<p>I believe &quot;yield volatility&quot; here means the realized variance in the interest rate.
Generally speaking, we think that the price of a bond will be the NPV of its cash flows:
<span class="math-container">$$ p = \sum_{\tau = 0} ^ {T} \frac{C}{\Pi_{s = 0} ^ {\tau} (1+r_s)} + \frac{FaceAmount}{\Pi_{s = 0} ^ {T} (1+r_s)}$$</span>
If we can assume that interest rates are constant, then this simplifies:
<span class="math-container">$$ p = \sum_{\tau = 0} ^ {T} \frac{C}{(1+r)^\tau} + \frac{FaceAmount}{(1+r)^T}$$</span></p>
<p>For a long term bond, T will be large and the discount factor of the later cashflows will be something like <span class="math-container">$(1+r)^10$</span> or <span class="math-container">$(1+r)^30$</span> That means that a small difference in r will be amplified many many times by the exponentiation, leading to the price being very sensitive to <span class="math-container">$r$</span>, especially compared to a short-term bond where we might have the largest terms of <span class="math-container">$(1+r)^{1/12}$</span>, <span class="math-container">$(1+r)^{1/4}$</span>, or <span class="math-container">$(1+r)^{1/2}$</span>. The math is more complex if interest rates change, but the idea is similar.</p>
<p>Why should &quot;yield volatility&quot; be higher for short term bonds? The basic intuition here is that some interest changes are temporary. A short term bond won't have time for those temporary changes to be undone. A long term bond will not. To see a very simple example of this, consider a situation where interest rates were expected to be 3% forever, but now are expected to be 4% for one month and then go back to 3% forever. That will make the yield on the one period bond change from 3% to 4%, but will make the yield on the 30 year bond change much less. As a counter-argument though, consider that interest rates can drift much more over long periods than over short ones, which would make long-term bonds have more volatile yields.</p>
<p>Empirically, looking over the last 20 years, it does indeed look like short-term yields are more volatile than long-term yields.
<a href="https://i.sstatic.net/ECtiv.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ECtiv.png" alt="yields on constant maturity" /></a></p>

CC BY-SA 4.0
2022-08-13T16:44:35.353Z
Thanks for the answer. On a side note, if you are risk-averse which one would you buy? Long-term or short-term bonds? On the other hand if you want to gain from the fluctuation in bonds, which one should you pick? Long-term or short-term bonds?
Disintegrators
CC BY-SA 4.0
2022-08-04T14:07:45.650Z
2022-08-04T14:07:45.650Z
BKay
<p>Why do short term bonds have higher yield volatility but lower price volatility?</p>
<p>I believe because of what we call duration prices of long terms bonds are more volatile. But what does yield volatility mean here?</p>

<interest-rate><finance><bonds>
Why are the yields on short-term bonds more volatile than those of long-term bonds?
Disintegrators
# Why are the yields on short-term bonds more volatile than those of long-term bonds?

(Asked by: username_0 on 2022-08-04)

Why do short term bonds have higher yield volatility but lower price volatility?

I believe because of what we call duration prices of long terms bonds are more volatile. But what does yield volatility mean here?

**Tags:** interest-rate, finance, bonds

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## 1 answers

### Answer by username_1 on 2022-08-04. Score: 1

I believe "yield volatility" here means the realized variance in the interest rate.
Generally speaking, we think that the price of a bond will be the NPV of its cash flows:
$$ p = \sum\_{\tau = 0} ^ {T} \frac{C}{\Pi\_{s = 0} ^ {\tau} (1+r\_s)} + \frac{FaceAmount}{\Pi\_{s = 0} ^ {T} (1+r\_s)}$$
If we can assume that interest rates are constant, then this simplifies:
$$ p = \sum\_{\tau = 0} ^ {T} \frac{C}{(1+r)^\tau} + \frac{FaceAmount}{(1+r)^T}$$

For a long term bond, T will be large and the discount factor of the later cashflows will be something like $(1+r)^10$ or $(1+r)^30$ That means that a small difference in r will be amplified many many times by the exponentiation, leading to the price being very sensitive to $r$, especially compared to a short-term bond where we might have the largest terms of $(1+r)^{1/12}$, $(1+r)^{1/4}$, or $(1+r)^{1/2}$. The math is more complex if interest rates change, but the idea is similar.

Why should "yield volatility" be higher for short term bonds? The basic intuition here is that some interest changes are temporary. A short term bond won't have time for those temporary changes to be undone. A long term bond will not. To see a very simple example of this, consider a situation where interest rates were expected to be 3% forever, but now are expected to be 4% for one month and then go back to 3% forever. That will make the yield on the one period bond change from 3% to 4%, but will make the yield on the 30 year bond change much less. As a counter-argument though, consider that interest rates can drift much more over long periods than over short ones, which would make long-term bonds have more volatile yields.

Empirically, looking over the last 20 years, it does indeed look like short-term yields are more volatile than long-term yields.
[![yields on constant maturity](https://i.sstatic.net/ECtiv.png)](https://i.sstatic.net/ECtiv.png)

> **Comments:**
> **username_0**: Thanks for the answer. On a side note, if you are risk-averse which one would you buy? Long-term or short-term bonds? On the other hand if you want to gain from the fluctuation in bonds, which one should you pick? Long-term or short-term bonds?  

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