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Bond Convexity Calculator

Created by Wei Bin Loo
Reviewed by Tibor Pál, PhD candidate and Adena Benn
Last updated: Jan 18, 2024


With this bond convexity calculator, we aim to help you calculate the effective convexity of a bond. This metric can help you measure the non-linear interest rate risks of your bond investments. Please check our bond yield calculator to understand more.

We have written this article to help you understand what bond convexity is and how to apply the bond effective convexity formula. We will also show some bond convexity examples to help you understand the metric.

What is bond convexity?

Bond convexity is one of the most commonly used metrics to assess the non-linear effect of interest rate changes. It is a crucial metric for analyzing the interest rate risk of your bond investments.

This is especially critical when analyzing bonds embedded with options. There are two types of bonds with embedded options. Callable bonds allow the issuers to buy back the bonds at a specified price, whereas putable bonds enable the bondholders to sell the bonds back to the issuers at a specified price.

This has made the cash flows of these bonds somewhat unpredictable, hence more susceptible to interest rate movements. For instance, the bond issuers can buy back the bonds by triggering the call options embedded when the interest rate drops.

Now that you understand what bond convexity is, let's discuss the metric's calculation.

How to calculate bond convexity? Bond convexity formula

To understand more about the effective convexity formula and how to find the convexity of a bond, let's take Bond Alpha with the following properties as the bond convexity example:

  • Face value: $1,000;
  • Annual coupon rate: 5%;
  • Coupon Frequency: Annual;
  • Years to maturity: 10 years;
  • Yield to maturity (YTM): 8%; and
  • Yield differential: 1%.

The bond convexity calculation requires four steps:

  1. Calculate the coupon per period.

    For the first step, we need to calculate the coupon per period using the formula below:

    coupon per period = face value × coupon rate / frequency

    As this is an annual bond, the frequency = 1. And the coupon for Bond A is:

    ($1,000 × 5%) / 1 = $50.

  2. Calculate the bond price

    Next, let's calculate the bond price using the bond price formula:

    where:

    • cf - Cash flows;
    • r - YTM; and
    • n - Years to maturity.

    The bond price calculation looks like this:

    bond price = $50 / (1 + 8%)1 + $50 / (1 + 8%)2 + $50 / (1 + 8%)3 + ... + $50 / (1 + 8%)9 + $1,050 / (1 + 8%)10 = $798.70

    You can use our bond price calculator to speed up the calculation.

bond price=k=1ncf(1+r)k\quad \small \text{{bond price}} = \sum_{k=1}^{n} \frac{{cf}}{{(1 + r)^k}}
  1. Calculate the bond price after shifting the bond yield

    The next step is to calculate the upward and downward bond price. These are the bond prices where the bond yield shift downwards and upwards, respectively. They will help us analyze the bond price's sensitivity to the change in interest rates.

    The upwards bond price is the bond price when the bond yield shift downwards by the amount of yield differential, whereas the downwards bond price is the bond price when the yield shift upwards.

    Thus, the bond yield for the upwards bond price calculation is 8% - 1% = 7%, and for the downwards bond price calculation, we will use 8% + 1% = 9% as the bond yield.

    Hence, the calculation of the upwards bond price is as follows:

    upwards bond price = $50 / (1 + 7%)1 + $50 / (1 + 7%)2 + $50 / (1 + 7%)3 + ... + $50 / (1 + 7%)9 + $1,050 / (1 + 7%)10 = $859.53

    On the other hand, the downwards bond price calculation looks like this:

    downwards bond price = $50 / (1 + 9%)1 + $50 / (1 + 9%)2 + $50 / (1 + 9%)3 + ... + $50 / (1 + 9%)9 + $1,050 / (1 + 9%)10 = $743.29

  2. Calculate the bond convexity using the effective convexity formula

    The final step is to calculate the bond convexity using the bond convexity formula below, where:

    • BP is the bond price; and
    • YD is the yield differential.

