Hand-In, Hand-Out
Course Works #1, please!
In return you get
–
–
–
1
Workshop 2 solutions w/ comments
Hull’s Chapter 4 on interest rates
Slides and an updated lecture plan
October 14, 2010
MATH 2510: Fin. Math. 2
Agenda




2
Recap: Zero coupon bonds, spot and forward
rates, the yield curve. CT1, Unit 13, Sec. 14.1.
In fact also recap: Yield to maturity (or: to
redemption). CT1, Unit 13, Sec. 4.2.
Par yield. CT1 Unit 13, Sec. 4.3. And the
solution of Q9 from the April 2009 CT1-exam.
A teaser: Something that isn’t the yield curve.
October 14, 2010
MATH 2510: Fin. Math. 2
Yield to Maturity (or: to Redemption)
Consider a bond with cash-flows ct at times t = 1, 2, …,
T, and price P. Its yield to maturity, i, is the solution to
the equation:
T
ct
P
.
t
t 1 (1  i )
This is a non-linear equation; must be solved
numerically.
3
October 14, 2010
MATH 2510: Fin. Math. 2
The yield to maturity can be interpreted as
measure of the average return of the bond
over its life-time.
But be careful about taking the intepretation too
far quantitatively. (”You don’t buy groceries
based solely on price per kilo”.) Do value
calcuations and comparions w/ the zero
coupon yield curve.
4
October 14, 2010
MATH 2510: Fin. Math. 2
The yield to maturity is a weigthed average of
forward rates, where more weight is given
bigger ct’s. (Imagine making forward rate
agreements to reinvest ct between t and T.)
But again: Not a quantitatively simple average.
5
October 14, 2010
MATH 2510: Fin. Math. 2
An Often Seen Case
For a (bullet) bond w/ n coupons c and full redemption
of notional (say 1) we have
P  c  ai  v  n
n
where as usual v=1/(1+i) and
6
October 14, 2010
ai  (1  v n ) / i
n
MATH 2510: Fin. Math. 2
Cont. Comp. Yield to Maturity
The continuously compounded yield to maturity, Y,
for a bond paying at dates tj is the solution to
P
n
 ct j e
t j Y
j 1
If payment dates are not completely regular, I’d
use continuous compunding.
7
October 14, 2010
MATH 2510: Fin. Math. 2
Par Yield
The par yield , ycn, is the bullet bond coupon rate
that makes a n-term bullet bond trade at par, i.e.
have a price of £1 per £1 notional. Or in
symbols:
1  Pn  ( yc n ) j 1 Pj
n
 ( yc n ) 
1  Pn

n
j 1
.
Pj
With a non-flat yield curve, par yield and yield to
maturity is not the same. (Artimetic vs. geometric
effect.)
8
October 14, 2010
MATH 2510: Fin. Math. 2
April 2009 CT1-Exam Q9
[see hand-out]
9
October 14, 2010
MATH 2510: Fin. Math. 2
Teaser: A Graph I Got From Bloomberg
Mid-Oct. 2010 yields to maturity of UK government bonds (y-axis)
for different maturities (x-axis).
10
October 14, 2010
MATH 2510: Fin. Math. 2
Questions
(0. How do you like the scaling?)
1. Why is that not the (zero coupon spot) yield
curve?
2. What do we do about that? How do we
estimate the yield curve?
11
October 14, 2010
MATH 2510: Fin. Math. 2