ACTSC 445: Asset-Liability Management

Department of Statistics and Actuarial Science, University of Waterloo

Unit 5 – Measuring Interest-Rate Risk

References (recommended readings): Chap. 9 of Fabozzi et al.

As mentioned before, the holder of a ﬁxed income security or portfolio is exposed to interest-rate risk.

That is, if interest rates increase, the price of the security will drop, which may result into a loss for

the investor if he/she needs to sell the security before maturity.

It is important for investors to assess the sensitivity of ﬁxed income securities to changes in interest

rates in a precise, quantitative way, so that the interest-rate risk can be better understood. In this

unit, we’ll see diﬀerent ways to do that:

1. The full-valuation approach;

2. Price value of a basis point;

3. Duration and convexity: we’ll spend most of our time on that approach.

We will also discuss alternative deﬁnitions of duration that can be used in more complex settings.

Before we go over these diﬀerent approaches, it is important to mention that in what follows, most

of the time we will look at interest rates changes as being observed on ytm’s. Also, unless otherwise

stated, we assume bonds pay semi-annual coupons.

As an example, we show in Table 1 how the price of diﬀerent bonds vary when the ytm goes from 4%

to diﬀerent values ranging between 2 and 6%. All rates below are assumed to be annual, compounded

semi-annually. Figure 1 gives a graphical depiction of the numbers given in Table 1.

Table 1: Bond prices as a function of yield

ytm 4%-4 year 4%-12 year 8%-4 year 8%-12 year

2% 107.65 121.24 122.96 163.73

3% 103.74 110.02 118.71 150.08

3.5% 101.85 104.87 116.66 143.79

3.9% 100.37 100.95 115.05 138.99

3.99% 100.04 100.09 114.69 137.94

4% 100 100 114.65 137.83

4.01% 99.96 99.91 114.61 137.71

4.1% 99.63 99.06 114.25 136.67

4.5% 98.19 95.40 112.68 132.18

5% 96.41 91.06 110.76 126.83

6% 92.98 83.06 107.02 116.94

Equivalently, we can compute the instantaneous percentage change of the bond’s price for the diﬀerent

changes in the ytm:

1

Figure 1: Bond prices as a function of yield

Table 2: Bond’s price percentage change as a function of yield

ytm 4%-4 year 4%-12 year 8%-4 year 8%-12 year

2% 7.65 21.24 7.24 18.79

3% 3.74 10.02 3.54 8.89

3.5% 1.85 4.87 1.75 4.32

3.9% 0.37 0.95 0.35 0.85

3.99% 0.04 0.09 0.03 0.08

4.01% -0.04 -0.09 -0.03 -0.08

4.1% -0.37 -0.94 -0.35 -0.84

4.5% -1.81 -4.60 -1.72 -4.10

5% -3.59 -8.94 -3.40 -7.98

6% -7.02 -16.94 -6.66 -15.16

Several comments about the sensitivity of the bond’s price to changes in interest rates can be observed

from Tables 1 and 2:

•For small changes in the ytm, the percentage price change for a given bond is roughly the same,

whether the ytm goes up or down.

•For large changes in the ytm, the percentage price change is usually larger (in absolute value) for

a decrease in the ytm (i.e., when the price goes up) than for an increase in the ytm (i.e., when

the price goes down). That is, the percentage price change is asymmetric.

•The degree of sensitivity of the bond’s price is aﬀected by diﬀerent properties of the bond:

–maturity: the longer maturities seem to have a greater sensitivity.

2

–coupon rate: the lower the coupon rate, the greater is the sensitivity. Thus, for a given ytm,

we’ll have the following ordering in terms of price sensitivity: premium bond <par bond <

discount bond <zero-coupon bond.

–yield level: the higher is the (original) ytm, the smaller is the sensitivity.

–embedded options: typically, a bond with embedded options is not as sensitive to interest

rate changes as an option-free bond. The reason is that for a bond with embedded options,

the price is given by

price of the option-free bond −value of the option.

Typically, the value of the option moves in the same direction as the (option-free) bond’s

price, so the two partially cancel out each other.

We are now ready to discuss three diﬀerent approaches for measuring the sensitivity to interest rate.

Full-valuation approach

As the name suggests, this method recomputes the value of the security for each possible interest-rate

change scenario. This is what we did to ﬁll out the entries in Table 1. The problems with this approach

are (1) can be very time-consuming for large portfolios of bonds; (2) choosing which scenarios to test is

not obvious. Its advantages are that it can deal with complex interest-rate changes (e.g., when changes

diﬀer for diﬀerent maturities), and can be useful for stress testing (looking at extreme scenarios to

assess exposure to interest-rate changes).

Price value of a basis point (PVBP)

•Given by the absolute change in the price of a bond for a 1 basis point (0.01%) change in the

yield.

•Also called Dollar value of an 01 (DV01).

•As mentioned before, for small changes in the yield, the price changes are about the same whether

the yield goes up or down. Thus we can use either case to compute the PVBP. For example, for

the four coupons in Table 1, based on a initial ytm of 4%, we have that

4% 100 100 114.6510 137.8279

3.99% 100.0366 100.0946 114.6908 137.9439

4.01% 99.9634 99.9055 114.6112 137.7119

PVBP

backward diﬀ. 0.0366 0.0946 0.0398 0.1160

forward diﬀ. 0.0366 0.0945 0.0398 0.1160

3

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