Okay, and then I'm going to visit one more
topic, quickly from the first week.
So I talked about something called bond
duration, and
it turned out that, was just a linear
approximation.
So I was using the derivative, to make a
linear approximation of a yield curve, for
a bond.
So this is the, the yield curve for a
bond.
The price formula, so the price of a bond
is the,
it's essentially, it's just the discounted
value of the futures stream.
So, what I'm going to get paid, is the
face value of the bond, in n periods.
And then sort of my reward, for giving
somebody money now, that they're going to
pay me back in the future, is they're
going to make coupon payments every every
month.
And so, this actually should have had
parentheses around it.
And so what, what I'm after here
is just the discounted value of this
stream of coupon payments,
plus the discounted value of the face
value when the bond matures.
So I have coupon payment, number of
periods remaining.
So N is just how many periods I have to
wait before I get my money back.
F is the face value, so that's what I get.
That's sort of usually it's how much I
have sold
the bond for.
And then I just get, my money back some
time in the future.
And then lambda, you can think of this as
the interest rate.
But it's, for a bond it's called yield for
maturity.
[COUGH]
And so last time, I said we can make a, a
linear approximation.
So I, I introduced this concept called
duration, but then I, I showed
that it was basically just a, a simple
transformation of the first derivative.
And so I could use that to make a linear
approximation to my bond.
And so if I did that at the point 10% and
100.
And you can see, you know, maybe between
8% and 12%, the linear approximation is
pretty good.
But you know, if you deviate a long way
from there, then
the distance between the actual price and
the approximation gets pretty big.
And I said, we could improve this by
adding a quadratic term.
And so we'll see that, that quadratic term
is just, a degree 2 Taylor polynomial.
So, the quadratic term is called
convexity.
And we just have to compute it, by looking
at the present value of the future stream
of payments.
So if I say ak is the amount that I'm
going to receive after, in the kth period.
Then I want to find the present value of
that amount.
Just by discounting it, by a factor that's
equal to 1 plus the yield
to maturity.
And then I'll say that, the price of the
current
price of the bond is just equal to the
sum.
Of the present values of all of the future
payments I'll receive.
And that's equal to the sum of all of
these amounts discounted by 1 plus lambda.
Then the term convexity is defined to be,
1 over P times the second derivative.
Of this price formula with respect to
lambda.
And, so I said the price was equal to this
thing over here.
And, so I'm going to just put 1 over p
times the price, and instead of having
this in the denominator, I just think I
make
fewer mistakes when I can use the power
rule.
So I write that as, 1 minus lambda to the
minus k instead.
And then the only thing that has a lambda
in
it is this term here.
So I'm going to move my derivative
operator across
the 1 over P and then across the sum.
So we have this linearity property,
remember, so the
derivative of a sum is the sum of the
derivative.
And then in 1 step I will take 2
derivatives, so
I add this to the minus k, so I get, k
here, and then a k plus 1, and since I'm
doing
it twice, the minus signs are going to
cancel each other out.
And then I end up with minus k plus 2 in
the denominator.
So the denominator, the power's getting
smaller.
And you can write this as 1 over p, so I'm
going to take this plus
2 here, and write it as 1 over p times the
quantity 1 plus lambda squared.
And then I have inside this sum, k times k
plus 1 times
these a sub k divided, so
that the discounted feature payments again
essentially.
And now I need a point, where I can make
my approximation.
So I all ready saw that in my graph it was
the point.
10% and 100 so that gives me a lambda note
and a P0,
so, so some point where I've actually
evaluated the, the price of the bond.
Then I have Dm, as something called the
modified duration and C as the convexity,
that we calculated on the last slide.
And so if you look in the investment
science textbook, you'll find that just,
the change in price
of the bond is approximately equal to
minus the modified
duration, times the price times the change
in the interest
rate, plus 1 half times the price times
the convexity.
Times the change in the interest rate
squared.
And so let's just try and write this out
into
things that look more familiar for, with
what we've just done.
So I'll say delta P, that's just going to
be the difference between P and P0.
So I'll leave the P on the left, lefthand
side
right here, but I'll move the P0 to the
righthand side,
so I have P0 minus the modified duration
times the price times lambda minus lambda
0.
So that's delta lambda, I'm just going to
think of that as lambda minus lambda 0.
And then 1 half price times convexity
times lamba minus lambda 0 squared.
And if I go back and substitute in how
we got these terms in the first place, the
modified
duration time price and price times
convexity, both of those just came from
the derivatives of this price formula for
the bond with respect to lambda.
So the first one is just dP d-lambda,
evaluated at P0.
And then price times convexity, that's the
same thing as the second derivative of
P with respect to lambda evaluated at P0.
So really all we've done,
when we use modified duration and
convexity
to approximate the, the price of a bond.
we've used order 2 Taylor polynomial to,
to approximate the same thing.
And so just to finish up this is the, the
picture of what that looks like.
So the, the blue curve is the actual price
curve, yield curve of the bond.
[COUGH]
And the dashed black line is the linear
approximation, and then
the red line is the order 2 approximation.
And so again in
maybe, maybe a little bit wider now maybe
6 or 7 to
about 13 or 14, we have a good
approximation.
But then as we get into more extreme
values, much further away from the point
we used to build the approximation.
The the true yield curve is deviating from
our approximation, and so on
this side it's higher than the
approximations, and on this side it's
lower.