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SEMINAR ON BONDS Day 1 • Some Preliminaries • The Time Value of Money • Compounding • Types of Bonds • Yields & Pricing • Yield Curves • Duration (Macauley’s duration , Modified Duration, PVBP) Day2 • Effective Duration. Callable/ Putable Bonds • Convexity • Bond Price Volatility • Inflation Linked Bonds George Stylianopoulos, Fixed Income 1 Present Value and Future Value (One Interest period) • For securities with remaining maturity < 1 year (one cash flow) FV PV days 1 y base PV= present value ( i.e. the cash amount we pay to buy the security) FV = future value ( i.e. the redemption amount of the security including coupon payment, if any) Days= no of days from purchase to maturity Base = 360, or 365, or actual, depending on the market convention Y= yield annualized. George Stylianopoulos, Fixed Income 2 Present Value and Future Value(…continued) Example: Investing 100 € now (PV) for one year at a yield of 10% we get on maturity 110 € (FV) assuming 365 days and 365 base Investing though, today 90 € (PV) for one year and getting on maturity 100 €,(FV) we realize a yield of ? 100 11.11% 1 90 George Stylianopoulos, Fixed Income 3 Present Value and Future Value Multiple interest periods The effect of compounding Example: We invest 100 € on a 3 years security paying an annual coupon 3% once a year each year. What will be the future value of our 100 € investment? FV PV * (1 Y ) n , FV 100 * (1 3%)3 109.2727 This is because each coupon payment we assume we reinvest at the initial yield. The same security paying a semiannual coupon yields us a future value of: y m n FV PV (1 ) m 3% 6 FV 100 (1 ) 109.3443 2 George Stylianopoulos, Fixed Income 4 Interest Calculations • Periodic vs. Continuous compounding Periodic Compounding: m n R FV PV 1 m FV=Future Value (Principle+interest) PV=Present Value (Principle) m=interest frequency per year n=years Continuous compounding: the limit of periodic compounding with m i.e. FV PV e Rn e=2.71828 George Stylianopoulos, Fixed Income 5 Interest Calculations (…continued) Converting periodic to continuously compounding interest rate Let R1= continuously compounded interest rate R2= periodic compounded interest rate m n R FV PV e R1 n FV PV 1 2 m m R e R1 1 2 m R R1 m R1 m ln 1 2 R2 m e 1 m George Stylianopoulos, Fixed Income 6 Interest Calculations (examples) Example 1: The effect of increasing the compounding frequency Interest Rate : 5% Compoundi ng FV at end PV Frequency of year 1 (m) 1 100 105.00 2 100 105.06 4 100 105.09 12 100 105.12 52 100 105.12 365 100 105.13 Example 2: Converting periodic interest rates to continuously compounded Periodic Continuously Periodic Continuously Period Frequency Rates Compounded Rates Compounded Overnight 365 2.10% 2.10% 10.00% 10.00% 1 week 52 2.10% 2.10% 10.00% 9.99% 1 month 12 2.12% 2.12% 10.00% 9.96% 2 months 6 2.14% 2.13% 10.00% 9.92% 3 months 4 2.18% 2.17% 10.00% 9.88% 6 months 2 2.22% 2.20% 10.00% 9.76% 12 months 1 2.33% 2.31% 10.00% 9.53% George Stylianopoulos, Fixed Income 7 Definitions and Concepts • BOND: a financial obligation for which the issuer promises to pay the bondholder a specified stream of future cash flows, including periodic interest payments (coupons) and a principal repayment. Default or Credit Risk BONDHOLDER RISK Market (interest rates risk) Liquidity Risk George Stylianopoulos, Fixed Income 8 Why buy bonds? Investors have traditionally held bonds in their portfolios for three reasons: • Income. Most bonds provide their holders with fixed income. On a set schedule, annually or semiannually or quarterly the issuer sends the bondholder a fixed payment • Diversification. Although diversification does not ensure against loss, an investor can diversify a portfolio across different asset classes that perform independently in market cycles to reduce the risk of low, or even negative, returns. • Protection against Economic Slowdown or Deflation. George Stylianopoulos, Fixed Income 9 Types of Bonds • Bonds are differentiated according to the issuer, and according to the type of their cash flows. • The major bond markets are: 1) sovereign bonds (issued by governments) 2) corporate bonds (issued by corporates) There are many