Risk Management
FINA20202A
Lecture 2
Interest Rate Risk Management
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Contents
I.
Bonds – Quick Review
II. Interest Rate Risk and Bond features
III. Duration as a measure of interest rate risk
IV. Portfolio duration
V. Hedging interest rate risk
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I. Bonds – Quick Review
Reminder
•
•
•
•
•
The price of a bond
Yield To Maturity (YTM)
Annual Periodical Rate (APR) and annual effective rate
Coupon rate
The term structure of interest rates (the yield curve)
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The equation of a bond’s price
The value of a bond is equal to:
PV of coupons + PV of face value
C 
1 
F
VA   1 

r  (1  r )t  (1  r )t
Where:





VA = PV = Present Value
F = Face value of bond
C = periodic coupon payment ( C = F × periodic coupon rate)
r = periodic interest rate
t = number of periods until maturity
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Bond Quotes
Source: Bloomberg June 30th, 2021
Issuer: The company, province (state) or country that is issuing the bond.
Coupon: Fixed interest rate that the issuer pays to the lender
Maturity date: date on which the borrower will pay the investors their principal back.
Bid price: the price someone is willing to pay for the bond. It is quoted in relation to 100, no
matter what the par value is. A bond with a bid of 93 means it’s trading at 93% of its par value.
Yield: indicates annual return until the bond matures.
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Some additional remarks
• Biannual coupons are standard in North America
• Settlement often occurs between coupon payments
• Market convention: Canada government bonds are quoted
on an actual/actual basis
• ‘Dirty price’ = ‘clean price’ + accrued interests
 Dirty price, also called ‘full price’
 Clean price, also called ‘flat price’ or ‘quoted price’
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The Yield Curve
Shows YTM for bonds with the same credit quality with different maturities
Shapes: Normal, Inverted and Flat
Source: CaixaBank Research, based on data from Bloomberg
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The Yield Curve of Canada government
bonds
Source: Refinitiv Datastream, BlackRock Investment Institute, as of June 15, 2020
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II. Interest rate risk and Bond
features
Interest rate risk
• It is the probability/risk of a decline in the value of an asset (Fixed-Income
Investment) as a result of unexpected fluctuations in interest rates.
• Risk free bonds are exposed to interest rate risk.
Risks resulting from changes in interest rates:
• Price risk
 Main risk
 Interest rate (YTM) increases -> price risk increases (price of bonds decreases)
• Reinvestment risk
 Remember: When we priced bonds we assumed that we reinvest the coupons
at the YTM. (This is not true!)
 Interest rate (YTM) increases -> reinvestment risk decreases (less likelihood to
be paid back before maturity)
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Example 1: Interest rate risk and bond features
Suppose we have 4 bonds with an annual coupon
and a YTM of 6%. Their features and prices are the
following:
Bond
Coupon
Residual time to
maturity
Price
A
B
C
D
6%
6%
2%
2%
5
20
5
20
100.00
100.00
83.15
54.12
Question: Which bond is the most sensitive to
interest rate risk?
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Price
Convex relationship between bond price and YTM
(Bond B)
(Bond A)
YTM
Holding all other features equal, longer maturity bonds
are more convex.
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Percentage Price Variations
What are the percentage price variations of the 4 bonds if the required rate of
return (YTM) changes?
Bond features
Variation
6%
6%
2%
2%
in bps
5
20
5
20
3.0%
-300
13.74%
44.63%
14.76%
57.28%
4.0%
-200
8.90%
27.18%
9.56%
34.55%
5.0%
-100
4.33%
12.46%
4.64%
15.69%
6.0%
0
0.00%
0.00%
0.00%
0.00%
7.0%
100
-4.10%
-10.59%
-4.39%
-13.10%
8.0%
200
-7.99%
-19.64%
-8.55%
-24.07%
9.0%
300
-11.67%
-27.39%
-12.48%
-33.30%
New
return
The percentage price variations tells us which bond is the most
sensitive to interest rate movements (YTM).
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Three important lessons to be learned!
1. When the interest rate increases, bond prices (in PV)
decrease.
2. Longer-term bonds are more sensitive to interest rate
risk.
3. Lower-coupon bonds are more sensitive to interest
rate risk.
Note: Just by comparing the features of the 4 bonds, we can conclude that
bond D is the most sensitive, without doing the price variation calculations.
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A few more (subtle) remarks
• An increase in the YTM leads to a smaller price
decrease than the price increase caused by a
decrease in YTM of the same importance.
(Rememeber: Convex Relationship!)
