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Let B(y) be the bond price as a function of yield to maturity. If I understood correctly, duration can (roughly) be understood as a measure of the change in $B in response to a change in y. What I don't understand is - how could anything cause a change in y that would be relevant to the concept of duration in the first place?

Suppose, for instance, Bond B has a duration of 5%, then a 1% change in interest rates would cause a 5% drop in bond prices. I'm struggling to understand what factors could possibly cause such a drop in the first place because as far as I know, the change in bond price is a result of the change in price, not the other way round.

EXAMPLE: As far as I understood, it is often said that rising inflation expectations cause a rise in yields and it is implied that such an increase in yields causes an drop in price, which can be measured/approximated/modelled using duration. However, this causal chain does not make sense to me - I always thought that rising inflation expectations cause bond prices to drop as my cash flows are now worth less, hence causing the YTM to rise as a result of the drop in the price. So why would duration be an appropriate measure in this situation? If it is not, are there any circumstances in which it is an appropriate measure in the first place?

In short, I can't think of any real-life situation in which duration the way it is commonly measured would have relevance as changes in YTM are caused by changes in price and not the other way round.

EDIT To phrase it differently: Duration is a measure of price change in response to change in YTM. However, YTM doesn't change without a change in bond price in the first place. Thus, the question arises: Why is it measured in this way?

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You're confusing Yield To Maturity (YTM) with a generic "interest rate".

Duration is a measure of bond price sensitivity to interest rates (see Investopedia's take on it).

Here, the interest rate in question is generally the policy rate set by the central bank that issues the currency in which that bond is denominated.

If there is an expectation that inflation will increase soon, then the implicit expectation becomes that the central bank will increase their target rate as part of their monetary policy response.

Increasing this target rate (Federal Funds Rate (FFR) in the US or Bank Rate in the UK) will cause investors to sell low yield bonds to maximise returns elsewhere. This then has the effect of decreasing the bond's market price and, as a result, pushing up its yield (and YTM).

Duration therefore gives you an indication of how sensitive a particular bond's market price might be to changes in this initial policy rate change, with various factors affecting it.

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    $\begingroup$ Hey, thanks for your comment. I thought so as well, but then I checked my professors note and he wrote the following (without much comment though): "Let B be the price of the bond, then y is the rate such that $B = \sum_{n = 1}^N c_n*e^{-y*t_n}$. Duration is a measure of sensitivity of B caused by changes in y." This would imply that $y$ is not necessarily the risk free rate, no? $\endgroup$
    – l337n00b
    Jan 15 at 16:11
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    $\begingroup$ This is also what is mentioned in the comment section of this thread: quant.stackexchange.com/questions/45078/duration-and-yield. $\endgroup$
    – l337n00b
    Jan 15 at 16:13
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    $\begingroup$ @l337n00b Okay yes, I see the issue I think. You're correct that a bond's YTM is determined BY the bond's market price relative to its face value (and its coupon payments) and not the other way around. Duration does technically show how a bond's price might change for a given change in its YTM using it's current YTM, even though that's a bit circular. The reason is the YTM will shift in the same direction as prevailing market interest rates (largely determined by the central bank). So practically, investors will want to know how a bond's price will change given a change in wider rates. $\endgroup$ Jan 15 at 16:36
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    $\begingroup$ So if I understand correctly, we don't really "care" about duration in the way it is commonly described because YTM and bond price are essentially equivalent, but use its formula as a proxy for the change in price caused by changing market interest rates because generally YTM and market interest rates move in the same direction? $\endgroup$
    – l337n00b
    Jan 15 at 16:45

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