Bonds V  and convexity: a continuation of Part IV
suggested by Keith B.

Here we consider Bond Convexity ...

When we first talked about Bonds, we considered how the Bond Value varied with the Years to Maturity, Yield, etc., as in TABLE 1:
V = B { 1/(1+R)N + (Cr/R) (1 - (1+R/m)-mN)}
    N = number of years to maturity
    Cr = annual Coupon rate
    m = number of coupons per year
    B = value of Bond at maturity,
    R = Annual Yield

and, in particular, the dependence upon Yield (as in Figure 1.)

Figure 1
Subsequently, (in Bonds-2), we talked about the rate of change of the Bond Value, V, when y, the Yield per Period, changed and got the following:
B = maturity value of the bond (in dollars)
y = R/m = yield per period (period may be years or months)
c = the coupon rate per period (which may be years or months)
n = mN = the number of periods to maturity (n = years x coupons_per_year)
then the Bond Value is just the Present Value of all the coupons PLUS the Present Value of the Value at Maturity, namely:

(1)     V = B { (1+y)-n + c Σ(1+y)-k }

where Σ means the sum of terms, from k = 1 to k = n.

>And what about TABLE 1?
There's a formula for the sum of the series in (1), namely
Σ (1+y)-k = 1/(1+y) + 1/(1+y)2 + ... + 1/(1+y)n = {1 - (1+y)-n} / y
and Table 1 uses that formula. Indeed, we can rewrite (1) like so:
V = B { (1+y)-n + (c/y) [1 - (1+y)-n] }

where y is the yield, n is the number of years to maturity, c is the coupon rate and B is the Maturity Value.

Anyway, we'll use (1) to determine the Rate of Change of V with respect to y (the first derivative, in Calculus-speak) and determine the relationship between this Rate of Change and the Macauley Bond Duration, BD, like so:

(2)     (1/V) dV/dy = -1/(1+y) { n(1+y)-n + c Σk (1+y)-k } / { (1+y)-n + c Σ(1+y)-k } and

(3)     dV/dy = -V/(1+y) BD
dV = -[V/(1+y)] BD dy
(which gives a change of V, that's dV, in terms of a change in Yield, that's dy.

and where

(4)     BD = Bond Duration = { n(1+y)-n + c Σk (1+y)-k } / { (1+y)-n + c Σ(1+y)-k }

As you may imagine, there's a magic formula for the sum of the series, so we'd get:
Macauley Bond Duration: BD = (1+y)/y - {1+y + n(c-y)} / {c[(1+y)n - 1] + y}

where y is the yield and n is the number of years to maturity and c is the coupon rate.

This is just a review of what we've already talked about in earlier tutorials. I haven't got the Convexity yet!
First we'll stick some numbers into the above equations, with an example ... like so:

Current Yield: y = 4.0%
Years to Maturity: n = 18 years
annual Coupon Rate: c = 6%
then we get from (1A) the Value of a $1K Bond, namely $1253.19 and from (4A) the Bond Duration: BD = 12.21 years.

We used equations:
(1A)     V = 1000*((1+y)^-n + (c/y) *(1 - (1+y)^-n))
(4A)     BD = (1+y)/y-(1+y+n*(c-y))/(c*((1+y)^n-1)+y)

all of which is contained in this calculator ... as well as dV/dy, from (3):

Current Yield = %
Years to Maturity =
Coupon Rate = % (annually)
Macauley Bond Duration = BD = years
Okay, now we'll use equation (3) which tells us how much V will change when we change the current Yield from 4% to 5%.
The current situation is shown is Figure 2.

>The blue dot?
Yes, the blue dot. However, if the yield changes to 5% ...

>The red dot?
Yes, of course ... and the Bond Value, V, changes from $1253.19 to $1116.90,
a change of -$136.29.

