Duration is a measure of the average (cash-weighted) term-to-maturity of a
bond. The are two types of duration, Macaulay duration and modified duration.
Macaulay duration is useful in immunization, where a
portfolio of bonds is constructed to fund a known liability. Modified duration
is an extension of Macaulay duration and is a useful measure of the sensitivity
of a bond's price (the present value of it's cash flows) to interest rate movements.
 Macaulay Duration
The calculation of Macaulay Duration is shown below:
Graphically, Macaulay Duration is the point of balance (in years) for the
cash flows from the bond (see below).
Modified Duration
Modified duration is a measure of the price sensitivity of a bond to
interest rate movements. It is calculated as shown below:
Modified Duration = Macaulay Duration /( 1 + y/n), where y = yield to
maturity and n = number of discounting periods in year ( 2 for semi - ann pay
bonds )
Then, % Price Change = -1 * Modified Duration * Yield Change
Modified duration indicates the percentage change in the price of a bond
for a given change in yield. The percentage change applies to the price of the
bond including accrued interest. In the section showing a bonds price as the
present value of its cash flows, the bond shown was priced initially at
par (100), when the YTM was 7.5%, with a Macaulay Duration of 4.26 years. The
bond was repriced for an increase and decrease in rates of 2.5%. The Modified
Duration for this bond will be: Dmod = -1 * 4.26 / (1 + .075/2) = 4.106 years.
Therefore, a change in the yield of +/- 2.5% should result in a % change in the
price of the bond of: -/+ 4.106 * .025 = +/- 0.10265 (+/- 10.265 %). Since the
bond was initially priced at par, the estimated prices are : $110.27 at 5.00%
and $89.74 at 10.00%. The actual prices were: $110.94 at 5.00% and $90.35 at
10.00%. The discrepancy between the estimated change in the bond price and the
actual change is due to the convexity of the bond, which must be included in the
price change calculation when the yield change is large. However, modified
duration is still a good indication of the potential price volatility of a bond.
 Convexity
The previous percentage price change calculation was inaccurate because it
failed to account for the convexity of the bond. Convexity is a measure of the
amount of "whip" in the bond's price yield curve (see above) and is so
named because of the convex shape of the curve. Because of the shape of the
price yield curve, for a given change in yield down or up, the gain in price for
a drop in yield will be greater than the fall in price due to an equal rise in
yields. This slight "upside capture, downside protection" is what
convexity accounts for. Mathematically Dmod is the first derivative of price
with respect to yield and convexity is the second (or convexity is the first
derivative of modified duration) derivative of price with respect to yield. An
easier way to think of it is that convexity is the rate of change of duration
with yield, and accounts for the fact that as the yield decreases, the slope of
the price - yield curve, and duration, will increase. Similarly, as the yield
increases, the slope of the curve will decrease, as will the duration. By using
convexity in the yield change calculation, a much closer approximation is
achieved (an exact calculation would require many more terms and is not useful).
Using convexity (C) and Dmod then: % Price Chg. = -1 * D mod * Yield Chg. +
C/2 * Yield Chg * Yield Chg.
Using the previous example, convexity can be calculated and results in the
expected price change being: $111.02 at 5.00% and $90.49 at 10.00% The actual
prices were: $110.94 at 5.00% and $90.35 at 10.00%
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