2 Outline Price/Yield Relationship for Option Free Bonds Bond Price TheoremsPrice Volatility of Option Free BondsMeasures of Bond Price TheoremsBond DurationBond Convexity
3 Price/Yield Relationship A fundamental property of an option free bond is that the price of the bond changes in the opposite direction of the change in the required yield for the bondBy how much bond price will change for a given change in yield will depend on the time to maturity, coupon, and interest rates
4 Bond Price TheoremsBond prices move inverse to the change in interest ratesIf all other factors are held constant, a bond’s interest rate risk rises with the length of time remaining until it maturesBond price volatility and time to maturity are directly relatedA bond’s interest rate risk rises at a diminishing rate as the time remaining until its maturity increasesThe price change that results from an equal sized increase/decrease in a bond’s YTM is asymmetrical.
5 A bond’s interest rate risk is inversely related to the coupon High volatilityLow coupon andHigh maturityLow volatilityHigh coupon andLow maturityHow do we measure bond’s volatility?
6 Bond Duration Macaulay’s duration Duration is defined as a weighted average time to recovery of all interest payments plus principalNumber of years needed to fully recover the purchase price of a bond, given present value of its cash flowsExamples
7 Modified DurationAn adjusted measure of Macaulay’s duration is called modified duration.Modified duration can be used to approximate bond price volatilityModified duration equals Macaulay’s duration divided by one plus the current YTM.Examples
8 Approximating the percentage price change using modified duration
9 Features of Bond Duration Duration of a bond with coupon payments will always be less than maturity of the bondInverse relationship between coupon and durationPositive relationship generally holds between term to maturity and durationDuration increases at a decreasing rate with maturityThe relationship between duration and maturity is not directShape of the duration/maturity curve depends on the coupon and the yield to maturity
10 All else being the same, there is an inverse relationship between YTM and duration More distant cash flows with smaller present value will receive less weight, because they are being discounted at a higher YTMSinking funds and call provisions can accelerate the total cash flows for a bond, and, therefore, significantly reduce the bond duration
11 Graphical depiction of duration and price/yield relationship Modified duration helps in approximating the bond price change due to a small change in the required yieldModified duration is a linear approximation of a curvilinear relationshipGraphical depiction of duration and price/yield relationshipWhat is yield changes are large?Does duration still provide a good approximation of bond price change due to change in required yield?
12 Bond ConvexityFor large changes in bond yields, duration can be supplemented with an additional measure to capture the curvature or convexity of a bondConvexity is a measure of the curvature of the price/yield curveMathematically, convexity is the second derivative of price with respect to yield divided by the priceConvexity is a measure of how much a bond’s price-yield curve deviates from the linear approximation of that curveFor noncallable bonds, convexity is always positive
13 Measuring ConvexityDuration attempts to estimate a convex relationship with a straight lineIf we add convexity to duration, we get a better approximation to the price of the bond due to change in required yield