Chapter 7
Introduction to the
Measurement of
Interest Rate Risk
1
Introduction
◼ In Week 1, we first discussed the interest rate risk
in bond investment. We know that:
 Bond price and interest rate move in opposite directions
 In particular, a rise in the interest rates will lead to a
decline in the bond value. This is known as the interest
rate risk to bond investors
◼ The key in measuring the interest rate risk is the
accuracy in estimating the value of the bond
portfolio after an adverse interest rate move
 We can do so using either the full valuation approach
or the duration/convexity approach
2
Full valuation approach
◼ This approach revalues a bond or bond portfolio for
a given interest rate change scenario. It is also
known as scenario analysis
 On the following two slides, we illustrate this approach
using a portfolio of two bonds, bond A and bond B
 We consider two scenarios:
in Scenario 1, there is a parallel increase in the
yields on both bonds by 50 basis points
◼ in Scenario 2, there is a parallel decrease in the
yields on both bonds by 100 basis points
◼
We can also do a similar analysis for a nonparallel
shift in yields (see Exhibit 3 on p.159)
3
Full valuation approach
◼ Both bonds A and B are option free
 Bond A is 8% coupon payable semiannually, 5 years,
$100 par, yields at 6%, and sells at $108.53
◼
(N = 10; PMT = 4; FV = 100; I/Y = 3; CPT PV = -$108.53)
 Bond B is 5% coupon payable semiannually, 15 years,
$100 par, yields at 7%, and sells at $81.61
◼
(N = 30; PMT = 2.5; FV = 100; I/Y = 3.5; CPT PV = -$81.61)
◼ The value of the bonds and the value of the
portfolio consisting of bonds A and B in each
scenario are summarized in the table on the next
slide …
4
Full valuation approach
◼
Scenario Yield change
Current
None
50 basis points ()
1
100 basis points ()
2
Bond A
$108.53
$106.32
$113.13
Bond B
$81.61
$77.71
$90.20
Porfolio
$190.14
$184.03
$203.33
Portfolio value change
None
3.21% ()
6.94% ()
In Scenario 1 (i.e., interest rates rise), bond A declines
by 2.04% [= ($106.32 - $108.53) / $108.53] in price,
and bond B drops by 4.78% in price
In Scenario 2 (i.e., interest rates drops), bond A increases
by 4.24% in price, and bond B rises by 10.53% in price
5
Full valuation approach
◼
Bond B is a longer maturity bond and has
a lower coupon rate: both of which mean
higher interest rate risk
 The changes in bond B price are 4.78% and 10.53%,
both higher than the corresponding changes in bond A
price (2.04% and 4.24%, respectively)
 Recall from Week 1:
◼ Higher (lower) coupon rate means lower (higher)
interest rate risk
◼ Longer (shorter) maturity (generally) means higher
(lower) interest rate risk
◼ Higher (lower) yield means lower (higher) interest
6
rate risk
Full valuation approach
◼
Advantages of this approach:
 It is more accurate than the duration/convexity
approach (which is only for parallel yield curve
changes, as we will see later in the chapter)
◼
Disadvantages of this approach:
 1) Can be very complex and time consuming
◼
In our example, we only consider a portfolio of two
option free bonds. Yet in reality bond portfolios
usually consist of hundreds or even thousands of
bonds, many of them have embedded options
7
Full valuation approach
◼
Disadvantages of this approach (continued):
 2) Modeling risk: the model analyzing embedded
options may produce wrong value because of the
incorrect assumptions made in the model (see also
p.15 and p.118)
 3) Which scenarios to use? There might be tens of
thousands of possible scenarios rather than just two!