types of bonds, depending on their cash flows among which are: 1) Fixed rate bonds (bonds paying periodically fixed coupon) 2) Floating rate notes (bonds that their coupons are linked to an index, e.g Libor) 3) Callables: I.e. bonds giving the right to the issuer to buy them back before their maturity (e.g. mortgages) 4) Bonds with Sinking Funds: for example a 7% 10 years corporate bond, paying down 10% of the principal amount annually beginning in year 3 5) Zero Coupon bonds: bonds that are just redeemed on maturity without any other periodic payment of interest George Stylianopoulos, Fixed Income 10 Market conventions • Bonds are quoted in the secondary market as a percentage of their face value (Clean or Flat Price) without including the accruals (if any). Example: Market quote for Hell Rep 3.10% due 20 April 10 on 7th Oct 2005 with settlement 12th Oct 2005 was: 101.12 - 101.14 This means that to buy Face value 1 million Euro of this bond we have to pay 1,011,400 Euro + 175 days (days from 20 April 2005 to 12 Oct 2005) accrued interest (3.10%*175*1,000,000/365) 14863.01 Euro = total 1,026,263.01 Euro • Dirty or Full Price of the Bond = Clean Price + Accruals. • Coupon frequency (annual or semiannual), and the days count convention (e.g. 30/360, Act/360, Act/Act, etc..) is predetermined by the issuer. • Most Euro zone government bonds have annual coupons and Actual/Actual year fraction • US Treasuries, Italian BTPs, U.K. Gilts have semiannual coupons and Actual/Actual year fractions. George Stylianopoulos, Fixed Income 11 Yields The main types of yields are the following: • Current Yield c Yc= current yield Yc c = annual coupon payment P P= Clean price • Simple Yield to maturity (or Japanese Yield): Ys=simple yield to maturity RP c R=redemption value (most cases 100) Tsm Ys 365 P=Clean Price P Tsm=days from settlement to maturity c=annual coupon • Yield to maturity: is the value of the discount rate in the bond equation that equates the present value of all future cash payments of the bond to the current market price George Stylianopoulos, Fixed Income 12 Yield to Maturity. The Bond Equation B= dirty price of the bond = Present Value of the bond’s cash flows discounted by the yield to maturity Ci= cash flow of the bond at time ti with 1<i<n Y=yield to maturity n B ci (1 y ) ti i 1 Analytically. Having a bond paying an annual coupon c for n years and redeems at par (100), yielding to maturity y its present value is given by: c c2 c c 100 B 3 3 n n (1 y) (1 y) (1 y) 1 2 (1 y) (1 y) n George Stylianopoulos, Fixed Income 13 Yield to Maturity. The Bond Equation (…continued) • For any coupon frequency the present value of the bond ( i.e. its dirty price is given by the formula: B= dirty price=PV c n R R= redemption value (usually 100) B DSC m DSC n 1 i 1 i 1 Y E Y E Y= Yield to maturity 1 1 m= coupon frequency m m DSC= days from settlement to next coupon payment E= no of days between coupon dates George Stylianopoulos, Fixed Income 14 Yields (…continued) Example: Bond Hell Rep. 3.10% due on 20th April 10. Market price 101.14 as at 7th Oct 2005, settlement 12th Oct 2005. Calculate its current yield, simple yield and yield to maturity. Setlement: 12-Oct-05 Current Yield: 3.065% Maturity: 20-Apr-10 Simple YTM: 2.816% Coupon: 3.10% Yield to Maturity: 2.825% Price: 101.14 Tsm 1651 days from settlement to maturity Tsm/365 4.5232877 remaining years to maturity R-P -1.14 premium/discount of the bond George Stylianopoulos, Fixed Income 15 Current Yield, Simple Yield, Yield to Maturity Examples Settlement: 1-Nov-05 Maturity: 26-Mar-08 20-Apr-08 21-Jun-08 Coupon: 8.60% 3.50% 2.90% Clean Price: 113.45 101.79 100.32 Tsm: 876 901 963 Tsm/365 2.400 2.468 2.638 R-P -13.45 -1.79 -0.32 Current Yield: 7.580% 3.438% 2.891% Simple (Japanese Yield) 2.641% 2.726% 2.770% Yield To Maturity: 2.729% 2.734% 2.769% George Stylianopoulos, Fixed Income 16 Bloomberg Yield Analysis screen for GGB 3.60% JUL 16 George Stylianopoulos, Fixed Income 17 Current vs. Japanese and Yield to Maturity • Current yield is the quickest check of the yield of a bond but not reliable as it ignores any change in the value of the capital invested. Investors today do not rely on current yield. • Simple Yield to