Revision of the
Fixed-Income Portfolio
1

 Introduction
 Passive versus active management strategies
 Duration re-visited
 Bond convexity
2

 Fixed-income security management is largely a matter of altering the level of risk the portfolio faces:
• Interest rate risk
•
Default risk
• Reinvestment rate risk
 Interest rate risk is measured by duration
3

 Passive strategies
 Active strategies
 Risk of barbells and ladders
 Bullets versus barbells
 Swaps
 Forecasting interest rates
 Volunteering callable municipal bonds
4

 Buy and hold
 Indexing
5

 Bonds have a maturity date at which their investment merit ceases
 A passive bond strategy still requires the periodic replacement of bonds as they mature
6

 Indexing involves an attempt to replicate the investment characteristics of a popular measure of the bond market
 Examples are:
•
Salomon Brothers Corporate Bond Index
•
Lehman Brothers Long Treasury Bond Index
7

 The rationale for indexing is market efficiency
•
Managers are unable to predict market movements and that attempts to time the market are fruitless
 A portfolio should be compared to an index of similar default and interest rate risk
8

 Laddered portfolio
 Barbell portfolio
 Other active strategies
9

 In a laddered strategy , the fixed-income dollars are distributed throughout the yield curve
 For example, a $1 million portfolio invested in bond maturities from 1 to 25 years (see next slide)
10

50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
1 3 5 7 9 11 13 15 17 19 21 23 25
Years Until Maturity
11

 The barbell strategy differs from the laddered strategy in that less amount is invested in the middle maturities
 For example, a $1 million portfolio invests
$70,000 par value in bonds with maturities of 1 to 5 and 21 to 25 years, and $20,000 par value in bonds with maturities of 6 to 20 years (see next slide)
12

50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
1 3 5 7 9 11 13 15 17 19 21 23 25
Years Until Maturity
13

 Managing a barbell portfolio is more complicated than managing a laddered portfolio
 Each year, the manager must replace two sets of bonds:
• The one-year bonds mature and the proceeds are used to buy 25-year bonds
• The 21-year bonds become 20-years bonds, and
$50,000 par value are sold and applied to the purchase of $50,000 par value of 5-year bonds
14

 Identify bonds that are likely to experience a rating change in the near future
•
An increase in bond rating pushes the price up
•
A downgrade pushes the price down
15

 Interest rate risk
 Reinvestment rate risk
 Reconciling interest rate and reinvestment rate risks
16

 Duration increases as maturity increases
 The increase in duration is not linear
• Malkiel’s theorem about the decreasing importance of lengthening maturity
• E.g., the difference in duration between 2- and
1-year bonds is greater than the difference in duration between 25- and 24-year bonds
17

 Declining interest rates favor a laddered strategy
 Increasing interest rates favor a barbell strategy
18

 The barbell portfolio requires a reinvestment each year of $70,000 par value
 The laddered portfolio requires the reinvestment each year of $40,000 par value
 Declining interest rates favor the laddered strategy
 Rising interest rates favor the barbell strategy
19

 The general risk comparison:
Rising Interest Rates Falling Interest Rates
Barbell favored Laddered favored Interest Rate Risk
Reinvestment Rate Risk Barbell favored Laddered favored
20

 The relationships between risk and strategy are not always applicable:
•
It is possible to construct a barbell portfolio with a longer duration than a laddered portfolio
–
E.g., include all zero-coupon bonds in the barbell portfolio
• When the yield curve is inverting, its shifts are not parallel
–
A barbell strategy is safer than a laddered strategy
21

 A bullet strategy is one in which the bond maturities cluster around one particular maturity on the yield curve
 It is possible to construct bullet and barbell portfolios with the same durations but with different interest rate risks
• Duration only works when yield curve shifts are parallel
22

 A heuristic on the performance of bullets and barbells:
•
A barbell strategy will outperform a bullet strategy when the yield curve flattens
•
A bullet strategy will outperform a barbell strategy when the yield curve steepens
23

 Purpose
 Substitution swap
 Intermarket or yield spread swap
 Bond-rating swap
 Rate anticipation swap
24

 In a bond swap , a portfolio manager exchanges an existing bond or set of bonds for a different issue
25

