Convexity & Barbell Strategy

Convexity describes the curvature of the relationship between the price of an obligation and changes in interest rates. Consequently, it’s an important concept because, during interest rate fluctuations, capital losses are usually smaller than capital gains. In other words, the price decreases more for a given increase in interest rates than it increases for the same decrease (this decrease in sensitivity is termed “convexity”).

The calculation of this effect is possible. It’s a bit complex mathematically, but I can simplify it:

Let’s start with the fact that there’s a Taylor series expansion function (which is material from school) describing the result in terms of a reason for a change (incremental) in interest rates.

In our case:

• Duration (D) – the first derivative of the bond’s price with respect to the interest rate – graphically, this is the slope of the tangent to the bond’s price curve;
• Convexity (CX) – the second derivative of the bond’s price with respect to the interest rate – graphically, this indicates the change in the slope of the tangent (which is not significant for small changes in interest rates);

We can discuss the 3rd and 4th derivative changes with Taylor’s formulas, but practically, it’s not necessary.

Why is the general consideration of CX necessary?

It’s because, practically, immunization (protection against interest rate risk) can be achieved by combining different combinations of obligations with different durations, but varying CX. So, the same D and different CX.

A positive CX gives better results for immunization in the face of small interest rate changes, while a large CX provides better results for a predetermined change in interest rates. (They say that the market often underestimates the effects of convexity). This means that with small percentage changes, immunization is achieved between assets and liabilities relative to Duration, while with large percentage changes, it’s based on the best result planned.

Barbell Strategy
Below is an image illustrating investment made with the goal of immunization, utilizing two alternatives with fixed Duration but different CX. This is called the barbell strategy.

In this moment’s context, pension, insurance, and other fund managers strive to obtain low CX in the liability part and high CX in the asset part. The formula is as follows:

CX = Scaling Factor * (Negative Effect + Positive Effect).

As a scaling factor, a fixed number of 10^8 is taken – which statistically corresponds to one percent change in interest rates. The negative effect is the loss of capital with an increase in one unit of interest rates, and the positive one is vice versa.

Source:

Financial Markets and Institutions – by A. Saunders, M. Cornett & O. Erhemjamts

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