comm W. Suo Slide 1

comm W. Suo Slide 2 Managing interest rate risk  Bond price risk  Coupon reinvestment rate risk  Matching maturities to needs  The concept of duration  Duration-based strategies  Controlling interest rate risk with derivatives

comm W. Suo Slide 3 Interest Rate Sensitivity BondCouponMaturityInitial YTM A12%5 years10% B12%30 years10% C3%30 years10% D3%30 years6% ABCDABCD Change in yield to maturity (%) Percentage change in bond price 0

comm W. Suo Slide 4  Inverse relationship between price and yield  An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield  Long-term bonds tend to be more price sensitive than short- term bonds  As maturity increases, price sensitivity increases at a decreasing rate  Price sensitivity is inversely related to a bond’s coupon rate  Price sensitivity is inversely related to the yield to maturity at which the bond is selling Bond Pricing Relationships

comm W. Suo Slide 5 Derivation of Formula For Macauley’s Duration  The slope of a bond’s price-yield relationship measures the bond’s sensitivity to YTM Modified duration (Mod) Macauley duration (MAC) Macaulay (1938) suggested studying a bond’s time structure by examining its average term to maturity  Approximating the bond price change using duration is equivalent to moving along the slope of the bond price-yield curve

comm W. Suo Slide 6 Example: Calculation of MAC and MOD  Given information A $1,000 par bond with a YTM of 10% has three years to maturity and a 5% coupon rate Currently sells for $

comm W. Suo Slide 7 Example: Calculation of MAC and MOD  MAC can be calculated using the previous present value calculations TPV of CF Each CFs PV as fraction of Price T weighted by CF × = × = × = MAC = MOD = MAC  (1+y) =  1.10 =

comm W. Suo Slide 8 Interest Rate Risk  Interest rate elasticity measures a bond’s price sensitivity to changes in interest rates

comm W. Suo Slide 9 Example: Evaluating a Bond’s EL  Given information: A bond has a 10% coupon rate and a par of $1,000. Its current price is $1,000 as the YTM is 10% If interest rates were to rise from 10% to 11%, what would the new price be?  The price drops by $17.13 or 1.713%  -$17.13  $1,000 = or –1.713%  The increase in YTM from 10% to 11% is a percentage change of  (0.11 – 0.10)  1.1 = or 0.9% Results in an EL of –  = -1.90

comm W. Suo Slide 10 Interest Rate Risk  MAC can also be used to calculate a bond’s elasticity MAC = [(t=1)($  $1,000] + [(t=2)($ )  $1,000] = 1.90 years  Interest rate elasticity and MAC are equally good measures of interest rate risk

comm W. Suo Slide 11  Price change is proportional to duration and not to maturity Duration/Price Relationship  Or, if we denote D * = modified duration

comm W. Suo Slide 12 Why is duration a key concept?  It’s a simple summary statistic of the effective average maturity of the portfolio;  It measures the sensitivity of bond price change relative to the change in yield  It is an essential tool in immunizing portfolios from interest rate risk;  It is a measure of interest rate risk of a portfolio

comm W. Suo Slide 13 Contrasting Time Until Maturity and Duration  MAC  T For zeros MAC = T For non-zeros MAC < T Earlier and/or larger CFs result in shorter MAC Macauley durations for a bond with a 6% YTM at various times to maturity Coupon Rate T2%4%6%8% 

comm W. Suo Slide 14 Contrasting Time Until Maturity and Duration

comm W. Suo Slide 15 Contrasting Time Until Maturity and Duration

comm W. Suo Slide 16 Contrasting Time Until Maturity and Duration Duration and convexity for a bond with a 5% coupon at YTM of 10% 2-year bond12-year bond22-year bond PV$913.22$659.32$ MAC MOD Convexity As horizon increases, bond’s MAC, MOD increase.

comm W. Suo Slide 17 Contrasting Time Until Maturity and Duration  A bond’s duration is inversely related to its coupon rate Duration and convexity for a bond with a 12 year maturity at YTM of 10% Zero coupon 5% coupon10% coupon 15% coupon PV$318.63$659.32$1,000$1, MAC MOD Convexity

comm W. Suo Slide 18 Rules for Duration Rule 1 The duration of a zero-coupon bond equals its time to maturity Rule 2 Holding maturity constant, a bond’s duration is higher when the coupon rate is lower Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower Rule 5 The duration of a level perpetuity is (1+y)/y Rule 6 The duration of a level annuity is equal to: Rule 7 The duration for a coupon bond is equal to:

comm W. Suo Slide 19 Duration and Convexity % Price Change Duration Pricing error from convexity Yield

comm W. Suo Slide 20 Convexity of Two Bonds 0 Change in yield to maturity (%) Percentage change in bond price Bond A Bond B

comm W. Suo Slide 21 Correction for Convexity Correction for Convexity:

comm W. Suo Slide 22 Convexity calculation 8% Bond Time Sem. PaymentPV of CF (10%) Weightt(t+1)x C sum

comm W. Suo Slide 23 Convexity calculation (cont.)  Convexity is computed like duration, as a weighted average of the terms (t 2 +t) (rather than t) divided by (1+y) 2  Thus, in the above example, it is equal to / = in semester terms.

comm W. Suo Slide 24 Other Measures  Effective duration  PVBP (also know as PV01, DVBP, DV01) How to calculate it?

comm W. Suo Slide 25 Duration and Convexity of Callable Bonds 0 Interest Rate Call Price Region of positive convexity Region of negative convexity Price-yield curve is below tangent 5% 10%