ACTSC 445: Asset-Liability Management
Department of Statistics and Actuarial Science, University of Waterloo
Unit 4 – The Term Structure of Interest Rates
Diﬀerent factors aﬀect the value of interest rates associated with ﬁxed income securities like bonds.
In what follows, we will focus on the term structure of interest rates, that is, how rates change with
the maturity of these securities. Note that each security leads to a diﬀerent set of interest rates. To
study the term structure of interest rates, it is useful to focus on one type of security. So which one
should we choose? A natural choice is to focus on the Treasury market (i.e., securities issued by the
government (T-bills, notes and bonds)). There are two reasons for this choice: (1) treasury securities
are considered default-free, so diﬀerences in assessment of the creditworthiness of the issuer cannot
aﬀect the yield estimates for these securities; (2) as the most active bond market, the Treasury market
oﬀers no illiquidity problems, and prices can readily be observed.
Types of interest rates
In this section, we discuss four diﬀerent ways of representing interest rates:
2. spot rate
3. forward rate
4. short rate.
• This measure is widely used for bonds (see Unit 2).
• Given by the constant interest rate that equates the discounted value of the future cash ﬂows
under the bond and its current market price.
• Also called internal rate of return.
• More precisely, using the same notation as in the notes for Unit 2, the ytm y (measured on the
same type of period as the coupon-paying period) is the value such that
P = cF 1 − (1 + y) + F(1 + y)−n ,
where we assume for simplicity that cF is the actual value of the coupon.
• The yield curve refers to the graphical depiction of the yield level as a function of time.
1 • See http://finance.yahoo.com/bonds for current and historical data on US Treasury Securities.
• Important to note: ytm is cash-ﬂow dependent... Another problem is that it does not reveal the
year by year information.
• When using ytm for the term structure of interest rates, presumably the same coupon rate is
used for all diﬀerent maturities.
First, we need to introduce some notation. Let P(t,t + k) be the price at time t of a zero-coupon (or
pure discount) bond with k periods until maturity (and face value F = 1).
Let t = 0. The spot rate for k periods to maturity—denoted s —iskthe ytm for a zero-coupon bond
with k periods to maturity, i.e., k is such that
P(0,k) = (1 + s k −k
(Note: just like in the above discussion of ytm, for simplicity we assume here that the spot rates are
measured on the same type of period as the coupon-paying period. Later on, we’ll work with annual
rates compounded at the same frequency as the coupons. E.g, right now if coupons are paid twice a
year, k = 2, and P(0,2) = 0.92, then we compute s = (120.92) − 1 = 0.0426 as a semi-annual
rate; later on, we’ll instead work with the annual rate, which here would be 2 × 0.0426 = 0.0851.)
Figure 1 shows the spot-rate curve from 0 to 30 years for diﬀerent maturities of Canadian Treasury
securities. (The data has been obtained from the Bank of Canada website www.bankofcanada.ca).
Figure 1: Canadian Treasury Spot Rates from 0 to 30 years at selected times
Example: an initial investment of 100$ accumulates to 106$ after 1 year, and to 113.42$ after two
years. What are the corresponding one-year and two-year spot rates?
2 −1 −2
Solution: we have that1s is such that 100 = 106(1 + s⇒ s1= 0.06, and s2is such that 100 = 113.42(1 2 s )⇒
Note: with a term structure of interest rates based on spot rates, we can ﬁnd the price P of a bond by
P = cF(1 + s ) −k + F(1 + s ) −n , (1)
and then ﬁnd its ytm.
Question: using the same data as in the previous example, assume you have the following alternative
strategy. Invest 100$ at time 0; at time 1, use the proceeds from this investment to reinvest at a rate
f, where f is determined at time 0. What should be f so that there is no arbitrage opportunity?
Solution: the two strategies should produce the same outcome at time 2, otherwise there would be an arbitrage opportunity.
Hence we must have that f satisﬁes
106(1 + f) = 113.42 ⇒ f = 113.42/106 − 1 = 7%.
Forward rates are the interest rates that would be used for contracts made today covering transactions
in future periods.
At t = 0, we denote by f j,kthe forward rate covering period j to k. For instance, if the unit of time is
years, then f corresponds to the forward rate between year 2 and year 4. Can think of the forward
rate fj,kas the rate agreed upon at time 0 for borrowing money at time j that will be repaid at time
The relationship with spot rates is as follows:
(1 + s )k
(1 + f )k−j = k
j,k (1 + sj)j
or in terms of prices for zero-coupon bonds, we have
(1 + fj,kk−j = .
The one-period forward rates f k,k+1 for k = 0,1,..., are denoted as f (wekdrop the k + 1 in the
Using this notation, we can give yet another relationship between forward rates and spot rates:
1 + sk= ((1 + f )01 + f )1..(1 + f k−1)) .
In other words, the spot rates are geometric averages of the forward rates. For example, previously
we had f 0 = s 1 = 0.06 and f 1 = 0.07 (denoted f above), corresponding to a two-year spot rate
s2= (1.06 × 1.07) = 1.06499. Figure 2 shows how the spot rates and the one-period forward rates
compare in two diﬀerent scenarios.
As shown by these relationships, spot rates uniquely determine forward rates and vice-versa.
Figure 2: One-period forward rates versus spot rates
Short rates refer to the one-period interest rates that may turn out to be available in the market during
a future period. More precisely, the short rate r is the rate that will apply between time k and k + 1.
The diﬀerence with the forward rate f is that khe latter is the rate agreed upon at time 0 that applies
between time k and k + 1, whereas r is the ackual rate (unknown at time 0) that will apply during
this period. Hence short rates are treated as random variables.
Continuous models are often used to represent s