Yield Curve Arbitrage
The most important prices in economics are the interest rates determining the
relative prices of money now vs money later (or for real interest rates, goods now vs
goods later). The driving force of economic progress is investment and production.
Any investment involves giving up money now, to buy the inputs, in the hope that
the inputs can be used to produce outputs in the future that can be sold for money.
No manager would undertake such an investment if he could achieve a better return
return by putting the initial outlay of money in the bank, or if he could use it to
buy riskless bonds with a higher yield. On the other hand, if he could make a higher
return by his real investment than the rate of interest he would be charged to borrow
the initial outlay, he would be foolish not to borrow and invest.
Thus the interest rates are the
f
rst things any businessman needs to know. How
does he
f
nd them? And why is there more than one?
We shall show how to proceed from the Treasury Yield Curve, which is generally
available in newspapers, to derive the "implied" money present value prices
π
t
that
every
f
rm must know. Along the way we will derive the implied zero yields and the
implied forward rates.
1P
r
e
s
e
n
t
V
a
l
u
e
We denote the price today (at time 0) of $1 at time t by
π
t
=
price today for $1 at time
t
π
t
=
1
1+
R
t
1
π
t
=1+
R
t
=
price at time
t
of $1 today
We say that the present value of a dollar at time
t
is
π
t
.
Sometimes
π
t
is called
the
t
period market discount factor.
1.1 Interest Rate
Since people are impatient (and since money is storable) it is always the case that
π
t
≤
1
Anybody who gives up $1 today can expect to get more than $1 tomorrow,
1
π
t
≥
1
.
Many people
f
nd it easier to think of how much extra is earned after the dollar is
repaid from having waited. Thus the interest rate is de
f
ned by the surplus gain
R
t
=
1
π
t
−
1
.
1
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View Full Document1.2 Annualized Interest
Impatient people will want a bigger surplus the longer they have to wait to get their
money back. To put bonds of all maturities on the same footing it is customary to
talk about annualized interest rates. The interest rate
r
t
corresponding to a tperiod
surplus of
R
t
satis
f
es
(1 +
r
t
)
t
=1+
R
t
If
t
corresponds to 6 months, then the annualized interest rate is
(1 +
r
t
)
.
5
=1+
R
t
1.3 Yield
An annualized interest rate is called a yield. We saw that there are di
f
erent kinds of
interest rates, and therefore di
f
erent kinds of yields.
1.4 Zero Coupon Prices and the Zero Yield Curve
A bond that promised $1 at time
t
and nothing else is called a zero coupon bond,
since it pays no coupons before maturity, and
π
t
=
1
1+
R
t
is called the
t
ma
tu
r
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 '09
 GEANAKOPLOS,JOHN
 Interest Rates

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