Bond Price Volatility

1. Bond Price Volatility

2. Outline • Price/Yield Relationship for Option Free Bonds • Bond Price Theorems • Price Volatility of Option Free Bonds • Measures of Bond Price Theorems • Bond Duration • Bond 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 bond • By 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 Theorems • Bond prices move inverse to the change in interest rates • If all other factors are held constant, a bond’s interest rate risk rises with the length of time remaining until it matures • Bond price volatility and time to maturity are directly related • A bond’s interest rate risk rises at a diminishing rate as the time remaining until its maturity increases • The 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 volatility • Low coupon and • High maturity • Low volatility • High coupon and • Low maturity • How 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 principal • Number of years needed to fully recover the purchase price of a bond, given present value of its cash flows • Examples

7. Modified Duration • An adjusted measure of Macaulay’s duration is called modified duration. • Modified duration can be used to approximate bond price volatility • Modified 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 bond • Inverse relationship between coupon and duration • Positive relationship generally holds between term to maturity and duration • Duration increases at a decreasing rate with maturity • The relationship between duration and maturity is not direct • Shape 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 YTM • Sinking funds and call provisions can accelerate the total cash flows for a bond, and, therefore, significantly reduce the bond duration

11. Modified duration helps in approximating the bond price change due to a small change in the required yield • Modified duration is a linear approximation of a curvilinear relationship • Graphical depiction of duration and price/yield relationship • What is yield changes are large? • Does duration still provide a good approximation of bond price change due to change in required yield?

12. Bond Convexity • For large changes in bond yields, duration can be supplemented with an additional measure to capture the curvature or convexity of a bond • Convexity is a measure of the curvature of the price/yield curve • Mathematically, convexity is the second derivative of price with respect to yield divided by the price • Convexity is a measure of how much a bond’s price-yield curve deviates from the linear approximation of that curve • For noncallable bonds, convexity is always positive

13. Measuring Convexity • Duration attempts to estimate a convex relationship with a straight line • If we add convexity to duration, we get a better approximation to the price of the bond due to change in required yield