There are several formulas for calculating the duration of specific bonds that are simpler than the above general formula. Note that modified duration is always slightly less than duration since the modified duration is the duration divided by 1 + the yield per payment period. Learn how to calculate convexity with a structured approach, breaking down key components, formulas, and interpretation for practical application.
Modified Duration
As per the above example, a negative convexity will see the bonds decline at a much faster rate (2%), compared to the respective rate (1.5%) of an increase. If you’re here reading an advanced topic like bond convexity, then it’s likely that you already have a firm grasp of how bond yields work. With that said, the yields of a bond can play a major role in determining its convexity, so let’s recap.
- This method is similar to the modified duration method, but does not require the calculation of the modified duration.
- Using the concept of duration, we can calculate that Bond A has a duration of four years while Bond B has a duration of 5.5 years.
- If a bond’s duration increases as yields increase, the bond is said to have negative convexity.
- Therefore, U.S. Treasuries should have higher durations than corporate bonds, and, therefore, change in price more when market interest rates change.
How convexity and duration are related and why convexity is a better measure of interest rate risk?
However, duration assumes that this relationship is constant, which is not always the case. In reality, the price-yield curve of a bond is not a straight line, but a convex curve. This means that as the yield changes, the slope of the curve also changes, indicating a different sensitivity of price to yield.
Should I Attempt to Calculate the Bond Convexity?
It means a bond with a maturity period of 20-years will see a significant change in its price if an interest rate increases in the early stages. When interest rates rise, newly issued bonds will offer a higher yield to investors. Thus, existing bonds with older interest rate arrangements would lose the faith of investors. As a result, it will cause a decline in their demand, and hence their prices will fall. Bond convexity looks at the relationship between interest rates and the bond duration.
Positive and Negative Convexity
Using the Price function also means that you don’t have to be on a payment date. Low-coupon and zero-coupon bonds, which tend to have lower yields, show the highest interest rate volatility. In technical terms, this means that the modified duration of the bond requires a larger adjustment to keep pace with the higher change in price after interest rate moves.
This is because new bonds offer higher interest rates, and sellers must accept a discount in order to sell older bonds. Bank liabilities, which are primarily the deposits owed to customers, are generally short-term in nature, with low duration statistics. By contrast, a bank’s assets mainly comprise outstanding commercial and consumer loans or mortgages. These assets tend to be of longer duration, and their values are more sensitive to interest rate fluctuations.
The timing of these payments affects convexity, as longer maturities or higher coupon rates produce different convexity characteristics than shorter-term or lower-coupon bonds. If a bond’s duration rises and yields fall, the bond is said to have positive convexity. As yields fall, bond prices rise by a greater rate or duration than if yields rise. If a bond has positive convexity, it would typically experience price increases as yields fall, compared with price decreases when yields increase.
You will get 10.90 years for modified duration, and 11.23 years for Macaulay duration. These are exactly the same answers that we got using the approximation techniques. The image below shows how we will set up the worksheet to calculate modified duration using the approximation technique that was just outlined.
- Duration can be a good measure of how bond prices may be affected due to small and sudden fluctuations in interest rates.
- If you’re here reading an advanced topic like bond convexity, then it’s likely that you already have a firm grasp of how bond yields work.
- We have previously demonstrated the Price function in the article “Bond Valuation Using Microsoft Excel,” so be sure to check that article if you aren’t familiar with the Price function.
- The straight line, tangent to the curve, represents the estimated change in price, via the duration statistic.
- If rates rise by 1%, a bond or bond fund with a five-year average duration would likely lose approximately 5% of its value.
As a result, Bond B has a higher price sensitivity to interest rate changes than Bond A. For example, when the interest rate decreases from 5% to 4%, Bond B’s price increases by 8.39%, while Bond A’s price increases by 7.89%. Conversely, when the interest rate increases from 5% to 6%, Bond B’s price decreases by 7.64%, while Bond A’s price decreases by 7.18%. Therefore, Bond B benefits more from interest rate decreases and suffers less from interest rate increases than Bond A, which is the effect of convexity. As expected, both bonds decrease in price when interest rates increase, but bond B decreases less than bond A, despite having a higher duration. This is because bond B has a higher convexity, which means that its price change is less sensitive to interest rate changes than bond A’s. Conversely, both bonds increase in price when interest rates decrease, but bond B increases more than bond A, for the same reason.
This shows how, for the same 1% increase in yield, the predicted price decrease changes if the only duration is used as against when the convexity of the price yield curve is also adjusted. With that said, it is important to remember that the specific bond convexity is nothing more than speculation. By this, we mean that there is no sure-fire way to know how much a bond will change in value in relation to an increase or decrease in interest rates. As you can see from the above example, if the bonds have a positive convexity, they will increase in value at a faster rate (2%) when interest rates go down, compared to the rate of decline when rates go up (1.5%). In Layman’s Terms, you as the bondholder will experience more gains when the yield falls, than you will lose when the yield increases. If you’ve mastered the ins and outs of how bonds work, things are about to get a lot more complicated; bond convexity.
No, because you would want to receive a higher return to reflect the increase in risk. As such, by demanding 4% instead of the fixed coupon rate of 3%, the yield has increased. As noted previously, Excel has built-in functions that can be used to calculate both modified convexity formula and Macaulay duration on any date. However, note that these two functions require dates, so if you don’t have them, then you will need to use the approximation technique shown above. Also, there is no function to calculate convexity, so you have to use the method shown in the previous section.
Nevertheless, the model does not account for the curvature in larger yield changes, and this is where convexity becomes particularly useful. Bond duration measures the change in a bond’s price when interest rates fluctuate. If the duration of a bond is high, it means the bond’s price will move to a greater degree in the opposite direction of interest rates.
This is because such calculations are based on highly advanced mathematical models. Financial institutions and hedge funds are well versed in the convexity of bonds, as they have the required resources to assess this at the click of a button. The final piece of the puzzle in understanding bond convexity is the bond duration. This is where things start to get a bit more complicated, as being able to calculate the duration of an asset requires highly advanced mathematical modelling, much like the convexity. Well, when interest rates go up, the yield on bonds will go up, meaning they are worth less than what you originally paid. If the Federal Reserve then increases interest rates, it will force the value of the bond yield to go up, let’s say to 3%.
The duration of a zero bond is equal to its time to maturity, but as there still exists a convex relationship between its price and yield, zero-coupon bonds have the highest convexity and its prices most sensitive to changes in yield. Convexity allows for other risk factors that count towards the final market price of a bond. The Duration of a bond that is used to calculate to recover the cash flows of a bond is one such factor. However, duration only considers a linear relationship between bond prices and interest rates. Because duration depends on the weighted average of the present value of the bond’s cash flows, a simple calculation for duration is not valid if the change in yield could change cash flow. Valuation models must be used to calculate new prices for yield changes when the cash flow is modified by options.
As you can see, the graph is curved which shows that the rate of change in price is different at different points on the graph. This method is similar to the modified duration method, but does not require the calculation of the modified duration. However, it may be less accurate if the change in the yield is too large or too small. Graphically, the duration of a bond can be envisioned as the fulcrum under a seesaw, placed so as to balance the weights of the present values of the coupon payments and the principal payment. This typically involves taking positions in financial instruments with negative correlations (i.e. move in the opposite direction) with the convexity of the assets or liabilities being hedged.
