Duration
An Introduction to Duration
Duration is a measure of interestrate risk. Or, stated differently, duration is a measure of how sensitive the price of a fixedincome instrument is to interestrate 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 semiannual period, we convert duration to years before quoting it (a duration of 8 semiannual 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 YieldtoMaturity 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.
Interactive App
With the following app, you can set the maximum yieldtomaturity, and time to maturity, and see the resulting 3D duration surface. You can also change the coupon rate and see the effect on the duration surface.

You can move the 3D surface around, and zoom in and out, with your mouse.

How does increasing the yieldtomaturity, or time to maturity, affect the bond’s duration?

How does increasing the bond’s coupon rate affect duration?

Can you see the case where duration is decreasing with an increased time to maturity?
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:

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

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

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 zerocoupon 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{11/(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, linebyline contributions to this presentation, and all the collaborators who have contributed to 5Minute Finance via GitHub.
Learn more about how to contribute here.
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