TechTheDay

On the other side of the balance sheet, the introduction of longer-term bank certificates of deposit (CDs) with fixed terms to maturity, serve to lengthen the duration of bank liabilities, likewise contributing to the reduction of the duration gap. In this paper, the authors show that single-asset trend strategies have built-in convexity, provided their returns are aggregated over the right time scale, ie, that of the trend filter. By monitoring convexity levels across various bonds and adjusting their holdings accordingly, these managers can exploit opportunities to enhance returns and manage risk more effectively.

Zero-coupon bonds have the highest convexity among fixed-rate bonds, as they do not have periodic coupon payments. Duration provides a linear approximation of a bond’s price change in response to interest rate changes. It represents the weighted average time until a bond’s cash flows are received, taking into consideration the present value of each cash flow. The higher the duration, the more sensitive a bond’s price is to changes in interest rates.

Conversely, when this figure is low, the debt instrument will show less movement to the change in interest rates. The opposite is true of low convexity bonds, whose prices don’t fluctuate as much when interest rates change. When graphed on a two-dimensional plot, this relationship should generate a long-sloping U shape (hence, the term “convex”). Incorporating convexity into bond portfolio management helps investors diversify their interest rate risk. Most regular bonds, such as fixed-rate bonds and zero-coupon bonds, exhibit positive convexity.

1. The rest of this article attempts to provide an intuitive look at how price changes for a bond (or an option) are determined by the first and the second derivative, what they mean, and how they are to be interpreted.
2. As the ten-year yield rose from 1.70 percent in early May to 2.90 percent in August, mortgage portfolio durations extended significantly, forcing MBS hedgers to sell duration, or to sell the underlying MBS.
3. When graphed on a two-dimensional plot, this relationship should generate a long-sloping U shape (hence, the term “convex”).
4. Bonds with higher convexity experience larger price increases in response to falling interest rates, providing the potential for greater capital gains.
5. It describes how the duration of a bond changes in response to changes in interest rates.

While duration would indicate a price of B when rates fall by 1%, the market price would be C, due to the convexity of the bond. The rise in the ten-year Treasury rate last summer was perhaps the most dramatic since the 2003 bond market sell-off. This post explains how major changes in the composition of agency mortgage-backed securities (MBS) ownership caused by the large-scale asset purchase programs (LSAPs) may have prevented a major convexity event triggered by MBS duration extension hedging. In fact, MBS hedging activity remained muted by historic standards and likely contributed only modestly to the rise in interest rates. There are mixed signals from policymakers and financial markets regarding potential inflationary and sharply rising interest rate scenarios.

Bonds with negative convexity have price decreases that are larger than the price increases when interest rates change by equal amounts. So why is the relationship between a bond’s yield and its price known as convexity? As yields change, the change in the price of the bond is not linear; it is curved in a convex fashion. To understand convexity more directly take a look at the following three graphs, all for a $1,000 par value bond, with a coupon rate of 3.452%, making payments twice per year, and with zero expectation of a yield change in the future.

Duration can be a good measure of how bond prices may be affected due to small and sudden fluctuations in interest rates. However, the relationship between bond prices and yields is typically more sloped or convex. Therefore, convexity is a better measure for assessing the impact on bond prices when there are large fluctuations in interest rates. If market rates rise by 1%, a one-year maturity bond price should decline by an equal 1%.

Negative and Positive Convexity

In a falling interest rate environment, bonds with higher convexity can also be beneficial, as they allow investors to capitalize on the positive impact of rate decreases on bond prices. In a rising interest rate environment, bonds with higher convexity are more desirable, as they provide a cushion against the negative impact of rate increases on bond prices. Conversely, when interest rates fall, the present value of a bond’s future cash flows increases, leading to a higher bond price. For a 1% change in interest rates, a bond’s price will change (inversely) by an amount roughly equal to its duration. For example, a 5-year bond with a coupon of 4.0% matures in 5 years and has a duration of 4.5 years. If interest rates fell 1%, that bond would rise approximately 4.5% in value, for a total return of 9.5% (4% coupon plus 4.5% price appreciation).

