Dr. Maddah ENMG 602 Intro to Financial Eng g 01/18/10. Fixed-Income Securities (2) (Chapter 3, Luenberger)

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1 Dr Maddah ENMG 60 Itro to Fiacial Eg g 0/8/0 Fixed-Icome Securities () (Chapter 3 Lueberger) Other yield measures Curret yield is the ratio of aual coupo paymet to price C CY = For callable bods yield to call is the IRR calculated assumig the bod is called at the earliest possible date Duratio Duratio gives a direct measure of bod price sesitivity to iterest rate Geerally for a cash flow stream with cash flows at times t 0 t t the duratio is give by D = = 0 V( t) t where V(t ) is the preset value of the cash flow at time ad = V( t ) is the preset value of the whole stream = Duratio is a weighted average of cash flow times Whe cash flows are all oegative t 0 D t

2 For a fiacial istrumet that geerates a cash flow stream of m paymets per year over a total of periods ad a paymet C i period the duratio is give by D = = ( / m) C /[ + ( λ / m)] = C /[ + ( λ / m)] where λ is the yield to maturity (iterest rate) This duratio is called Macaulay duratio For a bod with a price of ad a face value F maig m coupo paymets per year of C/m (with a total of paymets) ad YTM λ C = C/m for = ad C = C/m + F Defie the coupo rate per period as c = (C/m)/F ad the yield per period as y = λ/m The the Macaulay duratio of the bod is ( / m)( cf)/( + y) + ( / m) F /( + y) = D = F y cmf my y /( + ) + [( ) /( )] /( + ) ( c/ m) /( + y) + ( / m)/( + y) = = /( + y) + ( c/ y) /( + y) cy( + y) /( + y) + y = = my + cm ( + y)

3 It ca be show that = ( + y) ( + y+ y) /( + y) /( + y) = y Upo simplificatio the duratio of the bod is + y + y+ ( c y) D = my my + cm ( + y) If the bod is at par c = y + y D = my ( + y ) roperties of bod duratio D(c m y ) 5 D( ) D( ) D( ) D( ) Duratio is always less tha time to maturity As time to maturity gets large duratio teds to a fiite limit Duratio is ot too sesitive to coupo rate Log duratios are achieved with log maturities ad low coupo rates 3

4 Duratio ad sesitivity The bod cash flow at time is V C = [ + ( λ / m)] dv ( / m) C ( / m) V d [ + ( / m)] [ + ( / m)] The = = + λ λ λ Recall that the price of the istrumet is = V The = Therefore d dv ( / ) = = m V = D dλ dλ + ( λ/ m) + ( λ/ m) = = d D dλ = where D M D / [ + (λ/m)] is the modified duratio That is D M measures the relative chage of as λ chages For a small chage of Δλ the relative price chage is Δ DM Δ λ M Duratio of a portfolio Cosider a portfolio havig m b bods Let V be the preset value of the cash flow at time from bod Suppose there is a total of time periods 4

5 The the price of the portfolio is mb mb = = = = V = where is the price of bod The duratio of the portfolio is D mb mb tv D m = = = = = = b = wd where D is the duratio of bod ad w / is the weight of bod Immuizatio Immuizatio is the process of structurig a bod portfolio to protect agaist iterest rate ris Specifically suppose that a series of future obligatios is to be met from ivestmet i a bod portfolio Matchig the preset value of the obligatios with the preset value of the portfolio allows meetig the obligatios if the yield (iterest rate) does ot chage (I this chapter we assume all bods have the same yield) However if the yield chages the the preset values may ot match aymore Immuizatio approximately solves this problem by matchig both preset value ad duratios 5

6 Immuizatio implies that the preset values of the obligatios ad the bod portfolio will respod idetically (to the first order) to yield chage Immuizatio is widely used i practice Graphical illustratio of immuizatio The followig figure gives the preset value of the obligatio (V O ) ad of the immuized portfolio (V I ) as a fuctio of the iterest rate λ for Example V O V I λ Note that the immuized portfolio has the same value ad the same slope as the obligatio at the iitial yield (iterest rate) value λ = 009 As a result as λ chages by small amouts aroud 009 the values of the obligatio ad the portfolio remai close 6

7 To see the beefit of immuizatio cosider two alterate portfolios: ortfolio havig Bod oly with value V ad portfolio with Bod oly with value V The umber of Bods i i = i ortfolio i is selected so that V O = V i at the iitial yield λ = 009 (Recall that the immuized portfolio has both Bods ad with weights that match the obligatio value ad duratio) The followig figure show how V ad V vary as λ chages o the same graph as V O ad V I 700 V O 600 V I 500 V V λ Ulie the immuized portfolio ortfolios ad values do ot follows the value of the obligatio closely as λ varies 7

8 Issues with immuizatio If the yield chages the the portfolio will ot be immuized at the ew rate It is therefore desirable to rebalace the portfolio Assumig equal yields is problematic as log-maturity bods usually have higher yields tha short-maturity bods I additio it is uliely that yield o all bods will chage by the same amout Chapter 4 cosiders of bods with differet yields Covexity Duratio gives a first-order approximatio of bod price sesitivity to yield A better approximatio is obtaied by icludig a secodorder term called covexity Cv where d dv d ( / mc ) Cv = = = + dλ = dλ = dλ [ + ( λ/ m)] ( + ) C The Cv = [ + ( λ/ m)] m [ + ( λ/ m)] = Covexity is the weighted average of t t + modified by the factor /[(+(λ/m)] (Recall that the modified duratio is the weighted average of t modified by /[(+(λ/m)] ) For a small chage of Δλ the relative price chage is Δ Cv D Δ M λ + Δ 8

1 + r. k=1. (1 + r) k = A r 1

1 + r. k=1. (1 + r) k = A r 1 Perpetual auity pays a fixed sum periodically forever. Suppose a amout A is paid at the ed of each period, ad suppose the per-period iterest rate is r. The the preset value of the perpetual auity is A