At the end of this session, you should be able to understand the difference between duration and modified duration and how they are related as measures of the interest rate risk of a bond. Examples used throughout this session are based on fixed rate or nominal bonds for ease of calculation but floating rate note duration and modified duration are calculated using the same principles.

### Duration and modified duration - useful tools for comparing bonds

Duration measures the average term to maturity of a bond, weighted by the present value of each cashflow. It is also commonly referred to as “Macaulay Duration”. It is a more meaningful measure of a bond than simply looking at the term to maturity of the bond, as it takes into account the coupon payments that are made. If a bond is a zero-coupon bond, then the duration is equal to the term to maturity.

Mathematically, duration is represented as follows

Where

t = term in years to the coupon payment date

n = term in years to the maturity date

disct = discount factor for term t based on the yield to maturity

Modified duration gives you the ability to be able to quickly determine how the price of a bond will change in response to interest rate movements. For those mathematically inclined, it is the first derivative of price with respect to yield. That is, it represents the ‘rate of change’ of the price to changes in yields. Fortunately, the mathematics of this calculation means that modified duration can be calculated simply once you know the duration.

Where

i = yield to maturity

f = frequency (eg monthly, quarterly, semi-annually etc)

Calculating the modified duration of a bond or portfolio allows you to quickly estimate how much the value with rise/fall when there are changes in interest rates. For example, if a bond has a modified duration of 3 years, this means that the value of the bond will increase by around 3% if interest rates decrease by 1%. Conversely, the value of the bond will decrease by around 3% if interest rates increase by 1%.

The longer the duration, the more sensitive the security is to interest rate movements. Therefore, in a rising interest rate environment, investors would prefer a security with a lower duration as the value of the security would fall by less than one with a higher duration. Conversely, in a falling interest rate environment, investors would prefer a security with a higher duration as the value of the security would rise by more than one with a lower duration.

If an investor holds a security until maturity then the changes in prices along the way due to interest rate movements are not relevant as the yield is locked in from the start. While a bond’s value may change as prices change, gains and losses are only realised when the security is sold. The changes in prices, and therefore duration, are relevant to those investors who may choose to sell investments prior to maturity.

### Duration

One way to compare bonds is by measuring the average life of a bond including its cash flows and this is known as duration. It measures when, on average, the investor will receive cash back equal to their initial investment. It can also be described as measuring the present value of the weighted average cash flows of the bond. The coupons and the face value can all be treated as cash flows from an investment. Duration is typically expressed in years, as is the term to maturity of a security.

When calculating this average term of cash flow, the size of each cash flow is not used to weight the averages, rather the present value of each cash flow is used as the weighting factor. This is very important because it ensures the cash flows are weighted by their importance and value in today’s terms.

It is known that a dollar today is worth more than a dollar in ten years. If given the choice of having a dollar today or a dollar in ten years, an investor would naturally pick today due to the time value of money, where you can reinvest your dollar, and in ten years from now it will be worth more than one dollar. Similarly, in order to compare cash flows in the future, each term needs to be weighted by its present value.

As mentioned earlier, Duration is calculated as follows:

This may appear to be time consuming, however it is quite simple once you set up a spreadsheet to do the calculations for you.

#### Example 1

Face Value = $100,000

Coupon = 7%

Yield = 8%

### Duration graphically

Figure 7.1 below shows a graphical example of duration. There are five cash flows of equal size, which represent the coupon payments, and a larger cash flow at the end, which is the Face Value plus the final coupon.

The coupon amounts are all the same, however the present values of those cash flows are smaller the more distant in the future they are. The final cashflow, being the coupon and face value, is the main contributor to the duration, however duration is shortened due to the coupon payments.

Look at the present value rectangles or the dark blue sections in the chart, and imagine the X- axis represents the top of a seesaw and that the triangle below the X-axis is the fulcrum of the seesaw. The value (or duration) is the average of the cash flows. In this case it would be 2.75 years for this three year semi-annual coupon bond.

### The effect of coupon rate and yield

How does the coupon rate and yield affect duration? The coupon rate and yield are important inputs into pricing and duration calculation formulas. Let’s see how duration is affected by different coupon rates and yields.

Earlier above we calculated the duration of a three year, 7% coupon bond with a yield of 8%. What if the bond had a 6% coupon instead of 7% coupon? How would the duration be affected?

#### Example 2

Face Value = $100,000

Coupon = 6%

Yield = 8%

The duration in this example is 2.7831 (coupon = 6%) compared to a duration of 2.7537 (coupon = 7%). Therefore you can see that there is an inverse relationship between duration and coupon rate.

#### Example 3

We discovered above that there is an inverse relationship between duration and coupon. What is the relationship between duration and yield?

Face Value = $100,000

Coupon = 7%

Yield = 7%

The duration in this example is 2.7575 (yield = 7%) compared to a duration of 2.7537 (yield = 8%). Therefore you can see that there is an inverse relationship between duration and yield.

### Relationships between price, duration, interest rates and coupon

Therefore, if you want a security with a higher duration, you should purchase in a security with a lower yield or coupon. Conversely, if you want a security with a lower duration, you should purchase a security with a higher yield or coupon.

### Duration over time

Duration of a bond decreases over time, however it does increase slightly on each coupon payment date. Remember that the duration can be thought of as the balancing point of a see-saw. When a coupon payment is made, the fulcrum needs to move a little to the right to rebalance it.

Here is a graphical representation of the duration of a 3 year bond.

### Portfolio duration and modified duration

Duration and modified duration can also be calculated for a portfolio, not just for each individual security.

For duration, weight the duration of each bond in the portfolio by its size, in comparison to the portfolio, and likewise for modified duration. It is important to remember when doing these weightings and calculations, that the market value of the bond and the market value of the portfolio are used. The market value is the present value and it is essential to use this value to weight portfolios to ensure that like is being compared with like.

For a portfolio of bonds, duration of the portfolio is the present value weighted term to cash flow for every coupon flow and face value in the portfolio. It represents when, on average, the portfolio will return its cash flows to the investor.

In the same way that modified duration impacts a single bond, if our portfolio had a modified duration of four, and interest rates were to rise by 1% on every security in the portfolio, the portfolio would lose about 4% in value. Or if interest rates were to fall by 1%, the portfolio would gain 4% across the entire portfolio.

Investors can use this duration indicator to position their portfolios in accordance with their view on the underlying interest rate cycle.

#### Example 4

Here is a portfolio of bonds. The duration and modified duration are calculated as the weighted average based on the market value.

### Conclusion

By now you should have an understanding of the terms duration and modified duration, and that they are very important tools for fixed income investors and are often used in the description of managed bond funds. The duration measures the average term of all cash flows of the bond and is the present value of all that bond’s weighted average cash flows. The modified duration measures the price sensitivity of a bond to changes in interest rates.

### Review questions

#### 1. What does duration measure?

- The time to maturity of a bond
- The face value of the bond
- The value of coupon payments you can expect over the life of a bond
- The average term to maturity of a bond weighted by the present value of each cashflow

#### 2. Modified duration allows investors

- To estimate the change in price of a bond for a given change in interest payments
- The change in price of a bond given payment of a coupon

#### 3. If I have a zero coupon bond, then

- Duration will be shorter than the term to maturity
- Duration will be equivalent to the term to maturity
- Duration will be longer than the term to maturity