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where N is the maturity of the bond, and t is when each coupon comes due. Holding other factors constant, the duration of a bond will increase with the maturity of the bond and decrease with the coupon rate on the bond. For example, the duration of a 7%, 30-year coupon bond, when interest rates are 8% and coupons are paid each year, can be written as follows:

Duration of 30 - year Bond =

11 This measure of duration estimated above is called Macaulay duration, and it does make same strong assumptions about the yield curve; specifically, the yield curve is assumed to be flat and move in parallel shifts. Other duration measures change these assumptions. For purposes of our analysis, however, a rough measure of duration will suffice.

What does the duration tell us? First, it provides a measure of when, on average, the cash flows on this bond come due, factoring in both the magnitude of the cash flows and the present value effects. This 30-year bond, for instance, has cash flows that come due in about 12.41 years, after considering both the coupons and the face value. Second, it is an approximate measure of how much the bond price will change for small changes in interest rates. For instance, this 30-year bond will drop in value by approximately 12.41% for a 1% increase in interest rates. Note that the duration is lower than the maturity. This will generally be true for coupon-bearing bonds, though special features in the bond may sometimes increase duration.12 For zero-coupon bonds, the duration is equal to the maturity.

This measure of duration can be extended to any asset with expected cash flows. Thus, the duration of a project or asset can be estimated in terms of its pre-debt operating cash flows:

I -.t=1 (1+ r)t (1 + r)N Duration of Project/Asset = dPV/dr = 1-1-

where CFt is the after-tax cash flow on the project in year t, the terminal value is a measure of how much the project is worth at the end of its lifetime of N years. The duration of an asset measures both when, on average, the cash flows on that asset come due, and how much the value of the asset changes for a 1% change in interest rates.

One limitation of this analysis of duration is that it keeps cash flows fixed, while interest rates change. On real projects, however, the cash flows will be adversely affected by the increases in interest rates, and the degree of the effect will vary from business to business - more for cyclical firms (automobiles, housing) and less for non-cyclical firms (food processing). Thus the actual duration of most projects will be higher than the estimates obtained by keeping cash flows constant. One way of estimating duration without depending upon the traditional bond duration measures is to use historical data. If

12 For instance, making the coupon rate floating, rather than fixed, will reduce the duration of a bond. Similarly, adding a call feature to a bond will decrease duration, while making bonds extendible will increase duration.

the duration is, in fact, a measure of how sensitive asset values are to interest rate changes, and a time series of data of asset value and interest rate changes is available, a regression of the former on the latter should yield a measure of duration:

a Asset Valuet = a + b a Interest Ratet In this regression, the coefficient ' b' on interest rate changes should be a measure of the duration of the assets. For firms with publicly traded stocks and bonds, the asset value is the sum of the market values of the two. For a private company or for a public company with a short history, the regression can be run, using changes in operating income as the dependent variable -

a Operating Income, = a + b a Interest Ratet Here again, the coefficient "b" is a measure of the duration of the assets.

Illustration 9.5: Calculating Duration for Disney Theme Park

In this application, we will calculate duration using the traditional measures for the Disney Bangkok Theme Park that we analyzed in chapter 5. The cash flows for the project are summarized in Table 9.10, together with the present value estimates, calculated using the cost of capital for this project of 10.66%.

Table 9.10: Calculating a Project's Duration: Disney Theme Park

Year

Annual Cashflow

Terminal Value

Present Value

Present value *t

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