How to Apply the Effective Convexity Formula in Finance

The effective convexity formula is a powerful tool in finance, allowing you to measure the sensitivity of bond prices to interest rate changes.

To apply the effective convexity formula, you need to understand the relationship between bond prices and interest rates.

Effective convexity is a measure of the rate of change of duration with respect to interest rates, and it's calculated by multiplying the duration of a bond by the slope of the yield curve.

You can use the effective convexity formula to estimate the potential price impact of interest rate changes on a bond portfolio.

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Effective convexity is a curve convexity statistic that measures the secondary effect of a change in a benchmark yield curve.

Effective convexity is used to measure the change in price resulting from a change in the benchmark yield curve for securities with uncertain cash flows. This is in contrast to approximate convexity, which is based on a yield to maturity change.

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The formula for effective convexity is given by:

$$\text{Effective convexity}=\frac{PV_{-}+PV_{+}-2PV_{0}}{(ΔCurve)^2PV_{0}}$$

Where:

PV– = Price if yield curve declines by Yield (parallel shift)

PV+ = Price if yield curve increases by Yield (parallel shift)

PV0 = Initial bond price

For a bond with an embedded option, a yield to maturity based calculation of convexity does not consider how changes in the yield curve will alter the cash flows due to option exercise. To address this, an effective convexity must be calculated numerically.

Here are some key facts about effective convexity:

Callable bonds exhibit negative convexity, while putable bonds always have positive convexity.

Bond convexity is a measure of the sensitivity of a bond's duration as interest rates change. It's a more accurate measure of interest rate risk than duration alone.

Convexity is considered positive if a bond's duration increases when interest rates fall, and negative if it increases when interest rates rise. This means a bond with positive convexity will see its price fall by a smaller rate if rates rise than if they had fallen.

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The true relationship between a bond's price and its yield-to-maturity is a curved line, which duration estimates as a straight line. This is why convexity is used to improve the estimate of the percentage price change.

Here's a key formula to keep in mind:

$$\text{Convexity effect} ≈ \frac{1}{2}×\text{AnnConvexity}×(ΔYield)^2$$

This formula shows how to calculate the convexity adjustment, which adds to the linear estimate provided by duration alone to bring the adjusted estimate close to the actual price on the curved line.

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Approximate Convexity is a crucial concept in bond pricing, and it's essential to understand how it works.

The true relationship between bond price and yield-to-maturity (YTM) is a curved line, not a straight one. The duration, which is a common measure of bond price sensitivity, only estimates the change in bond price along a straight line that is tangent to the curved line.

For small changes in YTM, the difference between the curved line and the straight line is negligible. However, as the change in YTM grows larger, the difference becomes significant.

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To improve the estimate of the percentage price change, a convexity measure is used. This is done using a mathematical formula that includes a convexity adjustment.

The convexity adjustment is calculated as half of the annual convexity statistic, multiplied by the change in YTM squared. This amount adds to the linear estimate provided by the duration alone, bringing the adjusted estimate very close to the actual price on the curved line.

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Bond convexities vary due to differences in payment structures, such as zero-coupon bonds and amortizing bonds. These bonds have different sensitivities to parallel yield curve shifts, even if they have the same maturity.

The price sensitivity to parallel changes in the term structure of interest rates is highest with a zero-coupon bond, and lowest with an amortizing bond.

However, if two bonds have the same par value, coupon, and maturity, but are located at different points on the price-yield curve, their convexities may still differ.

Their prices will be affected equally by small, first-order, parallel yield curve shifts, but will start to change by different amounts with each further incremental parallel rate shift due to their differing payment dates and amounts.

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As you delve into the world of bond convexity, it's essential to understand how bond duration changes with interest rates. A key concept to grasp is that as interest rates increase, the present value of longer-dated payments declines in relation to earlier coupons.

This is because the discount factor between the early and late payments grows larger. As a result, bond price declines when interest rates increase, but changes in the present value of each coupon times timing are larger than changes in the bond price.

In fact, the modified duration D differs from the regular duration by the factor one over 1 + r, which also decreases as r is increased. This means that increases in r must decrease the duration, or in the case of zero-coupon bonds, leave the unmodified duration constant.

A zero-coupon bond has a convexity of 120. If there's a sudden interest rate change of 50 basis points, the convexity effect is approximately 0.5 times the annuity convexity times the squared change in yield.

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The positivity of convexity can also be proven analytically for basic interest rate securities. This implies the negativity of the derivative of duration by differentiating dB/dr=-DB.

Here's a summary of how bond duration changes with interest rates:

Note that the duration of all bonds approaches 0 when the maturity date gets near, with the exception of long-term discount bonds. For these bonds, the duration increases as the bond approaches the maturity date.

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Calculating effective convexity can be a bit tricky, but it's a crucial step in understanding how bonds with embedded options behave in different interest rate scenarios. To calculate effective convexity, you'll need to gather some specific data, including the initial bond price (PV0), the price if the yield curve declines by a certain yield (PV-), and the price if the yield curve increases by the same yield (PV+).

You can find these values by using an option pricing model or by analyzing the bond's cash flows. Once you have this data, you can plug it into the effective convexity formula: (PV- + PV+ - 2PV0) / ((ΔCurve)^2PV0). This formula will give you the effective convexity of the bond.

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A higher effective convexity value means that the price of the bond is more sensitive to changes in interest rates. This is because the bond's value is more affected by changes in the yield curve. To illustrate this, consider the following facts about bonds with embedded options:

These facts highlight the importance of considering the benchmark yield when analyzing the behavior of bonds with embedded options. By understanding how the benchmark yield affects the bond's value, you can better estimate the effective convexity of the bond.

To make things easier, here's a simple example of how to calculate effective convexity in Excel. Suppose you have the following values:

You can plug these values into the formula: =(C3 + C4 - 2*C2) / (2*C2*(B5^2)), where C2 is PV0, C3 is PV-, C4 is PV+, and B5 is ΔCurve. This will give you the effective convexity of the bond.

Convexity is a risk management figure used to manage market risk in bond portfolios. It's a number that helps investors understand how much their bond portfolio will be affected by changes in interest rates.

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The combined convexity and duration of a trading book are key indicators of risk. If they're high, the risk is higher, and if they're low, the book is hedged, and the loss will be minimal even with substantial interest rate movements.

Convexity is used to quantify interest rate risk. It's an adjustment made to the modified duration formula to account for the non-linear relationship between bond prices and interest rates.

The formula for the percentage change in bond price is: ΔP(%) = -ModD × Δr + 0.5 × C × (Δr)^2. This formula takes into account both the modified duration and the convexity adjustment.

Here's a breakdown of the formula's components:

Let's consider an example. Suppose we have a bond with 10 years until maturity, an annual coupon of 7%, and a par value of USD 100. If the convexity is 64.933 and the modified duration is 7.0236, and the yield increases by 45 basis points, what will be the value of the bond price?

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