The Calculus of Duration

We have been discussing bonds and spot rates in finance as a fairly "simple" investment vehicle (no default risk for government bonds and steady and predictable cash flows).

While I had technically learned the math behind bond duration, it was after the homework assignment that we were assigned in finance that I began to get a better understanding of exactly what this means and why it's important.

As I had mentioned before, duration is defined as the percent change in price for a given change in yield. You'll notice that the definition of the formula is strikingly similar to elasticity calculations.

However, rather that jump in at that level, let's talk about it from the first principles of calculus:

P = C + C/(1+y) + C/(1+y)^2 + C/(1+y)^3 + ... + C/(1+y)^n + M/(1+y)^n

Where P is the price of the bond, C is the coupon and y is the YTM and M is the face value. So far nothing new. But the next idea is to take the derivative of the Price, P, with respect to y to understand how the price will be affected for any give change in y. We can re-write the formula as:

P = C + C (1+y)^-1 + C (1+y)^-2 + C (1+y)^-3 + ... + C (1+y)^-n + M (1+y)^-n

dP/dy = - C (1+y)^-2 + -2 C (1+y)^-3 + -3 C (1+y)^-4 + ... + -n C (1+y)^-(n+1) + -n M (1+y)^-(n+1)
= -1/(1+y)[ C (1+y)^-1 + -2 C (1+y)^-2 + -3 C (1+y)^-3 + ... + -n C (1+y)^-n + -n M (1+y)^-n

Recall that dP/dy describes the change in P with respect to y. In order to get percentage change, we divide both sides by P. This gives us the term dP/dy * 1/P which is a precursor to understanding % change in P or modified duration:

dP/dy * (1/P) = (1/P)[ -1/(1+y)][ C (1+y)^-1 + -2 C (1+y)^-2 + -3 C (1+y)^-3 + ... + -n C (1+y)^-n + -n M (1+y)^-n]

There is a special definition for the monstrous summation term above called the Macaulay duration which is defined as:

Macaulay duration = [ C (1+y)^-1 + -2 C (1+y)^-2 + -3 C (1+y)^-3 + ... + -n C (1+y)^-n + -n M (1+y)^-n] / P

Modified Duration = -Maccaulay duration / (1 + y)
= dP/dy * (1/P)

That's a lot of complicated math with calculus. What is the value of this exercise? Well if you can understand how your bond liabilities will move with interest rates, you can construct a portfolio of bonds which can insulate (immunize) you from changes in the price due to changes in the yield even if the bond's structure you are using to immunize is not of the same construction as the original liability buy matching face values and durations.

Therefore any change in the percentage value of your liability will be offset by a similar change in the percentage value of your bond portfolio (a hedging asset).