﻿ Negative convexity

# Negative convexity

## Negative convexity

Negative convexity arises when the duration of a bond increases in conjunction with its yield. As the yield on the bond increases, its price will decrease. The price of a bond increases as interest rates decrease, whereas the value of a bond with negative convexity decreases in the same situation.

## What is Negative convexity?

A bond’s yield curve and its response to changes in interest rates are described by the phrase “negative convexity” in the context of fixed-income investment.

Under a negative convexity scenario, bond prices appreciate at a slower rate than they decline as interest rates rise. On the flip side, when rates go up, prices go down more sharply than when rates go down by the same amount.

This feature is most often observed in bonds that can be “called” by the issuer if interest rates decline before the bond’s maturity.

Understanding negative convexity is crucial for investors since it has the potential to influence bond portfolio risk and total return.

To make the most of your investment strategy planning and risk management efforts in an interest rate environment where rates are subject to fluctuations, it is helpful to have a firm grasp of this idea.

## Understanding Negative convexity

The length of time that interest rate fluctuations affect bond prices is called the bond’s duration. The concept of convexity illustrates the relationship between interest rate fluctuations and bond durations. The price of a bond usually goes up as interest rates go down. In contrast, a decline in interest rates results in a falling price for bonds with negative convexity.

A callable bond’s price won’t go up as fast as a non-callable bond’s because as interest rates go down, the issuer has more of an incentive to call the bond at par. A rise in the probability of calling a callable bond might cause its price to fall. For this reason, the price-yield curve of a callable bond will have a concave or negatively convex form.

## Formula for Negative convexity

Convexity may be expressed as the following formula:

Convexity = (Σ [ (t^2 + t) * PV(CF_t) ] ) / (P * (1 + YTM)^2), where Σ is the symbol for summation, t is the period, PV(CF_t) is the present value of the cash flow at time t, P is the bond price, and YTM is the yield to maturity.

## Calculation for Negative convexity

Investors, analysts, and traders determine a bond’s convexity since length is not a perfect indicator of price change. A portfolio’s exposure to market risk may be measured and managed with the use of convexity, a valuable tool in risk management. Predictions of future price movements have been improved.

Although the precise formula for convexity is somewhat involved, a simplified version of it may be determined using the following formula:

Convexity approximation = (P(+) + P(-) – 2 x P(0)) / (2 x P(0) x dy ^2)

In what context:

• P(+) = bond price when the interest rate is decreased
• P(-) = bond price when the interest rate is increased
• P(0) = bond price
• dy = change in interest rate in decimal form

Consider a bond that is now valued at \$1,000 as an example. The bond would be worth \$1,035 after a 1% reduction in interest rates. The bond would be worth \$970 after a 1% interest rate hike. We may estimate the convexity as:

The approximate convexity is \$2 x \$1,000 x 0.01^2, simplifying to \$5 / \$0.2, or 25.

A convexity adjustment is required when using this to predict the price of a bond based on its duration. To account for convexity, the formula is as follows:

Adjustment for convexity equals convexity multiplied by 100 times (dy)^2.

This case calls for a convexity adjustment of:

Convexity adjustment = 25 x 100 x (0.01)^2 = 0.25

The following formula allows an investor to estimate the price of a bond for a given change in interest rates by combining duration and convexity:

Bond price change = duration x yield change + convexity adjustment

## Example of Negative convexity

It is critical to grasp negative convexity, and further examples can help clarify:

Example 1: Mortgage-backed securities

One type of bond that exhibits negative convexity is mortgage-backed securities (MBS). Mortgage refinancing is common among homeowners who want to take advantage of historically low interest rates. Consequently, mortgage-backed securities (MBS) holders run the risk of receiving their expected interest payments early (or never). Since the predicted cash flows for MBS decline when rates fall, this shows that they exhibit negative convexity.

Example 2: Callable corporate bonds

Negative convexity is also possible for callable corporate bonds. Bonds can be “called” by their issuers to refinance them at a cheaper interest rate when rates fall. If the bond is redeemed before its maturity date, the investor may incur capital losses since they will not receive future interest payments.

Negative convexity is crucial in the complex realm of fixed-income instruments. Investors and portfolio managers must understand its complexities and repercussions to make educated decisions. It has advantages and disadvantages, but it is a useful instrument for interest rate risk management. If you are familiar with negative convexity and how it affects bond prices, you will be more equipped to handle the complicated world of fixed-income investing.

Bond prices are less affected by changes in interest rates when convexity is positive, which is why traders like it. When interest rates rise, negative convexity indicates that price swings will be bigger, which is bad news for traders.

The fixed interest rate that most bonds pay makes them more appealing as rates go down, which increases demand and the bond’s price. Bond prices fall as interest rates rise because investors no longer value the security of a bond’s lower fixed interest yield.

Bonds’ interest rate risk may be measured by looking at their duration, which takes into account their maturity, yield, coupon, and call characteristics. A bond’s potential sensitivity to changes in interest rates is determined by a single figure resulting from these several elements.

When bond prices are very convex, they are mostly unaffected by changes in interest rate markets. Thus, investors anticipating a gradual rise in interest rates would do well to favour high convexity.

This bond’s convexity is calculated as the second derivative of its price relative to its yield, divided by its price. The formula C = (P+ + P- – 2P)/(P dy^2) can be used to approximate it.

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