◼
Stress testing: risk managers look at extreme
scenarios (e.g., market crash, the entire industry
collapses etc.) to assess the exposure of the portfolio
to interest rate risk, especially for highly levered
investors such as hedge funds (see also p. 118)
8
Price volatility of bonds
◼
Option free bonds
 Consider an option free, 8% coupon semiannual pay,
20 year bond currently trading at par ($100). The price
yield relationship for this bond is shown as below
along with another bond price curve (solid blue line)
Price
The price yield curve for an
option free bond is convex
toward the origin
A
$110.68
$100
$90.80
0
A bond with same duration, same
yield but different convexity
B
7%
8%
Yield
9%
9
Price volatility of bonds
◼ Option free bonds: from the graph on the
previous slide, we can see the following four
properties of the price volatility of option free
bonds
 Property 1: the price yield curve is downward sloping,
but is not a straight line (compared to the straight line
AB), so the percentage price change is not the same
for all bonds
 Property 2: for small changes in yields, the
percentage change in bond price for a given bond is
roughly the same, no matter the yields increase or
decrease (see the next slide for an example …)
10
Price volatility of bonds
◼
Option free bonds
 Property 2:
◼ For the bond that we are considering,
suppose the yield rises by 10 basis points,
then bond price drops to $99.02, a decline
of 0.98% ( 1%)
◼ If the yield drops by 10 basis points, then
bond price rises to $101, an increase of 1%
11
Price volatility of bonds
◼
Option free bonds
 Property 3: for large changes in yields, the
percentage price change is not the same for
yield increase as it is for yield decrease
In our example, when yields rise by 1% (8%  9%),
the bond price drops by 9.2% ($100  $90.80)
◼ In contrast, when yields drop by 1% (8%  7%), the
bond price increases by 10.68% ($100  $110.68)
◼
9.2%  10.68%!
12
Price volatility of bonds
◼
Option free bonds
 Property 4: for a given large change in yields,
the percentage price increase is greater than
the percentage price decrease
◼
In our example, for a given 1% (i.e., 100 basis
points) change in yields, bond price may increase
by 10.68%, or drop by 9.2%, depending on the sign
of the yield change
10.68% > 9.2%
◼
This is known as the convexity (or positive
convexity) of the price yield curve. This property is
beneficial to bond investors (Why?)
13
Price volatility of bonds
◼
Callable or prepayable bonds
 The price yield curve for a callable or
prepayable bond usually looks like:
Price
Call option
value
Call price
Option free bond
Callable
bond
0
Negative y*
convexity
Yield
Positive convexity
14
Price volatility of bonds
◼
Callable or prepayable bonds
 In the previous graph, y* is usually equal to
the bond coupon rate
When yield is higher than y*, the embedded option
has almost no value, and the callable or prepayable
bond is roughly equal to the corresponding option
free bond
◼ When yield is lower than y*, the embedded option
has a positive value to the issuer, the price
appreciation potential of the callable bond is capped
by the call price (known as price compression),
and the callable bond is less valuable than the
comparable option free bond
◼
15
Price volatility of bonds
◼
Callable or prepayble bonds
 The region where yield is lower than y* is
known as the negative convexity (or
concave), as opposed to positive convexity
where yield is higher than y*
◼
In negative convexity, for a given change in yield,
the percentage price increase is less than the
percentage price decrease (see the next slide for an
illustration …)
16
Price volatility of bonds
◼
Callable or prepayable bonds
Price
P1
P0
P2
0
y 0 − y1 = y 2 − y 0
P1 − P0 P2 − P0
But
|
|
P0
P0
y1
y0
y2 y*
Negative convexity
Yield
Positive convexity
17
Price volatility of bonds
◼
Callable or prepayable bonds
 From the graph on Slide 16, the embedded call
option value is the difference between the price
of the corresponding option free bond and the
price of the callable bond
◼
Recall from Chapter 2:
Price of callable bond = price of option free bond
– value of the embedded call option
18
Price volatility of bonds
◼
Putable bonds
 The value of an embedded put option is larger
to the bondholders at higher yields and is
smaller at lower yields (Can you see why?),
opposite to the call option value
◼
Also typically the put price is set equal to par