• Price sensitivity with respect to YTM variations
increases with maturity, although at a decreasing rate.
• Bond prices are more sensitive to YTM variations when
the bond is sold at a lower initial YTM.
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III. Duration as a measure of
interest rate sensitivity
The duration of a bond measures the sensitivity of the
bond’s full price (including accrued interest) to changes
in interest rates.
Macaulay Duration (D)
• In1938, Frederick Macaulay suggests an approach to
determine the price volatility of a bond. He calls this
measure ‘duration’, which then became the commonly
accepted ‘Macaulay duration’
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• Macaulay duration is a weighted average of the time
to receipt of the bond’s promised cash flows
(coupons and face value).
• The weighting of each cash flow is determined by
dividing the present value of the cash flow by the
bond’s price
• The cash flows of a bond are:
Time
Cash flows
1
2
…
T
C1
C2
…
CT
• The Macaulay duration of the bond is:
C1 (1  r )
C2 (1  r ) 2
CT (1  r )T
D  1
 2
 ...  T 
P0
P0
P0
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Modified Duration (D*)
D
D 
1 y
*
y = yield per period
Example 2: Computing duration
Suppose we have a bond with 2-year maturity, 8%
annual coupon and $100 face value. The yield curve is
flat at 5%.
Question 1: What is the Macaulay duration of this bond?
Question 2: What is the Modified duration of this bond?
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Question 1: The following table helps to compute Macaulay
duration
Computing Macaulay Duration
Time
CF
PV of CF
Weights*
Time x
weights
(1)
1
2
(2)
8
108
(3)
7.62
97.96
(4)
0.072
0.928
(1) x (4)
0.072
1.856
105.58
1.000
1.928
* to get the weights divide each PV of CF by the bond’s price, 7.62/ 105.58 =0.072
• The fair price of the bond is 105.58 (PV of the Cash Flows
discounted at 5%)
• Macaulay duration (D) is 1.928 years
Question 2: Computing Modified duration
1.836
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Why is Modified Duration (D*) important?
1. Modified duration measures the first-order effect of yield variation,
it provides a linear estimate (approximation) of the percentage price
change for a bond given a change in its YTM.
P
*
 D  y
P
The percentage variation of a bond’s price caused by the variation in
YTM can be estimated by multiplying the modified duration by the
YTM’s variation
2. The Modified Duration (D*) is the only variable in the equation that
can be managed (by changing the composition of the portfolio),
which is why we can see D* as a tool for risk management.
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Graphical interpretation
Price/Yield Relationship for an Option-Free Bond with a Tangent Line
Source: Fabozzi(2012), Handbook of Fixed Income Securities (eighth edition), McGraw-Hill.
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Example 3: Approximating the Percentage price
Change and Price using duration
Consider a 5% 20-year bond trading at 113.6777 whose
Modified duration is 13.09.
What is the approximate percentage price change for a 10
basis point increase in yield and the estimated price?
P
 D*  y
P
Approximate percentage price change = -13.09 x (+0.001) = -1.309%
For a 10 basis point increase in yield, duration estimates
that the price will decline to 112.1897.
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Dollar Duration of a Bond (Money Duration)
• The Dollar duration of a bond is a measure of the price change in
units of the currency in which the bond is denominated.
• Dollar duration is calculated as the annual modified duration times
the full price of the bond (including accrued interest)
P
 D*  P0
y
Price Value of a Basis Point (PVBP)
• Dollar duration is sometimes expressed in basis points. The
PVBP is an estimate of the change in the full price given a 1bp
change in the YTM. (Multiply Dollar duration times 1bp)
P  (D*  P0 )  0.0001
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Approximate Modified Duration
(ApproxModDur)
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Example 4: Bond assessment and duration
Consider a zero-coupon bond with a 10-year maturity
and a 6% YTM (annual effective rate) and 1,000 face
value.
• What is the bond’s price?
Bond’s price =
• What is the bond’s Macaulay duration?
• What is the bond’s Modified duration?
1000
(1.06)10
= 558.39
D = 10
10
D*= 1+0.06 = 9.43
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Now, let’s calculate price approximations
• What is the approximate price variation if the YTM
increases to 6.5%? Hint: PP  D  y
*
Approximate percentage price change = -9.43 x (+0.005) = -4.715%
• What is the new approximate price of the bond?
Approximate price of the bond = 558.39 x (1 – 0.04715) = 532.06
• What is the error made in this approximation? (Hint:
Recompute the price of the bond)
1000
Bond’s new price = (1.065)10 = 532.73
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Graphical interpretation
558.39 =
532.73
532.06
6%
6.5%
Source: Fabozzi(2012), Handbook of Fixed Income Securities (eighth edition), McGraw-Hill.