    Thus, the Bond Alpha's bond convexity is ($859.53 + $743.29 - 2 × $798.70) / ($798.70 × (1%) 2) = 67.95.

Bond Convexity=BPup+BPdown2×BPBP×(YD)2\quad \small \text{Bond Convexity} = \frac{{\text{BP}_{up} + \text{BP}_{down} - 2 \times \text{BP}}}{{\text{BP} \times (\text{YD})^2}}

How to interpret the bond effective convexity?

Now that we have talked about how to find the convexity of a bond let's spend some time understanding how to interpret it.

Bond convexity measures the non-linear sensitivity of bond prices to changes in interest rates. It captures the curvature in the price-yield relationship.

When interest rates change, bond prices are affected by two factors: the linear effect (captured by effective duration) and the non-linear effect (captured by bond convexity). Effective duration measures the bond's linear sensitivity to interest rate changes, while bond convexity quantifies the bond's non-linear response.

A higher bond convexity indicates a stronger non-linear relationship between bond prices and interest rates. It implies that larger changes in interest rates will have a more pronounced impact on bond prices.

However, bond convexity is an approximation and relies on certain assumptions, such as a constant yield curve and small interest rate changes. It should be used in conjunction with effective duration and other risk measures for a comprehensive assessment of a bond's interest rate risk.

Keep in mind that bond convexity is a tool, not a precise predictor of bond price movements. Other factors like market liquidity, credit risk, and supply and demand dynamics also influence bond prices.

Therefore, when using bond convexity to assess the impact of interest rate changes, consider it alongside effective duration and conduct thorough analysis. Consulting with a financial professional or utilizing specialized bond analytics tools can provide deeper insights into the risks and rewards associated with bond investments. You can check out our effective duration calculator for this.

FAQ

What is the difference between effective convexity and effective duration?

The main difference between effective convexity and effective duration is the fact that effective duration measures the linear effects of interest rate changes, while effective convexity measures the non-linear effects.

How do I calculate the bond convexity?

You can calculate the effective duration using the following steps:

  1. Calculate the coupon per period.
  2. Determine the bond price.
  3. Calculate the bond price after shifting the bond yield.
  4. Apply the bond convexity formula:
    effective duration = (upwards bond price + downwards bond price - 2 × bond price) / (bond price × (yield differential) 2)

Is effective convexity used to assess non-linear interest rate effect?

Yes, unlike the effective duration, which measures the linear effect of the change in interest rates, the bond effective convexity is used to assess the non-linear effect of the change in interest rates.

What is the effective duration?

The effective duration is a metric used to assess the interest rate risk of bond, just like the effective convexity. However, instead of measuring the linear effect of interest rate changes, the metric focuses on the non-linear effects.

What is the bond convexity if the bond price is $100?

Assuming that the yield differential is 0.1%, the bond convexity will be 75. You can calculate it using this formula:

effective duration = (upwards bond price + downwards bond price - 2 × bond price) / (bond price × (yield differential) 2).

Disclaimer

Bond convexity is a useful concept in estimating the change in bond prices in response to yield fluctuations. However, it's important to note that bond convexity is an approximation and may not perfectly predict the actual bond price movements. Several factors can affect bond prices, such as market liquidity, credit risk, and changes in interest rates. Additionally, bond convexity assumes a constant yield curve and small changes in yield, which may not hold true in all market conditions. Therefore, while bond convexity provides valuable insights, it should be used as a tool alongside other fundamental and technical analyses to assess the potential impact on bond prices. It's always recommended to consult with a financial professional or conduct thorough research before making any investment decisions based on bond convexity calculations.

Wei Bin Loo
Coupon per period
Face value
$
Annual coupon rate
%
Coupon frequency
Annually
Coupon per period
$
Annual coupon
$
Bond price
Years to maturity
year(s)
Yield to maturity
%
Bond price
$
Bond convexity
Yield differential
%
Bond convexity
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