6-1 Chapter 6 The Risk of Changing Interest Rates

6-2 Short Horizon Investors n Maturity Time P 1, the price at Time 1, is important. P0P0 P1P1 y0y0 y1y1 10

6-3 Long Horizon Investors P0P0 CC + PAR Maturity Time Reinvest Value at some distant date n is important. 102 C n

6-4 Bond Price Interest Rates

6-5 Bond Price Interest Rates Actual Price Change P0P0 P1P1 y1y1 y0y0

6-6 =derivative of bond price as yield to maturity changes =slope of tangent of price curve

6-7 Duration as an Approximation of Price Change Price Interest rate Slope of tangent equals numerator of duration Actual price change equals P 0  P 1 Duration approximation of price change equals P 0  P´ 1 Price Tangent P0P0 P1P1 P´1P´1 y0y0 y1y1

6-8 Move along tangent to approximate price change. From calculus Divide both sides by price =a measure of sensitivity of bond prices to changes in yields =a measure of risk

6-9 Percent Price  [Duration][Yield Change]. Change is called “modified” duration.

6-10 Macaulay’s Duration (DUR) Often used by short horizon investors as a measure of price sensitivity. DUR= % change in price as yield changes DUR =. -[d P /d y ](1 + y) Price

6-11 This expression may be interpreted as the weighted average maturity of a bond. DUR =. 1c/(1 + y) 1 + 2c/(1 + y) 2 + … + n(c + PAR)/(1 + y) n Price

6-12 Macaulay’s Duration for Special Types of Bonds Bond Price Volatilities for Special Types of Bonds Type of bondDuration Zero-coupon n Par Perpetual(1 + y)/y

6-13 Simplified Way of Computing Macaulay’s Duration

6-14 Duration for Various Coupons and Maturities YTM of 8% Coupon Maturity Note: Perpetual bond has duration of 1.08/0.08 =

6-15 Bond Price Interest Rates P0P0 P H,2 y1y1 y2y2 y0y0 High Risk Bond Low Risk Bond P H,1 P L,1 P L,2  

6-16 Duration versus Maturity Duration Zero-coupon Discount Par Premium 1 1 Maturity 1 + y y 1 + y y.

6-17 (Risk) Feasible Low Risk High Risk 30 Duration versus Maturity Duration Zero-coupon Discount Par Premium 1 1 Maturity 1 + y y 1 + y y.

6-18 Duration Gap Bank Balance Sheet AssetsLiabilities & Equity CashDeposits LoanBonds BuildingsEquity DUR A DUR L GAP = DUR A – DUR L

6-19 Immunization at a Horizon Date Points in Time 0n The zero coupon strategy Buy zero coupon bond -$P Receive par value +$X

6-20 Points in Time 0 Maturity strategy Receive par + 1 coupon 2n Buy coupon- bearing bond 1 -$P+c c + Par Reinvest coupons... Receive coupons...

6-21 Points in Time 0n Duration strategy Buy coupon- bearing bond Maturity of bond 12 -$P+c c + Par Reinvest coupons m... Receive coupons + reinvest Sell original bond + reinvested coupons c