Transcript Introduction to Bond Markets, Analysis, and Strategies

Bond Price Volatility
Chapter 4
Price Volatility Characteristics
Exhibit 4-3. Instantaneous Percentage Price Change for 6 Hypothetical Bonds
Six hypothetical bonds, priced initially to yield 9%:
9% coupon, 5 years to maturity, price = $100.0000
9% coupon, 25 years to maturity, price = 100.0000
6% coupon, 5 years to maturity, price = 88.1309
6% coupon, 25 years to maturity, price = 70.3570
0% coupon, 5 years to maturity, price = 64.3928
0% coupon, 25 years to maturity, price = 11.0710
Yield Change
to:
6
7
8
8.5
8.9
8.99
9.01
9.1
9.5
10
11
12
Change
Percentage Price Change (coupon/maturity in years)
in BP
9%/5
9%/25
6%/5
6%/25
0%/5
0%/25
–300
12.8
38.59
13.47
42.13
15.56
106.04
–200
8.32
23.46
8.75
25.46
10.09
61.73
–100
4.06
10.74
4.26
11.6
4.91
27.1
–50
2
5.15
2.11
5.55
2.42
12.72
–10
0.4
1
0.42
1.07
0.48
2.42
–1
0.04
0.1
0.04
0.11
0.05
0.24
1
–0.04
–0.10
–0.04
–0.11
–0.05
–0.24
10
–0.39
–0.98
–0.41
–1.05
–0.48
–2.36
50
–1.95
–4.75
–2.05
–5.09
–2.36
–11.26
100
–3.86
–9.13
–4.06
–9.76
–4.66
–21.23
200
–7.54
–16.93
–7.91
–18.03
–9.08
–37.89
300
–11.04
–23.64
–11.59
–25.08
–13.28
–50.96
Price Volatility of Option-Free Bond




Although the prices of all option-free bonds move in
opposite direction from the change in yield required,
the % price change is not the same for all bonds.
For very small changes in the yield required, the %
price change for a given bond is roughly the same,
whether the yield required increases or decreases.
For large changes in the required yield, the % price
change is not the same for an increase in the
required yield as it is for a decrease in the required
yield.
For a given large change in basis points, the % price
increase is greater than the % price decrease.
Price Volatility



For a given term to maturity and initial yield,
the price volatility of a bond is greater, the
lower the coupon rate.
For a given coupon rate and initial yield, the
longer the term to maturity, the greater the
price volatility.
The higher the YTM at which a bond trades,
the lower the price volatility.
Measures of Price Volatility

price value of a basis point – gives dollar
price volatility not %
Bond
5-year 9% coupon
25-year 9% coupon
5-year 6% coupon
25-year 6% coupon
5-year zero-coupon
25-year zero-


Initial Price
(9% Yield)
100
100
88.1309
70.357
64.3928
11.071
Price
at 9.01%
99.9604
99.9013
88.0945
70.2824
64.362
11.0445
Price Value
of a BP
yield value of a price change
duration
0.0396
0.0987
0.0364
0.0746
0.0308
0.0265
Duration
Duration
Exhibit 4-5. Calculation of Macaulay Duration and Modified Duration
Coupon rate: 9.00%
Term (years): 5
Initial yield: 9.00%
Period, t
CF PV of $1 at 4.5%
PV of CF
t x PVCF
1
$4.50
0.956937
4.30622
4.30622
2
4.50
0.915729
4.120785
8.24156
3
4.50
0.876296
3.943335
11.83
4
4.50
0.838561
3.773526
15.0941
5
4.50
0.802451
3.61103
18.05514
6
4.50
0.767895
3.455531
20.73318
7
4.50
0.734828
3.306728
23.14709
8
4.50
0.703185
3.164333
25.31466
9
4.50
0.672904
3.02807
27.25262
10
104.50
0.643927 67.290443
672.90442
100
826.87899
Duration
Exhibit 4-6. Calculation of Macaulay Duration and Modified Duration
Coupon rate: 6.00%
Term (years): 5
Initial yield: 9.00%
Period, t
CF
PV of $1 at 4.5%
PV of CF
1
$3.00
0.956937
2.870813
2
3.00
0.915729
2.74719
3
3.00
0.876296
2.62889
4
3.00
0.838561
2.515684
5
3.00
0.802451
2.407353
6
3.00
0.767895
2.303687
7
3.00
0.734828
2.204485
8
3.00
0.703185
2.109555
9
3.00
0.672904
2.018713
10
103.00
0.643927
66.324551
88.130923
t x PVCF
2.87081
5.49437
7.88666
10.06273
12.03676
13.82212
15.43139
16.87644
18.16841
663.24551
765.8952
Modified Duration
Duration