 The price yield curve for a putable bond is
shown on the next slide …
◼
In the graph, y* is usually equal to the bond
coupon rate
19
Price volatility of bonds
◼
Putable bonds
Price
Putable bond
Option free
bond
Put option value
0
y*
Yield
Positive convexity
20
Price volatility of bonds
◼
Putable bonds
 The embedded put option value is given by the
sum between the price of the putable bond and the
price of the comparable option free bond
◼ Recall from Week 1:
Price of putable bond = price of option free bond
+ value of the embedded put option
 The price yield curve for a putable bond is flatter
than the curve for the corresponding option free
bond because embedded options tend to reduce
the interest rate risk (see Week 1)
21
Duration/convexity approach
◼
Duration
 The formula for the approximate percentage
price change for a 100 basis point change in
yield is:
price if yield declines - price if yield rises
2  (initial price)  (change in yield in decimal)
 The above measure of interest rate risk is called
duration
22
◼
Duration/convexity
approach
Duration
 If we let
y = change in yield in decimal
◼ V0 = initial bond price
◼ V- = bond price if yields decline by y
◼ V+ = bond price if yields increase by y
◼
 then the duration formula becomes:
V- − V+
2  (V0 )  ( y)
23
◼
Duration/convexity
approach
Duration
 Example: y = 0.0016 (i.e., 16 basis points),
V0 = $1,020, V- = $1,022.50, and V+ = $1,019
$1,022.50 − $1,019
= 1.07
Duration =
2  $1,020  0.0016
◼
Implication: the approximate price change when
yield changes by 100 basis points is 1.07%; for a 50
basis point change in yield, price changes by
roughly 0.535% (= 1.07% / 2) etc.
24
Duration/convexity
approach
◼
Approximate percentage price change
 Given duration, we can approximate the
percentage change in bond price using:
approximat e percentage price change = - duration  y *
where y* is the yield change (in decimal) for which
the estimated percentage price change is sought
Note: do not confuse the y in the duration formula on Slide 25
with the y* in the above formula. y is the (typically small) change
in yields used to estimate duration (we refer to y as “rate shock”).
On the other hand, y* is the specific change in yields for which the
approximate percentage price change is sought
25
Duration/convexity
approach
◼
Approximate percentage price change
 Example: given a duration of 1.07, the
approximate percentage price change for a
200 basis point decrease in yield is
- duration  y * = −1.07  (−0.02) = 0.0214 = 2.14%
26
Duration/convexity
approach
◼
Modified duration is the duration
measure in which it is assumed that yield
changes do NOT alter the expected cash
flows
 In the duration formula on Slide 25, the
values of V- and V+ are calculated using the
same cash flows as those used to compute
the value of V0
 Modified duration is appropriate for option
free bonds such as noncallable Treasury
securities etc.
27
Duration/convexity
approach
◼
Effective duration (or option adjusted
duration) is the duration measure that
recognizes the fact that yield changes may
alter the expected cash flows on a bond as
well
 In the duration formula on Slide 25, the cash
flows used to calculate the value of V- and V+
are different from those used to compute the
value of V0
28
Duration/convexity
approach
◼
Effective duration is more appropriate for
bonds with embedded options (e.g., call, and
put options and mortgage backed securities
etc.)
◼
The difference between modified duration and
effective duration for bonds with embedded
options can be huge
 For example, for some mortgage backed
securities, the modified duration could be 7 and
the effective duration could be 20!
29
Duration/convexity
approach
◼
Duration
The sensitivity of a bond’s price (as a percentage
of initial price) to a change in yield
Modified duration
Effective duration
Assumes yield changes
will not alter the expected
cash flows
Recognizes that yield
changes may alter the
expected cash flows
30
Duration/convexity
approach
◼
Macaulay duration
 The price of an option free bond is given as
follows:
C
C
C + Par
P =[
+
+ ... +
],
2
n
1 + y (1 + y)
(1 + y)
where P = price of the bond, C = semiannual coupon
in $, y = one half of the YTM, n = number of
semiannual periods (= number of maturity in years 
2), and Par = face value of the bond in $
31
Duration/convexity
approach
◼
Macaulay duration (continued)
Macaulay duration =
C
C
(C + Par )
1
+ 2
+ ... + n 
2
(1 + y)
(1 + y)
(1 + y) n
[
].
