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Chapter 6

Chapter 6. The Risk of Changing Interest Rates. Short Horizon Investors. Maturity. 0. 1. n. Time. P 0. P 1. y 0. y 1. P 1 , the price at Time 1, is important. Long Horizon Investors. Maturity. 0. 1. 2. n. Time. P 0. C. C. C + PAR. Reinvest.

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Chapter 6

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  1. Chapter 6 The Risk of Changing Interest Rates

  2. Short Horizon Investors Maturity 0 1 n Time P0 P1 y0 y1 P1, the price at Time 1, is important.

  3. Long Horizon Investors Maturity 0 1 2 n Time P0 C C C + PAR Reinvest Value at some distant date n is important.

  4. Bond Price Interest Rates

  5. Bond Price P0 Actual Price Change P1 y0 y1 Interest Rates

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

  7. Duration as an Approximation of Price Change Price Price Slope of tangent equals numerator of duration Actual price change equals P0 P1 Tangent Duration approximation of price changeequals P0 P´1 P0 P1 P´1 Interest rate y0 y1

  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

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

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

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

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

  13. Simplified Way of Computing Macaulay’s Duration

  14. Duration for Various Coupons and Maturities YTM of 8% Coupon Maturity 0 0.04 0.06 0.08 0.10 0.12 1 1 1 1 1 1 1 5 5 4.59 4.44 4.31 4.20 4.11 10 10 8.12 7.62 7.25 6.97 6.74 15 15 10.62 9.79 9.24 8.86 8.57 20 20 12.26 11.23 10.60 10.18 9.88 25 25 13.25 12.15 11.53 11.12 10.84 30 30 13.77 12.73 12.16 11.80 11.55 Note: Perpetual bond has duration of 1.08/0.08 = 13.50.

  15. Bond Price High RiskBond  PH,2 PL,2 P0 LowRisk Bond PL,1  PH,1 y2 y0 y1 Interest Rates

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

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

  18. Duration Gap

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

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

  21. Points in Time . . . 0 1 2 n m Buy coupon-bearingbond Receive coupons + reinvest Sell original bond + reinvested coupons Maturity of bond . . . -$P +c +c c c + Par Reinvest coupons Duration strategy

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