Interest Rate Sensitivity - 1
Bond prices and yields are inversely related:
Chapter 11
Bond
A
B
C
D
™Interest rate sensitivity
™Duration and Convexity
™Portfolio Duration
¾
As yields fall, bond price rise.
Coupon
12%
12%
3%
3%
Maturity
(years)
5
30
5
30
Initial
YTM
10%
10%
10%
10%
Face
value
$1,000
$1,000
$1,000
$1,000
Price
$1,077.22
$1,189.29
$729.74
$337.47
The graph on the next page shows the price-yield relation of these
four coupon bonds.
™Horizon Analysis and Immunization
1
As yields increase, bond prices fall.
Consider the following four bonds with different coupon rate and
maturity, assuming currently all are offering a yield of 10%.
Bond Valuation: Part II
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Bond Valuation: Part II
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Bond Valuation: Part II
Interest Rate Sensitivity - 2
Bond Duration - 1
We want to assess the risk of possible changes in bond portfolio
values.
From the graph, we see that:
The relation between bond price and yield is non-linear.
Bond price movements are a combination of two factors:
An increase in a bond’s yield to maturity results in a smaller
price change than a decrease in yield of equal magnitude.
Interest Rate Sensitivity
¾
Exposure to interest rates
¾
Movements in interest rates
$5,000
Can we summarize risk in a simple fashion?
$4,500
Duration is a measure of linear exposure to changes in yield.
$4,000
A: 5 year 12% coupon bond
B: 30 year 12% coupon bond
C: 5 year 3% coupon bond
D: 30 year 3% coupon bond
$3,500
If we write bond prices as:
N
$3,000
P=∑
Price_A
Price
Price_B
t =1
Price_C
$2,500
Price_D
Then Duration is defined as:
$2,000
N
1 N Ct
t = ∑wt t
D= ∑
P t =1 (1 + y)t
t =1
$1,500
$1,000
where
$500
$0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
Yield to maturity
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Ct
(1 + y ) t
3
Bond Valuation: Part II
22%
PV (C t )
1 Ct
=
t
P (1 + y )
P
y is the yield to maturity (or semi-annual yield if coupons are paid
semi-annually), P is the current price, N is maturity and Ct is the
cash flow at time t.
wt =
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Bond Valuation: Part II
Bond Duration - 2
Bond Duration - 3
Duration is a concept of “effective maturity”: it is the weighted
average of the times to coupon and principal payments.
It turns out that duration D also measures exposure to yield
changes.
∆P
D
=−
∆y
P
1+ y
Modified duration: D* directly measures the interest rate
sensitivity.
D* =
Both Duration and modified duration are reported in years.
¾ This computation must use the same units as the
compounding period.
¾ Similarly, with semi-annual coupon payment, we
should use ½ of the yield-to-maturity as the y in
calculating modified duration.
Example: 8% 2-year bond with semi-annual coupon selling at a
yield of 8%.
Period
(half year)
1
2
3
4
Sum:
D
1+ y
Modified duration measures the sensitivity of bond rate of return
to instantaneous yield movements
∆P
= − D * × ∆y
P
Cash flow
PV factor
$4
$4
$4
$104
0.962
0.925
0.889
0.855
PV(CF)
t*PV(CF)
$3.846
$3.698
$3.556
$88.900
$100.000
D (1/2 yr)
D (years)
Modified Duration
1.89/1.04 =
If yield goes up by 10 basis points, how much will the price of the
bond change using duration approximation?
∆P
Answer: P = − D * ×∆y = −1.81 × 0.1% = −0.181%
Bond price will go down by 0.181%.
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Bond Valuation: Part II
3.846
7.396
10.668
355.599
377.509
3.78
1.89
1.81
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Bond Valuation: Part II
Bond Duration - 4
Bond Duration - 5
Example: Consider a 25-year 6% bond with semi-annual coupon
payments. Currently the price is $70.357 per $100 par. The duration
of the bond is 11.10 years.
1) What is the yield to maturity of this bond?
2) What is its modified duration?
3) If the yield goes down by 20bp, what will be the percentage
change in bond price using duration approximation?
Properties of Duration:
For regular coupon-paying bonds, duration is less than maturity.
D < N.
For zero-coupon bonds, duration is equal to its maturity. D = N.
For perpetuities, duration is D = (1+y)/y.
