Investment Science: Bond Price Sensitivity

Tuesday, September 14, 2010

Before looking into a practical example on immunizing a bond portfolio, let's look at an important concept related to bonds, the Bond - price sensitivity, as we know that the price of a bond is affected by changes in yield and in this post we will look into the mathematical formula for computing the change in price of a bond with respect to a given change in yield.

The bond price sensitivity formula is given below


Where

P = Price of the bond
D = Macaulay Duration of the bond
λ = Yield to Maturity (YTM)
m = Number of coupon periods
dP = Change in price
dλ = Change in yield

The above formula can also be written as


Where DM is the modified duration of the bond.


As we said before, zero coupon bonds are more sensitive to changes in yield than the coupon bonds, let's see how the price of the bond gets affected for a small change in yield.

For this example, first we will consider a 30-year, 8% bond that is trading at a yield of 10% with yearly coupon payments.

For this bond, based on the bond price formula discussed in the previous article, we get,

P = $81.146 and D = 10.646 (refer to the duraton formula)

Assume that the yield increases by 1% (dλ = 0.01) in the above scenario in which case based on the above formula for Bond - sensitivity, we have

dP = -1/1.1 * 10.646 * 81.146 * 0.01 = -7.853 or the price reduces by 9.68% (-7.853/81.146) if the yield increases by 1%.

Consider a 10-year, zero coupon bond that is trading at a yield of 10%, for this bond,

P = $5.7308 and D = 30

For a 1% increase in yield (dλ = 0.01), we have

dP = -1/1.1 * 30 * 5.7308 * 0.01 = -1.562 or the price reduces by 27.25% (compared to 9.68% for the coupon bond)

Hence zero coupon bonds of longer duration are more sensitive to changes in yield, with this knowledge, we are all set to explore a real world example on bond portfolio immunization in the next article.

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