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.

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•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.

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)1/2−1=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?

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Solution: we have that s1is such that 100 = 106(1 + s1)−1⇒s1= 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

using

P=

n

X

k=1

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

k.

The relationship with spot rates is as follows:

(1 + fj,k)k−j=(1 + sk)k

(1 + sj)j

or in terms of prices for zero-coupon bonds, we have

(1 + fj,k)k−j=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

notation).

Using this notation, we can give yet another relationship between forward rates and spot rates:

1 + sk= ((1 + f0)(1 + f1). . . (1 + fk−1))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.

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