Let's examine McCauley duration in detail.

This measure reflects two key observations.

Bonds with longer maturities tend to be more sensitive to interest rate changes while bonds with higher coupons show less sensitivity.

By combining these observations, we reach an important insight.

The sooner a bond's payments are made on average, the less sensitive it'll be to interest rate fluctuations.

So to compare two bonds in terms of their interest rate sensitivity, we can calculate the weighted average time to maturity for all the bonds Cash flows, and this is exactly what Macauley duration does.

As shown in the formula, Macauley duration is calculated by multiplying the present value PV of each cashflow by the number of years until the payment is received.

For simplicity, we'll assume annual payments throughout.

If the payments are semi-annual or follow another frequency, the formula adjusts slightly, but the concepts remain the same.

Next, we sum all the maturity weighted present values pvs and divide this total by the unweighted Present values of the bond's cash flows.

Essentially, this is the bond's price.

The result is the weighted average maturity, which is expressed in years because we're calculating an average, the Macaulay duration might be, for example, 9.2 years for a 10 year bond, even though no actual payment occurs exactly at 9.2 years.

This is important to remember.

McCauley duration represents the average time to receive all cash flows, not the specific payment dates.

The interpretation of McCauley duration is relatively straightforward.

The higher the number, the longer it takes on average for the bond holder to receive the bond's cash flows.

As a result. The longer the macauley duration, the higher the bond's sensitivity to interest rate changes.

This measure takes into account both, the bonds time to maturity and its coupon level.

And lastly, here are a few useful rules of them.

For a zero coupon bond, the Macaulay duration will always be equal to the bonds time to maturity, As there is only one cashflow, which occurs at maturity.

For fixed coupon bonds with a coupon greater than 0%, the Macaulay duration will be shorter than the time to maturity because a portion of the bond is repaid before maturity through coupon payments.

The higher the coupon, the larger the portion that is repaid earlier, and thus, the shorter the macauley duration.

You can think of a fixed coupon bond as a basket of zero coupon bonds each with different maturities.

Each zero coupon bond in this basket will have a macauley duration equal to its time to maturity.

The macauley duration of the entire basket is simply the weighted average of the individual Zero coupon bonds durations based on the proportion of each cash flow relative to the bond as a whole.

Let's take a look at a practical example to calculate macauley duration using the bond details provided on screen.

We're looking at a five-year bond with an annual coupon rate of 2.4% and a yield to maturity YTM of 2.43%.

Let's look at the calculation process step by step.

First, we consider the cash flows.

The bond pays an annual coupon of 2.4% of the face value every year for five years.

In the final year, year five, the bond holder receives both the final coupon, the 2.4%, and the face value 100% resulting in a total cash flow of 102.4% in year five.

The next step is to calculate the present value PV of these cash flows.

Each cash flow is discounted by the yield to maturity YTM of 2.43%.

The PV is determined by dividing each cash flow by one plus YTM to the power of T, where T represents the year in which the payment is received.

For example, the present value of the first coupon 2.4% is approximately 2.3431%, and the present value of the final cashflow in year five is 90.8164%.

Now let's move on to the maturity weighted pvs, the PV times T column in the table.

This is where we take the PV of each cashflow and multiply it by the number of years until the payment is received.

For instance, the PV of the first year cashflow is Multiplied by one while the PV of the year five cashflow is multiplied by five.

These values give us the time weighted contribution of each cashflow.

Once we've calculated these, the next step is to sum all the maturity weighted pvs.

This gives us a total of 476.4204%.

Now, to calculate the Macaulay duration, we simply divide the total maturity weighted pvs.

The 476.4204% by the total PV of the bonds cash flows the 99.8603%.

The result is a McCauley duration of 4.7709 years.

What does this mean? The McCauley duration of this bond is 4.7709 years, which indicates that the bond holder will receive the bond's cash flows on average in approximately 4.77 years.

This reflects the bond sensitivity to interest rate changes.

The higher the Macaulay duration, the more sensitive the bond's prices to changes in interest rates.

Since this is a fixed coupon bond, the McCauley duration is shorter than the bond's five year time to maturity as part of the bond holder's investment is returned before maturity through the annual coupon payments, reducing the average time to receive the bonds.

Cash flows below the time to maturity.