Interest rate sensitivity is typically measured using duration, but there are a number of different types of duration.

The type of duration which is often just referred to as duration can be more technically referred to as Macaulay duration.

Macaulay duration or more generally duration measures the weighted average time to the reception of each of the cash flows within that Bond.

So it's the weighted average maturity of the bonds cash flows.

Now the formula for calculating duration or Macaulay duration looks super complicated.

But if we break it down, it's not so bad.

So if we're looking at the left hand formula here, we need to take the cash flow for each time period and then multiply it by one plus the yield to maturity to the power of minus the number of time periods away into the future, but that cash flow is received.

This is essentially calculating the present value of that cash flow.

And then back to the first element in the numerator, we multiply this by the number of the time period when that cash flow is received.

We then add that up for every single cash flow in the bond and then divide this by the sum of the present value of all of the cash flows, which is on the right hand side another way of expressing the current price of the bond.

the higher the Macaulay duration of a bond the more sensitive that bond is going to be to changes in interest rates.

This measure captures the time to maturity the size of the coupon cash flows and also the yield which will have an impact on the sensitivity of that Bond's price to a change in interest rates or yields.

So let's have a look at an example of calculating Macaulay duration.

We've got an example Bond here that's got five years to maturity. We've got an annual coupon of 2% and the yield to maturity on this Bond at the moment is also 2% In order to calculate Macaulay duration, we need to look at what's happening in every single year of the Bond's life.

So for each of the five years of the Bond's life, we're going to look at a cash flows individually now here the cash flows are being expressed as a percentage of the par value. But if we've got a hundred dollars of par value, then this 2% will be two dollars. So we've got a 2% of a par value as our cash flow in the first year. We need to calculate its present value.

By dividing by 1 plus the yield to the power.

of one so 1.02 to Power of One that will give us the present value of 1.96% of the par value or 1.69. If we've got a hundred dollars of par value and then for the final column on the right hand side, we need to multiply this by the time period where that cash flow is received and for the year one cash flow, we multiply by one so it doesn't change it at all.

For the year two cash flow 2% of the par value. We calculate its present value. So a lower present value because that's more discounting over two years than over one.

And then we multiply that present value the 1.92% by the number of the year that we're in being two. So that turns Us in the final column into 3.84.

We then need to replicate this for every single cash flow present value at first of all, then multiply it by the number of the Year where the cash flow is received.

So all the way down there for the fifth year in the fifth year, we get the par value and the coupon. So that's 102% of the par value here. The present value of 102% is the 92.38 and then we multiply that by 5 to get the 461.92.

To calculate duration. What we need to do is add up.

So sum.

The present value times the number of the time period for every single Year's cash flows and then divide that by the sum of the cash flows, not multiply by the number of the time period so we're going to take a total the 480.77.

Divided by the 100% and hopefully not a huge surprise that a bond that has the same yield as its coupon rate will be priced at its par value.

So as a result, we'll take the 480.77 and divide it by a hundred to give us a Macaulay duration of 4.81.

the units of the Macaulay duration are years This is the average number of years until we receive the cash flows on the bonds.

So we're receiving most of the cash flows of this bond in year five, but we're getting some of those cash flows earlier in year one to four. We're getting the coupon cash flows only so a smaller amount of the cash flows, but we're receiving an early which drags the average.

Time to receive the cash flows below the time to maturity.

For a coupon paying Bond the Macaulay duration must be less than the time to maturity because some of the cash flows on the bond will have been received before the maturity on that Bond.

So Macaulay duration the weighted average at the time to receive those cash flows for a coupon payment Bond must be less than the time to maturity.