####
Professor Ian Giddy's Foundations of Finance

##
**Duration and Convexity**

##
*The Price-Yield Relationship*

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 its
cash flows) to interest rate
movements.

##
Macaulay Duration

The calculation of Macaulay Duration is shown 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%

**More about**

The Stern School of Business
The Instructor
*Global Financial
Markets*
Prof Giddy's International
Financial Management course
Prof Giddy's Corporate
Finance course
Prof Giddy's Debt
Instruments and Markets course
Prof Giddy's short courses and seminars

Go to Giddy's Web Portal • Contact Ian Giddy
at igiddy@stern.nyu.edu