Maturity although more accurate than current yield is still not the ideal measure of yield because 1)it assumes constant capital gain (for bonds trading in discount) or capital loss (for bonds trading at premium), and 2) it ignores the time value of money. It is simple. • Yield to Maturity is the most accurate measure of a bond’s yield as it explicitly recognizes the importance of points in time at which different cash payments from a bond are to be received. Implicit in the definition of the yield to maturity is the assumption that the investor will be able to reinvest all coupon payments at at a rate equal to yield to maturity at which he bought his bond. This risk is known as the reinvestment risk. George Stylianopoulos, Fixed Income 18 ZERO COUPON BONDS • Zero coupon bonds are bonds that pay no coupons, or bonds of which their coupons have been striped Cash flows wise are the simplest fixed rate bonds as they have a single cash flow on maturity. The bond equation is simplified as there are no coupon payments to: B=Bond price (dirty=clean) R=redemption value (usually 100) R Y= yield to maturity B DSM DSM= days from settlement to maturity (1 Y ) E E= base of the year (360,365) George Stylianopoulos, Fixed Income 19 ZERO COUPON BONDS (examples) A. Calculating the price of a zero coupon bond from its yield to maturity Settlement: 12-Oct-05 Maturity: 20-Apr-10 Yield: 5% Price: 80.1964 DSM: 1651 Base: 365 Redemption: 100 B. Calculate what is the yield to maturity of the following zero coupon bond Settlement: 12-Oct-05 Maturity: 12-Oct-10 Yield: 5.159% Price: 77.75 DSM: 1826 Base: 365 Redemption: 100 George Stylianopoulos, Fixed Income 20 The Price:Yield Function Price vs Yield 180.00 150.00 Price 120.00 90.00 60.00 30.00 0% 5% 10% 15% 20% Yield George Stylianopoulos, Fixed Income 21 Yield Curves • A yield curve plots the yields to maturity of a series of bonds of the same quality ( i.e. same credit) against their respective terms to maturity. A yield curve can exhibit the following 4 shapes: • Normal or positively sloped yield curve: A curve in which short-term interest rates are lower than longer term interest rates • Inverted yield curve or negatively sloped: A curve in which long -term interest rates are lower than that of shorter term interest rates. • Flat yield curve: Short-term and long term interest rates are roughly equal. • Humped yield curve: A humped yield curve is positively sloped from the short maturity sector to the intermediate sector, but negatively sloped from the intermediate to the long sector. George Stylianopoulos, Fixed Income 22 Normal Yield Curve 4 3.8 3.6 3.4 Yields (%) 3.2 3 2.8 2.6 2.4 2.2 2 0 5 10 15 20 25 30 35 Time to maturity George Stylianopoulos, Fixed Income 23 Inverted Yield Curve 8 7.5 7 Yields (%) 6.5 6 5.5 5 4.5 4 0 5 10 15 20 25 30 Time to maturity George Stylianopoulos, Fixed Income 24 Flat Yield Curve 5 4.5 4 Yields(%) 3.5 3 2.5 2 0 5 10 15 20 25 30 Time to Maturity George Stylianopoulos, Fixed Income 25 Humped Yield Curve 4.5 4 Yields (%) 3.5 3 2.5 2 0 5 10 15 20 25 30 Time to maturity George Stylianopoulos, Fixed Income 26 Theories explaining the yield curve shape • Liquidity preference theory. This theory states that that the yield curve will be upward sloping because of the preference of investors for liquidity. Liquidity here is defined as the ability to recover the principal of the bond in a reasonably short period of time. • Market segmentation theory. This theory views the fixed income market as a series of distinct markets, segregated by maturity. Individual investors and issuers are restricted to specific maturity sectors. Thus investors and issuers do not have complete maturity flexibility. • Expectations theory. According to the expectations theory the shape of the yield curve reflects the market consensus forecast of future interest rate levels. George Stylianopoulos, Fixed Income 27 Summing up theories of the yield curve shape • Yield curves tend to exhibit a modestly positive slope over long periods of time, reflecting the market participants desire for liquidity. Liquidity preference • Market segmentation shows up as an influence on yield curve shape, particularly over short term horizons (e.g. auction