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ACTSC445 Lecture Notes - United States Treasury Security, Spot Contract, Yield Curve

Actuarial Science
Course Code
Jiahua Chen

of 7
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:
1. yield-to-maturity
2. spot rate
3. forward rate
4. short rate.
Yield-to-maturity (ytm)
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)n
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.
See 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.
Spot rates
First, we need to introduce some notation. Let P(t, t +k) be the price at time tof a zero-coupon (or
pure discount) bond with kperiods until maturity (and face value F= 1).
Let t= 0. The spot rate for kperiods to maturity—denoted sk—is the ytm for a zero-coupon bond
with kperiods to maturity, i.e., skis such that
P(0, k) = (1 + sk)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 s2= (1/0.92)1/21=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
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?
Solution: we have that s1is such that 100 = 106(1 + s1)1s1= 0.06, and s2is such that 100 = 113.42(1 + s2)2
s2= 0.06499.
Note: with a term structure of interest rates based on spot rates, we can find the price Pof a bond by
cF (1 + sk)k+F(1 + sn)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 fis determined at time 0. What should be fso 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 fsatisfies
106(1 + f) = 113.42 f= 113.42/106 1 = 7%.
Forward rates
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 fj,k the forward rate covering period jto k. For instance, if the unit of time is
years, then f2,4corresponds to the forward rate between year 2 and year 4. Can think of the forward
rate fj,k as the rate agreed upon at time 0 for borrowing money at time jthat will be repaid at time
The relationship with spot rates is as follows:
(1 + fj,k)kj=(1 + sk)k
(1 + sj)j
or in terms of prices for zero-coupon bonds, we have
(1 + fj,k)kj=P(0, j)
P(0, k).
The one-period forward rates fk,k+1 for k= 0,1, . . . , are denoted as fk(we drop the k+ 1 in the
Using this notation, we can give yet another relationship between forward rates and spot rates:
1 + sk= ((1 + f0)(1 + f1). . . (1 + fk1))1/k.
In other words, the spot rates are geometric averages of the forward rates. For example, previously
we had f0=s1= 0.06 and f1= 0.07 (denoted fabove), corresponding to a two-year spot rate
s2= (1.06 ×1.07)1/2= 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.