 Bond swaps are intended to:
•
Increase current income
•
Increase yield to maturity
•
Improve the potential for price appreciation with a decline in interest rates
•
Establish losses to offset capital gains or taxable income
26

 In a substitution swap, the investor exchanges one bond for another of similar risk and maturity to increase the current yield
•
E.g., selling an 8% coupon for par and buying an 8% coupon for $980 increases the current yield by 16 basis points
27

 Profitable substitution swaps are inconsistent with market efficiency
 Obvious opportunities for substitution swaps are rare
28

 The intermarket or yield spread swap involves bonds that trade in different markets
•
E.g., government versus corporate bonds
 Small differences in different markets can cause similar bonds to behave differently in response to changing market conditions
29

 In a flight to quality , investors become less willing to hold risky bonds
•
As investors buy safe bonds and sell more risky bonds, the spread between their yields widens
 Flight to quality can be measured using the confidence index
• The ratio of the yield on AAA bonds to the yield on BBB bonds
30

 A bond-rating swap is really a form of intermarket swap
 If an investor anticipates a change in the yield spread, he can swap bonds with different ratings to produce a capital gain with a minimal increase in risk
31

 In a rate anticipation swap, the investor swaps bonds with different interest rate risks in anticipation of interest rate changes
•
Interest rate decline: swap long-term premium bonds for discount bonds
• Interest rate increase: swap discount bonds for premium bonds or long-term bonds for shortterm bonds
32

 Few professional managers are consistently successful in predicting interest rate changes
 Managers who forecast interest rate changes correctly can benefit
•
E.g., increase the duration of a bond portfolio is a decrease in interest rates is expected
33

 Callable bonds are often retied at par as part of the sinking fund provision
 If the bond issue sells in the marketplace below par, it is possible:
•
To generate capital gains for the client
•
If the bonds are offered to the municipality below par but above the market price
34

 We already saw that the concept of duration can be seen as a time-weighted average of the bonds discounted payments as a proportion of the bond price, or as a weighted average of the cash flows “times”.
 Duration can also be interpreted as a risk measure for bonds, however.
35

As originally defined by Macaulay (1938), the duration is:
D=
1
P t
N 

1
(1 t C t
 r ) t
where Bond Price P= t
N 

1
(1
C t
 r ) t
However, most bonds provide semi-annual payments:
D=
  

1
P t
2 N 

1
(1 t C t
 r
/ 2
/ 2) t

 where Bond Price P= t
2 N 

1
(1
C t
 r
/ 2
/ 2) t or D=
1
P t

N 
0.5
(1 t C t

/ 2
EAR ) t
where Bond Price P
 t
N 

0.5
(1
C t

/ 2
EAR ) t t

0.5,1,1.5,..., N t

0.5, 1, 1.5, ..., N

2 N " 6-month periods". Also note that EAR=(1+r/2)^21.
Notice that the two expressions given for the bond price are equivalent. One uses periods of 6 months, while the other converts the periods in years. The formulas given for the duration are thus also equivalent (i.e. they yield the same result).
36

Differentiating the bond price with respect to r yields: d P dr
= t
N 

1
 tC t
(1
 r ) t

1
 
DP
1
 r
This last expression provides the percentage change in bond value for a given percentage change in discount factor: dP
P
 
D dr
1
 r or dP
 
DP dr
1
 r
D
Defining now MD=
1
 r
as the "Modified Duration", we have: dP
 
MD dr
P or dP
   
37

In the case of semi-annual payments, differentiating the bond price with respect to the interest rate yields: dP dr

 
  t
2 N 

1
(1

 t C r t
/ 2
/ 2) t

1
 
DP
1
 r / 2
This last expression provides the percentage change in bond value for a given percentage change in discount factor: dP
P
 
D dr
1
 r / 2 or dP
 
DP dr
1
 r / 2
D
Defining now MD=
1
 r / 2
as the "Modified Duration", we hav e: dP
 
MD dr
P or dP
 
MD P dr
38

Example: Bond A has a 10-year maturity, and bears a 7% coupon rate. Bond B has 10 years left to maturity, and a coupon rate of 13%. The current market interest rate is 7%.
The price of bonds A and B are $1,000 and $1,421.41 respectively. What happens to these prices if the market rate changes from 7% to 7.7% ?
39

