Global Edition. Drawbacks of Traditional Yield Spread Analysis. Static Spread: Discount the Cash flows with Zero Rates

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1 Drawbacks f Traditinal Yield Spread Analysis Glbal Editin Chapter 17 Analysis f Bnds with Embedded Optins Traditinal analysis calculating the difference f the yield t maturity (r yield t call) f the bnd in questin and a cmparable-maturity Treasury btained frm the Treasury yield curve. Fr example, cnsider tw 8.8% cupn 25-year bnds: I s s u e Treasury Crprate P r i c e $ Yield t Maturity (%) The yield spread fr these tw bnds= 10.24%-9.15%=109 bps The drawbacks are (1) the yield fr bth bnds fails t take int cnsideratin the term structure f interest rates, (2) in the case f callable and/r putable bnds, expected interest rate vlatility may alter the cash flw f a bnd Pearsn Educatin Static Spread: An Alternative t Yield Spread In traditinal yield spread analysis, cmpares the yield t maturity f a bnd with that f a similar maturity n-the-run Treasury security. It makes little sense! The cash flws f the crprate bnd will nt be the same as that f the benchmark Treasury. The prper way t cmpare nn-treasury bnds: Cmpare with a prtfli f Treasury securities that have the same cash flw. The crprate bnd s value is equal t the PV f all the cash flws. Static Spread: Discunt the Cash flws with Zer Rates The crprate bnd s value, given the cash flws are riskless, will equal the PV f the replicating prtfli f Treasury securities. These cash flws are valued at the Treasury spt rates. See next slide (Exhibit 17-1) The price wuld be $ The crprate bnd s price is $ Investrs in fact require a yield premium fr the risk assciated with hlding a crprate bnd rather than a riskless package f Treasury securities Pearsn Educatin 2013 Pearsn Educatin

2 Exhibit 17-1 Calculatin f Price f a 25-Year 8.8% 8% Cupn Bnd Using Treasury Spt Rates Treasury Present Perid Cash Flw Spt Value Rate (%) Theretical price Pearsn Educatin Definitins f Static Spread Als called the zer-vlatility spread, a measure f the spread ver the entire Treasury spt rate curve if the bnd is held t maturity. Nt a spread ver ne pint n the Treasury yield curve, l like the traditinal yield spread. Cash flws frm the crprate bnd discunted at the Treasury spt rate plus the static ti spread equal lt the crprate bnd s price. Exhibit 17-2 (next slide) illustrates the calculatin f the static spread fr a 25-year 8.8% cupn crprate bnd Pearsn Educatin Exhibit 17-2 Calculatin f the Static Spread fr a 25-Year 8.8% 8% Cupn Crprate Bnd Perid Cash Flw Treasury Spt Rate (%) Present Value if Spread Used Is: 100 BP 110 BP 120 BP Ttal present value Pearsn Educatin Cmpare Yield Spread with Static Spread Exhibit 17-3 (Next slide) shws the static ti spread and the traditinal yield spread fr bnds with varius maturities and prices. The shrter the maturity, the less the static ti spread will differ frm the traditinal yield spread. The magnitude f the difference between the yield spread and the static spread als depends n the shape f the yield curve. The steeper the yield curve, the mre the difference fr a given cupn and maturity. Anther reasn fr the small differences is that the crprate bnd makes a bullet payment at maturity Pearsn Educatin

