An Introduction to Duration

Duration is a measure of interest-rate risk. Or, stated differently, duration is a measure of how sensitive the price of a fixed-income instrument is to interest-rate changes.

When we say, “The duration of the bond is 4 years,” we mean:

• “If the interest rate on the bond goes up by 1%, the bond’s price will decline by 4%.”

Duration is quoted in “years.”

• If a bond has a semi-annual period, we convert duration to years before quoting it (a duration of 8 semi-annual periods is 4 years).

Duration is Approximate

Duration is a linear approximation of a nonlinear relationship.

• Duration is more accurate as the change in the interest rate becomes smaller.
• The error when using duration to estimate a bond’s sensitivity to interest rates is often called convexity.

Determinants of Duration

Duration is affected by the bond’s coupon rate, yield to maturity, and the amount of time to maturity.

• Duration is inversely related to the bond’s coupon rate.
• Duration is inversely related to the bond’s yield to maturity (YTM).
• Duration can increase or decrease given an increase in the time to maturity (but it usually increases). You can look at this relationship in the upcoming interactive 3D app.

For a review of bond coupon rates and yields see these presentations: An Introduction to Bond Valuation; For What is the Yield-to-Maturity Used?.

Average of Time Payments are Received

Duration can be thought of as the weighted average of when the bondholder receives payment. The weights are the proportion of the bond’s total value received in each period.

• In the following app, change the bond’s coupon, YTM, and maturity and see how it affects the weights. Check that the way the weights react is consistent with the relationships on the previous slide.

• This will give you an intuitive understanding of how these variables affect duration.

Calculating Duration

Duration is the slope of the line tangent to the bond’s price at the bond’s present YTM.

• Remembering our calculus, we immediately see we need to calculate the derivative of the bond’s price with respect to the YTM. See this video for a complete derivation.

However, it is customary to first calculate what is called Maucaulay Duration, and then use this to calculate Modified Duration.

• Modified Duration is equivalent to the derivative of the bond’s price with respect to the YTM.

• Once we have Modified Duration, we can use it to calculate the bond’s price (or % change) given a change in YTM.

Using Duration

Specifically, the steps in using duration are:

1. Calculate ‘Macaulay Duration’ which is the weighted average of when the bondholder receives their payments.

2. Divide this by `(1 + YTM)` to get Modified Duration (which is the derivative).

3. Calculate the % change in the bond’s price as a linear function of modified duration.

Macaulay Duration

Let `BP` be the bond price, `CF_i` the cash flow from the bond in period `i`, and `n` the number of periods until maturity. Say the par value is $1000. The Macaulay Duration is:

`\text{Macaulay Duration} = \sum_{i = 1}^n {(i)\frac{(\frac{CF_i}{(1+YTM)^i})}{BP}}`

Note that since `BP = \sum_{i = 1}^n {\frac{CF_i}{(1+YTM)^i}}` the second term in the summation is the proportion of the bond received at time `i`. These are weights (and sum to 1). Denoting them `w_i` we have:

Macaulay Duration = `\sum_{i = 1}^n {(i)w_i}`

Macaulay Duration

From our definition above, we can make the following observations:

• Duration cannot exceed the number of periods to maturity of the bond.
• The Duration of a zero-coupon bond is the number of years until maturity.

Also note, we can calculate the duration of a bond portfolio as the weighted average of the duration of all of the individual the bonds in the portfolio.

Modified Duration

Macaulay Duration is a bit off however, so we adjust it by dividing it by `(1+YTM)`.

The result (Modified Duration) then matches the derivative of the bond’s price with respect to YTM.

Modified Duration = `\frac{\text{Maucaulay Duration}}{(1+YTM)}`

Simple Example

Say we have a 5% coupon bond with annual payments and 8 years until maturity. Let the bond’s YTM be 3%.

The bond’s price is: `P = \$50\frac{1-1/(1.03)^8}{0.03} + \frac{\$1000}{(1.03)^8} = \$1140.39`

`\text{Macaulay Duration} = \sum_{i = 1}^7 (i)\frac{(\frac{50}{(1.03)^i})}{\$1140.39} + 8\frac{(\frac{1050}{1.03^8})}{\$1140.39} = 6.87\ yrs`

`\text{Modified Duration} = \frac{6.87}{1.03} = 6.67\ yrs`

Now You Try

Calculate the Macaulay and Modified Durations for the following bonds. You can check your answers with the interactive app on the following slide. All bonds have annual payments in the interactive app.

• A 15% coupon bond with 20 years to maturity and a 3% YTM.

• A 4% coupon bond with 10 years to maturity and a 7% YTM.

• A 0% coupon bond with 10 years to maturity and a 2% YTM.

You can also use the following app to see duration decrease when maturity increases.

• Set the coupon to 3%, the YTM to 18%, and increase years to maturity from 17.

Modified Duration Calculation

Using Modified Duration

We use Modified Duration to approximate the change in the bond’s price for a give change in yield. In terms of percent, we can say:

`\%\Delta P = -(\text{Modified Duration}) \Delta YTM`

• For example, if a bond has a Modified Duration of 8, then given a 0.5% increase in yield, the bond is expected to decline by 4%.

`\%\Delta P = -8(0.5\%) = 4\%`

Improving on Duration

If we want to improve our estimate of the % change in the bond’s price, we can add a convexity adjustment. - This is covered in the 5MinuteFinance interactive presentation on Bond Convexity.

Credits and Collaboration

Click the following links to see the code, line-by-line contributions to this presentation, and all the collaborators who have contributed to 5-Minute Finance via GitHub.

Learn more about how to contribute here.