Agenda
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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.
Messing w/ youd head: Something that isn’t
the yield curve.
Estimating the yield curve by bootstrapping:
Hull Chapter 4, Sec. 5
October 19, 2010
MATH 2510: Fin. Math. 2
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2
More practical complications in yield curve
estimation.
CT1 Unit 13, Sec. 5: Risk or sensitivity
measures; duration.
October 19, 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.
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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.)
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MATH 2510: Fin. Math. 2
April 2009 CT1-Exam Q9
[different slides, old hand-out]
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Teaser: A Graph I Got From Bloomberg
Mid-Oct. 2010 yields to maturity of UK government bonds (y-axis)
for different maturities (x-axis).
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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?
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MATH 2510: Fin. Math. 2
Estimating the Yield Curve
Consider the small principal/notional-100 bullet bond
market from Hulls’s Table 4.3:
Time to maturity
Coupon
Price
0.25
0%
97.5
0.50
0%
94.9
1.00
0%
90.0
1.50
8%
96.0
2.00
12%
101.6
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MATH 2510: Fin. Math. 2
Let’s
 assume that coupons are paid semi-anually.
 work with continuously compounded zero
coupon (spot) rates
We can now determine – ”estimate” - the yield
curve by working from short to lond
maturities. This is called bootstrapping.
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The Bootstrap Method
The 3 month rate solves the equation
100e
R0.25 0.25
 97.5
This means 10.127% with continuous
compounding.
Similarly the 6 month and 1 year rates are 10.469%
and 10.536% with continuous compounding.
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MATH 2510: Fin. Math. 2
To calculate the 1.5 year rate we solve
4e
0.104690.5
 4e
0.105361.0
 104e
 R1.5 1.5
 96
to get R1.5 = 0.10681 or 10.681%. Notice how
the previously calculated rates are used.
Similarly the 2 year rate is 10.808%
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The Resulting Yield Curve
Determining Zero Coupon Rates:
Hull's Bootstrapping Example in Chapter 4, Sec. 5
0.109
0.108
Interest rate
0.107
0.106
0.105
Zero coupon rate
0.104
0.103
0.102
0.101
0.1
0
0.5
1
1.5
2
2.5
Maturity
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MATH 2510: Fin. Math. 2
Yield Curve Estimation in Practice
Yield curve estimation (also known as ”yield
curve stripping”) is the back-bone/the ”meat
and potatoes” of any bank’s fixed income
department.
Complication: More cash-flow dates than
bonds. Solution: Use some interpolation
scheme. (Piecewise constant, linear, NelsonSiegel.)
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Complication: Which products to use?
Particularly in focus ”post Credit Crunch”.
This will be the topic of Course Work #2.
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Duration
Consider this situation: You …
 … are a pension fund.
 … have to pay out £30m in 20 years
 … have collected £12m in premiums and
invested the money in 5-year (zero coupon)
bonds
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MATH 2510: Fin. Math. 2
Assume the interest rate is 5% per year.
Your assets (the zero coupon bonds) are worth £12m;
you own 15.32m of them (15.32 * (1.05)-5=12)
Your liabilities (the future pension payments) have a
present value of £30m* (1.05)-20=£11.31m
So all is well; you are nice and solvent. (You might
report a solvency percentage of (12 – 11.31)/11.31
~6%.)
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Now the interest rate changes to 4%.
You assets are now worth £15.32m * (1.04)-5=£12.59m,
whereas the present value of your liabilities is £30m*
(1.04)-20=£13.69m. Thus: You are no longer solvent.
What just happened here? And how do we avoid such
nasty surprices?
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Calculate not only the present values of our
positions, but also their sensitivity to interest
rate changes.
This is what duration (CT1, Unit 13, Sec. 5) is
about.
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Set-up:
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Cash-flows Ct at tk
Yield curve flat at i (or continuously
compounded/on force form: )
Present value of cash-flows:
k
A  k Ct k (1  i) tk
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Effective duration
The effective duration (or: volatility, or: modified
duration) is defined as
t C (1  i)
1 

k k tk
v : 
A
A i
A
 ( t k 1)
So: Duration is a sensitivity to shifts in the yield
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MATH 2510: Fin. Math. 2
Macauley Duration
The Macauley duration (or: discounted mean term ) is
defined by
 :
k tk Ctk (1  i)
A
tk
 (1  i)v
So duration can also be intepreted as a weighted
average of payment dates.
(Also: Sensitivity to shifts in the force of interest.)
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