3 Exhibit 17-3 Cmparisn f Traditinal Yield Spread and Static ti Spread fr Varius Bnds a Spread (basis pints) Bnd Price Yield t Maturity (%) Traditinal Static Difference 25-year 8.8% 8% Cupn Bnd Treasury A B C year 8.8% Cupn Bnd Treasury J K L a Assumes Treasury spt rate curve given in Exhibit Callable Bnds and Their Investment Characteristics Tw disadvantages f callable bnds t the bndhlder: i. Expse bndhlders t reinvestment risk ii. price appreciatin ptential fr a callable bnd in a declining interest-rate envirnment is limitedi This phenmenn fr a callable bnd is referred t as price cmpressin. If the investr receives sufficient i ptential ti cmpensatin in the frm f a higher ptential yield, an investr wuld be willing t accept call risk Pearsn Educatin 2013 Pearsn Educatin Traditinal Valuatin Methdlgy fr Callable Bnds Price-Yield Relatinship fr a Callable Bnd The practice has been t calculate a yield t wrst, the smallest f the yield t maturity and the yield t call fr all pssible call dates. The yield t call assumes that all cash flws can be reinvested at the cmputed yield until the assumed call date. The yield t call assumes that i. the investr will hld the bnd t the assumed call date ii. the issuer will call the bnd n that date. These underlying assumptins abut the yield t call are unrealistic because they d nt take int accunt hw an investr will reinvest the prceeds if the issue is called. Exhibit 17-4 (next slide) shws the price yield relatinship fr bth a nncallable bnd and callable bnd. The price yield relatinship fr an ptin-free bnd is cnvex. See the cnvex curve a a' The unusual shaped curve a b is the price yield relatinship fr the callable bnd. The reasn fr the unusual shape fr the callable bnd: When the prevailing market yield is higher than the cupn interest, it is unlikely that the issuer will call the bnd. If a callable bnd is unlikely t be called, it will have the same cnvex price yield relatinship as a nncallable bnd when yields are large enugh 2013 Pearsn Educatin 2013 Pearsn Educatin

4 Exhibit 17-4 Price-Yield Relatinship fr a Nncallable and Callable Bnd Price-Yield Relatinship fr a Callable Bnd (cntinued) Price a b Callable Bnd a - b Nncallable Bnd a - a a As yields in the market decline, the likelihd that the issuer will call the bnd increases. The exact yield level at which the issue is likely t be called may nt be actually estimated: called it y*. At yield levels belw y*, the price-yield relatinship fr the callable bnd departs frm that t fr the nncallable bnd. Fr a range f yields belw y*, there is price cmpressin Limited price appreciatin as yields decline. The prtin f the callable bnd price-yield relatinship belw y* is said t be negatively cnvex. y* Yield 2013 Pearsn Educatin 2013 Pearsn Educatin Price-Yield Relatinship fr a Callable Bnd(cntinued) Negative cnvexity implies price appreciatin will be less than the price depreciatin fr a given change in yield. Fr a bnd that t is ptin-free: psitive cnvexity, price appreciatin will be greater than the price depreciatin fr a given change in yield. The price changes resulting frm bnds exhibiting psitive cnvexity and negative cnvexity are in Exhibit 17-5 (see next slide). ) It is imprtant t understand that a bnd can still trade abve its call price even if it is highly likely t be called. Exhibit 17-5 Price Vlatility Implicatins f Psitive and Negative Cnvexity Abslute Value f Percentage Price Change Change in Interest Rates Psitive Cnvexity Negative Cnvexity -100 basis pints X% Less than Y% +100 basis pints Less than X% Y% 2013 Pearsn Educatin 2013 Pearsn Educatin

5 Cmpnents f a Bnd with an Embedded Optin (Callable Bnds) Decmpse abnd with ptins int its cmpnent parts. In a callable bnd: Bndhlder has sld the issuer a call ptin that allws the issuer t repurchase the bnd frm the first callable until the maturity date. The callable bnd hlder enters int tw separate transactins: i. buys a nncallable bnd frm the issuer ii. sells the issuer a call ptin A callable bnd is equal t the price f the tw cmpnents parts: callable bnd price = nncallable bnd price call ptin price Graphically, this can be seen in Exhibit 17-6 (see next slide). The difference between the price f the nncallable bnd and the callable bnd at any given yield is the price f the embedded call ptin. Price Exhibit 17-6 Decmpsitin f a Price f a Callable Bnd Nte: At y** yield level: P NCB = nncallable bnd price P CB = callable bnd price P NCB -P CB = call ptin price b P CB Callable Bnd a - b a P NCB Nncallable Bnd a - a a y** y* Yield 2013 Pearsn Educatin 2013 Pearsn Educatin Cmpnents f a Bnd with an Embedded Optin (Putable Bnds) The abve lgic can be applied t putable bnds. The bndhlder has the right t sell the bnd t the issuer at a designated price and time. A putable bnd can be brken int tw separate transactins. i. The investr buys a nncallable bnd. ii. The investr buys an ptin frm the issuer that allws the investr t sell the bnd t the issuer. The price f a putable bnd is then putable bnd price = nn-putable bnd price + put ptin price Valuatin Mdel The bnd valuatin prcess requires that we use the theretical spt rate t discunt cash flws. Equivalent t discunting at a series f frward rates. Fr an bnd with embedded ptin, the valuatin mdel cnsiders hw interest-rate t t vlatility affects the bnd value thrugh h its effects n the ptins. Three mdels can be used t accunt fr the valuatin effect f embedded ptins. i. Nt a mrtgage-backed security r asset-backed security and which can be exercised at mre than nce. ii. A bnd with an embedded d ptin where the ptin can be exercised nly nce. iii. A mrtgage-backed security r certain types f asset-backed securities Pearsn Educatin 2013 Pearsn Educatin