What is your current financial priority?

In the image below, the curved line represents the change in prices, given a change in yields. The straight line, tangent to the curve, represents the estimated change in price, via the duration statistic. The shaded area reveals the difference between the duration estimate and the actual price movement.

If you think about it, convexity reflects the error in the estimation of a bond’s price if modified duration alone were to be used in such an estimate. I don’t need to do any approximations and mess with convexity and modified duration. When you have a large portfolio of say thousands of bonds (consider the Lehman or now Barclays Global Aggregate), then this full calculation of each bond’s price becomes too intensive an exercise. It is easier to calculate the duration of the portfolio, and its convexity, and estimate price changes and risk using these rather than a full computation. However, the relationship between bond prices and interest rates is non-linear, leading to inaccuracies in duration-based price change estimates.

So an additional constraint is added which would ensure that the present value of assets would increase by more (or decrease by less) than the present value of liabilities when interest rates change. The additional constraint that would make this possible is ensuring that the convexity convexity risk of assets is greater than the convexity of liabilities. When interest rates increase, the price of an MBS tends to fall at an increasing rate and much faster than a comparable Treasury security due to duration extension, a feature known as the negative convexity of MBS.

Convexity in Bond Portfolio Management

Modified duration, on the other hand, is an adjusted version of Macaulay duration that directly measures a bond’s price sensitivity to changes in interest rates. Bonds with higher duration and convexity tend to experience more significant price changes in response to interest rate shifts. While it is generally agreed that a modest increase in inflation and interest rates would be beneficial to the life insurance industry, a sharp increase in interest rates and the resulting tail risk can be quite the opposite. This article is intended to provide insights to the life insurance industry on how to prepare for what lies ahead.

Convexity-adjusted duration combines duration and convexity to accurately measure a bond’s price sensitivity to interest rate changes. Look at how curved — i.e., how convex — the graph of the price-yield relationship is! Notice also that there are no capital gains/changes in price at the exact yield of the bond, 3.45%, where the line actually touches https://1investing.in/ the horizontal axis. This means that if yields stay the same as the coupon rate there should be no change in the price of the bond. Similarly, should rates rise, duration alone would predict a loss greater than would actually occur. For a bond or portfolio of bonds, duration will always underestimate performance over large changes in yields.

A financial professional will offer guidance based on the information provided and offer a no-obligation call to better understand your situation. Multiply the present value of each cash flow by the corresponding time period squared plus the time period. Jason Voss, CFA, tirelessly focuses on improving the ability of investors to better serve end clients. He is the author of the Foreword Reviews Business Book of the Year Finalist, The Intuitive Investor and the CEO of Active Investment Management (AIM) Consulting. Previously, he was a portfolio manager at Davis Selected Advisers, L.P., where he co-managed the Davis Appreciation and Income Fund to noteworthy returns.

When you are driving a car your speed is the rate of change in the car’s location. Then you either give the car more gas with the accelerator or press down on the brakes to slow the car down. Speeding up means that there is a positive second derivative, while slowing down means that there is a negative second derivative. All things equal, the larger a bond’s coupon, the shorter its duration because a greater proportion of the cash payments are received earlier. A zero-coupon bond’s duration is equal to its maturity, as nothing is paid until maturity.

The duration accomplishes this, letting fixed-income investors more effectively gauge uncertainty when managing their portfolios. Think mortgage-backed securities (MBS) – when interest rates fall, mortgages are refinanced and MBS will pay back par value a lot earlier than at maturity. When interest rates rise, homeowners are less likely to refinance their mortgage, leading to a longer duration of cash flows for the MBS. In this case, the fixed cash flows become less valuable, and investors would demand a higher yield to compensate them for the risk of owning the security.