Measurement of Interest Rate Risk
• The most widely used measure of interest
rate risk is the “duration”. A bond with a
higher duration has higher interest rate risk.
• The duration approximately measures the
sensitivity of the bond price to a given
change in yield.
– The duration is an approximate measure as it
assumes that the bond price is linear in yield
changes. Because, in reality, the bond price is
convex in the yield, the duration will be
accurate only for small changes in yields.
Definition of Duration
• We will define the duration as the
percentage change in the bond price for a
given change in yield.
• Let D = Duration, then:
• D = (% change in bond price)/ change in
yield
Mathematical Derivation of Duration for
Zero Coupon Bond (1/2)
• We can easily derive the duration from the bond
price formula by differentiating it.
• Consider a zero coupon bond with face value
equal to 100.
• Let B = bond price, y = yield to maturity, and
T=maturity of bond.
• Then, assuming semi-annual compounding, we get
the bond price as: B = 100/(1 + y/2)2T
Mathematical Derivation of Duration for
Zero Coupon Bond (2/2)
• Differentiating the bond price with respect to y, we
get:
• dB/dy = (-T) B/(1 + y/2)
• Therefore, from the definition of D,
• D = (dB/B)/dy = -T/(1+y/2)
• We normally express the duration as a positive
number, and the negative sensitivity of the bond to
the change in yield is assumed.
• Notes: This measure of duration is also called the
“modified duration” to differentiate it from the
“Macaulay duration”. The Macaulay duration for
the zero coupon bond is simply equal to (T).
Duration and the Maturity of the Bond
• Thus, the duration of the zero coupon bond
(except for the adjustment in the
denominator of (1+y/2)) is almost exactly
equal to the maturity of the bond.
– In other words, a bond of maturity 10 years
will be twice as risky as a bond of maturity 5
years, all else equal.
– The above statement will be approximately true
even if bonds have different yields and coupons
(although this may not be true if bonds have
embedded options).
Duration for Coupon Bonds
• We can extend the analysis to coupon bonds by
considering each cash flow of the bond as a
separate zero coupon bond.
• Thus, viewing a coupon bond as a portfolio of
zero coupon bonds, the duration will be the
weighted average of the individual durations of
each of the cash flows.
– The weights will be equal to [PV(cash flow)/Total
Value of Bond]
• The spreadsheet provides a detailed example of
the computations involved.
Example of Use of Duration
• We can now use the duration to estimate the impact
of a change in yield on the bond price.
• Consider a bond of maturity of exactly 2 years,
coupon and yield equal to 5%. The price of this
bond is 100.
• What would be the new bond price if the yield
increases by 6%?
• This bond has a (modified) duration of 1.869. Thus:
the % change in bond price is equal to D x increase
in yield = -1.869 x 1% = -1.869%.
• If the yield increases from 5% to 6%, we expect the
bond price to decline by 1.869% to 98.13.
– The exact bond price for a yield of 6% is 98.14, so the
duration gave us a reasonable answer.
Convexity (1/2)
• As noted earlier, the duration is an approximate
measure of the actual % change bond change for a
given change in yield.
• To get a more accurate measure of the price
change, we can use the second risk measure called
the “convexity” of the bond.
• The convexity measure the curvature of the bond,
and when used along with the duration, allows for
a more accurate measurement of bond interest rate
risk. This is done as follows (denote convexity =
C):
• % change in bond price = -D x (increase in yield)
+ 0.5 x C x (increase in yield)^2.
Convexity (2/2)
• Using the spreadsheet, the bond used in our
earlier example has a convexity of 2.2656.
• Thus, the impact of the convexity on the bond
price is equal to 0.5 x 2.2545 x (change in
yield)^2.
• For an increase in yield of 1%, this works out to
0.011%.
• Thus (combining both the duration and
convexity), for an increase in yield of 1%, we
expect the % price change to be = -1% x
(D=1.869%) + 0.5 x 2.2545 x (1%)^2 = -1.869%
+ 0.011% = -1.858%.
• Thus, if interest rates increase from 5% to 6%, we
expect the bond price to decrease to 98.14.