6 Valuatin f Optin-Free Bnds PV f the cash flws discunted at the spt rates. Given the fllwing hypthetical yield curve: M a t u r i t y Y e a r s Y i e ld t M a t u r i t y (%) M a r k e t V a l u e Assuming annual-pay bnds. Using the btstrapping t methdlgy, the spt rates and the ne-year frward rates can be btained. Y e a r s S p t R a t e (%) O ne Y e a r F r w a r d R a t e Valuatin f Optin-Free Bnds (cntinued) EXAMPLE. Cnsider an ptin-free bnd with three years remaining t maturity and a cupn rate f 5.25%. The price f this bnd can be calculated in ne f tw ways, bth prducing the same result. i. The cupn payments can be discunted at the zer-cupn rates: $5.25 $5.25 $100 + $ = $ ( ) ( ) ii. The secnd way is t discunt by the ne-year frward rates: $5.25 $5.25 $100 + $ = $ ( 1.035) ( ) ( 1.035) ( ) ( ) 2013 Pearsn Educatin 2013 Pearsn Educatin Intrducing Interest-Rate Vlatility When we allw fr embedded d ptins, cnsideratin must be given t interest-rate vlatility. This can be dne by intrducing an interest-rate tree,, r an interest-rate lattice. This tree is nthing mre than a graphical depictin f the ne-perid future spt rates ver time based n sme assumed interest-rate mdel and interest-rate vlatility. Interest-Rate Mdel It is a prbabilistic descriptin f hw interest rates can change ver the life f a derivatives. Mdel the relatinship between the level l f shrt-term interest rates and interest-rate vlatility cmmnly use arbitrage-free mdels t describe hw shrt-term interest rates can evlve ver time. The interest-rate mdels based slely n mvements in the shrt-term interest rate are referred t as ne-factr mdels. Mre cmplex mdels wuld cnsider hw mre than ne interest rate changes ver time Pearsn Educatin 2013 Pearsn Educatin

7 Interest-Rate Tree Exhibit 17-7 (next slide) shws a basic type f interest-rate rate tree, a binmial interest-rate rate tree. Referred t as the binmial mdel. The interest rates can realize ne f tw pssible rates in the next perid. Valuatin mdels that assume that interest rates can take n three pssible rates in the next perid are called trinmial mdels. Mre cmplex mdels exist that assume that mre than three pssible rates in the next perid can be realized. Exhibit 17-7 Three-Year Binmial Interest-Rate t t Tree r 0 N r 1H r 1L N L r 2HH r 2HL L r 3HHH H r 3HHL L r 3HLL LL r 2LL N LL r 3LLL N LLL Tday 1 Year 2 Years 3 Years 2013 Pearsn Educatin 2013 Pearsn Educatin Interest-Rate Tree (cntinued) Interest-Rate Tree (cntinued) Each nde (bld blue circle:) represents a time perid that is equal t ne year frm the nde t its left. Labeled with an N, representing nde, Subscript indicates the path that ne-year future spt rates tk t get t that nde. H represents the higher f the tw future spt rates L the lwer f the tw future spt rates. Ex: nde NHH means that t get t that nde the fllwing path fr neyear rates ccurred: The ne-year rate Realized the higher f the tw future spt rates twice. We can simplify the ntatin by letting r t be the lwer ne-year future spt rate t years frm nw because all the ther future spt rates t years frm nw depend n that rate. Exhibit 17-8 (next slide) shws the interest-rate tree using this simplified ntatin. Befre we g n t shw hw t use this binmial interest-rate tree t value bnds, we first need t fcus n i. what the vlatility parameter (σ ) represents ii. hw t find the value f the bnd at each nde 2013 Pearsn Educatin 2013 Pearsn Educatin

