Morningstar Magazine - October/November 2016 - 52

Cashﬂows Final Cashﬂow Strategies $ $ i on on a h a ow o as h is o a o on on 1 Year a lan In the last issue of Quant U, I discussed how a l on s a i o o a in i l s an how o hos i l s w an i h i s an i l s on both zero-coupon bonds and coupon bonds ha a a in s on a a s n his iss o an show how o as h is o a o on on Review of Coupon Bond Math all ha a o on on a s la a n s coupons n il h on a s a whi h i i a s i s face value o anal a o on on w s n o n h a so h o on a n s s a in wi h h s o on n h a ha a on is iss i all h a o h s o on a n is on o on io in h his is on a i o ons a ai ann all an si on hs i h a ai s iann all as on s a a on s on a a s h a o sal an o an i in a o on io i l s a in w sho l n o on a n a s is l sin a al n a o os s h s an a o i a ion Fi s w n o al la h n o o on a n s in T no h a s o a i o h bond and q no h n o o ons ai a h n o a n s is n = qT $ $ a in on a s on a o now wha a ion o a 2 n il h s 3o on /q will ass a his as ollows Coupon Bond Yields n h las iss o an n h i l on a o on on is n as h ann ali in nal a o no is a ns i nis n = qT i n o h wo s i is h n y ha sol s h ollowin a ion f = n- qT n P= c q Final- nCashﬂowf -j ∑ (1 + ) + (1 + ) (1 + ) y q y q j =1 $ ∑( $ ) ( -j j =1 y q -n ) ( )( ) ) ( ) ( ∑( n f- j ( ) ) - qT ∑( ) ( ) ( )( ) P = that ( ) (1 + ) + T (1 + ) 1 + )and (on 1 +each ∑ the (1 +date) +of(issue ) coupon Note Macaulay Duration D ( y; 1c,+T ) = + c,is)T )F 1( y; ( ) In h Cana ian (Pono i a n a f= = introduced the concept of aPa( y;lac, T( ) - P ) y; ( c, )(1 + ) P ( y; c, T ) = (1 + al) o+ (on1 + a ) i h on a ion as h w i h a a o h Let c no h∑ann a o n ai a h a in o ons as a a ion o Pi ( y; c,oT(1) a+= P)i( y;oc, a) (h1 +a ) n o a on = ( ) (1 + ) + T (1 + ) P ( y; c, ) D ( y; c, T ) = P ( y; c, T ) (1 + ) = (1 + ) = c, (T1) + ) P y; c, P ( Py )( y; ( ) (1 + ) Physical Analogy for the Macaulay Duration P ( y; c, T ) = P ( y; c, ) (1 + ) D ( y ) =(1 +(1 +) ) - = Value of Face Value (1 + ) = Present (1 +Values ) of Coupon Payments P( y; c, P) Present ( y; c, T ) ) (y; c, T = D P ( y; c, T ) P ( y; c, ) (1 + ) P ( y ) = (1 +P ()y; c, T ) (1 + ) D ( y; c, T ) y ) =c, T )(= D ((y; 1+ ) - = 1+ P ( y; c, ) c q n c q -j y q y q -n y f q y q j =1 n c q y q f- j y q c q - qT j =1 n j =1 j -f q c q y q f- j y q - qT c f- j y q y q c q y f q m c cm n q ( ( ) y f Pc ( y ) =1y1 1 + q Year 1 y 1 y y q -j y ) 1+ q - y f q Dm (y; c, T ) = f- j y q c y qf q Z (T ) s (T ) - qT Z (T ) y f q y f q n q n q -j y q y q f- j n q n f- jq y q y q y q f q f- j ys (T ) ys (T ) y f q s (T ) n q 1 y y f q 1 y y q s (T ) -j y q c q n q f- j c q c q j -f q c q 1 y y f q n q Dc ( y ) = y q n j =1 f- j y q c q c q c EXHIBIT 1 c q y q + 1f + 1+ Duration P ( y; c, T ) = y n =1 c, P ( y; c, T ) = P jy; 1+ q q f = n- qT ∑nin(1s+a ) +o(1 + o ) (1 + ) y q y q n - qT f- j n = qT c y where c is the P ( y; c, TP )is=theq price1 +ofqythe +bond 1 +and q j =1 ann al qTo on a Fo h os o f = n- this discussion, nI treat the price as f- j a function of - qT c y j -f y n + ayT ion -nq f1+ q j =1-ij q h q y1i+ cl an yw h i in PD (=y; c,q T ) =1 + q +5 (Maturity) 1+ q 1+ q as 4ollows P ( y; c, T ) j =1 2 f q 3 f yq q D ( y; c, T ) y + 3 P ( y; c, T ) DmP(y; c,- TDm) = y 4P P ( y;5c,(Maturity) T) D ( y; c, T ) DV 01R AT = TIPO) =N-s D y = P Dm c, D Dm U(y; y ( ) + Z T = e1 (T)T q1 + ys (T ) * q n c qo 3 $ on h n i in a a w n 1 coupon period ( Year is si n =ai qT P ( y; c, T ) Dm (y; c, T ) = P ( y; c, T ) f = n- qT P= 52 2 $ $ h on s a al o i a on has a a an a s a s iann al o on al o 5a (Maturity) a h s iann al o on at a 4rate of os s h s n h is Fo o Durationo la o h i o a o on on ni o a al wh h a s ans o a h ceiling Cashﬂows o h n his ans ha i x is an x is x, but if x is no an in x is in h in s hi h han x Fo a l 3 = , but = Q UA N T U a l $ ln [ Z (T )] P s (T ) = - - Dm y T P ( y; c, T ) P C (y; c, T ) = P ( y;ys (c,T )T ) -qT Z (T ) = 1+ D q DV 01 = P =P D ( ) * 1+

Table of Contents for the Digital Edition of Morningstar Magazine - October/November 2016