>$1116.90? Where'd that come ...?
Just use the above calculator, with Yield = 5%.
However, according to (3A), we'd expect a change in Value of:
(3A)     dV = -[V/(1+y)] BD dy = -[(1253.19)/(1.04)] (12.21)(0.01),
and that's a change of -$147.13 (as noted in the above calculator)
and that'd bring the Bond Value down to $969.77, instead of $1116.90.

Figure 2
>Down with Macauley! Down with Bond Duration! Down with ...!
Behave! That brings us to ...

>Don't tell me! Convexity!
You got it. Stare intently at Figure 3 and tell me why the difference between the first-derivative estimate of the change in Bond Value, as per (3A), and the actual change
... and why the first-derivative estimate gives a value (namely $969.77) which is too low ??

Figure 3
>The curve is dropping fast at Yield = 4% and, continuing at this rate gives $969.77, but ... uh ...
Yes, at 4% it's dropping at the rate of $147.13 for each percentage increase in Yield.
But that rate decreases above 4% and, at 5%, is only $126.05 (which we got with that calculator, above).
That curvature is Bond Convexity, which describes the convex shape of the curve.

>That makes an average rate of (147.13 + 126.05)/2 which is ... uh ...
An average of $136.59 which is much closer to the actual drop of $136.29.

>So why not use the actual Value at 5% rather than estimating 5% by extrapolating from 4%?
Beats me. However, if one wanted to see how convex the curve is at, say 4%, then that calculator says
dV/dy = 14713 (so a 1% increase would mean dy = 0.01 so dV = 14713*0.01 = $147.13).

Now try the calculator with Years = 18, 12, 8, 6, 1 and watch the convexity decrease.
>Or I can look at Figure 1 and see that the curve gets straighter as Years to Maturity is reduced.
Yes, that's true. Do you want a formula for convexity?
>Sure. Why not?
Okay, it's the rate at which dV/dy changes. Indeed, it's d{dV/dy}/dy = d2V/dy2, so we need to differentiate

dV/dy = -V/(1+y) BD = -V/(1+y) [ (1+y)/y - {1+y + n(c-y)} / {c[(1+y)n - 1] + y} ]

>I changed my mind. Let's forget the differentiation, okay?
I'm with you ... but here's a picture for our example above, where we plot Bond Value vs Yield (for several Years to Maturity)
... and indicate, at Yield = 4%, the values of V ' = dV/dy and V '' = d2V/dy2

Figure 4

>Aah, I see an error. Do you?
>For Years = 18, the value of V ' is shown as 14,717 so that ...
So for a 1% change (that's dy = 0.01), we'd get dV = $147.17 and ...
>Yet you got dV = $147.13, earlier.
Mea culpa. Earlier, I went with the calculator values (to two decimal places), as in BD = 12.21, but in Figure 4, I used Excel and retained more decimal places, using BD = 12.21355 and that accounts for the difference. Besides it's only four cents. Your 2 cents and my 2 cents.
>Don't blame me for your 4 cent error!

Okay, here's something interesting. We can expand V(y+dy) in a series like so:
V(y+dy) = V(y) + V '(y) dy + V ''(y) dy2 / 2! + V '''(y) dy3 / 3! + ...
and, if we retain only the terms up to the second derivative, we'd get the approximation:
V(y+dy) - V(y) = V '(y) dy + V ''(y) dy2 / 2
and, using y at 4% and y+dy at 5% (as in our example, above), we get (picking V ' = 14,717 and V '' = 230,638 from Figure 4):
dV = V(0.05) - V(0.04) = V '(0.04) (0.01) + V ''(0.04) (0.01)2 / 2 = (-14717)(0.01) + (230638)(0.01)2/2 = -147.17 + 11.532 = -135.64
which agrees pretty well with the actual change in Bond Value (namely -$136.29)
... and is a mite better than just using the first derivative term (namely -$147.17).

>So, what's the formula for Bond Convexity?
If you MUST know, you'll find it here

... or you can use this calculator  

Current Yield = %
Years to Maturity =
Coupon Rate = % (annually)
Convexity =

Figure 5