# Bond Convexity: The Relationship Between Bond Yields and Interest Rates

If you’ve mastered the ins and outs of how bonds work, things are about to get a lot more complicated; bond convexity. In a nutshell, the convexity of a bond refers to the relationship between bond yields and interest rates.

Although not an exact science, if the ‘duration’ on a bond increases and the yield falls, bonds have a positive convexity. Similarly, if the yield increases and the duration falls, its convexity is negative.

Confused?

If so, we would suggest reading this in-depth guide on bond convexity. We explain in plain English what bond convexity is, how it works, and how financial institutions can mitigate the risks of a bond investment by analyzing its convexity.

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## What is Bond Convexity?

Understanding how bond convexity works is no easy feat. In its most basic form, the convexity of an asset is a tool that allows us to assess the relationship between bond yields and interest rates. For those unaware, when interest rates go up, the price of a bond goes down. Similarly, when interest rates go down, prices increase. More on this later.

The bond convexity goes one step further than simply measuring the relationship between interest rates and yields, as it also takes into account the bond’s duration. The duration measures the sensitivity of an asset in relation to external market forces, such as interest rates.

So, in order to understand how the convexity of a bond actually works, and why it’s a crucial tool in mitigating your risks, we need to first understand its influencing factors.

This includes:

- Bond Yield
- Interest Rates
- Bond Duration

### Bond Yield

If you’re here reading an advanced topic like bond convexity, then it’s likely that you already have a firm grasp of how bond yields work. With that said, the yields of a bond can play a major role in determining its convexity, so let’s recap.

- When bonds are issued by a government state, they will come with a fixed coupon rate. For example, if the coupon rate is 3%, this means that you will receive interest payments of 3% annually until the bonds mature. This coupon rate will never change if you hold on to the bonds until they mature.
- If you decided to sell the bonds before they matured, you would need to do this on the secondary market. In doing so, the value of the bond would likely differ to the fixed coupon rate. This is known as the bond yield.
- So, if the risks of the bonds go down, maybe because the underlying economy is performing well, then you should expect to receive a premium. In yield terms, this means that the yield will go down for new investors.
- At the same time, if the risks of the bonds increase, maybe because the economy is performing worse than expected, then you would need to sell the bonds at a discount. As such, the yield on the bonds would go up.

If you’re confused by the fact that bond yields increase when the risk goes up, think of it like this. If you were buying US Treasury bonds with a fixed yield of 3%, but the risk of default had increased, would you be happy getting the same 3% that investors got when they were first issued?

No, because you would want to receive a higher return to reflect the increase in risk. As such, by demanding 4% instead of the fixed coupon rate of 3%, the yield has increased.

### Interest Rates

So now that we’ve covered bond yields, we need to look at how interest rates have a major influence on the value of bonds, and in turn, its convexity. To clarify, we are referring to the interest rates set by central banks like the Federal Reserve, Bank of England, and European Central Bank.

- The aforementioned institutions will increase or decrease interest rates to mirror the strength or weakness of the wider economy.
- In simple terms, if the economy is performing worse than expected, the central bank will likely decrease interest rates. This is why global interest rates are now so low.
- Similarly, if the economy is performing well, the interest rates go up.

How does this impact bonds? Well, when interest rates go up, the yield on bonds will go up, meaning they are worth less than what you originally paid. For example, let’s say that you have US government bonds at a yield of 2%. If the Federal Reserve then increases interest rates, it will force the value of the bond yield to go up, let’s say to 3%. This means that because investors will now get a yield of 3% on newly issued bonds, they won’t be interested in your 2% bonds. As such, the value of your bonds would go down as you would be required to sell them at a discount.

The same is true when government interest rates go down but in reverse.

### Bond Duration

The final piece of the puzzle in understanding bond convexity is the bond duration. This is where things start to get a bit more complicated, as being able to calculate the duration of an asset requires highly advanced mathematical modelling, much like the convexity. With that said, let’s explain the basics.

- Firstly, don’t be confused by the term ‘duration’, as this has nothing to do with the bond’s maturity or expiry date. It relates to the sensitivity of the bonds. More specifically, the bond duration attempts to measure how much a bond will change in price if and when interest rates move.
- In most cases, the higher the duration, the greater a bond’s value will drop when interest rates go up. Similarly, as the bond duration gets lower, this means that a change in interest rates will have less of an impact on the value of the bond.
- Interest rates are not the only thing that can impact the duration of a bond. The bond’s coupon rate and the amount of time until maturity can also play a part.

Ultimately, the higher the bond duration, the higher the risks for the investor. This is because the bonds will react in a more volatile manner if interest rates change.

## How Does Bond Convexity Work?

Now that you know the three main factors that can influence the convexity of a bond, we can now explain how bond convexity works in practice. First and foremost, the bond convexity is very closely aligned with the bond’s duration. Remember, the duration looks at how the value of a bond will change in relation to a move in government interest rates.

However, the convexity goes one step further by exploring how both the bond price and bond duration will change when interest rates go up or down. In doing so, this allows financial institutions to gauge how much risk their government bond portfolios present. Interestingly, the convexity of a bond can be either positive or negative.

### Positive Bond Convexity

If the convexity of a bond is positive, the following conditions are typically in play:

- If interest rates fall, the value of the bond will rise at a faster rate than it would do if the opposite happens.
- To keep things simple, let’s say that interest fall by 1%. As such, a positive convexity means that the value of the bonds increases by 2%.
- Sticking with the same simplified model, let’s say that we’ve got a positive bond convexity, but interest rates increase by 1%. In turn, the value of the bonds decreases by just 1.5%.

As you can see from the above example, if the bonds have a positive convexity, they will increase in value at a faster rate (2%) when interest rates go down, compared to the rate of decline when rates go up (1.5%). In Layman’s Terms, you as the bondholder will experience more gains when the yield falls, than you will lose when the yield increases.

### Negative Bond Convexity

If a bond has a negative convexity, it operates much like its positive counterpart, but in reverse.

This means that:

- If interest rates rise, the value of the bond will decline at a faster rate than it would do if the opposite happens
- So, let’s say that interest rise by 1%. As such, a negative convexity means that the value of the bonds decreases by 2%.
- Sticking with the same simplified model, let’s say that we’ve got a negative bond convexity, but interest rates decrease by 1%. In turn, the value of the bonds increases by just 1.5%.

As per the above example, a negative convexity will see the bonds decline at a much faster rate (2%), compared to the respective rate (1.5%) of an increase.

## Should I Attempt to Calculate the Bond Convexity?

In short, if you’re an everyday retail trader buying and selling bonds on a DIY basis, it’s highly unlikely that you will have the necessary tools to calculate the bond convexity. This is because such calculations are based on highly advanced mathematical models. Financial institutions and hedge funds are well versed in the convexity of bonds, as they have the required resources to assess this at the click of a button.

With that said, it is important to remember that the specific bond convexity is nothing more than speculation. By this, we mean that there is no sure-fire way to know how much a bond will change in value in relation to an increase or decrease in interest rates. After all, the financial markets can be irrational at times.

## Conclusion

We hope that by reading our guide from start to finish, you now have a firm understanding of what the convexity of a bond is. As we have discussed in great depth, you will first need to get to grips with three key metrics in order to understand convexity. This includes the bond yield, bond duration, and how the movement of government interest rates can impact the value of a bond.

Once you’ve got your head around these three metrics, you can begin to unravel what a positive or negative bond convexity looks like. With that being said, calculating the bond convexity is best left to large-scale financial institutions who have the required mathematical models.

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