Modified Duration as the Yield Derivative of Bond Price
Summary
The document derives modified duration from the present value of a bond’s coupon payments and principal. It differentiates the bond price with respect to yield to obtain the absolute price sensitivity, then divides by price and changes the sign to express sensitivity as a percentage. This gives the local relationship that a small yield move implies an approximate percentage price move of the opposite sign, scaled by modified duration.
The derivation uses a fixed coupon bond with annual discounting and a ten-year maturity to illustrate the calculation. It is an intuition for first-order sensitivity, rather than a complete pricing model: the approximation is local, and the example does not discuss convexity, different compounding conventions, or how cash flows may change for bonds with embedded options. The quoted derivative is with respect to yield expressed as a decimal, so yield changes must use consistent units.
Key ideas
- Bond price is the present value of its promised coupons and principal, discounted at yield.
- Differentiating price with respect to yield gives the absolute price sensitivity.
- Modified duration scales that derivative by bond price and reverses the sign to express percentage sensitivity.
- For a small yield change, duration gives a first-order estimate of the percentage price move in the opposite direction.
- The approximation omits higher-order effects such as convexity.
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Full text
# Modified Duration and how it explains bond price sensitivity to changes in the yield to maturity
# Modified Duration and how it explains bond price sensitivity to changes in the yield to maturity
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
## Answer by Jan Stuller (score 6)
https://quant.stackexchange.com/a/60422
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):
$$MD:=\frac{-1}{PV}*\frac{\partial PV}{\partial y}=\frac{\left( \sum_{i=1}^{10} \left( \frac{iC}{(1+y)^i} \right) + \frac{10N}{(1+y)^{10}} \right)}{(1+y)*PV}$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.