• prevailing interest rates directly affect the required rate of return from a bond
• when prevailing interest rates rise bond prices fall; and when prevailing interest rates fall bond prices rise
• the change in price is the price sensitivity of the bond
• the longer the years to maturity the greater the price sensitivity of the bond (verify this using the bond valuation formula)
• the higher the coupon rate the lower the price sensitivity of the bond (verify this using the bond valuation formula)
• problem: need a single index of bond price sensitivity

• for a zero coupon bond you have to wait n years for your money but if there is a coupon, some of the money comes back earlier
• how long does it take on average? compute weighted average of time (weighted by the present value of each cash flow increment)
• this measure of modified maturity is called duration
• a single measure of bond price sensitivity to interest rate changes
• numerically equal to elasticity of price to (1+r) where r is the interest rate (approximately)
• duration of zeroes = n, duration of perpetuities = (1+y)/y
• many bond investing strategies boil down to duration matching
• bulldozer method of determining bond price sensitivity: use calculator and compute prices at two different interest rates

• the relationship between bond price and yield is not linear but curved
• the use of duration as elasticity especially over a large value of dk will therefore be in error by an amount that is proportional to the curvature
• you may correct your curvature error by computing the convexity
• but it is easier to just compute the price of the bond at the other rate

• consider a 6% 30-year strip
• at k = [6,5,7]% the price of the strip is P = [17.4, 23.13, 13.14]
• a 1% fall from 6% to 5% yields a gain of 32.9% while an equal rise in the rate to 7% incurs a loss of only 24.5%
• price sensitivity of bonds to changes in k are asymmetric: price increases are greater than price decreases
• the smaller the value of k the greater the asymmetry

face 1000, coupon 15%, ytm 10%

• yr pmt pv(pmt) t*pv
• 1 150 136.36 136.364
• 2 150 123.97 247.934
• 3 150 112.70 338.092
• 4 150 102.45 409.808
• 5 150 93.14 465.691
• 6 150 84.67 508.027
• 7 150 76.97 538.816
• 8 150 69.98 559.809
• 9 150 63.61 572.532
• 10 1150 443.37 4433.75
• sums: 1307.23 8210.82
• duration: 6.28

• yr pmt pv(pmt) t*pv
• 1 196 178.18 178.182 196.00 196.00 196.00
• 2 196 161.98 323.967 411.60 401.80 421.40
• 3 196 147.26 441.773 648.76 617.89 680.61
• 4 196 133.87 535.483 909.64 844.78 978.70
• 5 196 121.70 608.503 1196.60 1083.02 1321.51
• 6 196 110.64 663.821 1512.26 1333.17 1715.73
• 7 196 100.58 704.053 1859.49 1595.83 2169.09
• 8 196 91.44 731.484 2241.43 1871.63 2690.46
• 9 196 83.12 748.108 2661.58 2161.21 3290.03
• 10 1196 461.11 4611.1 4123.74 3465.27 4979.53
• 1589.88 9546.47 15761.09 13570.61 18443.05
• duration: 6.00
• bond value at t=6: 1304.31 1517.71 1131.33
• reinvestments+sale: 2816.57 2850.88 2847.06