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Bond Price Volatility

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**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

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**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

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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.

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**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?

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**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

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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

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**Approximating the percentage price change using modified duration**

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**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

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**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

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**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 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?

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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

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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

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