<p>The normal yield curve is usually drawn concave. Why is this typically the case?</p>
<p>My current understanding is that long term bonds have higher interest rate risk because a higher percentage of its outflows come from coupon payments. For example, a 1 month bill price won't change much for a change in interest rates since its price is dominated by the par value which will be paid back in a month.</p>
<p>Applying the logic to convexity vs. concavity: as time to maturity increases, the percentage of a bond's price which comes from the final par value payout decreases slower and slower. Therefore, the risk premium should increase slower and slower creating a concave yield curve.</p>
<p>Is this logic correct or is there more to it?</p>

CC BY-SA 4.0
2021-01-03T02:01:08.407Z
This is probably better suited for the quantitative finance stack exchange. However, the suggested logic is not particularly easy to follow, and probably detracts from the question. The explanation that I am aware of is that the ratio of convexity to duration rises at the ling end; you get more convexity per unit of duration. Convexity is valuable, hence the yield premium.
Brian Romanchuk
CC BY-SA 4.0
2021-01-03T02:51:00.157Z
If you mean something like [this](https://www.investopedia.com/thmb/hQnF5caoqth9MxK9eQ1g_XYm6z8=/3274x0/filters:no_upscale():max_bytes(150000):strip_icc():format(webp)/YieldCurve2-362f5c4053d34d7397fa925c602f1d15.png) I would call that concave.  But the idea is that the intermediate points get from short-term interest rates to essentially flat long-term interest rates and if short-term interest rates are relatively lower than long-term ("normal") then you may get that stylised shape.  Reality can often be rather different
Henry
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2021-01-03T20:08:18.707Z
@Henry my bad I swapped concave and convex. I just rewrote the question more clearly. I understand in reality tons of factors influence this curve including liquidity. I'm interested in the shape in the absence of these factors. I've included in the question my current understanding for concavity. Is this correct?
curiousgeorge
CC BY-SA 4.0
2021-01-04T00:08:58.540Z
You can get the same yield curve with completely different coupon "conventions". That's why it's called a yield curve; not a coupon curve. This does not disprove what you are saying. It just makes what you're saying less likely to be the explanation.
H2ONaCl
CC BY-SA 4.0
2021-01-04T00:16:16.397Z
It looks like this was already answered in great detail in quantitative finance stack: https://quant.stackexchange.com/questions/32147/shape-and-geometry-of-the-yield-curve. That answer discusses a lot of other stuff but also concavity
csilvia
<yield><treasuries>
Why is the yield curve typically concave?
curiousgeorge
# Why is the yield curve typically concave?

(Asked by: username_0 on 2021-01-03)

The normal yield curve is usually drawn concave. Why is this typically the case?

My current understanding is that long term bonds have higher interest rate risk because a higher percentage of its outflows come from coupon payments. For example, a 1 month bill price won't change much for a change in interest rates since its price is dominated by the par value which will be paid back in a month.

Applying the logic to convexity vs. concavity: as time to maturity increases, the percentage of a bond's price which comes from the final par value payout decreases slower and slower. Therefore, the risk premium should increase slower and slower creating a concave yield curve.

Is this logic correct or is there more to it?

**Tags:** yield, treasuries
> **Comments:**
> **username_1**: This is probably better suited for the quantitative finance stack exchange. However, the suggested logic is not particularly easy to follow, and probably detracts from the question. The explanation that I am aware of is that the ratio of convexity to duration rises at the ling end; you get more convexity per unit of duration. Convexity is valuable, hence the yield premium.  
> **username_2**: If you mean something like [this](https://www.investopedia.com/thmb/hQnF5caoqth9MxK9eQ1g_XYm6z8=/3274x0/filters:no_upscale():max_bytes(150000):strip_icc():format(webp)/YieldCurve2-362f5c4053d34d7397fa925c602f1d15.png) I would call that concave.  But the idea is that the intermediate points get from short-term interest rates to essentially flat long-term interest rates and if short-term interest rates are relatively lower than long-term ("normal") then you may get that stylised shape.  Reality can often be rather different  
> **username_0**: @Henry my bad I swapped concave and convex. I just rewrote the question more clearly. I understand in reality tons of factors influence this curve including liquidity. I'm interested in the shape in the absence of these factors. I've included in the question my current understanding for concavity. Is this correct?  
> **username_3**: You can get the same yield curve with completely different coupon "conventions". That's why it's called a yield curve; not a coupon curve. This does not disprove what you are saying. It just makes what you're saying less likely to be the explanation.  
> **username_4**: It looks like this was already answered in great detail in quantitative finance stack: https://quant.stackexchange.com/questions/32147/shape-and-geometry-of-the-yield-curve. That answer discusses a lot of other stuff but also concavity  

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