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Interest Rate Sensitivity

• We can summarize changes in bond prices with respect to changes in interest rates in a few

statements, known as Malkiel’s Bond-Pricing Theorems:

1. Bond prices and yields are inversely related; as yields increase, bond prices fall; as yields fall,

bond prices increase.

2. An increase in a bond’s yield to maturity results in a smaller price change than a decrease in

yield of equal magnitude.

3. Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of short

term bonds.

4. The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity

increases. In other words, interest rate risk is less than proportional to bond maturity.

5. The interest rate risk is inversely relation to the bond’s coupon rate. Prices of low coupon bonds

are more sensitive to changes in interest rates than prices of high-coupon bonds.

Duration

• Bond maturity and cash flow timing pay key roles in price changes due to interest rates

• Ideally, we’d have a method to measure this using a single stat that summarized the relative

maturity of all of a bond’s cash flows, rather than just its face value

• To do this, we’ll measure the average timing of all bond cash flow, Duration

Macaulay Duration

• Most common form of duration measurement

• Macaulay’s duration weights the time-to-maturity of a bond by the relative, discounted cash

flow in each time period

• Macaulay’s Duration:

• Where t is the timing of each cash flow in years.

• The weight for each year t is given by:

Modified Duration

• Another important concept is the relative change in a bond’s price in response to a change in

yields

• We can show that for a yield change of , the percentage change in price

can be linearly

approximated as:

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• To make this calculation easier, we can modify our duration measure to a second measure as

modified duration:

• Replacing D with D* and with , we get:

Duration in Practice

• There are a few rules for how a bond’s characteristics affect its duration

1. The duration of a zero-coupon bond equals its time to maturity.

2. Holding maturity constant, a bond’s duration is lower when its coupon is higher.

3. Holding coupon rate constant, a bond’s duration generally increases with time to maturity.

Duration always increases with maturity if the bond is trading at or above par.

a. NOTE: A bond selling at a very high discount may have a duration that decreases with

maturity.

4. All else equal, the duration of a coupon bond is higher when the bond’s YTM is lower.

5. The duration of a perpetuity with a yield of y is given by

.

Changes in Duration: Convexity

• An issue with the duration formula is that it’s a linear approximation of the change in bond price

as a result of a yield change

• When yields change, the bond’s price responds non-linearly to this change

• Thus, duration doesn’t fully represent the potential change in a bond’s price

• To improve the approximation of a bond’s price change we need to use another measure that

we refer to as convexity

Convexity

• We measure the change in the slope of the pricing curve using a measure called convexity.

• Convexity can be calculated as:

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