Relation between Duration and Price Sensitivity

As we already discussed, price of a bond can be given by present value of its future cash flows.
P = CF1/(1+r)1 + CF2/(1+r)2 + …….+ CFn/(1+r)tn

From the above formula, it’s evident that r, the market yield or discount rate, affects price of a bond in a big way. Now let’s examine how prices change in response to a small change in yield (r).

Differentiate the above equation with respect to r

dP/dr = -CF1/(1+r)2 -2*CF2/(1+r)3 -3*CF3/(1+r)4 -………-n*CFn/(1+r)n+1

The above equation can be written as

dP/dr = -1/(1+r) [CF1/(1+r) + 2*CF2/(1+r)2 + ….+ n*CFn/(1+r)n ]

This is absolute change in price due to a small change in yield. To get the percentage price change divide the equation by price (P)

dP/dr * [1/P] = -1/(1+r) [CF1/(1+r) + 2*CF2/(1+r)2 + ….+ n*CFn/(1+r)n ] * [1/P]

Let’s go back to the calculations that we used for Duration. Duration is nothing but present value of future cash flows per Total PV of cash Flow multiplied by year number.

So the above equation can be written as

dP/dr * [1/P] = -1/(1+r) [Duration]

The figure, -1/(1+r) [Duration], is called modified duration of a bond.

% change in price of a bond, dP/P *100 = Modified Duration * dr * 100

% change in price of a bond = Modified Duration * % change in yield.

Modified duration of 3 implies that price of the bond will change by 3% in response to 1% change in yield. The negative sign indicates that the change in the price of the bond will be in the opposite direction in the movement of the interest rate.

In the example we assumed that coupon is paid out annually, if this is not the case

r should be calculated as

Yield/n

Where n is number of coupon periods per year

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June 14, 2008 | Filed Under Bonds 

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