P
➢ To understand Macaulay duration, think of the coupon bond as a
package of zero coupon bonds.
➢ Macaulay duration is a weighted average of the maturities of the
underlying zero coupon bonds, where the weight on each maturity is the
present value of the corresponding zero coupon bond calculated using
the coupon bond’s YTM (see the next slide for an illustration …)
32
Duration/convexity
approach
◼
Macaulay duration (continued)
Maturity
1
2
3
…
n
Present value of
zero-coupon bond
C / (1 + y)
2
C / (1 + y)
3
C / (1 + y)
…
(C + Par) / (1 + y)n
33
Duration/convexity
approach
◼
Macaulay duration (continued)
 Note that Macaulay duration is measured in
periods
 For zero coupon bonds, maturity measures the
length of time that a bondholder has invested
money.
 But for coupon bonds, maturity is an imperfect
measure of this length of time because much of a
coupon bond’s value comes from coupon
payments made before maturity
34
Duration/convexity
approach
◼
Macaulay duration (continued)
 Example: a 4-year $100 par 3% coupon payable
annually option free Treasury bond yielding at
2.5%
◼
Bond price P = $101.88, C = $3, y = 2.5%, n = 4, and
Par = $100
Macaulay duration =
$3
$3
$3
$(3 + 100)
1
+ 2
+ 3
+ 4
2
3
(1 + 2.5%)
(1 + 2.5%)
(1 + 2.5%)
(1 + 2.5%) 4
[
]
$101.88
= 3.8305
35
Duration/convexity
approach
◼
Macaulay duration (continued)
 Investors often refer to the ratio of Macaulay
duration to (1+y) as modified duration:
Macaulay duration
Modified duration =
1+ y
Example: for the bond on the previous slide, its
modified duration is given by: [3.8305 / (1 + 2.5%)] =
3.7371 periods or 1.8686 years
36
Duration/convexity
approach
◼
Portfolio duration
 A portfolio’s duration is the weighted
average of the duration of the bonds in the
portfolio, where the weight is the proportion of
the portfolio that a bond comprises. That is:
w1D1 + w 2 D 2 + ... + w K D K ,
where wi is the weight of bond i in the portfolio, i.e.
wi = (price of bond i) /(market value of the portfolio), Di is
duration of bond i, and there are a total of K bonds in the
portfolio
37
Duration/convexity
approach
◼
Portfolio duration
 Example: consider the following 4 bond
portfolio
Bond
Price ($) Duration
1
1,156
4.50
2
978
10.70
3
915
0.52
4
1,362
3.67
Market value of the portfolio
= $1,156 + $978 + $915 + $1,362 = $4,411
 Portfolio duration =
$1,156
$978
$915
$1,362
 4.5 +
10.7 +
 0.52 +
 3.67
$4,411
$4,411
$4,411
$4,411
38
= 4.79
Duration/convexity approach
◼
Portfolio duration
 On the previous slide, a portfolio duration of 4.79
means that for a 100 basis point change in the yield on
every bond, the portfolio value changes by roughly
4.79%
◼ Be careful: the yields on the four bonds must ALL
change by 100 basis points for the above measure
of portfolio duration to be useful (i.e., the yield curve
must undergo a parallel shift). This is the critical
assumption made!
