Interest Rates: The Big Picture
So far, we have assumed a single, constant interest
rate.
We may quote it discretely (i in CT1) or
continuously (”as a force”  in CT1).
That is not realistic.
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October 12, 2010
MATH 2510: Financial
Mathematics 2
Danish Interest Rates
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October 12, 2010
MATH 2510: Financial
Mathematics 2
Interest rates are ”non-constant in two
dimensions”: Term (or time to maturity) and
calendar time.
In CT1, Units 13 and 14 we relax the
assumption
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October 12, 2010
MATH 2510: Financial
Mathematics 2
The Term Structure of Interest Rates
CT1, Unit 13 Sec. 1-3: Definitions and concepts
 (n-year) zero coupon bonds and (discrete)
spot rates
 Discrete forward rates; intepretation and
realtion to spot rates
 Continuous versions of the rates
 Instantaneous forward rates
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October 12, 2010
MATH 2510: Financial
Mathematics 2
Zero coupon bonds
An n-term zero coupon bond (ZCB) pays £1 after n
years – and nothing else.
Its price is denoted by Pn.
ZCBs differ by term.
If we know ZCB prices, we can easily price anything
else w/ deterministic payments:
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October 12, 2010
MATH 2510: Financial
Mathematics 2
Spot Rates/Zero Coupon Rates
The discrete spot rate, yn, is the yield on an n-term
zero coupon bond:
1
Pn 
(1  yn ) n

yn  Pn

1
n
1
The function that maps term n to spot rate yn is
called the (zero coupon) yield curve or the term
structure of interest rates
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October 12, 2010
MATH 2510: Financial
Mathematics 2
Forward Rates
The discrete forward rate ft,r is the interest rate
that can be agreed upon today for a future
loan that runs between time t and time t+r.
At time t, we recieve £1, at time t+r we pay
back £1*(1+ ft,r)r. (And no money changes
hands today.)
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October 12, 2010
MATH 2510: Financial
Mathematics 2
An (absence of) arbitrage argument shows that forward
rates and zero coupon bond prices are related
(1  f t ,r ) r 
Pt
Pt  r
We write ft for the one-period forward rates ft,1 ,by
repeatedly using the formula above we see that
(1  yt )t  (1  f0 )  (1  f1 )   (1  ft 1 )
Spot rates are (geometric’ish) averages of forward rates.
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October 12, 2010
MATH 2510: Financial
Mathematics 2
The Yield Curve
CT1, Unit 13, Sec. 4.1
Real-world yield curves come in many shapes
 Increasing w/ term (”normal”; Fig. 2, Unit 13)
 Decreasing w/ term (”inverted”; Fig. 1)
 Humped (Fig. 3)
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October 12, 2010
MATH 2510: Financial
Mathematics 2
Five Danish Yield Curves
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October 12, 2010
MATH 2510: Financial
Mathematics 2
Why do Yield Curves Look the Way the
Do?
Popular explanations (not ”models”)

Expectations Theory. Rates will fall. I’d better
lock in a good rate on my long term
investments. Higher demand for long term
bonds -> higher prices -> lower yields.
Good for inverted yield curves.
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October 12, 2010
MATH 2510: Financial
Mathematics 2
Liquidity Preference: Higher long term rates
is a compensation for risk. (Or: Lower short
term rates is a reward for short term funding.)
Does actually have ”theoretical merit”.

Market Segmetation.
Supply and demand are important. But also
”Humped curve? I give up!”

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October 12, 2010
MATH 2510: Financial
Mathematics 2