What is Duration?
Duration is a measure of a bond's sensitivity to interest rate changes. The higher the bond's duration, the greater its sensitivity to changes in interest rates (also known as volatility) and vice versa.
Macaulay Duration Formula and Example
There is more than one way to calculate duration (which we'll get to below), but the Macaulay duration (named after Frederick Macaulay, an economist who developed the concept in 1938) is the most common.
The formula for Macaulay duration is:
Macualay duration = [tC / (1+y)t] + [(nM / (1+y)n] / P
t = period in which the coupon is received
C = periodic (usually semiannual) coupon payment (in $)
y = the periodic yield to maturity or required yield
n = number periods
M = maturity value (in $)
PV = market price of bond (in $)
As an example, let's calculate the duration of a three-year, $1,000 Company XYZ bond with a semiannual 10% coupon. By using the present value formula, we can find PV of Cash Flows for each period. So for the first row, we'd figure in ($50) / (1 + .05)^1 = $47.62. For the second row, we'd calculate ($50) / (1 + 0.05)^2 = $45.35. And so on until we make up the following table.
Notice in the table above that we first weighted the cash flows by the periods in which the occurred and then calculated the present value of each of these weighted cash flows. There are six periods (three years x 2 semiannual payments per year) and a measure of 5% is used instead of 10% because payments are semiannual.
To calculate the Macaulay duration, we then divide the sum of the present values of these cash flows by the current bond price (which we are assuming is $1,000):
Macaulay duration = $5,329.48 / $1,000 = 5.33
As mentioned earlier, duration can help investors understand how sensitive a bond is to changes in prevailing interest rates.
By multiplying a bond's duration by the change, the investor can estimate the percentage price change for the bond. For example, consider the Company XYZ bonds with a duration of 5.33 years. If market yields increased by 20 basis points (0.20%), the approximate percentage change in the XYZ bond's price would be:
(- Macaulay Duration x Change in Yield) = Approximate Change in Price
-5.33 x 0.002 = -0.01066 or -1.066%
In other words, if the bond had originally sold for $1,000 with a 5% yield it would now be selling for $1,000 x (1 - 0.01066) = $989.34 -- a discounted rate (Note that this is an approximation and is not as precise as a present value of a bond calculator).
Why would the bond be selling for a discount? Because as interest rates go up, other interest-bearing investments become more attractive to investors and thus the bond's value (or price) goes down.
In economics, the simple rule is that yields move in the opposite direction of prices. As yields (i.e. interest rates) go up, bond prices move down. Conversly, as yields go down, bond prices move up.
Expanding on that, there are five factors affect a bond's duration:
Coupon: The higher a bond's coupon, the more income it produces early on and thus the shorter its duration. The lower the coupon, the longer the duration (and volatility). Zero-coupon bonds, which have only one cash flow, have durations equal to their maturities.
Maturity: The longer a bond's maturity, the greater its duration (and volatility). Duration changes every time a bond makes a coupon payment. Over time, it shortens as the bond nears maturity.
Yield to Maturity: The higher a bond's yield to maturity, the shorter its duration because the present value of the distant cash flows (which have the heaviest weighting) become overshadowed by the value of the nearer payments.
Sinking Fund: The presence of a sinking fund lowers a bond's duration because the extra cash flows in the early years are greater than those of a bond without a sinking fund.
Modified Duration Formula and Example
The modified duration formula can produce more accurate results than the traditional Macaulay duration formula. To calculate the modified duration of a bond, use the formula for modified duration:
Modified Macaulay Duration = Macaulay Duration / (1 + y)
By plugging in the figures from our earlier example, we'd use the modified duration formula to come up with this:
Modified Macaulay Duration = 5.33 / (1 + 0.05) = 5.076
As we did with the last example, we apply this result to come up with our change in price (or discount rate in this case):
Approximate Change in Price = 5.076 x 0.002 = 0.010152
That means if the bond had originally sold for $1,000 with a 5% yield, it would now be selling for $1,000 x (1 - 0.010152) or $989.85 -- a discount from its original $1,000 price.
(Note: In terms of accuracy, this formula comes out only $0.06 different from the actual price of $989.91 calculated on a present value of a bond calculator.)
Effective Duration Formula and Example
The Effective Duration formula is another way to calculate bond duration. The formula uses the bond's currenty yield to maturity (YTM) along with two more present values (a slightly higher YTM and a slightly lower yield YTM).
The formula for Effective Duration is:
Effective Duration = (PL - PH) / (2 x P0 x Change in Yield)
PL = Price of the bond for the lower yield
PH = Price of the bond for the higher yield
P0 = Price of the bond at its current yield
Using the same example from our earlier table with interest rates rising 20 basis points, we can use discounted present values for each period to utlimately calculate effective duration:
Using the sum of cash flows under each PV we calculated, we can plug in the numbers into our effective duration formula:
Effective duration = ($1005.09 - $994.94) / (2 x 1,000 x 0.001) = 5.075
As we did with the last examples, we apply this result to come up with our change in price (or discount rate in this case):
5.075 x 0.002 = 0.0105
Again, think of this number as a discount off the original bond price (If yields fell, this number would be a premium on the bond).
So, $1,000 x (1 - 0.0105) = $989.85
Thus, the estimated price of the original bond using effective duration and assuming a 0.20% raise in interest rates (or yield) would be $989.85 -- the same as our modified duration estimate.
Why Duration is So Important
Understanding the duration formula is not nearly as important as understanding that duration is a measure of risk because it has a direct relationship with price volatility. The greater duration of the bond, the greater its percentage price volatility.
By providing a way to estimate the effect of certain market changes on a bond's price, duration can help you choose investments that might better meet your future cash needs. Duration also helps income investors who want to take on minimal interest rate risk (that is, they believe interest rates might rise) understand why they should consider bonds with high coupon payments and shorter maturities.
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