periods) and within specific issuer sectors. Imbalances of supply and demand create bumps on the yield curve at various maturity points for specific periods of time. • Finally there are an adequate number of investors with maturity flexibility (e.g. mutual funds) to validate a degree of expectations reflected in the yield curve shape. Inflation fears tend to steepen the slope of the yield curve while disinflation expectations act to flatten or invert the yield curve. George Stylianopoulos, Fixed Income 28 The higher credit quality, the flatter the yield curve German, Italian and Greek Curve 4.5 4.25 4 3.75 3.5 Yields (%) Germany (AAA) 3.25 Italy (AA) 3 Greece (A) 2.75 2.5 2.25 2 0 5 10 15 20 25 30 Maturity (years) George Stylianopoulos, Fixed Income 29 Germany (AAA)- Belgium (AA+) – Greece (A) George Stylianopoulos, Fixed Income 30 Determinants of the absolute level of the yield curve A nominal interest rate can be dissected into three basic components . Nominal interest rate = real interest rate + inflation premium + risk premium 1. The real interest rate is the compensation to the investor for deferring consumption to a future period. Even if inflation is stable at 0% a risk less investment (e.g US Treasury) must offer a positive rate of return. 2. The inflation premium is intended to preserve the purchasing power of the investor over time. This premium reflects an expectation of the future inflation level over the lifespan of the investment. 3. The risk premium protects the investor against all other potential negatives, including a) credit or default risk, b) call or early redemption risk, c) liquidity or marketability risk, d) risk of unexpected changes in inflation (i.e. the degree of unpredictability in assessing future inflation. George Stylianopoulos, Fixed Income 31 DURATION • The weighted average maturity of a bond’s cashflows, where the present values of the cash flows serve as weights • the term to maturity of the equivalent zero coupon bond • the balancing point of a bond’s cash flow stream, where the cash flows are expressed in terms of present value Cash flows 0 Time in years Duration George Stylianopoulos, Fixed Income 32 Calculation of duration of a 10 year bond with coupon 4% priced at par to yield 4% on maturity (4)=(3)/Price (1) (2) (3) (5)= (4)x(1) PV Weighting Years Cashflow PV=(2)/(1+4%)^(1) Weighted t Factor 1 4.00 3.846 0.0385 0.0385 2 4.00 3.698 0.0370 0.0740 3 4.00 3.556 0.0356 0.1067 4 4.00 3.419 0.0342 0.1368 5 4.00 3.288 0.0329 0.1644 6 4.00 3.161 0.0316 0.1897 7 4.00 3.040 0.0304 0.2128 8 4.00 2.923 0.0292 0.2338 9 4.00 2.810 0.0281 0.2529 10 104.00 70.259 0.7026 7.0259 100.000 1.0000 8.4353 Bond Price Duration George Stylianopoulos, Fixed Income 33 Factors influencing Duration • Term to maturity • Coupon rate • Accrued interest • Market yield level • Sinking fund features • Call provisions • Passage of time George Stylianopoulos, Fixed Income 34 The influence of each factor on duration Duration Behavior Factor Low Duration High Duration Term to maturity Shorter Longer Coupon Rate Higher Lower Accrued Interest Large Small Market yield Higher Lower Sinking funds Many Few Call provisions Many Few George Stylianopoulos, Fixed Income 35 Factors influencing duration- A rule of thumb As a rule of thumb: Long Maturity, Low Coupon, Low Yield= High Duration George Stylianopoulos, Fixed Income 36 Duration of 7% coupon bonds of various maturities Each bond is priced at par YTM: 7% Term to Duration Term to Duration Maturity Maturity 1 1 11 8.82 2 1.94 12 9.49 3 2.83 13 10.14 4 3.67 14 10.78 5 4.48 15 11.42 6 5.26 16 12.04 7 6.01 17 12.65 8 6.74 18 13.26 9 7.45 19 13.86 10 8.15 20 14.46 George Stylianopoulos, Fixed Income 37 The duration:term to maturity relationship for coupon bearing bonds 15 10 Duration 5 0 1 15 30 40 50 Term to maturity (years) George Stylianopoulos, Fixed Income 38 The duration:term to maturity relationship for zero coupon bonds 45 30 Duration 15 0 0 10 20 30 40 50 Term to maturity (years) George Stylianopoulos, Fixed Income 39 The durations of 15 year Govt. Bonds for several coupon rates. Yield environment 7% Coupon rate Duration Coupon rate Duration 0 15.00 10 9.10 1 13.34 13 8.69 2 12.24 15 8.49 3 11.46 16 8.40 