8 Exhibit 17-8 Three-Year Binmial Interest t Rate Tree with One-Year 6 Future Spt Rates r 0 N r 1 e 2σ r 1 N L r 2 e 4σ N r 2 e 2σ L r 2 N LL r 3 e 6σ r 3 e 4σ r 3 e 2σ LL Vlatility and the Standard Deviatin In the binmial i mdel, it can be shwn that t the standard d deviatin f the ne-year future spt rate is equal t r 0 σ. The standard deviatin is a statistical measure f vlatility. Nte that the vlatility is measured relative t the current level f rates. EXAMPLE. If σ is 10% and the ne-year rate (r 0 )is4%, what is the standard deviatin f the ne-year future spt rate? What is if r 0 = 12%? r 0 σ =4% 10% = 0.4% r 40 basis pints r 0 σ = 12% 10% = 1.2% r 120 basis pints r3 r 3 N LLL Tday 1 Year 2 Years 3 Years Lwer 1-yr spt rate r 1 r 2 r Pearsn Educatin 2013 Pearsn Educatin Determining the Bnd Value at a Nde Exhibit 17-9 Calculating a Value at a Nde In the binmial mdel, we find the bnd value as illustrated in Exhibit 17-9 (in next slide). The future cash flw frm present nde will be either i. the bnd s value if the shrt rate is the higher rate plus the cupn payment ii. the bnd s value if the shrt rate is the lwer rate plus the cupn payment. One-Year Rate at Nde Where Bnd s Value Is Sught V r* Bnd s Value in Higher-Rate State One Year Frward V H +C Cash Flw in Higher-Rate State The value at present nde is the PV f the expected cash flws Price with the backward inductin methd the apprpriate discunt rate t use is the ne-year future spt rate at the nde. V = [( V + C)0.5 + ( V + C)0.5 ] H 1 + r * L V L +C Bnd s Value in Lwer-Rate State One Year Frward Cash Flw in Lwer-Rate Statet 2013 Pearsn Educatin 2013 Pearsn Educatin

9 Cnstructing the Binmial Interest- Rate Tree Cnstructing the Binmial Interest- Rate Tree (Cnstruct First Time Step) T cnstruct the tree, we use current n-the-run yields and assume a vlatility, σ. The rt rate fr the tree, r 0, is simply the current ne-year rate. In the first year there are tw pssible ne-year rates, the higher rate and the lwer rate. What we want t find is the tw future spt rates that will be cnsistent with the vlatility assumptin, the prcess that is assumed t generate the bserved market value f the bnd. The steps are described in Overheads 17-36, 17-37, and and illustrated in Exhibits and (see Overheads and 17-40). Step 1: Arbitrarily Pick a r 1 : the lwer ne-year future spt rate ne year frm nw, says 45% 4.5%. Step 2: Determine the higher ne-year future spt rate with: r 1 (e 2σ ). This value is reprted at nde. Step 3: Cmpute the 2-year bnd s value ne year frm nw: 3a. The bnd s value tw years frm nw must be determined. 3b. Calculate the PV f the bnd s value fund in 3a using the higher rate: V H. 3c. Calculate the PV f the bnd s value fund in 3a using the lwer rate: V L. 3d. Add the cupn t V H and V L t get the cash flw at and N L, respectively. 3e. Calculate PV f V Hthe + tw C values using V L + Ccurrent spt r*, s we can cmpute: 1 + r* and 1 + r* Pearsn Educatin 2013 Pearsn Educatin Cnstructing the Binmial Interest- Rate Tree (Cnstruct First Time Step) Step 4: Calculate l the average PV f fthe tw cash hflws in step 3. This is the value at a nde is 1 V H + C V L + C r* 1 + r* Step 5: Cmpare the value in step 4 with the bnd s market value. If the tw values are the same, r 1 is the rate we seek. If the value fund in step 4 is nt equal t the market value f the bnd, this means that the value r 1 is incnsistent with (1) the vlatility assumptin f 10%, (2) the bserved market value f the bnd. Find a different value fr r 1. and repeat the afrementined prcess. [Nte. If we get a value less than $100, then the value fr r 1 is t large. Thus we try a lwer value fr r 1.] Ex: when r 1 is 4.5% we get a value f $ in step 4, which is less than the bserved market value f $100 (See Exhibit 17-10) 10).Therefre, 4.5% is t large and the five steps must be repeated, trying a lwer value fr r 1, says 4.074% (See Exhibit 17-11) N Exhibit Finding the One-Year Frward Rates fr Year 1 Using the Tw-Year 4% On-the-Run: First Trial V = C=0 r 0 = 3.500% L V = C= r 2,HH =? r 1,H = 5.496% r 2,HL =? V = N L r 1,L = 4.500% N LL r 2,LL =? 2013 Pearsn Educatin 2013 Pearsn Educatin