Answer:
33
34
35
36
30
31
32
24
25
26
27
28
29
40
41
42
43
37
38
39
A
Bond
A
B
D
B
Actual
P
-47.61
-60.92
MDuration of bond B
D r
Bond price
-DP
D r/(1+r)
C D
P
$ 1,000
$ 1,421
E
Approximating Price Changes Using Duration
D
7.5152
6.7535
D r
0.007
0.007
=-(-PV(7%,10,70)+1000/(1.07)^10-(-PV(7.7%,10,70)+1000/(1.077)^10))
Using Excel's MDuration formula:
MDuration of bond A
D r
Bond price
-DP
D r/(1+r)
6.3117 <-- =MDURATION(DATE(1999,10,31),DATE(2009,10,31),13%,7%,1)
0.007
1,421
62.80 <-- =C42*C41*C40
-DP
D
F r/(1+r)
-49.17
-62.80
G H I
<-- =-C28*D28*E28/(1.07)
7.0236 <-- =MDURATION(DATE(1996,12,3),DATE(2006,12,3),7%,7%,1)
0.007
1000
49.17 Product of 3 terms above = -DP
D r/(1+r)
40

 The duration of a portfolio is the weighted average of the durations of the individual assets making up the portfolio.
 Proof: suppose you hold N
1
1 and N
2 units of security units of security 2. Let P
1 and P
2 be the prices of the two securities, and let
D
1 and D
2 be their respective durations.
41

The value of the portfolio is
 
N P
1 1

N P
2 2
We can then compute:
D

 
1
 d
 dr

1

(

D

 w D
1 1
 w D
2 2
N
1 dP
1

N
2 dr dP
2 ) dr where w i

N P i i
/

is the fraction of wealth invested in bond i
42

 The importance of convexity
 Calculating convexity
 General rules of convexity
 Using convexity
43

 Convexity is the difference between the actual price change in a bond and that predicted by the duration statistic
 In practice, the effects of convexity are relevant if the change in interest rate level is large.
44

 The first derivative of price with respect to yield is negative
•
Downward sloping curves
 The second derivative of price with respect to yield is positive
•
The decline in bond price as yield increases is decelerating
•
The sharper the curve, the greater the convexity
45

Greater Convexity
Yield to Maturity
46


As a bond’s yield moves up or down, there is a divergence from the actual price change
(curved line) and the duration-predicted price change (tangent line)
•
The more pronounced the curve, the greater the price difference
•
The greater the yield change, the more important convexity becomes
47

Current bond price
Yield to Maturity
Error from using duration only
48


The percentage change in a bond’s price associated with a change in the bond’s yield to maturity: dP



1
 dP
P

P dR dR



 


2
1

2 d P
P dR 2 where P

bond price
( dR 2 )



Error
P
R

yield to maturity
49

 The second term contains the bond convexity:
Convexity

1

2
2 d P
P dR
2 dR
2
( )
50

 There are two general rules of convexity:
•
The higher the yield to maturity, the lower the convexity, everything else being equal
• The lower the coupon, the greater the convexity, everything else being equal
51

 Recall the previous immunization example.
 Bond 2 (asset) has the same duration as the liability.
 However, there are other ways to select a portfolio of assets with a duration matching the liability’s duration.
52

1
11
12
13
14
7
8
9
10
15
16
4
5
6
2
3
BASIC IMMUNIZATION EXAMPLE WITH 3 BONDS
Yield to maturity
Coupon rate
Maturity
Face value
Bond price
Face value equal to $1,000 of market value
Duration
A B
6%
Bond 1
6.70%
10
1,000
$1,051.52
$1,095.96
$ 951.00
$ 912.44
7.6655
C
Bond 2
6.988%
15
1,000
10.0000
=dduration(B7,B6,$B$3,1)
D
Bond 3
5.90%
30
1,000
$986.24
$ 1,013.96
14.6361
53

