Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond

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1 What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixed-icome security that typically pays periodic coupo paymets, ad a pricipal paymet at maturity Cash Flows of A Typical Bod 2 3 t $C $C $C $C $F Bod Valuatio Importat characteristics of a bod: Face Value ($ F) Pricipal amout. Coupo ($ C) The periodic iterest paymet. (t periods) Number of periods to maturity. Required rate (r ) Give the risks associated with the bod (e.g., default risk, iterest rate risk), the appropriate required rate. Coupo Rate ad Curret Yield Cash Flows of A Typical Bod Coupo rate is C/F Curret yield is the aual coupo of a bod expressed as a percet of its market price, P. CY C P 2 3 t $C $C $C $C $F Curret yield gives the retur o your ivestmet oly i terms of the coupo iterest paymets.

2 Bod Valuatio Give face value, coupo paymets, umber of periods to maturity ad required rate of retur, we ca write dow a bod s itrisic value as: V B C r + F ( + ( + Bod Valuatio I geeral, the price of a bod is the preset value of its future cash flows, which typically cosists of coupos ad pricipal value. Usig the above priciple, you should be able to value bods with ueve cash flows or ueve iterest rates. Example of a Coupo Bod A coupo bod promises to pay at the ed of each year for the ext 28 years ad to retur the pricipal of 0 after 28 years. The appropriate required rate of retur o the bod, give its risk characteristics, is 9.5%. What is the bod s itrisic value? Bod Valuatio Example The pricipal paymet is worth: The coupo paymets are worth: Thus, Bod Value Example Wall Street Joural reports the followig details o a corporate bod: Face Value Coupo 8% aual coupo 0 years Example Wall Street Joural reports the followig details o a corporate bod: Face Value Coupo 8% 0 years Cash flows from the bod are: $8 $8 $8 $8 2

3 Example Wall Street Joural reports the followig details o a corporate bod: Face Value Coupo 8% aual coupo 0 years If the appropriate discout rate is 6%, what is the value of the bod? Example Wall Street Joural reports the followig details o a corporate bod: Face Value Coupo 8% aual coupo 0 years Solutio: The Bod Price-Retur Relatioship Prices ad returs o bods are iversely related $200 Price $50 $50 $0 C P r + ( + ( + 0% 2% 4% 6% 8% 0% 2% 4% 6% Rate F Discout ad Premium Bods The face value serves as a bechmark price to defie discout or premia o bods: Premium Bod: P > Face Value ad r < Coupo Rate At Par Bod: P Face Value ad r Coupo Rate Discout Bod: P < Face Value ad r > Coupo Rate Bod Maturity ad Price Covergece As the maturity date approaches, the price of a bod approaches its face value. Price Premium Bod Discout Bod Bod Prices ad Returs: Yield-to-maturity (YTM) Years to Maturity YTM is the required rate of retur that equates the market price of a bod to its itrisic value. P V B C r + ( + ( + YTM reflects the market s required rate of retur give the bod s features ad give its perceived riskiess. F 3

4 Example Wall Street Joural reports the followig details o a corporate bod: Face Value Coupo 8% aual coupo 0 years Suppose the market price of the bod is $ Fid the curret yield ad YTM of the bod. Example Wall Street Joural reports the followig details o a corporate bod: Face Value Coupo 8% aual coupo 0 years Suppose the market price of the bod is $ Fid the curret yield ad YTM of the bod. Example Wall Street Joural reports the followig details o a corporate bod: Face Value Coupo 8% aual coupo 0 years Suppose the market price of the bod is $ YTM is 7% because we have: Usig Equatios to Fid YTM You eed to solve for r such that: $8 r + ( + r) ( + r) 0 0 $07.02 Usig Equatios to Fid YTM You eed to solve for r such that: $8 r ( ) + + r ( + r) Trial ad error gives us: r 6%: $8 0 0 $ $07.02 Usig Equatios to Fid YTM You eed to solve for r such that: $8 r ( ) + + r ( + r) Trial ad error gives us: r 6%: r 8%: $8 0 0 $8 $ $

5 Usig Equatios to Fid YTM Usig a Fiacial Calculator to Fid YTM You eed to solve for r such that: $8 r ( ) + + r ( + r) Trial ad error gives us: r 6%: r 8%: r 7%: $8 0 0 $8 $ $ $ $07.02 PV$07.02 PMT$8 0 FV Calculate r% 7.00% Bod Risk Bod Risk The coupo paymet ad face value of a bod are fixed. However, bods are ot riskfree. Default risk is risk from the possibility that the issuer will ot be able to pay coupo iterest ad face value as promised. As with ay asset, risk icreases the required returs for bods. Bods are rated by several ratig agecies such as Moody s, Stadard ad Poor ad Fitch Ivestor Services. Lower ratigs by these agecies mea more risk ad higher required returs. Bod Ratigs Quality Prime S&P AAA Moody s Aaa High AA Aa Upper Medium A A Medium BBB Baa Bod Risk Iterest rate risk is risk from iterest rate chages durig a bod s term. It is easier to quatify the iterest rate risk tha the default risk Use duratio Juk I Default BB, B, CCC, CC,C D Ba, B, Caa, Ca, C D 5