10 N Exhibit One-Year Frward Rates fr Year 1 Using the Tw-Year 4% On-the-Run Issue C = 0 r 0 = 3.500% V = r 2HH 2,HH =? r 1,H = 4.976% L r 2,HL =? V = r N 1,L = 4.074% L N LL r 2,LL =? Cnstructing the Binmial Interest- Rate Tree (Fr Fllwing Time Steps) Next, we will use the three-year n-therun issue t get r 2. The same five steps are used iteratively ti prcess t find the ne-year future spt rate tw years frm nw. Object is t find the value fr r 2 that prduce the value matching the bserved market price. The binmial interest-rate tree cnstructed is said t be an arbitrage-free tree. It is s named because it fairly prices the n-the-run issues Pearsn Educatin 2013 Pearsn Educatin Exhibit One-Year Future Spt Rates fr Year 2 Using the Tw-Year 4.5% On-the-Run Issue C = % V = V = C = 4.50 C = 4.50 r 2,HH = 6.757% r 1,H = 4.976% C = 450 r 2,HL = 5.532% V = C = 4.50 N L r 1,L = 4.074% V = C = 4.50 N LL r 2,LL = 4.530% N r 0 = 3.500% L C 4.50 Applicatin t Valuing an Optin-Free Bnd Cnsider a 5.25% crprate bnd that has tw years remaining t maturity and is ptin-free. Let the issuer s n-the-run yield curve is the ne given earlier, and hence the apprpriate binmial interest-rate tree is the ne in Exhibit (the last slide). Exhibit (next slide) shws the backward inductin prcedure prduces a bnd value f $ This value is identical t the bnd value fund earlier when we discunted at either the zer-cupn rates r the ne-year frward rates. The valuatin mdel is cnsistent with the standard valuatin mdel fr an ptin-free bnd Pearsn Educatin 2013 Pearsn Educatin

11 Exhibit Valuing an Optin-Free Crprate Bnd with Three Years t Maturity and a Cupn Rate f 5.25% V = C = % N r 0 = 3.500% V = r 1,H = 4.976% V = r 2,HH = 6.757% V = C = L r 2,HL = 5.532% V = N L r 1,L = 4.074%.689 N LL r 2,LL = 4.530% Valuing a Callable Crprate Bnd The valuatin prcess fr a callable crprate bnd prceeds in the same fashin with ne exceptin: When the call ptin may be exercised by the issuer, the bnd value at a nde must be changed t reflect the lesser f its value if it is nt called (i.e., the cntinuus value ) and the call price. Fr example, cnsider a 5.25% crprate bnd with three years remaining t maturity that is callable in ne year at $100. Exhibit (next t slide) shws s the values at each nde f the binmial interest-rate tree Pearsn Educatin 2013 Pearsn Educatin Exhibit Valuing a Callable Crprate Bnd with Three Years t Maturity and a Cupn Rate f 5.25%, and Callable in One Year at 100 V = C = % V = V = r 2,HH = 6.757% r 1,H = 4.976% V = C = 525 r 2,HL = 5.532% ( ) r N 1,L = 4.074% L V =100 ( ) N LL r 2,LL = 4.530% N r 0 = 3.500% L C 5.25 Impact f Expected Interest Rate Vlatility n Price Expected interest rate vlatility is a key in the valuatin f bnds with embedded ptins. Exhibit (next slide) shws the price f fur 5%, 10- year callable bnds with different deferred call structures (six mnths, tw year, five years, and seven years) based n different expected vlatility f shrt-term interest rates. 1) The price f the ptin-free bnd is the same regardless f the interest rate vlatility. N embedded ptin that is affected by interest rate vlatility. 2) Fr any given level f interest rate vlatility, the lnger the deferred call, the higher the price. The value f the ptin-free bnd has the highest price. 3) The price f a callable bnd mves inversely t the interest rate vlatility Pearsn Educatin 2013 Pearsn Educatin