Building Portfolio of given Duration
 Instead of using Bond 2 (with duration of 10) to match the obligation’s liability, let us build a portfolio made up of bond 1 and 3.
 We want duration =10, therefore we need:
 wD
1
+(1-w)D
2
=10, (D
1
=7.665 and D
2
=14.636)
 This implies w=0.66509
54

If interest rates change to, say, 7%:
A
16
17
18
23
24
25
26
19
20
21
22
27
28
New YTM
Bond price
Reinvested coupons
Total
Multiply by percent of face value bought
Product
Portfolio of bonds 1 and 3
Proportion of bond 1
Proportion of bond 3
B
7%
Bond 1
$1,000.00
$925.70
$1,925.70
C D
Bond 2 Bond 3
$999.51
$965.49
$883.47
$815.17
$1,965.00
$1,698.64
E F bond 1 & 3 portfolio
95.10% 91.24%
$ 1,792.95
101.40%
$ 1,722.34
$ 1,794.84
0.6651 <-- =(10-D13)/(B13-D13)
0.3349 <-- =1-B27
The Portfolio’s payoff remains more or less intact, just like Bond
2, and would thus allow us to meet the $ 1,790.85 obligation.
55

$2,100
$2,050
$2,000
$1,950
$1,900
$1,850
$1,800
$1,750
0% 2%
Performance of Bond 2 versus Bond Portfolio
4%
Bond 2 Bond portfolio
6% 8%
Interest rate
10% 12% 14% 16%
56

Bond 2 vs. Portfolio

Last slide’s graph of Terminal values shows that both Bond 2 and the carefully chosen
Portfolio (of Bonds 1 and 3) have a slope of zero around 6%.
 This indicates that both have been immunized, i.e. they both have a duration of
10 in this case.
 However, their curvature is different: the
Portfolio is more convex than Bond 2.
57

 Since the graph represents terminal values, convexity here is a good thing. We get more over funding (extra $ after having paid the obligation) from the portfolio if the interest rate departs from 6% than we get from
Bond 2.
 Therefore, when comparing two immunized portfolios, the portfolio whose terminal value is more convex with respect to a change in interest rates is more desirable.
58

Making a Portfolio Completely
Insensitive to Changes in Yields
 There are situations when it may be desirable to render a portfolio as insensitive to interest rate changes as possible.
 The way to achieve this is to not only match the assets and liabilities durations, but to also match their convexities.
59

 Recall our earlier immunization problem where interest rates change from r to r+
D r. The new values of the future obligation and of the bond are:
V
0
V
0

V
0
 D r
V
0
 dV dr
0



NQ
(1
 r )
N

1

 r
1
2

1
D
2
2 d V
0 dr
2
( r )
2
(
D r )
2


  
1) Q
(1
 r )
N

2


V
B
 D
V
B

V
B


V
B
 D r t
M 

1 dV
B dr
1
2
2 d V
B dr
2
(1
 tP t
 r ) t

1

1
2
(
D r )
2
(
D r )
2

 t
M 

1
  
1) P t
(1
 r ) t

2


60

 Equating the two and recalling that we have already matched the first and second terms in the expansion yields the following requirement:
V
1
B t
M 

1
(

1) P t
(1
 r ) t

(

1)
 This is the constraint that must be met in order for the assets (bonds) and the liabilities (obligations) to have matching convexities (in addition to already having matching durations)
61

 You need to immunize an obligation whose present value V
0 is $1,000. The payment is to be made 10 years from now, and the current interest rate is 6%. The payment is thus the future value of
1,000 at 6%, therefore it is:
1,000(1.06) 10 = $1,790.85
 The Excel spreadsheet on the next slide shows four bonds that you have at your disposition to immunize the liability.
62