4 10.87 17 8.32 5 10.41 18 8.25 6 10.04 19 8.19 7 9.75 20 8.13 George Stylianopoulos, Fixed Income 40 The duration:coupon rate relationship 30 25 DURATION 20 15 10 5 0 0 5 10 15 20 25 COUPON RATE(%) George Stylianopoulos, Fixed Income 41 The Duration:accrued interest relationship A bond’s duration is inversely related to the amount of accrued interest attached to the bond. Hellenic Republic due 20/5/13, 7.50% price: 111, settlement 19/5/99, YTM:6.29%, Duration: 8.80 price: 111, settlement: 20/5/99, YTM:6.29%, Duration:9.39 George Stylianopoulos, Fixed Income 42 The Duration:market yield relationship The Durations of 15 year Govt. bonds priced at par in a variety of yield environments Yield environment (%) Duration Yield environment (%) Duration 1 14.00 11 7.98 2 13.11 12 7.63 3 12.30 13 7.30 4 11.56 14 7.00 5 10.90 15 6.72 6 10.29 16 6.47 7 9.75 17 6.23 8 9.24 18 6.01 9 8.79 19 5.80 10 8.37 20 5.61 George Stylianopoulos, Fixed Income 43 The duration-market yield relationship 26 21 16 Duration 11 6 1 0 5 10 15 20 25 -4 Market Yield (%) George Stylianopoulos, Fixed Income 44 Calculation of the duration of a 7% coupon, 10-year corporate bond with a sinking fund paying down10%of the principal amount annually, beginning in year 3. The bond is priced at par to yield 7% to maturity (1) Redeem (2) (3) PV (4)=(3)/Price (5)= (1) x (4) Coupon Year ed Cashflow CF Weight PV weighted 1 0 7 7 6.54 0.0654 0.0654 2 0 7 7 6.11 0.0611 0.1223 3 10% 7 17 13.88 0.1388 0.4163 4 10% 6.3 16.3 12.44 0.1244 0.4974 5 10% 5.6 15.6 11.12 0.1112 0.5561 6 10% 4.9 14.9 9.93 0.0993 0.5957 7 10% 4.2 14.2 8.84 0.0884 0.6190 8 10% 3.5 13.5 7.86 0.0786 0.6286 9 10% 2.8 12.8 6.96 0.0696 0.6266 10 30% 2.1 32.1 16.32 0.1632 1.6318 100% 100.00 1.0000 5.76 Bond Price Duration George Stylianopoulos, Fixed Income 45 The relative contributions to the price of a 7% coupon, 10year corporate bond with (1) no sinking fund provisions and (2) a 70% sinker. Each bond is priced at par to yield 7% to maturity. Percentage of Bond Price Attributable to: Bond Issue Coupon Payments Principal Payment Duration No Sinker 49.17 50.83 7.52 70% Sinker 37.8 62.2 5.76 Sinking fund provisions lower the duration of a bond by reducing the average maturity of the principal repayment. George Stylianopoulos, Fixed Income 46 The duration:passage of time relationship • As time passes a bond’s duration falls 13 at an increasing rate. The duration 12 decline is a natural consequence of the 11 progressively smaller set of remaining 10 coupon cash flows and the approaching 9 Duration principal repayment. 8 7 6 5 4 3 2 1 0 0 5 10 15 20 25 30 Remaining Time to Maturity George Stylianopoulos, Fixed Income 47 Modified Duration • Macaulay’s duration can be used as a measure of a bond’s risk. A longer duration implies a higher degree of price sensitivity and therefore, greater market risk. • Macaulay’s duration in order to be more accurate as a measure of bond risk requires a modification. This revised version of duration is called modified duration and is calculated as follows: MD=modified duration D MD D=duration 1 Y Y=yield to maturity George Stylianopoulos, Fixed Income 48 Modified Duration (..continued) • Modified duration calculates the percentage change in a bond price for one basis point change in yield. B MD Y B • B=dirty price of the bond • Y=yield to maturity George Stylianopoulos, Fixed Income 49 Price Value of a Basis Point (PVBP,or PV01) • Price Value of a Basis point is a measure that shows how much the price of the bond (or a portfolio) will change for a shift of 1 b.p. in yield . It is simply the Modified Duration times the dirty price of the bond • PV01 is widely used as it shows with good approximation how much money a bond or a portfolio of bonds will profit (loose) from a favorable (adverse) shifts in yield(s). George Stylianopoulos, Fixed Income 50 Mathematics of Duration n B ci (1 y ) ti i 1 n t i ci (1 y ) ti n ci (1 y ) ti D i 1 ti B i 1 B B n B D t i ci (1 y ) ti 1 B y i 1 y (1 y ) D B B MD B MD MD y (1 y ) y B • B=dirty price of the bond • ci= cash flow of the bond at time ti, 1< i < n • y=yield to maturity • D= Duration, MD= Modified Duration George Stylianopoulos, Fixed Income 51 …continued Mathematics of