12 rice (% %) P Exhibit Effect f Interest Rate Vlatility and Years t Call n Prices f 5%, 10-Year Callable Bnds Optin free 7y deferred 5y deferred 2y deferred 6m deferred Determining the Call Optin Value (r Optin Cst) The value f a callable bnd is expressed as the difference between the values f a nncallable bnd and the call ptin. value f a call ptin = value f a nncallable bnd value f a callable bnd The value f a nncallable bnd and the value f a callable bnd can be determined: The difference between the tw values is therefre the value f the call ptin. In ur previus illustratin, the value f the nncallable bnd is $ and the value f the callable bnd is $ , s the value f the call ptin is $ Vlatility f Shrt-Term Interest t Rate (%) 2013 Pearsn Educatin 2013 Pearsn Educatin Extensin t Other Embedded Optins Exhibit Valuing a Putable Crprate Bnd with Three Years t Maturity and a Cupn Rate f 5.25%, and Putable in One Year at 100 This framewrk can be used t analyze ther embedded ptins, such as put ptins, caps and flrs n flating-rate ntes Exhibit (next t slide) ) shws the binmial i interest-rate t t tree fr pricing putable bnds. Because the value f a nn-putable bnd can be expressed as the value f a putable bnd minus the value f a put ptin n that bnd, this means that value f a put ptin = value f a nn-putable bnd value f a putable bnd N V = C = 0 r 0 = 3.500%.261 (98.588) r 2,HH = 6.757% r 1,H = 4.976% (99.732) L r 2,HL = 5.532% V = N L r 1,L = 4.074% V = C= N LL r 2,LL = 4.530% 2013 Pearsn Educatin 2013 Pearsn Educatin

13 Remarks n Valuatin Mdel Optin-Adjusted Spread Incrprating Default Risk Can be extended t incrprate default risk. Adjusting the prbability f a payment default Adjusting the cash flw that will be recvered when a default ccurs. Mdeling Risk The risk that the utput f the mdel is incrrect due t incrrect assumptins. Implementatin Challenge T transfrm the basic interest rate tree int a practical tl requires refinements. The spacing f the time step in the tree must be much finer. Intrduce time-dependent nde spacing culd distrt the term structure f vlatility. The spread ver the Treasury curve that make the theretical price f an interest rate derivative equal t the market price. The reasn that the resulting spread is referred t as ptin-adjusted is because the cash flws f the security whse value we seek are adjusted t reflect the embedded ptin Pearsn Educatin 2013 Pearsn Educatin Optin-Adjusted Spread (cntinued) Translating OAS t Theretical Value Fr a specified OAS, the valuatin mdel can determine the theretical value f the security that is cnsistent with that OAS. As with the theretical ti value, the OAS is affected by the assumed interest rate vlatility. The higher (lwer) the expected interest rate vlatility, the lwer (higher) the OAS. Determining the Optin Value in Spread Terms The ptin value in spread terms is determined as fllws: ptin value (in bps) = static spread OAS Effective Duratin and Cnvexity Apprpriate fr estimating the mdified d duratins fr bnds with embedded ptins In general, the duratin fr any bnd can be apprximated as fllws: duratin = P_ P+ ( )( ) 2 P 0 dy P_ = price if yield is decreased by x bps P+ = price if yield is increased by x bps P 0 = initial price (per $100 f par value) y (r dy) ) = change in rate used t calculate l price (x basis pints in decimal frm) 2013 Pearsn Educatin 2013 Pearsn Educatin

14 Effective Duratin and Cnvexity (cntinued) When the apprximate duratin frmula is applied t a bnd with an embedded ptin, the bnd prices at the higher and lwer yield levels are evaluated frm the valuatin mdel. Duratin calculated in this way is called effective duratin r ptin-adjusted duratin. The differences between mdified duratin and effective duratin are summarized in Exhibit (next slide). The standard cnvexity measure may be inapprpriate fr a bnd with embedded ptins because it des nt cnsider the effect f a change in interest rates n the bnd s cash flw. Exhibit Mdified Duratin Versus Effective e Duratin Duratin Interpretatin: Generic descriptin f the sensitivity f a bnd s price (as a percent f initial price) t a parallel shift in the yield curve Mdified Duratin Effective Duratin Duratin measure in which h it is assumed Duratin measure in which h recgnitin that yield changes d nt change is given t the fact that yield changes may the expected cash flw change the expected cash flw 2013 Pearsn Educatin 2013 Pearsn Educatin