Approximate Modified Duration

Dear all,

It has been stated in Schwesers’ Notes (" SN") page 88 that modified duration (" ModDur") is a linear estimate of the relation between bond’s price and YTM. In addition on the same page, there is a Price vs YTM graph drawn which show a convex curve and 2 straight line ( 1 at a tangent to the curve and another one that cuts through the curve).

The formula for approx modified duration is: (V- less V+)/ (2 X Vo X change in YTM)

Question1: The “modified duration” refers to approx modified duration or modified duration? The formula for both are different.

Question 2: The formula above refers to the convex curve or the straight line? I am confused as the formula seems to be referring to the curve rather than the straight line.




Modified duration is calculated as Macaulay duration / (1+YTM) . Suppose YTM of the bond changes by 1% then modified duration provides appx. change in the price of the bond.

appx % change in the bond price = -Mod duration * Change in YTM

Whereas Appx modified duration is calculated by directly using bond values for same %age point increase or decrease in YTM.

In your formula above : there is a convexity in the price yield relationship so the price decreases to V_ and Price increases to V+ for similar change in YTM .

Here the prices are being averaged so it takes into a/c the convexity as compared to the simple formula above in which convexity is not taken into a/c. YTM in the denominator must be in the decimal form in the formula above.

Hope it is clear…

Thanks for explaining.

So am I right to say that the formula i wrote above is non-linear and it follows the actual price-yield curve?

This is a calculus thing. The modified duration is the slope of a tangent line (though it’s not the slope of the line tangent to the price-yield curve as the illustration would suggest); we approximate that slope with the slope of a secant line (a line going through two points on the curve). The closer those two points are to the point at which we want to compute the modified duration (i.e., the smaller Δy is), the better the secant approximation is to the tangent.

Take a look at this article I wrote about duration; it may help: Macaulay Duration, Modified Duration, and Effective Duration | Financial Exam Help 123.

Usually when finance people refer to modified duration, they mean the real value, but they calculate the approximate value. Finance people are weird.

The values V-, V0, and V+ are taken from the price-yield curve. The straight line is the approximation.

You’re quite welcome.