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What about convexity?
• Convexity is good!
• More Convexity is desirable. Why?
Source: Fabozzi(2012), Handbook of Fixed Income Securities (eighth edition), McGraw-Hill.
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Duration: The bottom line
Benefits
Easy to compute
Drawbacks
Acceptable only with bonds
generating fixed cash flows. Does
not work with bonds with
embedded options
Easy to understand
Quite acceptable and reasonable
in many circumstances (small
move in YTM)
Supposes parallels moves in the
yield curve
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IV. Portfolio Duration
Portfolio duration (whether Macaulay or Modified) is the
weighted sum of individual durations (linear concept)
N
DP 
wi Di  w1D1  w2D2  ...  wN DN

i 1
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Example 5: Long and Short durations
James Bond has a portfolio of 100,000 bonds (10-year maturity) with
5% annual coupons and $1,000 face value, partially financed through
the short sale of 1,000,000 30-year STRIPs with $100 face value. The
yield curve is flat at 6%.
Question 1: What is the duration of James Bond’s portfolio?
Question 2: What is the price variation estimated through duration if
the interest rate increases by 1%?
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Question 1: First, let’s compute the individual durations (D).
• The duration of a 10-year maturity bond with 5% annual coupon and
$1,000 face value is
Time
(1)
1
2
3
4
5
6
7
8
9
10
CF
(2)
$50.00
$50.00
$50.00
$50.00
$50.00
$50.00
$50.00
$50.00
$50.00
$1,050.00
Sum
M acau lay D u r at io n
M o d if ied D u r at io n
PV of CF
(3)
47.17
44.50
41.98
39.60
37.36
35.25
33.25
31.37
29.59
586.31
926.40
Weighted PV
(1) ´ (3)
47.17
89.00
125.94
158.42
186.81
211.49
232.77
250.96
266.35
5,863.15
7,432.07
8 .0 2
7 .5 7
 Macaulay duration:
D=
7,432.07
926.4
= 8.02
 Modified duration:
D*=
8.02
(1 +0.06)
= 7.57
• The duration of a 30-year STRIP. (A STRIP bond is also known as a zero
coupon bond)
 Macaulay duration: D = 30
 Modified duration: D*=
30
(1 +0.06)
= 28.3
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Question 2:
D*p=
2.94
(1 +0.06)
= 2.77
Also, since each bond is option free, we could directly
use the modified duration of the assets.
For a 1% increase in the yield of both bonds, the market
value of the portfolio will reduce by approximately 2.77%.
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Factors determining duration
• The duration of a coupon bond increases when the YTM
decreases
• For a given maturity, duration increases when coupon
rate decreases.
• For a given coupon rate, duration generally increases
with maturity. It always increases with maturity for bonds
sold at par or at a premium.
• The duration of a zerocoupon bond is its time to
maturity.
• The
duration
of
a
(1+𝑦)
perpetuity is
𝑦
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V. Hedging interest rate risk
Example 5: Changing interest rate exposure
Suppose James Bond wants to decrease the
interest rate risk of his bond portfolio.
His daughter, Finance student at HEC Montréal,
discusses duration with him. He then decides to sell
more 30-year STRIPs. He sells 425,000 additional
contracts and keeps the cash.
Question 1: What is now James Bond’s portfolio
duration?
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Question 1
The portfolio’s duration has decreased – a lot!
In fact, portfolio duration is close to (practically at) zero.
• What is another name for this technique (bringing portfolio
duration to zero?
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Example 6: Portfolio restructuring over time
Suppose that one year later, James Bond
recomputes his portfolio’s duration. What
happened?
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The limits of hedging
• Matching durations allows the (approximate) hedging
to protect against changes in the yield curve.
• Model risk:
 What happens if the yield curve steepens or
inverts?
 The yield curve changes assumed in the model we
have used does not allow such changes.
 This means we are not protected (hedged) against
such changes.
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Credit Risk
Investment Grade Bonds
Speculative Grade Bonds
High Yield Bonds
“Junk” Bonds
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Conclusion
• The relationship between bond prices and YTM is
convex.
• Duration is an approximate measure of interest
rate risk.
• We have studied portfolio duration, as well as long
and short durations positions in a portfolio.
• We have analyzed interest rate risk hedging and
portfolio restructuring over time.
• Takeaway: Bond risk management means knowing
the duration and adjusting it to the current risk
environment.
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Thanks
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