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Tutorial 5 Tutorial Textbook: Fundamentals of Futures and Options Markets by John C. Hull. Pearson ne !nternational "dition. "d #. !$%& number: '(#)*)+'+,-)*',)+ '(#)*)+'+,-)*',)+ Note: Questions with * must be covered covered in tutorial class class
Problem (.'. Companies X and Y have been offered off ered the following rates per annum on a $5 million 10-year investment:
Company X Company Y
Fied !ate "#0 "#"
Floating !ate %&'(! %&'(!
Company X re)uires a fied-rate investment* +ompany Y re)uires a floating-rate investment# ,esign a swap that will net a ban. a+ting a+ting as intermediary. intermediary. 0#/ per annum annum and will appear appear e)ually attra+tive to X and Y#
The spread between the interest rates offered to X and Y is 0.8% per annum on fixed rate investments and 0.0% per annum on floating rate investments. This means that the total apparent benefit to all parties from the swap is 0.8% per annum. Of this 0.2% per annum will go to the bank. This leaves 0.% per annum for ea!h of X and Y. "n other words# !ompan$ X should be able to get a fixedrate return of 8.% per annum while !ompan$ Y should be able to get a floatingrate return &"'O( ) 0.% per annum. The re*uired swap is shown in +igure ,-.. The bank earns 0.2%# !ompan$ X earns 8.%# and !ompan$ Y earns &"'O( ) 0.%.
Fi/ure $(.* ,wap for /roblem -. Problem (.*,. finan+ial institution institution has entered entered into an interest interest rate rate swap with +ompany +ompany X# nder nder the terms of the swap. it i t re+eives 10 per annum and pays si-month %&'(! on a prin+ipal of $10 million for five fi ve years# 2ayments are made every si months# 3uppose that +ompany X defaults on the sith payment date 4end of year 6 when the %&'(!7swap interest interest rate 4with semiannual +ompounding6 +ompounding6 is " per annum for all maturities# 8hat is the the loss to the finan+ial institution9 ssume ssume that si-month si-month %&'(! was per annum halfway through through year # se %&'(! dis+ounting = 0.3 ×.0
1t the end of $ear the finan!ial institution was due to re!eive 300#000 4 =
% of 0
0.3 ×
million5 and pa$ 630#000 4 % of 0 million5. The immediate loss is therefore 30#000. To value the remaining swap we assume than forward rates are reali7ed. 1ll forward rates are 8% per annum. The remaining !ash flows are therefore valued on the assumption
0.3 × 0.08 × .0, 000, 000 = $600, 000
that the floating pa$ment is
and the net pa$ment that
300, 000 − 600, 000 = $.00, 000
would be re!eived is !ost of foregoing the following !ash flows $ear 30#000 .3 $ear 00#000 6 $ear 00#000 6.3 $ear 00#000 3 $ear 00#000
. The total !ost of default is therefore the
9is!ounting these !ash flows to $ear at 6% per six months# we obtain the !ost of the default as 6#000. Problem (.*+. Companies and ' fa+e the following interest rates 4ad;usted for the differential impa+t of taes6:
3 ,ollars 4floating rate6 Canadian dollars 4fied rate6
%&'(!<0#5 5#0
' %&'(!<1#0 =#5
ssume that wants to borrow #3# dollars at a floating rate of interest and ' wants to borrow Canadian dollars at a fied rate of interest# finan+ial institution is planning to arrange a swap and re)uires a 50-basis-point spread# &f the swap is e)ually attra+tive to and '. what rates of interest will and ' end up paying9
:ompan$ 1 has a !omparative advantage in the :anadian dollar fixedrate market. :ompan$ ' has a !omparative advantage in the ;.,. dollar floatingrate market. 4This ma$ be be!ause of their tax positions.5
provides 1 with ;.,. dollars at &"'O( 0.23% per annum# and ' with :anadian dollars at =.23% per annum. The swap is shown in +igure ,-.2.
Problem (.*5. 8hy is the epe+ted loss from a default on a swap less than the epe+ted loss from the default
on a loan with the same prin+ipal9
"n an interestrate swap a finan!ial institution>s exposure depends on the differen!e between a fixedrate of interest and a floatingrate of interest. "t has no exposure to the notional prin!ipal. "n a loan the whole prin!ipal !an be lost. Problem (.++. >he one-year %&'(! rate is 10 with annual +ompounding# ban trades swaps where a fied rate of interest is e+hanged for 1/-month %&'(! with payments being e+hanged annually# >wo- and three-year swap rates 4epressed with annual +ompounding6 are 11 and 1/ per annum# ?stimate the two- and three-year %&'(! @ero rates when %&'(! dis+ounting is used#
The two$ear swap rate implies that a two$ear &"'O( bond with a !oupon of % sells for !2
par. "f
is the two$ear 7ero rate
?.0 + ? 4 + !5 2 = 00 !2 = 0.03
so that
The three$ear swap rate implies that a three$ear &"'O( bond with a !
so that The two and three$ear rates are therefore .03% and 2.-% with annual !ompounding.
Problem (.+-. &n an interest rate swap. a finan+ial institution pays 10 per annum and re+eives threemonth %&'(! in return on a notional prin+ipal of $100 million with payments being e+hanged every three months# >he swap has a remaining life of 1A months# >he average of the bid and offer fied rates +urrently being swapped for three-month %&'(! is 1/ per annum for all maturities# >he three-month %&'(! rate one month ago was 11#" per annum# ll rates are +ompounded )uarterly# 8hat is the value of the swap9 se %&'(! dis+ounting#
The swap !an be regarded as a long position in a floatingrate bond !ombined with a short position in a fixedrate bond. The !orre!t dis!ount rate is 2% per annum with *uarterl$ !ompounding or .82% per annum with !ontinuous !ompounding. "mmediatel$ after the next pa$ment the floatingrate bond will be worth 00 million. The next floating pa$ment 4 million5 is 0...8 × .00 × 0.23 = 2.3
The value of the floatingrate bond is therefore .02 .3e
−0. . .8 2× 2/ . 2
= .00. 6.
The value of the fixedrate bond is 2. 3e
−0...82× 2 / .2
+2.3e
+ 2 . 3e
−0...82×../ .2
− 0. ..82× 3/ .2
+ .02.3e
+ 2.3e
− 0. ..82×.6/ .2
− 0. ..82× 8/ .2
= 8 .=-8
The value of the swap is therefore .00.6. − 8.=-8 = $2 .2=million
1s an alternative approa!h we !an value the swap as a series of forward rate agreements. The !al!ulated value is 42 .3 − 2 .35 e −0...82 ×2 /.2 + 4 .0 − 2 .35 e −0 ...82 ×3 /.2 + 4 .0 − 2 .35 e
0...82 ×8 /.2
+ 4.0 − 2 .35 e
+ 4 .0 − 2 .35 e
−0 ...82 ×.6 /.2
−0 ...82 ×.. /.2
= $2 .2=million
whi!h is in agreement with the answer obtained using the first approa!h.