◼ Yet in practice there is no reason to believe that
there are only parallel shifts in yield curves
 Recall the yield curve risk in Week 1. This is
indeed one limitation of duration
39
Duration/convexity
approach
◼
Convexity adjustment
 The limitations of duration: let’s use the bond
example on Slide 10 to illustrate
◼
Recall the bond is option free, 8% coupon payable
semiannually, 20 years, currently trading at par
($100)
When yield rises by 10 basis points, then bond price
drops to $99.02, a decline of 0.98% ( 1%)
 If yield drops by 10 basis points, then bond price rises to
$101, an increase of 1%

$101 − $99.02
Duration =
= 9.9
2  $100  0.0010
40
Duration/convexity
approach
◼
Convexity adjustment
 A duration of 9.9 means that for a 100 basis point
change in yield, the bond price changes by roughly
9.9%; or equivalently, for a 10 basis point change in
yield, the absolute percentage price change is
approximately 0.99% (= 9.9  0.001, see Slide 28 for the
formula), which is close enough to the 1% actual change
in bond price
◼ So duration works well for small changes in yields
◼
But what if the yield on bond changes by e.g., 250
basis points?
41
Duration/convexity
approach
◼
Convexity adjustment
 When yield rises by 250 basis points, bond price
drops to $79.27, a decline of 20.73%
When yield drops by 250 basis points, bond price
rises to $130.10, an increase of 30.10%
◼ But according to duration, the bond price should
rise and drop equally at 24.75% (= 9.9  0.025)
◼ So duration underestimates the price increase
(24.75% < 30.10%) and overestimates the price
decrease (24.75% > 20.73%)!
◼
42
Duration/convexity
approach
◼ Convexity adjustment
 Moreover, duration incorrectly treats percentage
price increase as equal to percentage price
decrease for a given change in yield
Yet we know from both Property 3 and Property 4 of
price volatility of bonds that for a given large change in
yield, the percentage price increase should be higher
than the corresponding percentage price decrease due to
the positive convexity of the price yield curve (see Slide
10)
◼ Duration is therefore only a first (linear) approximation
for a small change in yield
◼ The approximation can be improved by using a second
approximation called the “convexity adjustment”
43
◼
Duration/convexity
approach
◼
Convexity adjustment
 The formula for the convexity adjustment to
the percentage price change is:
C  (y* ) 2 ,
V+ + V− − 2V0
where C =
2  V0  (y ) 2
Note: 1) the notations are explained on Slide 25;
2) y is different from y* (see Slide 28);
3) C is known as the convexity of the bond
44
Duration/convexity
approach
◼
Convexity adjustment
 Example: we continue with the bond example.
The convexity adjustment (in %) is calculated
as:
C  (y* ) 2
V+ + V− − 2V0
2
=[
]

(

y
)
*
2  V0  (y ) 2
=[
$99.0177 + $100.9970 − 2  $100
2
]

(
0
.
025
)
= 4.59%
2
2  $100  (0.0010)
In the above calculation, y = 0.0010 (i.e., 10 basis points),
yet y* = 0.025 (i.e., 250 basis points)
45
Duration/convexity approach
◼
The approximate percentage price change based on both
duration and convexity is found by adding these two
estimates
 Example: we continue with our bond example on the
previous slide
When yield rises by 250 basis points, the estimated change
using duration is -24.75%, the convexity adjustment is 4.59%, so
the total estimated percentage price change = (-24.75%) +
4.59% = -20.16%, which is closer to the actual decrease of 20.73% than using duration alone (-24.75%)
◼ When yield drops by 250 basis points, the estimated change
using duration is 24.75%, the convexity adjustment is 4.59%, so
the total estimated percentage price change = 24.75% + 4.59%
= 29.34%, which is closer to the actual increase of 30.10% than
using duration alone (24.75%)
◼
46
Duration/convexity
approach
For bonds exhibiting positive convexity in
their price yield curve, the convexity
adjustment is positive, as for option free
bonds (e.g., the bond example on slides)
◼ For bonds in the negative convexity region
of their price yield curve, the convexity
adjustment is negative, as for callable or
prepayable bonds
◼
47
Duration/convexity
approach
◼ Modified convexity adjustment assumes that
when yield changes the cash flows used to
compute convexity C won’t change