Duration B MD y , B y 1 B PV01 , PV01 MD B • B=dirty price of the bond, • MD=Modified Duration, • PV01=Present Value of a basis point. George Stylianopoulos, Fixed Income 52 Duration, Modified Duration, PV01 (example) Bond Data Settlement: 15-Oct-05 a) Acrruals? , Dirty Price?, Yield? , Duration?, M D?, PVBP? Maturity: 15-Apr-08 b) PVBP for a 5 million Eur Face Value? Coupon: 6.00% Annual, 30/360 Clean Price: 105 Accruals: 3.0000 Dirty Price: 108.00 Yield: 3.85% Duration Calculation Price Dates Cashflows PV=C/(1+y)^n Weighting PVWeighted t factor 15-Oct-05 15-Apr-06 6.00 5.888 0.05451 0.02726 15-Apr-07 6.00 5.669 0.05249 0.0787 15-Apr-08 106.00 96.446 0.89299 2.2325 108.00 1.00000 2.33848 Duration Modified Duration 2.25177 PVBP 2.4319 Face Value: 5,000,000 Market Value: 5,400,000.00 PVBP 1,216 Market Value for +1bp= 5,398,784.87 Loss: -1,215.13 George Stylianopoulos, Fixed Income 53 Effective Duration Effective Duration is… • For an option-free bond the bond’s modified duration • A measure of the average maturity of a bond’s cash flows. For a callable bond the average maturity of the cash flows is shortened by the possibility of early redemption of the issue. • A sophisticated weighted average of the modified durations that a callable (putable) bond can have. • A simple weighted average of the modified duration of the option-free component and the modified duration of the option component • For a callable (putable) bond effective duration lies between the modified duration to call (put) and the modified duration to maturity. It approaches the MD to call(put) as yields fall(rise), and the MD to maturity as yields rise(fall). George Stylianopoulos, Fixed Income 54 Callable Bonds •A callable bond is a bond that gives the right to the issuer to buy it back at a pre-specified price over a predetermined period •The price of a callable bond is calculated as the price of a an equivalent non callable bond of similar structure less the value of the call option(s) attached •The call option value is subtracted because the bondholder implicitly sells the call to the issuer of the bond. Noncallable Call Option Callable Bond Price = - Bond Price Value •The crossover price is the price at which the Yield to Call and the yield to Maturity are equal. The yield level is termed crossover yield George Stylianopoulos, Fixed Income 55 The price yield curves for a callable bond and for two non callable counterparts: a non callable maturing on the first call date and the a non callable maturing on the final maturity date M MM=Maturity Date Bond C1C1=Call Date Bond C2C2=Callable Bond Crossover Yield, Crossover Price Bond price C1 C2 C1 M C2 Market Yield (%) George Stylianopoulos, Fixed Income 56 Assessing the Duration of a Callable Bond There are 3 ways of assessing a callable bond duration 1. Calculate the duration to call (DTC) and duration to maturity (DTM). Use DTC if the bond’s market price exceeds the crossover price, and use DTM if the bond’s market price exceeds the crossover price 2. Calculate a weighted average duration based on a subjective assessment of the probability of call. 3. Calculate an effective, or option- adjusted duration by using an option valuation model. George Stylianopoulos, Fixed Income 57 Factors influencing a callable bond’s duration A. Selecting either DTC or DTM Market price > Crossover price Modified DTC Market price<=Crossover price Modified DTM Under this (more naive) approach the factors influencing duration are the market price and the bonds crossover price George Stylianopoulos, Fixed Income 58 Factors influencing duration of a callable bond B. The weighted average duration approach 1. The modified DTC and the modified DTM that form the lower and upper boundaries respectively of the bond’s modified duration. 2. The current yield environment. A low yield environment increases the probability of future calls. 3. The expected trend in interest rates. A trend to lower rates increases the probability of early redemption 4. The expected variability in interest rates. The greater the variability the more probable it is that the bond will be redeemed early. DTC DTM AVGDUR George Stylianopoulos, Fixed Income 59 Factors influencing the duration of a callable bond C. Effective duration approach 1. Call date(s). An early call date reduces a bond’s effective duration. 