### Duration

V: Bonds
15: Duration
Duration
 Concept
 Calculation
 Duration and Price Volatility
Chapter 15: Duration
Fundamental Risk
 Reinvestment Risk:
 The risk that coupons, paid out of the bond,
cannot be reinvested at the same YTM.
 Price Risk:
 The risk that the price of the bond will fall
 Note that this is a risk only if we sell the bond before
it matures. There is no price risk if we hold the bond
to maturity.
Chapter 15: Duration
Duration
 Duration:
Weighted by Net Present Value average
term to maturity.
 Duration can be calculated
on any cash flow structure.
Chapter 15: Duration
A Tale of Two Bonds
\$1000
5% Annual Coupon
\$1000
9% Annual Coupon
\$50 \$50 \$50 \$50 \$50
\$90 \$90 \$90 \$90 \$90
Price: \$918.00
Price: \$1,082.00
Chapter 15: Duration
A Tale of Two Bonds
Capital Gain -1.56%
Capital Gain 1.73%
5
\$1082.00 
Income Yield: 5.45%
5
\$918.00 
t 1
t 1
\$50
.07 

1 

1 

t

\$1000
.07 

1 

1 

5 year capital gain = 8.93%
Annual capital gain = 1.73%
Yield to Maturity: 7.00%
Chapter 15: Duration
Income Yield: 8.32%
5
\$90
 .07 
1 

1


t

\$1000
 .07 
1 

1


5 year capital gain = - 7.58%
Annual capital gain = - 1.56%
Yield to Maturity: 7.00%
5
A Tale of Two Bonds
How much of my investment faces a
reinvestment risk every year?
\$50
\$50
\$50
\$50
\$1050
\$90
\$90
\$90
\$90
\$1090
Chapter 15: Duration
Calculating Duration I
T
1 year
2 years
3 years
4 years
5 years
Chapter 15: Duration
5 year 5% Annual Coupon Bond at 7%
Cash
NPV
NPV/P
Flow
\$50
\$46.73
5.09%
\$50
\$43.67
4.76%
\$50
\$40.81
4.45%
\$50
\$38.14
4.16%
\$1050
\$748.64 81.55%
Total NPV =\$918.00 100.00%
A Tale of Two Bonds
 How much of each bond must be reinvested
after 1,2,3,4 and 5 years?
5.1%
5% Bond
4.8%
4.4%
9% Bond
4.2%
7.8%
7.3%
6.8%
6.3%
81.6%
Chapter 15: Duration
71.8%
Oltheten
& Waspi 2012
Waspi 2012
Calculation
5 year 5% Annual Coupon Bond at 7%
T
1 year
2 years
3 years
4 years
5 years
Chapter 15: Duration
Cash
Flow
\$50
\$50
\$50
NPV
\$46.73
\$43.67
\$40.81
NPV/P
Duration Convexity
T*NPV/P D*(T+1)
5.09% .050903
.101806
4.76% .095146
.285439
4.45% .133383
.533530
\$50
\$38.14
4.16% .166209
.831044
\$1050
\$748.64 81.55% 4.077553 24.465317
Total NPV =\$918.00 100.00% 4.523 yrs 26.217 yrs2
A Tale of Two Bonds
Chapter 15: Duration
Duration & Price Risk
 Volatility:
Change in the price of the bond due to a
change in market yield.
Δ Price Duration
Volatility

* Δ Yd
1  Yd
Price
Chapter 15: Duration
Duration & Volatility
 5% annual bond:

4.523 yrs * 1% =
4.227%
1.07
 Modified Duration is
4.227 years


If Yd1%
then
P4.227%
If Yd 1%
then P4.227%
Chapter 15: Duration
 9% annual bond:

4.272 yrs * 1% =
3.993%
1.07
 Modified Duration is
3.993 years


If Yd1%
then
P3.993%
If Yd 1%
then P3.993%
Price Yield Curve
150
140
130
120
110
100
90
80
70
60
1%
Chapter 15: Duration
3%
5%
7%
9%
11%
13%
Price Yield Curve
 5 year 5% annual coupon
 7% yield
120
110
100
90
80
70
1%
Chapter 15: Duration
3%
5%
7%
9%
11%
13%
Price Yield Curve
 20 year 6% semi-annual coupon
 8% yield
200
175
150
125
100
75
50
25
0
1%
Chapter 15: Duration
3%
5%
7%
9%
11%
13%
15%
Calculating Duration II
 Calculate the duration and convexity of a
semi-annual bond
 \$10,000
 6% coupon
 December 31, 2017
 Settles March 2, 2014
 102.000
Chapter 15: Duration
Calculating Duration II
 Base Price:
 62/180 days Accrued Interest:
 Invoice Price:
 \$10,200.00
 \$103.33
 \$10,303.33
 YTM: 5.41186%
Chapter 15: Duration
Calculating Duration II
Chapter 15: Duration
Exercise
 Calculate the duration and convexity of a
semi-annual bond
 \$1000
 6% coupon
 2.5 years to maturity
 Priced to yield 8%
Chapter 15: Duration
Semi-Annual Bonds
1 1/2 year 6% Semi-Annual Coupon Bond at 8%
T
Cash
Flow
NPV
NPV/P
Duration
T*NPV/P
Convexity
D*(T+1)
1
2
3
4
5
Chapter 15: Duration
Volatility
 Duration: First derivative of the Price Yield
Curve
 .D = dP/dY
 Slope of the Yield Curve
 Convexity: Second derivative of the Price
Yield Curve
 .C = dP2/d2Y
 Curvature of the Yield Curve
Chapter 15: Duration
Volatility
 Taylor Expansion:
ΔP
Duration
1 Convexity 2

Δy 
Δy
2
y
P
2 

y
1  
1 

2

2

Modified Duration
Modified Convexity
Yield at which duration was calculated
Chapter 15: Duration
Volatility
 \$1000
 6% semi-annual
coupon
 2 ½ years to
maturity
 Duration:
 Modified D:
 Convexity:
 Modified C:
2.355 yrs
2.355 =2.264
(1.04)
6.922 yrs2
6.922 = 6.400
(1.04)2
 Priced to Yield
8%
Chapter 15: Duration
Δ Yield +200 basis points
Duration only
-2.355 (+.02)
(1.04)
Chapter 15: Duration
Convexity
Correction
+ 1
2
Total
6.922 (+.02)2 =
(1.04)2
Δ Yield -200 basis points
Duration only
-2.355 (-.02)
(1.04)
Chapter 15: Duration
Convexity
Correction
+ 1
2
6.922 (-.02)2
(1.04)2
Total
=
Price Yield Curve
200
Convexity
corrections are
always positive
175
150
125
100
Price effect is
asymmetric
75
50
25
0
1%
Chapter 15: Duration
3%
5%
7%
9%
11%
13%
15%
Volatility
 Yields increase by 2%
- 4.5288% + 0.128% = - 4.4009%
 Yields decrease by 2%
+ 4.5288% + 0.128% = + 4.46568%
Convexity
corrections are
always positive
Chapter 15: Duration
Price effect is
asymmetric