Agenda
Recap:
- A useful duration formula (CT1, Unit 13,
Sec. 5.3)
- Redington immunisation conditions (CT1,
Unit 13, Sec. 5.5)
A warning about immunisation
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October 28, 2010
MATH 2510: Fin. Math. 2
Comments on Course Work #2:
- Exercise 4.4 and other useful Excel things
- Accrued interest, clean vs. dirty prices
- Treating time-shifted payments
- General Q&A
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October 28, 2010
MATH 2510: Fin. Math. 2
New Hand-Outs
These slides
Exercises for Workshop #5
Updated Course Plan (blue)
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October 28, 2010
MATH 2510: Fin. Math. 2
A Bond Duration Formula
The duration of a bond w/ coupon payments D and
notional R (R=0: annuity) is
D( Ia) n|  Rnvn
Dan|  Rvn
where
v  (1  i ) 1
1 vn
an| 
i
(1  i )an|  nvn
( IA) n| 
i
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October 28, 2010
MATH 2510: Fin. Math. 2
Redington Immunisation
3 conditions on assets (A) and liabilities (L)
PVA (i0 )  PVL (i0 )
 A (i0 )   L (i0 )
c A (i0 )  cL (i0 )
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October 28, 2010
MATH 2510: Fin. Math. 2
Operational Immunization
Liabilities are fixed. We buy x units of bonds #1, y units of
bond #2, where x and y solve the equations
xPV (bond#1)  yPV (bond#2)  PVL (i0 )
PV (bond#1) (bond#1)
PV (bond#2) (bond#2)
x
y
  L (i0 )
PVL (i0 )
PVL (i0 )
The convexity condition is treated by conderisering how
spread out assets and liabilities are around the duration;
the more spread out, the higher the convexity.
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October 28, 2010
MATH 2510: Fin. Math. 2
Immunisation isn’t the be-all-and-endall of interest rate risk management
Note that an immunised portfolio is looks very
much like an arbitrage.
That tells us that considering only parallel shifts
to flat yield curves isn’t the perfect way to
model interest rate uncertainty.
And models with genuinely random behaviour
is next week’s topic.
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October 28, 2010
MATH 2510: Fin. Math. 2
Exercise 4.4: Useful Excel Stuff
Formulas: Start w/ ”=”. (And LOG is base 10.)
Copying: Relative (”default”) vs. absolute ($)
references.
Solver
Yearfrac
[playing around w/ file]
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October 28, 2010
MATH 2510: Fin. Math. 2
Accrued Interest
When you buy a bond you must compensate the seller
for interest since last payment date, i.e. you have to
pay (now) accured interest of
coupon * #years since last payment.
(You can guess what happens w/ other payment
frequencies. And here we ignore the fine points of
day-counting.)
You receive the next coupon in full.
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October 28, 2010
MATH 2510: Fin. Math. 2
Accrued interest has to be added to the quoted
(so-called ”clean”) price to get the left-hand
side (the so-called ”dirty price”) for yield (and
other) calculations.
Why do you think the convention is to quote
clean prices, i.e. prices without accrued
interest?
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October 28, 2010
MATH 2510: Fin. Math. 2
Duration for a Standard Bond with
Shifted Payment Dates
See and solve exercise 5.4 for the Wed. Nov. 3
Workshop.
[scribbling on whiteboard]
Can save you some copying and pasting in
Questions 1 and in Course Work #2.
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October 28, 2010
MATH 2510: Fin. Math. 2
Other Questions Re. Course Work #2?
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October 28, 2010
MATH 2510: Fin. Math. 2