2. Maturity date. A short remaining term to maturity lowers a bond’s effective duration. 3. Call (i.e. strike) price. A low call price shortens effective duration. 4. Market price. A high market price (i.e. low yield) decreases duration. 5. Market yield volatility. A high degree of yield volatility reduces duration. George Stylianopoulos, Fixed Income 60 Example of Calculating either DTC or DTM A 10 year 6% corporate bond NC5 at 103.00 with a market price at 106.00. At market price 106.00 the modified duration of a non callable bond maturing in 10 years is: 7.48 At the same price (106.00) the modified duration of a non callable bond with maturity 5 years and redeeming at 103.00 is: 4.26 Yield Price to Price to Environment Maturity Call 5.21% 106.00 105.71 5.30% 105.32 105.32 Crossover price and Yield 6% 100.00 102.24 Market price (106.00) > than crossover price (105.32) therefore the duration of the callable bond is 4.26 George Stylianopoulos, Fixed Income 61 Example of calculating the weighted average duration A 10 year 6% corporate bond NC5 at 103.00 with a market price at 106.00 mod ified ( AVGDUR ) mod ifiedDTC Pc mod ifiedDTM (1 Pc ) Pc = probability of call Yield Bond Price Probability of Environment Call DTC= 4.26 2% 135.93 1 DTM = 7.48 3% 125.59 0.9 4% 116.22 0.8 5% 107.72 0.7 5.21% 106.00 0.600 6% 100.00 0.4 7% 92.98 0.3 8% 86.58 0.2 9% 80.75 0.1 10% 75.42 0 AVGDUR=4.26X0.60+7.48X0.40=5.55 George Stylianopoulos, Fixed Income 62 Example of calculating the effective duration A 10 year 6% corporate bond NC5 at 103.00 with a market price at 106.00 Effective Duration=(modified duration of a noncallable X Wb)+( modified duration of the call option X Wc) Wb = the market value weight of the bond component (expressed in decimal form) Wc = 1-Wb = the market value weight of the call option (expressed in decimal form) Wb + Wc =1 A non- callable bond with same maturity of the callable trades at 110.00 yielding 4.72% and having a modified duration of 7.56 The value of the call option is: Callable bond price = noncallable bond price – call option value 106 = 110 – call option value, therefore call option value = - 4 An option is extremely price sensitive and therefore has a large duration. If the call option has a modified duration of 50 then: Effective duration of the callable bond =7.56X110/106+50X(-4/106)=7.56X1.037736-50X1.88679= 5.96 George Stylianopoulos, Fixed Income 63 Putable Bonds •A putable bond is a bond that gives the right to the holder to sell it back at a pre-specified price on a predetermined put date. •The price of a putable bond is calculated as the price of a an equivalent non putable bond of similar structure plus the value of the put option attached •The put option value is added because the bondholder implicitly buys the put option from the issuer of the bond. Putable Bond Price = Option-free bond price + put option value •A putable bond is attractive because the holder (not the issuer) has the discretion to exercise the put option. •In a bull market it behaves like an option – free bond, (with significant price gains) whilst in a bear market downside price losses are limited. George Stylianopoulos, Fixed Income 64 Comparison of callable and putable bonds Feature Callable Putable Option definition A call option gives its holder the A put option gives its holder the right to buy the bond at a pre- right to sell a bond at a pre- specified price on a pre-specified specified price at a pre-specified date. date. Option holder Issuer Investor Market availability Widespread Limited Supply Option Strike Price Typically at a premium Typically at par Option Exercise A series of call dates and call A single put date prices In the money when Market price > call price Market price < put price Out the money when Market price < call price Market > put price Yield vs. an option-free bond Higher yield Lower yield George Stylianopoulos, Fixed Income 65 The Limitations of Modified Duration 1. Instantaneous yield change. Although it is possible to experience sizable intraday, yield shifts in turbulent financial markets, yield shifts typically occur