Bonds V and convexity:
a continuation of Part IV 
suggested by Keith B.
Here we consider Bond Convexity ...
>Huh?
Patience!
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} + (C_{r}/R)
(1  (1+R/m)^{mN})}
where
N = number of years to maturity
C_{r} = annual Coupon rate
m = number of coupons per year
B = value of Bond at maturity,
R = Annual Yield
 TABLE 1
and, in particular, the dependence upon Yield (as in Figure 1.)
 Figure 1

Subsequently, (in Bonds2),
we talked about the rate of change of the Bond Value, V, when y, the Yield per Period, changed and
got the following:
Letting
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:
(1A)
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 Calculusspeak) 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
or
(3A)
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:
(4A)
Macauley Bond Duration: BD = (1+y)/y  {1+y + n(cy)} /
{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.

>zzzZZZ
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.
>Huh?
We used equations:
(1A) V = 1000*((1+y)^n + (c/y) *(1  (1+y)^n))
(4A) BD = (1+y)/y(1+y+n*(cy))/(c*((1+y)^n1)+y)
all of which is contained in this calculator ... as well as dV/dy, from (3):
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 firstderivative estimate of the
change in Bond Value, as per (3A), and the actual change
... and why the firstderivative 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 = d^{2}V/dy^{2}, so we need to differentiate
dV/dy = V/(1+y) BD = V/(1+y) [
(1+y)/y  {1+y + n(cy)} /
{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 '' = d^{2}V/dy^{2}
 Figure 4

>Aah, I see an error. Do you?
Where?
>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) dy^{2} / 2! + V '''(y) dy^{3} / 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) dy^{2} / 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
 Figure 5

