unit5 measuring interest rate risk
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:
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.
–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.
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
•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
backward diﬀ. 0.0366 0.0946 0.0398 0.1160
forward diﬀ. 0.0366 0.0945 0.0398 0.1160