   interest   rate   to   start   the   process.   First,   we   know   the   YTM   has   to   be   lower   than   the   coupon   rate   since   the   bond   is   a   premium   bond.   That   still   leaves   a   lot   of    interest   rates   to   check.   One   way   to   get   a   starting   point   is   to   use   the   following   equation,   which   will   give   you   an   approximation   of    the   YTM:   Approximate   YTM   =   [Annual   interest   payment   +   (Par   value    –   Price)   /   Years   to   maturity]   /   [(Price   +   Par   value)   /   2]   Solving   for   this   problem,   we   get:      Approximate   YTM   =   [\$70   +   (–\$38.50   /   9)]   /   [(\$1038.50   +   1000)   /   2]   Approximate   YTM   =   0.0645,   or   6.45%   This   is   not   the   exact   YTM,   but   it   is   close,   and   it   will   give   you   a   place   to   start.   5.   Here   we   need   to   find   the   coupon   rate   of    the   bond.   All   we   need   to   do   is   to   set   up   the   bond   pricing   equation   and   solve   for   the   coupon   payment   as   follows:   P   =   \$963   =   C  (PVIFA 7.5%,12 )   +   \$1000(PVIF 7.5%,12 )   Solving   for   the   coupon   payment,   we   get:   C    =   \$70.22   The   coupon   payment   is   the   coupon   rate   multiplied   by   par   value.   Using   this   relationship,   we   get:   Coupon   rate   =   \$70.22   /   \$1000   Coupon   rate   =   0.0702,   or   7.02%   6.   To   find   the   price   of    this   bond,   we   need   to   realise   that   the   maturity   of    the   bond   is   19   years.   The   bond   was   issued   one   year   ago,   with   20   years   to   maturity,   so   there   are   19   years   left   on   the   bond.   Also,   the   face   value   is   \$200   000   and   the   coupons   are   semi ‐ annual,   so   we   need   to   use   the   semi ‐ annual   interest   rate   and   the   number   of    semi ‐ annual   periods.   The   price   of    the   bond   is:   The   coupon   is   \$200   000   x   6.1%/2   =   6100   P   =   \$6100(PVIFA 2.65%,38 )   +   \$200   000(PVIF 2.65%,38 )   P   =   \$219   014.80    7.   Here,   we   are   finding   the   YTM   of    a   semi ‐ annual   coupon   bond.   The   bond   price   equation   is:   P   =   \$188   000   =   \$6900(PVIFA R% ,26 )   +   \$200   000(PVIF R% ,26 )   Since   we   cannot   solve   the   equation   directly   for   R ,   using   a   spreadsheet,   a   financial   calculator,   or   trial   and   error,   we   find:   R   =   3.818%   Since   the   coupon   payments   are   semi ‐ annual,   this   is   the   semi ‐ annual   interest   rate.   The   YTM   is   the   APR   of    the   bond,   so:   YTM   =   2   3.818%   YTM   =   7.64%   8.   To   find   the   price   of    the   bill,   we   need   to   realise   that   the   maturity   of    the   bill   is   105   days.   Also,   the   face   value   is   \$500   000.   The   price   of    the   bill   is:   P   =   \$500   000   /   (1   +3.5%x105/365)   P   =   \$495   015.94   9.   To   find   the   price   of    this   bill,   we   need   to   realise   that   the   maturity   of    the   bill   is   now   50   days   (105–55).   The   face   value   remains   unchanged   at   \$500   000   The   price   of    the   bill   is:   P   =   \$500   000   /   (1   +3.25%x50/365)   P   =   \$497   783.84   10.   Here,   we   need   to   find   the   coupon   rate   of    the   bond.   All   we   need   to   do   is   to   set   up   the   bond   pricing   equation   and   solve   for   the   coupon   payment   as   follows:
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