## Sunday, February 20, 2011

### Why the duration of a Zero is the years to maturity?

Why the duration for a zero is the same as the maturity in years? I have never figured it out since L2 although I memorized it just for the sake of passing the exam. Here is a simple calculation on a zero coupon of maturity 10 years for \$100 par. The discount rate, yield to maturity is the same. The various present values for the respective discount rates are:

4.50% 64.3927682
5.00% 61.39132535
5.50% 58.54305794

Duration = (V- - V+)/(2 * Vo * (Delta Y) )

Where
V- = 64.3927682
V+ = 58.54305794
Vo = 61.39132535
Delta Y = 0.005

Thus, duration = 9.528561611

Hence, this demonstrate that duration of a zero is NOT equal to years. In fact, duration itself has no dimension. How could it be compared with a variable with dimension?

#### 1 comment: Anonymous said...

Dear Friend,

I came across your blog by accident. I think I can answer your question about duration. The formula you are using is an approximation that does not use calculus. The exact formula is

D = - 1/V0 * dV/dY . Now V = F *(1+Y)^-t , where F is the redemption price and dV/dY is the derivative (calculus meaning) of the present value with respect to the interest rate (or yield). Then dV/dY = F * -t * (1+Y)^(-t-1) . Since V0 = F * (1+Y)^-t, the formula gives exactly D = t for a zero coupon bond. The units are correct too because dimensionally the result is 1/Y, and Y is in units of 1/Years. So 1/Y is in years. I hope that helps.