6 Iterest Rate Risk Bod Valuatio II Ja-34 Ja-40 Ja-46 Ja-52 Ja-58 Ja-64 Ja-70 Ja-76 Ja-82 Ja-88 Ja-94 Ja-00 Overview of Term Structure of Iterest Rates Relatioship betwee yield to maturity ad maturity Iformatio o expected future short term rates ca be implied from yield curve Three major theories are proposed to explai the observed yield curve Yields Yield Curves Upward Slopig Flat Dowward Slopig Years to Maturity Yields (%) Yield Curve U.S. Treasury Yield Curve (3/3/2002) Years to Maturity Expected Iterest Rates i Comig Years (Table 5.) Expected Oe-Year Rates i Comig Years Year Iterest Rate 0 (today) 8% 0% 2 % 3 % 6

7 Pricig of Bods usig Expected Rates PV ( + r )( + r )...( + r ) 2 PV Preset Value of $ i periods r Oe-year rate for period r 2 Oe-year rate for period 2 r Oe-year rate for period Log-Term Rates ad Bod Prices usig Expected Rates Time to Maturity Price of Zero* Yield to Maturity $ % * $,000 Par value zero coupo bod Forward Rates A forward rate is the iferred short-term rate for a future period that makes the expected total retur of a log-term bod equal to that of rollig over short-term bods. Uder the expectatios hypothesis, the market s expectatio of future short-term iterest rates may be iferred from forward rates. Forward Rates Example: Suppose that two-year maturity bods offer yields to maturity of 5%, ad three-year bods have yields of 6%. What is the forward rate for the third year? If we buy a three-year bod, i 3 years per dollar ivested we will have Forward Rates If we buy a two-year bod ad re-ivest all proceeds i a oe-year bod i the third year at a rate of r 3, i 3 years per dollar ivested we will have Sice the two strategies must be equally attractive, Therefore, Forward Rates from Observed Log-Term Rates ( + y ) ( ( + f ) ) + y f oe-year forward rate for period y yield for a security with a maturity of ( + y ) ( + y ) ( + f ) 7

8 Example of Forward Rates usig Table 5.2 Numbers 4 yr 9.993% 3yr 9.660% f? (.0993) 4 (.0966) 3 (+f ) (.46373) / (.3870) (+f ) f.0998 or % Theories of Term Structure Expectatios Forward rates are expected future iterest rates Liquidity Preferece Upward bias over expectatios Market Segmetatio Preferred Habitat The Bod Price-Retur Relatioship Prices ad returs o bods are iversely related $200 Price $50 $50 $0 C P r + ( + ( + 0% 2% 4% 6% 8% 0% 2% 4% 6% Rate F Bod Duratio ad Iterest Rate Sesitivity Duratio measures the resposiveess of a bod s price to iterest rate chages. Duratio is computed as: t Ct t t ( + r) Duratio D P * D Modified Duratio D + r 0 Bod Duratio ad Iterest Rate Sesitivity Duratio gives the average time at which the bod s cash flows are received. Modified duratio gives the approximate percetage chage i a bod s price for a 00 basis poit chage i iterest rates: P r D P + r * D r Duratio for Bod The Wall Street Joural reports the followig details o a corporate bod: Face Value Coupo 8% aual coupo 0 years 8

9 Duratio for Bod The Wall Street Joural reports the followig details o a corporate bod: Face Value Coupo 8% aual coupo 0 years Suppose market price equals $4.72 ad the required rate is 6%. The duratio is give by: Duratio for Bod The Wall Street Joural reports the followig details o a corporate bod: Face Value Coupo 8% aual coupo 0 years Suppose market price equals $4.72 ad the required rate is 6%. The duratio is give by: Duratio 0 t t $8 0 t + ( ) ( ) $ years Duratio of a Zero Coupo Bod Similar to the Bod Cosider the followig zero-coupo bod which is similar to the bod: Face Value Iterest paid at maturity $ years A required rate of 6% gives P 0 $4.72 ad a duratio of: Duratio ad Price Chages Example: A bod sellig for $,84 has a yield to maturity of 6% ad a duratio of 7.37 years. If iterest rates rise by.50%, what is the ew price of the bod? Bod Maturity ad Iterest Rate Sesitivity Log-term bods have greater iterest rate risk tha short term bods. Size of Coupo ad Iterest Rate Sesitivity Low coupo bods have greater iterest rate sesitivity tha high coupo bods. Price $250 $200 $50 $50 $0 0 Year 20 Year 5 Year 0% 2% 4% 6% 8% 0% 2% 4% 6% Rate Price $200 $50 $50 $0 % 3% 5% 7% 9% Rate % Coupo Zero 3% 5% 9

10 Rules for Duratio Rule : The duratio of a zero-coupo bod equals its time to maturity. 2 3 T Rules for Duratio Rule 2: Holdig maturity costat, a bod s duratio is higher whe the coupo rate is lower. 2 3 T 0 C C C C 0 Rules for Duratio Rule 3: Holdig coupo rate costat, a bod s duratio geerally icreases with its time to maturity. Duratio always icreases with maturity for at par bod or premium bod. 2 3 T Passive Bod Maagemet Two classes of passive bod maagemet. The first is a idexig strategy that attempts to replicate the performace of a give bod idex. The secod class is immuizatio. C C C C 0 Portfolio Duratio The duratio of a portfolio is the valueweighted average of the idividual asset duratios: D P i w D i i Active Bod Maagemet There are basically two ways to profit from active bod maagemet: Correctly forecast iterest rate chages ad trade accordigly. Idetify bods which are mispriced, give their perceived level of default risk. Empirical evidece suggests that prices i the bod market are extremely efficiet. 0

11 Covexity The curvature of the price yield curve is called the covexity of the bod. Covexity is the rate of chage of the slope of the priceyield curve. Higher covexity meas that prices of bods will icrease more whe yields decrease ad fall less whe yields icrease. P D y + Covexity P 2 ( y) 2

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