Answer:
1) Using your financial calculator, semi-annual yield y = 4.5%.
Therefore, YTM=9%.
2) Modified duration D* = D/(1+y) = 11.10/(1+4.5%) = 10.62.
3) ∆p / p = − D * ×∆y = −10.62 × ( −0.20%) = 2.124 %
Bond price will go up by 2.1% if yield goes down by 20bp.
Example 1
Calculate the duration of a perpetuity that pays $100 once per year
with a yield of 15%.
Answer
D = (1+y)/y = (1+15%)/15% = 7.67 years.
¾ What if yield is 8%?
With 8% yield, D = 1.08/0.08 = 13.5 years.
Example 2
A 15-year Treasury STRIPS currently has yield-to-maturity of 5%.
How much will the price change if yield goes up by 10bp?
Answer
Duration of the zero-coupon bond is its maturity: D = 15 year.
Modified duration = D/(1+y) = 15/(1+2.5%) = 14.63 year.
∆p / p = − D * ×∆y = −14.63 × (0.10%) = −1.463%
Price will go down by -1.46% when yield goes up by 0.1%.
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Bond Valuation: Part II
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Bond Valuation: Part II
Bond Duration - 6
Bond Duration and Convexity
Properties of Duration:
Limitations of Duration:
Holding coupon rate and yield to maturity constant, duration
generally increases with maturity.
The term structure is flat and movements in yields are parallel
Movements in yields are small
Holding time to maturity and yield to maturity constant, duration
decreases with coupon rate.
For large changes in yields, a second-order term may be added:
(in Taylor series expansion) convexity
Holding other factors constant, duration is higher when the
bond’s yield to maturity is lower.
Convexity
Let us go back to the four coupon bond listed on page 2:
Bond
A
B
C
D
Coupon
12%
12%
3%
3%
Maturity
(years)
5
30
5
30
Initial
YTM
10%
10%
10%
10%
Price
$1,077.22
$1,189.29
$729.74
$337.47
Duration
(years)
3.9
9.8
4.6
12.2
The bond price is a non-linear function of the yield
¾ For full revaluations with yield change of ∆y, exact change in
value is ∆P = P(y+∆y) – P(y).
This price function can be approximated by linear and quadratic
terms:
∆P
= −D * ×∆y + 0.5 × C × ∆y 2
P
Several observations:
¾
Duration of B is higher than that of A because of its longer
maturity. (Also compare C and D.)
¾
Duration of A is lower than C because of higher coupon rate.
(Also compare B and D)
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Bond Valuation: Part II
¾ C stands for convexity
¾ The second term in the “Taylor” expansion defines convexity
¾ Convexity is reported in year-squared
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Bond Valuation: Part II
Bond Convexity - 1
Bond Convexity - 2
Example
Example (continued)
Consider an 8%, 30-year bond selling at yield to maturity of 8%.
(with annual coupon payment and a par value of $1,000).
The modified duration is 11.26 years, and convexity is 212.4 year2.
When yield goes up by 2% (200bp):
Duration approach: ∆P / P = − D * ×∆y = −11.26 × 2% = −22.52%
Calculate the change in price using the following three methods:
1. duration
2. duration and convexity
3. full pricing
When yield goes up by 10bp:
Duration + Convexity:
∆P / P = − D * ×∆y + 0.5C∆y 2 = −11.26 × 2% + 0.5 × 212.4 × ( 2%) 2 = −18.27%
Full repricing:
At 8% yield, P = $1,000.
At 10% yield, P = $811.46
Therefore, ∆P/P = -18.85%
Duration approach: ∆P / P = − D * ×∆y = −11.26 × 0.1% = −1.126%
Duration + Convexity:
∆P / P = − D * ×∆y + 0.5C∆y 2 = −11.26 × 0.1% + 0.5 × 212.4 × (0.1%) 2 = −1.115%
Full repricing:
At 8% yield, P = $1,000.
At 8.1% yield, P = $988.85 (use your financial calculator to verify this)
Therefore, ∆P/P = -1.115%
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Bond Valuation: Part II
When yield goes down by 2% (200bp):
Duration approach: ∆P / P = − D * ×∆y = −11.26 × −2% = 22.52%
Duration + Convexity:
∆P / P = − D * ×∆y + 0.5C∆y 2 = −11.26 × −2% + 0.5 × 212.4 × ( 2%) 2 = 26.77%
Full repricing:
At 8% yield, P = $1,000.
