The bond convexity statistic is the second-order effect in the Taylor series expansion. Getting an equation for convexity is just a matter of more calculus and algebra; see the Technical Appendix for all the details. However, the results are complicated enough to warrant separate equations for coupon payment dates and between coupons. Equation 6.16 is the formula that applies to a coupon payment date such that t/T = 0.
Granted, there are a lot of terms in the equation, but just three variables: c, the coupon rate per period; y, the yield to maturity per period; and N, the number of periods to maturity. One simplification emerges for a zero- coupon bond for which c = 0. Then much of equation 6.16 drops out and the convexity reduces to N * (N+ 1)/(1 + y)2.
Let's work on a 4%, semiannual payment, 25-year bond priced at a discount to yield 4.40% (s.a.). First, use equation 6.13 or 6.15 to get its Macaulay duration (t/T = 0), using y = 0.0220, c = 0.02, and N = 50.
That's the Macaulay duration that corresponds to a change in the yield per period; annualized it is 15.7156 (= 31.4312/2). The annual modified duration is 15.3773 (= 15.7156/1.0220). Note that we divide by one plus the yield per period, not by the annualized yield.
Okay, now enter the same inputs for y, c, and N into equation 6.16.
If you tried and got that result – congratulations! That's the convexity statistic that links the change in the yield per period to the change in market value. As in equation 6.11, it is annualized by dividing by the periodicity squared. So, this bond has an annualized yield convexity of 320.2689 (= 1,281.0757/4). Unfortunately, Excel does not have a financial function for convexity even though it uses the same inputs as duration.
Suppose three months go by and the bond is still priced to yield 4.40% (s.a.). Let t/T = 0.50 because we are halfway through the semiannual period. The Macaulay duration is easily calculated: 31.4312 – 0.50 = 30.9312. Annualized, it is 15.4656 (= 30.9312/2) and the modified duration is 15.1327 (= 15.4656/1.0220).
The convexity statistic between coupon payment dates is shown in equation 6.17.
The first term is the convexity that would prevail at the beginning of the period (hence t/T = 0) if the current yield per period y is used in the calculation in equation 6.16. Then we need to subtract the term in brackets, which
contains, as a bit of a surprise, the Macaulay duration (MacDur) in equation 6.15 calculated for t/T = 0 using the yield per period y.
As time passes, yields inevitably do change, and equations 6.15 and 6.16 have to be calculated using the new yield to maturity. For convenience, I assume that the yield remains the same at 4.40% (s.a.), so we can use the already obtained results. The convexity after the three months is 1,250.7438, using Convexity (t/T =0) = 1,281.0757, t/T = 0.50, y = 0.0220, and MacDur (t/T = 0) = 31.4312.
The annual convexity statistic is 312.6859 (= 1,250.7438/4).
In working through this convexity calculation, I have kept more precision (four decimals) than really is needed. That's because I want to illustrate how the modified duration and convexity statistics can be approximated quite accurately using numerical methods. The idea is to estimate the values for the first and second partial derivatives in equations 6.4 and 6.6. In calculus, dy is an infinitesimal change in the yield per period. I use a discrete change in this approximation, here chosen to be 20 basis points up and down.
The approximation formulas for (annual) modified yield duration and yield convexity are defined in equations 6.18 and 6.19.
MV(down) and MY(up) are the market values calculated using a pricing model (or equation) assuming the same decrease and increase in the yield.
The initial market value for the bond, MV(initial), is 94.999558 (percent of par value), using equation 3.11 from Chapter 3 and PMT = 2, y = 0.0220, N = 50, FV = 100, and t/T = 0.50.
This is the combined flat price and accrued interest. You also can get this result on Excel using the financial function PRICE for the flat price and then ACCRINT to get the accrued interest. The sum of the flat price obtained with PRICE and the accrued interest from ACCRINT is the full price, 94.999558. Assume arbitrarily that the 4%, 25-year bond is issued on July 15,2014, matures on July 15, 2039, and now on October 15, 2014 (three months into the semiannual period using a 30/360 day count), the yield is 4.40% (s.a.).
If the yield goes up by 20 basis points to 4.60% (s.a.), the market value is 92.182875, found by repeating the calculation for y = 0.0230.
If the yield goes down to 4.20% (s.a.), the market value is 97.935084, using y = 0.0210.
In Excel, repeat the PRICE calculations using 0.0460 and 0.0420 for the fourth entries. The accrued interest is the same and is just 1.00 (= 90/180 * 2). Note that the general bond math formula directly uses the periodic variables – the payment per period, yield per period, and number of periods to maturity – whereas Excel (like other software programs) allows you to enter the annual variables and the periodicity that adjusts them in the formulas embedded in the programming.
Substitute these results for the MVs into equations 6.18 and 6.19 and use 0.0020 for the change in yield.
These are really good approximations; the more accurate numbers calculated above using the mathematically derived formulas are 15.1327 and 312.6859. In fact, the approximations become even better for a smaller change in yield. In practice, the differences between the approximations and the exact results are not likely to be material. For most purposes the information content for the modified duration and convexity of this bond is 15.14 and 312.7 – precision beyond that likely is just data.
Now let's look at how you probably observe bond duration and convexity statistics in practice.