University of Connecticut
Math 3615: Financial Mathematics Problems
Fall 2008
Summary – Module 13 (McDonald Chapter 7) (Lombardi text, Unit 9)
- INTEREST RATE NOTATION -
- INTEREST RATE FORWARDS AND FUTURES -
- EURODOLLARS -
- not part of syllabus for Exam FM/2 -
Interest Rate Notation
Problems that involve interest rate swaps or forward rate agreements may utilize a
different notation for spot rates and forward rates than what we have learned from
studying compound interest.
P(0,n) represents the price for an n-year zero-coupon bond whose maturity value is 1.
This is equal to the present value of a payment of 1 in n years.
Thus, we have:
(0, ) (1 )
n
n
P n s
−
= + , where s
n
is the n-year spot rate.
The implied forward rate at time 0 for the future time interval (t
1
, t
2
) is represented by
0 1 2
r (t ,t )
. This leads to the relationship:
0
(0, 1) (0, )(1 ( 1, ))
P n P n r n n
− = + −
1
Note that if t
1
= 0, then
0 1 2
( , )
r t t
is simply the t
2
-year spot rate, which gives the relationship:
0
(0, ) [1 (0, )]
n
P n r n
−
= +
For a continuously compounded interest rate on the interval (0, n), the derivatives text
uses
r
cc
(0, n)
.
2
Thus the n-year accumulation at the n-year spot rate can be represented
by either
0
(1 (0, ))
n
r n
+
or
(0, )
cc
r n
e. And the price of an n-year zero-coupon bond can be
represented by:
(0, )
(0, )
cc
r n
P n e
−
=
Forward Rate Agreements
A forward rate agreement (FRA) is a contract relating to the interest rate for a defined
future period (for example, the 3-month period beginning 4 months from the date of the
agreement). The current forward rate for that period as of the date of the agreement
serves as a reference point for determining the payoff between the two parties to the
agreement. One party agrees to compensate the other for any positive difference between
the actual interest rate
3
and the rate specified in the agreement. Conversely, that party
will be compensated by the other for any negative difference between the actual rate and
the rate specified in the agreement.
1
Note that this is equivalent to the more familiar formula:
1
1 0
(1 ) (1 ) [ ( 1, )]
n n
n n
s s r n n
−
−
+ = + −
2
This is like an n-year force of interest, and in fact is numerically equal to n times the force of interest
(
n
δ
⋅
, where
δ
is expressed as an annual rate).
3
The actual rate is the spot rate for the specified (3-month) period, determined on the date that marks the
beginning of that period (4 months after the date of the forward rate agreement, in this example).