over time. 2. Small change in yield. Modified duration is a better approximation of the price behavior of the bond for small yield changes (10 bps or less) 3. Parallel shifts in yield. It would be a rarity to find all bond yields moving in tandem. Short-term bond yields fluctuate more than long- term bond yields. Nonparallel yield shifts often occur. George Stylianopoulos, Fixed Income 66 The Limitations of Modified Duration (example) Yld Duration Predicted Actual Bond Bond Data YTM(%) Error Change(bps) Bond Price Price Settlement: 15/10/2005 0.00 -400 132.44 140.00 7.56 Maturity: 15/10/2015 1.00 -300 124.33 128.41 4.08 Coupon: 4% 2.00 -200 116.22 117.97 1.74 Price: 100 3.00 -100 108.11 108.53 0.42 Yield to Maturity 4.00% 3.50 -50 104.06 104.16 0.10 Modified Duration 8.111 3.90 -10 100.81 100.81 0.00 3.99 -1 100.08 100.08 0.00 4.00 0 100 100 0.00 4.01 1 99.92 99.92 0.00 4.10 10 99.19 99.19 0.00 4.50 50 95.94 96.04 0.10 5.00 100 91.89 92.28 0.39 6.00 200 83.78 85.28 1.50 7.00 300 75.67 78.93 3.26 8.00 400 67.56 73.16 5.60 George Stylianopoulos, Fixed Income 67 CONVEXITY 210.00 180.00 Positive Convexity 150.00 Region Price 120.00 90.00 60.00 30.00 Modified Duration 0.00 0% 5% 10% 15% Yield George Stylianopoulos, Fixed Income 68 Definitions of Convexity • Convexity is the second derivative of the price:yield function and it shows the rate of change of modified duration as yields shift • Convexity can be defined as the difference between the actual bond price and the bond price predicted by the modified duration line. • The term convexity arises from the fact that price:yield curve is convex to the origin of the graph. This curvature creates the convexity effect • Convexity enhances a bond’s price performance in both bull and bear markets but not in a uniform manner. • The larger the changes in yield the greater the convexity effect. • A decline in yields creates stronger convexity impacts than does an equivalent rise in yields. George Stylianopoulos, Fixed Income 69 Convexity and the Price:Yield function The nonlinear price:yield function can be analyzed as a Taylor series of derivatives dP 1 d P 1 d 3P 1 d nP 2 P ( Y ) ( Y ) 2 ( Y ) 3 ( Y ) n dY 2! dY 2 3 n 3! dY n! dY dP mduration dY P 2 d P convexity dY 2 P P dP 1 1 d P 1 2 1 d 3P 1 3 1 d nP 1 2 ( Y ) ( Y ) ( Y ) ( Y ) n dY P 2! dY 2 3 n P P 3! dY P n! dY P P mdurationY convexityY residuals 1 2 P 2 George Stylianopoulos, Fixed Income 70 Factors influencing Convexity 1. Duration. Convexity is positively related to the duration of the underlying bond. It is also an increasing function of duration 2. Cash flow distribution. Convexity is positively related to the degree of dispersion in a bond’s cash flows. 3. Market yield volatility. High Volatility in interest rates creates large convexity effects. 4. Direction of yield change. Convexity is more positively influenced by a downward movement in yields than by an upward surge in yields George Stylianopoulos, Fixed Income 71 The Convexity:Duration relationship Price,yield curves for 3,10, 30 years bonds with a coupon 7% and price at par to yield 7% 350 300 250 P30, 30 year bond P3 200 Price P10 150 P10, 10 year bond P30 P3, 3year bond 100 50 0 0 5 10 15 20 25 30 Market Yield(%) George Stylianopoulos, Fixed Income 72 Convexity:cash flow distribution relationship The price:yield curves of a 30year Bond with a coupon 7% priced at par to yield 7% and a duration of 13.25 and a zero coupon bond of same duration(I.e. 13.25 years to maturity) at 7% yield 350 300 250 P30,30years coupon bond 200 Price P0 150 P 30 100 P0,zero coupon bond 50 0 0 5 10 15 20 25 MarketYield (%) George Stylianopoulos, Fixed Income 73 The convexity:yield volatility relationship High Volatility in interest rates creates large convexity effects 2.50 2.00 Volatility 1.50 1.00 0.50 0.00 0.00 20.00 40.00 60.00 80.00 100.00 Convexity George Stylianopoulos, Fixed Income 74 The Convexity:direction of yield change relationship • Convexity effects are greater in declining yield environment than in rising yield environment. 35 30 25 Convexity 20 15 10 5 0 -400 -300 -200 -100 0 100 200 300 400 Yield Change (in bps) George Stylianopoulos, Fixed Income 75

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