While McCauley duration may be intuitive, it does not allow us to directly calculate profit and loss impacts of changes in interest rates.

However, the ability to do this is very useful, especially in the context of risk management.

So let's take a look at some other ratios which will allow us to do this.

First, let's look at modified duration.

Modified duration is based on the concept of mathematical derivatives.

You may recall from school math classes that the first derivative of a function measures how fast the function changes as its input changes.

In simpler terms, it gives us the slope of the function at any given point.

For bonds, the bond price is essentially a function of its yield.

As yields change the bond price changes as well.

So the first derivative of the bond price with respect to its yield gives us a measure of the bond's interest rate sensitivity.

But what does modified duration actually tell us? It's defined as the relative change in bond price caused by a change in the yield to maturity.

YTM. This is a very useful measure because by comparing the modified durations of two bonds, we can quickly determine which bond will lose a larger percentage of its current price for a given change in yield.

And here's the good news, we don't have to go through the complex process of calculating the derivative ourselves.

There's a shortcut.

If you divide the McCauley duration by one plus the yield to maturity YTM, at least for bonds. With annual coupon payments, you get the modified duration of the bond.

Let's look at an example.

To calculate modified duration, we divide the macauley duration by one plus the yield to maturity.

In our example, the Macauley duration is 4.7709 years, and the YTM is 2.43%.

So dividing 4.7709 by 1.0243, which is one plus oh 0.0243 or 2.43%, gives us a modified duration of approximately 4.6577.

Now, what does this number tell us? It means that for every 1% change in the yield to maturity, the bond's price will change by approximately 4.6577% of its current price. How did We get there? We just need to multiply the modified duration by the assumed yield change, which was 1%.

In this case, as you can see from the calculation on screen, we are using a negative value for modified duration.

This reflects the inverse relationship between bond prices and yields.

When yields go up, bond prices go down and vice versa.

We will continue using a negative value for the same reason in all further calculations.

To put this into perspective, if the bonds yield were to increase from 2.43% to 3.43%, an increase of exactly one percentage point, we would expect the bond's price to decrease by 4.6577% of its current level.

However, this only gives us the relative price change.

If we want to calculate the absolute price change, we need to multiply the relative price change by the current bond price.

For our bond, the current price is 99.8603%, so multiplying the relative change, the 4.6577% by the current price, we get an absolute change of 4.6512% of the par value.

In other words, the bond price is expected to fall by 4.6512% in absolute terms.

If yields increase by one percentage point, starting from a bond price of 99.8603%, modified duration predicts the bond price will drop to approximately 95.2091%.

That's the 99.8603%, minus the absolute price change of 4.6512%.

When the yield to maturity reaches 3.43%.