FIN30021 Fixed Income and Debt Markets Topic 4 Suggested Solutions Note: If you have any questions relating to the solution, please place a question on Canvas Question 7.1 The price value of a basis point will be the same regardless if the yield is increased or decreased by 1 basis point. However, the price value of 100 basis points (i.e., the change in price for a 100-basis- point change in interest rates) will not be the same if the yield is increased or decreased by 100 basis points. Why? The convex relationship explains why the price value of a basis point (i.e., the change in price for a 1-basis-point change in interest rates) will be roughly the same regardless if the yield is increased or decreased by 1 basis point, while the price value of 100 basis points will not be the same if the yield is increased or decreased by 100 basis points. When the price-yield relationship for any option-free bond is graphed, it displays a convex shape. When the price of the option-free bond declines, we can observe that the required yield rises. However, this relationship is not linear. The convex shape of the price-yield relationship generates four properties concerning the price volatility of an option-free bond. First, although the prices of all option-free bonds move in the opposite direction from the change in yield required, the percentage price change is not the same for all bonds. Second, for very small changes in the yield required (like 1 basis point), the percentage price change for a given bond is roughly the same, whether the yield required increases or decreases. Third, for large changes in the required yield (like 100 basis points), the percentage price change is not the same for an increase in the required yield as it is for a decrease in the required yield. Fourth, for a given large change in basis points, the percentage price increase is greater than the percentage price decrease. Module 4 Solutions Page 1
FIN30021 Fixed Income and Debt Markets Question 4.2 c). Time Cf Yield 6% PV(cf) Pv(cf)/Price Pv(cf)xt/Price (b) Time Cf PV(cf) 1 2 3 4 5 6 7 8 9 10 Price 9 8.490566 0.069549049 9 8.009968 0.06561231 9 7.556574 0.061898406 9 7.128843 0.058394722 9 6.725324 0.055089361 9 6.344645 0.051971095 9 5.985514 0.049029335 9 5.646711 0.04625409 9 5.327086 0.043635934 109 60.86503 0.498565699 122.0803 Duration Pv(cf)/Price (t+t^2)/(1+r)^2 0.069549049 1.77999288 0.123797 0.13122462 5.33997864 0.350368 0.185695217 10.67995728 0.661072 0.23357889 17.7999288 1.039422 1.47088 0.275446804 26.6998932 0.311826571 37.37985048 1.942672 0.343205345 49.83980064 2.443612 0.370032717 64.07974368 2.96395 0.392723403 80.0996796 3.495224 4.985656986 97.8996084 48.80939 7.298939602 Convexity 63.30038 Pv(cf)xt/Price (t+t^2)/(1+r)^2 a). Time 1 2 3 4 5 6 6 4.229763 7 8 9 10 Price 6 5.660377 0.056603774 6 5.339979 0.053399786 0.050377157 6 5.037716 6 4.752562 0.04752562 6 4.483549 0.04483549 0.042297632 6 3.990343 0.039903427 6 3.764474 0.037644742 6 3.551391 0.035513908 106 59.18985 0.591898464 100 Duration 0.056603774 1.77999288 0.100754 0.106799573 5.33997864 0.285154 0.151131471 10.67995728 0.538026 0.190102479 17.7999288 0.845953 0.224177452 26.6998932 1.197103 0.253785795 37.37985048 1.581079 0.279323988 49.83980064 1.988779 0.301157938