## When yields go down, bonds go up, and when yields go up, bonds go down. Simple.

Does that mean that this relationship is always valid, and does that mean that when the one goes up by one, the goes down by one? No, Welcome to Convexity, one of the more mysterious concepts in bonds, but you should know this concept if you want to understand the risks one takes with bonds.

Let us unpack it!

The term “convexity” describes how a bond’s duration changes when the interest rate increases. Bond convexity is a measure of the non-linear relationship between bond prices and interest rate changes in finance.

In general, the further away from the maturity (the end date), the more vulnerable the bond price is to interest rate changes. Bond convexity[1] is one of the most fundamental and hardest to understand concepts in the financial world.

I copied this picture from Investopedia that explains it pretty well:

If you ask yourself: “Why does Unhedged still buy bonds while the bonds’ yield is so close to zero?” You want to know, read on.

The relationship between Yield and Bond price is not linear because of the risk premium and the duration left on the Bond. The price development can be almost logarithmic!

It basically says that the closer the yield comes to zero, the price will go up exponentially, or the higher the yield becomes, the prices don’t increase much…. How weird is that?

In the case of a bond yield of 30% or so, the default risk is already so high that the prices of the Bond are already close to zero. Getting more interest won’t make the Bond more valuable!

When the interest goes to zero or negative, the pricing in terms of duration must be very high, and the more negative the interest rate goes, the faster the price goes up.

It’s particularly essential to think about convexity during times of market volatility. Convexity is a measure of the amount of “whip” in a bond’s price yield curve. You can instantly lose or gain more than you thought was possible with bonds.

So, when somebody asks at a BBQ: why are you holding bonds, the interest is already so low? You can say: because of the convexity, of course! Turn around, open another cold one and smile.

## Bond and Convexity things you always wanted to know but never dared to ask

### Bond Duration

Bond duration quantifies the change in the price of a bond due to interest rate fluctuations. If the duration is long, the Bond’s will move in the opposite direction of the change in interest rates to a greater extent. When this value is low, the debt instrument will exhibit less movement in response to changes in interest rates. Essentially, the longer the term of a bond, the greater the price shift associated with interest rate changes. In other words, the higher the interest rate risk, the higher the yield. Therefore, if an investor feels that a significant shift in interest rates would negatively affect their bond portfolio, they should select shorter-duration bonds.

### Negative & Positive Convexity

When the length of a bond rises as the yield on the bond increases, the Bond is said to have negative convexity. In other words, as rates rise, the bond price declines at a faster pace than when yields fall. As a result, if a bond’s convexity is negative, its duration increases as the price decreases and vice versa. When the length of a bond increases while the yield decreases, the Bond is said to have positive convexity. In other words, when rates decline, bond prices climb faster—or for a longer duration—than they would if yields increased—positive convexity results in more significant bond price rises. If a bond’s convexity is positive, it usually experiences more price gains when rates fall than price declines as yields rise.

### Convexity & risk

Convexity extends the notion of duration by determining the sensitivity of a bond’s duration to changes in yields. Convexity is a more accurate indicator of interest rate risk when the bond length is included. Whereas duration implies a linear connection between interest rates and bond prices, convexity incorporates additional variables and generates a slope.

The duration may be a helpful indicator of how minor and abrupt interest rate changes may impact bond values. The connection between bond prices and yields, on the other hand, is usually more slanted or convex. As a result, convexity is a more appropriate metric for evaluating the effect of significant interest rate changes on bond values.

As convexity rises, the portfolio’s exposure to systemic risk increases. The phrase “systemic risk” gained popularity during the 2008 financial crisis when the collapse of one financial institution posed a danger to the stability of others. This risk, however, is present in all companies, industries, and the economy as a whole.

Due to the risk inherent in a fixed-income portfolio, current fixed-rate securities become less attractive when interest rates increase. The bond portfolio’s exposure to market interest rates diminishes with decreasing convexity, and it may be termed-hedged. Generally speaking, the greater the coupon rate or yield, the smaller the Bond’s—or market risk. This risk reduction occurs because market rates would have to rise significantly to exceed the coupon on the Bond, implying that the investor has less interest rate risk. However, other hazards, such as default risk, may still exist.

[1] Stanley Diller popularised the concept of convexity, which was based on Hon-Fei Lai’s work.

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