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Okay, and then I'm going to visit one more
topic, quickly from the first week.

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So I talked about something called bond
duration, and

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it turned out that, was just a linear
approximation.

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So I was using the derivative, to make a

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linear approximation of a yield curve, for
a bond.

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So this is the, the yield curve for a
bond.

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The price formula, so the price of a bond
is the,

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it's essentially, it's just the discounted
value of the futures stream.

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So, what I'm going to get paid, is the
face value of the bond, in n periods.

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And then sort of my reward, for giving
somebody money now, that they're going to

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pay me back in the future, is they're

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going to make coupon payments every every
month.

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And so, this actually should have had
parentheses around it.

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And so what, what I'm after here

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is just the discounted value of this
stream of coupon payments,

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plus the discounted value of the face
value when the bond matures.

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So I have coupon payment, number of
periods remaining.

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So N is just how many periods I have to
wait before I get my money back.

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F is the face value, so that's what I get.

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That's sort of usually it's how much I
have sold

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the bond for.

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And then I just get, my money back some
time in the future.

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And then lambda, you can think of this as
the interest rate.

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But it's, for a bond it's called yield for
maturity.

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[COUGH]

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And so last time, I said we can make a, a
linear approximation.

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So I, I introduced this concept called
duration, but then I, I showed

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that it was basically just a, a simple
transformation of the first derivative.

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And so I could use that to make a linear
approximation to my bond.

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And so if I did that at the point 10% and
100.

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And you can see, you know, maybe between

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8% and 12%, the linear approximation is
pretty good.

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But you know, if you deviate a long way
from there, then

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the distance between the actual price and
the approximation gets pretty big.

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And I said, we could improve this by
adding a quadratic term.

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And so we'll see that, that quadratic term
is just, a degree 2 Taylor polynomial.

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So, the quadratic term is called
convexity.

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And we just have to compute it, by looking

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at the present value of the future stream
of payments.

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So if I say ak is the amount that I'm
going to receive after, in the kth period.

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Then I want to find the present value of
that amount.

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Just by discounting it, by a factor that's
equal to 1 plus the yield

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to maturity.

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And then I'll say that, the price of the
current

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price of the bond is just equal to the
sum.

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Of the present values of all of the future
payments I'll receive.

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And that's equal to the sum of all of
these amounts discounted by 1 plus lambda.

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Then the term convexity is defined to be,
1 over P times the second derivative.

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Of this price formula with respect to
lambda.

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And, so I said the price was equal to this
thing over here.

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And, so I'm going to just put 1 over p
times the price, and instead of having

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this in the denominator, I just think I
make

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fewer mistakes when I can use the power
rule.

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So I write that as, 1 minus lambda to the
minus k instead.

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And then the only thing that has a lambda
in

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it is this term here.

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So I'm going to move my derivative
operator across

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the 1 over P and then across the sum.

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So we have this linearity property,
remember, so the

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derivative of a sum is the sum of the
derivative.

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And then in 1 step I will take 2
derivatives, so

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I add this to the minus k, so I get, k

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here, and then a k plus 1, and since I'm
doing

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it twice, the minus signs are going to
cancel each other out.

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And then I end up with minus k plus 2 in
the denominator.

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So the denominator, the power's getting
smaller.

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And you can write this as 1 over p, so I'm
going to take this plus

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2 here, and write it as 1 over p times the
quantity 1 plus lambda squared.

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And then I have inside this sum, k times k
plus 1 times

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these a sub k divided, so

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that the discounted feature payments again
essentially.

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And now I need a point, where I can make
my approximation.

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So I all ready saw that in my graph it was
the point.

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10% and 100 so that gives me a lambda note
and a P0,

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so, so some point where I've actually
evaluated the, the price of the bond.

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Then I have Dm, as something called the
modified duration and C as the convexity,

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that we calculated on the last slide.
And so if you look in the investment

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science textbook, you'll find that just,
the change in price

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of the bond is approximately equal to
minus the modified

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duration, times the price times the change
in the interest

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rate, plus 1 half times the price times
the convexity.

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Times the change in the interest rate
squared.

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And so let's just try and write this out
into

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things that look more familiar for, with
what we've just done.

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So I'll say delta P, that's just going to
be the difference between P and P0.

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So I'll leave the P on the left, lefthand
side

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right here, but I'll move the P0 to the
righthand side,

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so I have P0 minus the modified duration

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times the price times lambda minus lambda
0.

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So that's delta lambda, I'm just going to
think of that as lambda minus lambda 0.

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And then 1 half price times convexity
times lamba minus lambda 0 squared.

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And if I go back and substitute in how

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we got these terms in the first place, the
modified

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duration time price and price times
convexity, both of those just came from

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the derivatives of this price formula for
the bond with respect to lambda.

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So the first one is just dP d-lambda,
evaluated at P0.

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And then price times convexity, that's the
same thing as the second derivative of

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P with respect to lambda evaluated at P0.
So really all we've done,

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when we use modified duration and
convexity

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to approximate the, the price of a bond.

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we've used order 2 Taylor polynomial to,
to approximate the same thing.

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And so just to finish up this is the, the
picture of what that looks like.

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So the, the blue curve is the actual price
curve, yield curve of the bond.

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[COUGH]

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And the dashed black line is the linear
approximation, and then

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the red line is the order 2 approximation.
And so again in

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maybe, maybe a little bit wider now maybe
6 or 7 to

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about 13 or 14, we have a good
approximation.

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But then as we get into more extreme
values, much further away from the point

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we used to build the approximation.

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The the true yield curve is deviating from
our approximation, and so on

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this side it's higher than the

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approximations, and on this side it's
lower.