Modified Duration and Bond Price Changes as Time Passes

Summary

The document asks how modified duration relates to observed bond price and yield changes when time passes and a bond’s remaining maturity shrinks. It defines modified duration as the negative proportional sensitivity of price to yield, using a derivative that changes yield while holding the bond’s maturity and observation time fixed. The question contrasts this local sensitivity with a ratio calculated between prices and yields at two different dates and maturities.

The document raises a useful distinction between a partial derivative and a realized change across time, but provides no answer, analysis, or empirical evidence. Its proposed ratio combines the effect of a yield move with the passage of time and maturity roll-down, so it does not isolate modified duration. Readers need further explanation to separate those effects; the text does not discuss convexity, carry, or a practical estimation method.

Key ideas

• Modified duration measures local price sensitivity to yield at a fixed maturity and time.
• A price change observed across dates includes the effect of time passing and maturity shortening.
• The document poses the distinction but does not provide a resolution or supporting evidence.

Tags

# Modified Duration vs. Real-World Bond Price and Yield Changes


# Modified Duration vs. Real-World Bond Price and Yield Changes












We know that modified duration at time $t$ of a bond with maturity $n$ is defined as:

$$ D_{nt} = - \frac{1}{P_{nt}} \frac{\partial{P_{nt}}}{\partial y_{nt}} $$

And the definition of a derivative is:

$$ \frac{\partial{P_{nt}}}{\partial y_{nt}} = \lim_{\Delta y_{nt} \rightarrow 0} \frac{P_{nt}(y_{nt} + \Delta y) - P_{nt}(y_{nt})}{\Delta y_{nt}} $$

I understand that $\Delta y_{nt}$ is a small change in $y_{nt}$, a change in yield at the same maturity $n$ and time $t$. However, in real life, time passes and maturity decreases. That is, it would be more realistic to talk about something like this:

$$ \frac{\Delta P_{nt}}{\Delta y_{nt}} = \frac{P_{n-1, t+1} - P_{nt}}{y_{n-1, t+1} - y_{nt}} $$

Is this a thing? Am I missing something?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.