At 6% yield, P = $1275.30
Therefore, ∆P/P = 27.53%
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Bond Valuation: Part II
Bond Convexity - 3
Portfolio Duration - 1
More on Convexity
¾ Convexity measures the curvature of the price-yield
relationship of a bond.
¾ Bond with larger convexity is more attractive to investors, and
therefore sells at a higher price (or lower yield).
Conclusion
¾ Duration measures the sensitivity of a bond’s price to a change
in its yield. It is best used when interest rate change is small.
¾ Duration is a measure of the effective maturity of a bond,
defined as the weighted average of the times until each
payment, with weights proportional to the present value of the
payment.
¾ Duration measure can be refined with convexity.
¾ Duration plus convexity provides a more accurate measure of
bond’s price sensitivity to interest rate change, especially when
the yield change is large.
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Bond Valuation: Part II
The duration of a portfolio is simply the weighted average
duration of the bonds in the portfolios.
N
D P = ∑ wi Di
i =1
DP: Duration of portfolio P
wi: the value of bond i divided by the value of the portfolio
Di: Duration of bond i
Example:
Consider the following four bond portfolio with a total market value
of $100 million:
Bond
A
B
C
D
Market Value
$10m
$40m
$30m
$20m
Portfolio Weight
0.10
0.40
0.30
0.20
Duration
4 years
7 years
6 years
2 years
Portfolio duration = 0.1*4 + 0.4*7 + 0.3*6 + 0.2*2 = 5.4 years
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Bond Valuation: Part II
Portfolio Duration - 2
Horizon Analysis - 1
Interpretation:
If all the yields affecting the four bonds in the portfolio change
by 100 basis points, the portfolio’s value will change by
approximately 5.4%.
Duration only considers the effect of instantaneous moves
More generally, consider an investor with a fixed investment
horizon
Suppose there is a permanent interest rate shock
Contribution to portfolio duration: Portfolio managers look at
their interest rate exposure to a particular issue in terms of its
contribution to portfolio duration.
Assuming the yields go up, there will be two offsetting effects
on the value of the bond:
¾ The bond market price fall now and at the horizon
(duration effect)
¾ Future reinvestment of coupon accrue at higher rate
Contribution = weight of issue in portfolio * duration of the issue
For the four bond portfolio above, the contribution of each issue is:
Example:
Bond
Market Value
A
B
C
D
Total
$10m
$40m
$30m
$20m
$100m
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Portfolio
Weight
0.10
0.40
0.30
0.20
1.00
15
Duration
4 years
7 years
6 years
2 years
Contribution
to Duration
0.40
2.80
1.80
0.40
5.40
Bond Valuation: Part II
Consider a 5-year 12% coupon bond with semi-annual coupon
payment and face value of $1000. Currently the yield is 8%.
Calculate a one-year holding period return if the interest rate is 4%,
8%, and 12% for the next year.
If interest rate remains at 8%
P0 = $1162.22, FV(coupon) = 122.40, P1 = $1134.65
Assuming the semi-annual return is y, then
1162.22 (1+y)2 = 1134.65+122.40
Therefore, y = 4% and total return is 2y = 8%.
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Bond Valuation: Part II
Horizon Analysis - 2
Horizon Analysis - 3
Example (continued)
Example (continued)
If interest rate goes down to 4%
P0 = $1162.22, FV(coupon) = 121.20, P1 = $1293.02
Therefore, y = sqrt[(1293.02+121.2)/1162.22] – 1 = 10.31%.
total return is 2y = 20.6%.
With four year investment horizon:
If interest rate goes up to 12%
P0 = $1162.22, FV(coupon) = 123.60, P1 = $1000
Therefore, y = sqrt(1123.60/1162.22) – 1 = -1.68%.
total return is 2y = -3.35%.
If interest rate goes down to 4%
P0 = $1162.22, FV(coupon) = 514.98, P1 = $1077.66
Therefore, y = [1592.64/1162.22]1/8 – 1 = 4.02%.
total return is 2y = 8.04%.
If interest rate remains at 8%
P0 = $1162.22, FV(coupon) = 552.85, P1 = $1037.72
Therefore, y = [1590.57/1162.22]1/8 – 1 = 4.00%.
total return is 2y = 8%.
