My questions is the following: why is it that the modified duration can explain so well the change in price of bonds? I don't get the mathematical relationship behind this, that is between modified duration and the percentage change in price of the bond that it predicts. Can anyone give me the intuition behind this? Thank you

  • $\begingroup$ Modified duration measures the sensitivity of the bond price (present value) with respect to yield-to-maturity of the bond. If you differentiate the price of a coupon paying bond wrt. yield-to-maturity you would end up with the formula for modified duration. $\endgroup$
    – Pleb
    Jan 9 '21 at 11:36
  • $\begingroup$ I answered below, but also I'd encourage you to implore google and search for "modified duration definition": there's plenty of stuff online (although arguably, a lot of it is confusing). The wikipedia page is not too bad. $\endgroup$ Jan 9 '21 at 11:47

Bond price in terms of yield (denoted "$y$") is just the Present Value (PV) of the Bond coupons (denoted "$C$") and the final Notional (denoted $N$), discounted at the yield. Suppose the bond matures in 10 years time, then the present value can be written as (yields expressed as annualized for simplicity):

$$PV=\sum_{i=1}^{10} \frac{C}{(1+y)^i}+\frac{N}{(1+y)^{10}}$$

Modified duration (denoted $MD$) is actually defined as the percentage change of the bond price with respect to yield.

How do we compute the change of the bond price with respect to yield? Just take the derivative, specifically we can write:

$$\frac{\partial PV}{\partial y}=\frac{\partial }{\partial y}\left( \sum_{i=1}^{10} \left( \frac{C}{(1+y)^i} \right) + \frac{N}{(1+y)^{10}} \right)=(-1)(1+y)^{-1}\left( \sum_{i=1}^{10} \left( \frac{iC}{(1+y)^i} \right) + \frac{10N}{(1+y)^{10}} \right)$$

The above gives the absolute change in the bond price per 1 unit of yield. So if the yield changes by (say) 0.01 (which is equal to 1%), then you plug $y=0.01$ into the formula above and you will get the absolute (i.e. in dollars, in case the bond is denominated in USD) change in the bond price.

Modified duration is then just the above, divided by $PV$ and multiplied by (-1) (i.e. to turn the absolute change in the bond price with respect to yield into a percentage change of the bond price with respect to yield):

$$MD:=\frac{-1}{PV}*\frac{\partial PV}{\partial y}=\frac{\left( \sum_{i=1}^{10} \left( \frac{iC}{(1+y)^i} \right) + \frac{10N}{(1+y)^{10}} \right)}{(1+y)*PV}$$

  • $\begingroup$ thanks so much for taking the time to explain this! $\endgroup$
    – adriano
    Jan 9 '21 at 14:46

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