Formula of Convexity (Table of Contents)
What is Convexity Formula of Bond?
Convexity of a bond can be defined as a measurement of the curvature in relation to changes in interest rates which shall affect the changes in the price of a bond and further it does so by measuring the duration changes, as the market interest rates fluctuate. The formula for calculating Convexity of a bond is as follows:
Where,
 P_{+} is the price of the bond when there is a fall in interest rate
 P_{–} is the price of the bond when there is a rise in interest rate
 PO is the current market price of the bond
 ΔY is the change in the yield curve
Explanation of the Convexity Formula
In the capital market where bonds are traded, convexity can be referred to the relationship between yield and the bond price. When this relationship is plotted on the graph, it shall be forming a Ushaped curve and the relationship will not be linear.
A bond that has more convexity, and in case of interest rate movements, it shall experience comparatively fluctuations which shall be dramatic in nature. The formula is selfexplanatory and easy to calculate as well.
Examples of Convexity Formula (with Excel Template)
Below are some simple to advanced examples to do a calculation of convexity formula with excel template.
Example #1
The price of the bond increases to $1,162 when the market interest rates when down by 2% and the price of the bond decreases to $888 when the market interest rates fall by 2%. Suppose that bond is currently trading at face value. You are required to calculate the Convexity of the bond.
Solution:
We can use the abovegiven formula to calculate Convexity when there is a change in the market rate for 2% and the face value of the bond is $1,000:
 Convexity of Bond = $1,162 + $888 – (2 * $1,000) / 2 * ($1,000 x 2%^{2})
Convexity of Bond will be –
 Convexity of Bond = 62.50
Therefore, the convexity of the bond at par is 62.50
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Example #2
Fannie Mae has issued new bonds just last year and now they are trading at $998.45 in the market. A fed move is expected, and the market is not sure whether it is going up or going down. A survey was conducted and 50% of the experts said there would be upward movement in fed rate which shall affect the bond yields by 200 basis points and there were rest 50% of the expert who was anticipating fall in fed rate which shall affect the bond yields by 200 basis point again.
If that happens, the price of the new bond will be $836.45 if the rate rises and $1389.92 if the rate falls. Mr. P the portfolio manager wants to know the convexity of this bond as this would help in determining the overall portfolio’s estimated valuation.
Based on the above details you are required to calculate the Convexity of the Fannie Mae bond.
Solution:
We can use the abovegiven formula to calculate Convexity when there is a change in the market rate for 2% which is 200 basis points and the current market price of the bond is 998.45:
 Convexity of Fannie Mae bond = $1,389.92 + $836.45 – (2 * $998.45) / 2 * (998.45 * 2%^{2})
Convexity of the Fannie Mae bond will be –
 Convexity of Fannie Mae bond = 287.28
Therefore, the convexity of the Fannie Mae bond is 287.28
Example #3
Company X and Company Y are two competitors in the market and they both are trying to lure the market to invest in their new bond. The investors are studying their existing bonds in the market and as per recent prices, the bond price of Company X is prevailing at $956.72 and the bond price of Company Y is trading at $1,122.56. Investors want to prefer which is less convex as the yields would be higher.
The investors anticipated a 1% change in yield and the prices for Company X and Company Y anticipated were $1,098.33 and $1,322.88 when yield falls versus $871.33 and $1,001.39 when the rate rises.
You are required to advise where should investors invest by using Convexity as a deciding factor.
Solution:
We can use the abovegiven formula to calculate Convexity when there is a change in the market rate for 1% and the current market price of the bond for X is 956.72 and for Y it is $1,122.56:
Calculation of Convexity for Company X is as follows,
 Convexity of Company X = ($1,098.33) + ($871.33) – (2 * $956.72) / 2 * ($956.72 * 2%^{2})
Convexity for Company X is –
 Convexity of Company X = 293.82
Calculation of Convexity for Company Y is as follows,
 Convexity of Company Y = ($1,322.88) + ($1,001.39) – (2 * $1,122.56) / 2 * ($1,122.56 * 2%^{2})
Convexity for Company Y is –
 Convexity of Company Y = 352.54
Hence, if investors prefer less convexity, then Company X would be preferred as it would have higher yields.
Convexity Calculator
You can use the following convexity calculator.
P_{+}  
P_{}  
PO  
ΔY^{2}  
Convexity Formula  
Convexity Formula= 
 

Relevance and Use
For a portfolio of the bond, the convexity shall measure the bond’s risk together and it can be said that this is the weighted average of each bond with the bond’s market value being calculated as the weights.
Being a second derivative, it fails to predict exact price changes even though after considering the nonlinear shape of the priceyield curve which does an adjustment for estimating the changes in the price of the bond.
The more convex the bond is lesser risky it shall be because for the reduction in market interest rates the price changes is less. Henceforth, due to the market price in the lower risk, a bond that is comparatively more convex it shall have a lower yield.
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