What if we plan to hold the bond for four years?
If interest rate goes up to 12%
P0 = $1162.22, FV(coupon) = 593.85, P1 = $1000
Therefore, y = [1593.85/1162.22]1/8 – 1 = 4.03%.
total return is 2y = 8.05%.
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Bond Valuation: Part II
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Bond Valuation: Part II
Portfolio Immunization - 1
Portfolio Immunization - 2
When we set the investment horizon to equal the portfolio’s
duration, the portfolio return is immunized from interest rate
change.
For a horizon equal to the portfolio’s duration, price risk and
reinvestment risk exactly cancels out. The portfolio is
immunized.
In the previous example, the duration of the 5-year 12% coupon
at yield to maturity of 8% is exactly 4 years.
In this case, the portfolio total return over four years is always at
around 8%.
Example:
An insurance company must make a payment of $19,487 in
seven years. The market interest rate is 10%, so the present value of
the obligation is $10,000.
The company’s portfolio manager wishes to fund the obligation
using three-year zero-coupon bond and perpetuities paying annual
coupons. Both securities are selling at a yield of 10%. How can the
manager immunize the obligation?
Answer
Immunization requires
Duration (assets) = Duration (liabilities)
1. Calculate duration of the liability: D = 7 year
2. Calculate the duration of the assets in the portfolio.
D(3-year zero) = 3 year
D(perpetuity) = (1+y)/y = 1.10/0.10 = 11 year
3. Calculate the mix of assets that will give the same duration as the
duration of obligation.
Let the fraction of 3-year zero be w, then the fraction invested for
perpetuities is (1-w). Therefore,
w*3 + (1-w)*11 = 7
Î w = 0.5
4. Fully fund the obligation. This means investing $5,000 in 3-year
zero-coupon bond, and investing $5,000 in the perpetuity.
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Bond Valuation: Part II
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Bond Valuation: Part II
3. A bond has a current yield of 9% and yield to maturity of 10%. Is the
Readings and Homework
bond selling above or below par value? Explain.
™ Readings: Chapter 11
4. A nine-year bond has a yield-to-maturity of 10% and duration of 7.194
™ Exercises: Chapter 11 (Will NOT be collected)
ƒ Text website: Self-assessment quiz
ƒ End-of-Chapter CFA questions (page 493)
ƒ Problems: #5, 9, 11, 15, 17, 19.
™ Homework: (Will be collected and graded, due date to be
announced)
years. If the bond’s yield increases by 50 basis points, what is the
percentage change in the bond’s price? (Assume this bond makes annual
coupon payment)
5. An annual coupon bond has duration of 5 years and yield of 8%. If the
interest rate goes down by 10 basis points, how much will the price of the
bond change using duration approximation?
6. A pension plan is obligated to make disbursements of $1 million, $2
ƒ Chapter 11: Case Problem 11.2 (Page 498)
million, and $1 million at the end of each of the next three years,
respectively. Find the duration of the plan’s obligations if the interest rate
Additional Exercises (Ch.10 and Ch.11)
is 10% annually.
1. A 20-year maturity bond with par value $1,000 has a coupon rate of 8%.
7. The manager of the above-mentioned pension plan wants to fully
Coupons are paid semi-annually. Find the bond-equivalent yield and
immunize the portfolio. He can invest in perpetuity with yield of 10% and
effective annual yield of the bond is the bond price is: a) $950; b) $1,000;
a one-year zero-coupon bond with yield of 10% as well. How much
c) $1,050
should he allocate in each of these two instruments?
2. A 30-year 8% coupon bond (coupons are paid semi-annually) is callable
8. A 30-year 12% coupon bond (coupons paid annually) has duration of
in 5 years at a call price of $1,100. The bond currently sells at a yield to
11.54 years and convexity of 192.4. The yield-to-maturity is currently 8%.
maturity of 7%.
Find the price change if yield falls to 7% or rises to 9%.
a. What is the yield-to-call?
a. Do a full-repricing using financial calculator.
b. What is the yield-to-call if the call price is only $1,050?
b. Find the price change using duration approximation.
c. What is the yield-to-call if the call price is $1,100, but the bond can
c. Find the price change using duration + convexity approximation.
be called in two years instead of five years?
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Bond Valuation: Part II
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Bond Valuation: Part II