➢ It is appropriate for option free bonds
◼ Effective convexity adjustment (will see in
Week 6) assumes that when yield changes the
cash flows used to compute convexity C do
change
➢
This adjustment is more appropriate for bonds with
embedded options
48
Price value of a basis
point
◼
Another measure of price volatility to quantify
the interest rate risk is the price value of a
basis point (PVBP, also known as the dollar
value of an 01, DV01). That is:
PVBP = |Initial price – price if yield changes by 1 basis point|
Example : Initial price = $954,
price if yield drops by 1 basis point
= $954.8
 PVBP =| $954 − $954.8 |= $0.8
49
Price value of a basis
point
In the PVBP formula, it does not matter if yield
increases or decreases by 1 basis point, the
resulted change to bond value (in absolute
value) will be the same, according to Property 2
of price volatility of bonds
◼ The PVBP is related to duration. In fact, the
PVBP is almost equal to the dollar price change
for a 1 basis point change in yields estimated
using duration
◼
50
Yield value of a price
change
◼
An alternative measure of the price
volatility of a bond is the yield value of a
price change
 This is estimated by first calculating the
bond’s YTM if the bond’s price is decreased
by, say, X dollars. Then the difference
between the initial YTM and the new YTM is
the yield value of an X dollar price change
51
Yield value of a price
change
◼ The smaller the difference (in yields), the greater the
dollar price volatility, since it would take a smaller
change in yields to produce a price change of X
dollars
 Example: An option free $100 par value 3 year maturity
5% coupon rate payable semiannually bond is selling at
$97.2914
The bond’s initial YTM is 6%
◼ When the bond price drops by $1 to $96.2914, the bond’s
new YTM is 6.377781%. So the bond’s yield value of a $1
price change is (6% - 6.377781% =) 0.377781% in absolute
value
52
◼
Convexity Formula
◼ The convexity (measured in periods squared) of a
bond is defined as:
Convexity =
1 
C
C
(C + Par ) 
1

2

+
2

3

+
...
+
n

(
n
+
1
)



2P 
(1 + y ) 3
(1 + y ) 4
(1 + y ) n +2 
◼ To obtain the convexity in years squared, simply
divide by m2 where m is the number of periods per
year
◼ The percentage price change using duration and
convexity is:
P
 − Duration  (y ) + Convexity  (y ) 2
P
53
The importance of yield
volatility
◼ The greater the expected yield volatility, the
greater the interest rate risk for a given
duration and current value of a bond portfolio
 Consequently, to measure the exposure of a
bond portfolio to interest rate changes, it is
necessary to measure yield volatility
 The measure of yield volatility can be estimated
as the standard deviation of yield changes (as
we will see in Week 5)
54
Value-at-risk (VaR)
◼ Value at risk (VaR) is defined as the maximum
loss in the market value of a given portfolio that
is expected to occur with a specified probability
(see the next slide for a graphical illustration …)
 In finance VaR can be used for risk
management, risk measurement, and computing
regulatory capital etc.
 Back testing a VaR calculation methodology
involves looking at how often exceptions (i.e.,
loss > VaR) occur
55
Value-at-risk (VaR)
◼
Assume a standard normal distribution
VaR at an e.g. ,99% confidence level
56
Value at risk (VaR)
Regulators base the capital they require
banks to keep on VaR
◼ The market risk capital is k times the 10
day 99% VaR where k is at least 3.0
◼ Under Basel II, capital for credit risk and
operational risk is based on a one year
99.9% VaR
◼
57
Value-at-risk (VaR)
◼
VaR asks: “What loss level is such that
we are X% confident it will not be
exceeded in N business days?”
 It captures an important aspect of risk in a
single number
 It is easy to understand
 It asks the simple question: “How bad can
things get?”
58
Value-at-risk (VaR)
◼
VaR example: a bond portfolio of $180
million, a portfolio volatility of 0.1407, then
for a confidence level of 99% the portfolio
VaR for a one day holding period is
1
VaR = $180,000,000  0.1407 
 2.33 = $3,732,094
250
There is only 1% probability that the portfolio losses will
exceed $3,732,094 over one trading day
Note: in the above calculations, we assume:
1) a standard normal distribution; and
2) there are 250 trading days in a year
59