duration is less than (coupon bond) or equal to (zero
coupon bond) the term to maturity
all else equal,




the lower the coupon, the larger the duration
the longer the maturity, the larger the duration
the lower the yield, the larger the duration
the longer the duration, the greater the price
volatility
Duration


dollar duration = (-MD) * P
spread duration – measure of how a non-Treasury
bond’s price will change if the spread sought by the
market changes




spread duration = 0 for Treasury
for fixed rate security it is the approximate change in the
price of a fixed-rate bond for a 100 bp change in the spread
for a floater, a spread duration of 1.4 means that if the
spread the market requires changes by 100 bp, the
floater’s price will change by about 1.4%
portfolio duration – weighted average of bonds’
durations
Portfolio Duration
Bond
10% 5yr
8% 15yr
14% 30yr
Par Amt Owned
$4 million
$5 million
$1 million
Bond
10% 5yr
8% 15yr
14% 30yr
Price ($)
100.0000
84.6275
137.8586
Market Value
$4,000,000
$4,231,375
$1,378,586
Yield (%)
10
10
10
Duration
3.861
8.047
9.168
Portfolio Duration
Bond
Market Value
10% 5yr
8% 15yr
14% 30yr
$4,000,000
$4,231,375
$1,378,586
Duration
Change in Value for
50bp Change in Yield
3.861
$77,220
8.047
170,249
9.168
63,194
Total $310,663
Measures of Bond Price Volatility
Price-Yield Relationship
Price Approximation using Duration
Convexity

second derivative of price-yield is dollar
convexity measure of bond

convexity measure

convexity measure in terms of periods
squared so to convert to annual figure, divide
by 4
Calculation of Convexity for 5 Year, 9%,
Selling to Yield 9% (Price = 100)
Period, t
1
2
3
4
5
6
7
8
9
10
CF
$4.50
$4.50
$4.50
$4.50
$4.50
$4.50
$4.50
$4.50
$4.50
$104.50
t(t + 1)CF
0.876296
0.838561
0.802451
0.767895
0.734828
0.703185
0.672904
0.643927
0.616198
0.589663
9
27
54
90
135
189
252
324
405
11,495
12,980
7.886
22.641
43.332
69.11
99.201
132.901
169.571
208.632
249.56
6,778.19
7,781.02
Calculation of Convexity for 5 Yr, 6%,
Selling to Yield 9% (P=88.1309)
Period, t
1
2
3
4
5
6
7
8
9
10
CF
$3.00
$3.00
$3.00
$3.00
$3.00
$3.00
$3.00
$3.00
$3.00
$103.00
t(t + 1)CF
0.876296
0.838561
0.802451
0.767895
0.734828
0.703185
0.672904
0.643927
0.616198
0.589663
6
18
36
60
90
126
168
216
270
11,330
12,320
5.257
15.094
28.888
46.073
66.134
88.601
113.047
139.088
166.373
6,680.89
7,349.45
Convexity
consider the 25-year 6% bond selling at 70.357 to yield 9%
% Price Change

consider a 25 year 6% bond selling to yield 9%



MD = 10.62, convexity = 182.92
required yield increases 200 bp from 9% to 11%
estimated price change due to duration and convexity is 21.24% + 3.66% = -17.58%
Convexity


implication of convexity
for bonds when yields
change
market takes convexity
into account when
pricing bonds

but to what extent should
there be difference?
Convexity
1.
2.
3.
As the required yield
increases (decreases),
the convexity of a bond
decreases (increases).
This property is referred
to as positive convexity.
For a given yield and
maturity, the lower the
coupon, the greater the
convexity of a bond.
For a given yield and
modified duration, the
lower the coupon, the
smaller the convexity.
Approximating Duration
1.
Use the 25 year, 6% bond trading at 9%. Increase
the yield by 10bp from 9% to 9.1%. So ∆y = 0.001.
The new price is P+ = 69.6164.
2.
3.
Decrease the yield on the bond by 10 bp from 9%
to 8.9%. The new price is P- = 71.1105.
Because the initial price, P0, is 70.3570, the
duration can be approximated as follows
Approximating Duration
1.
2.
3.
Increase the yield on the bond by a small number
of bp and determine the new price at this higher
yield level. New price is P+.
Decrease the yield on the bond by the same
number of bp and calculate the new price. PLetting P0 be the initial price, duration can be
approximated using the following where ∆y is the
change in yield used to calculate the new prices. This gives the
average % price change relative to the initial price per 1-bp change
in yield.
Approximating Convexity