1
10
11
12
13
14
15
16
7
8
5
6
9
2
3
4
17
18
19
20
21
22
23
24
25
Yield to maturity
Coupon rate
Maturity
Face value
New yield to maturity
Bond price
Reinvested coupons
Total
A
Duration
Second derivative Constraint multiply by percent of face value bought
Product
B
BOND CONVEXITY
Bond price
Face value equal to $1,000 of market value
6%
Bond 1
4.50%
20
1,000
$827.95
6%
Bond 1
$889.60
$593.14
$1,482.73
120.78%
C
Bond 2
6.988%
15
1,000
$1,095.96
$ 912.44
Bond 2
$1,041.62
Bond 3 Bond 4
$913.37
$1,000.00
$921.07
$461.33
$1,449.89
$1,962.69
$1,374.70
$2,449.89
91.24%
$ 1,790.85
D
Bond 3
3.50%
14
1,000
Bond 4
11.00%
10
1,000
$767.63
$1,368.00
$ 1,302.72
$ 730.99
12.8964
10.0000
10.8484
7.0539
229.0873
136.4996
148.7023
67.5980
( secondDur(E7,E6,$B$3)/bondprice(E7,E6,$B$3) )
130.27%
$ 1,790.85
E
73.10%
$ 1,790.85
63

What happens if rates go up to 7%?
A
21
22
23
24
25
17
18
19
20
New yield to maturity
Bond price
Reinvested coupons
Total multiply by percent of face value bought
Product
B C D E
7%
Bond 1
$824.41
$621.74
$1,446.15
Bond 2
$999.51
Bond 3 Bond 4
$881.45
$1,000.00
$965.49
$483.58
$1,519.81
$1,965.00
$1,365.02
$2,519.81
120.78% 91.24%
$ 1,792.95
130.27%
$ 1,778.24
73.10%
$ 1,841.96
We notice that only Bond 2 preserves its terminal value close to $1,791: it is the only bond with matching duration.
64

 It worked because the change in interest rate was small. What happens if rates go up to
10% ? (a large shift)
A
19
20
21
22
23
24
25
Bond price
Reinvested coupons
Total multiply by percent of face value bought
Product
B
Bond 1
$662.05
$717.18
$1,379.23
C D E
Bond 2
$885.82
Bond 3 Bond 4
$793.96
$1,000.00
$1,113.71
$557.81
$1,753.12
$1,999.53
$1,351.77
$2,753.12
120.78%
$ 1,665.84
91.24%
$ 1,824.46
130.27%
$ 1,760.97
73.10%
$ 2,012.51
None of the bonds maintained their terminal values now. The change in interest rate was too large.
65

How to build a portfolio of bonds with matching convexity?
 Set up the following system (Example with three bonds):
 where




1
D
1
D
1

1
D
3
D
3

1
D
4
D
4








 w w w
4
1
3










1
10
110




D

1
V
B t
M 

1
( 1 tP t
 r ) t

Duration (of bonds 1, 3, 4)
 and
D
 
1
V
B t
M 

1 t ( t

( 1

1 ) P t r ) t

Convexity Constraint (of bonds 1, 3, 4)
66

 The numbers 1, 10 and 110 on the right-hand side come from the fact that the weights must sum to one, that the weighted average duration must match the liability (obligation) duration of 10, and finally that the weighted average convexity constraint must match the liability convexity value of N(N+1), i.e. 10(10+1)=110.

Using the “secondDur” Visual Basic function in
Excel (for convenience, but not required) for the convexity constraints and solving for the weights by inverting the matrix yields the weights of a portfolio that is fully immunized.
67

Solution for the Weights
17
18
19
20
21
22
23
24
25
26
27
I J K
Calculating the bond portfolio:
Matrix of coefficients
1 1
12.8964
10.8484
1
7.0539
229.0873 148.7023
67.5980
L M
Vector of constants
1
10.0000
110.0000
Solution
-0.5619
1.6415 <-- =MMULT(MINVERSE(I19:K21),M19:M21)
-0.0797
N
68

Verifying that it Works by Computing
Portfolio Terminal Values for Various Rates
C D E
Bond 2 Bond portfolio
39
40
41
42
43
44
45
32
33
34
35
28
29
30
31
36
37
38
0% $ 1,868.87
1% $ 1,844.71
2% $ 1,825.14
3% $ 1,810.05
4% $ 1,799.35
5% $ 1,792.97
6% $ 1,790.85
7% $ 1,792.95
8% $ 1,799.26
9% $ 1,809.76
10% $ 1,824.46
11% $ 1,843.37
12% $ 1,866.53
13% $ 1,893.98
14% $ 1,925.77
15% $ 1,961.98
$ 1,774.63
$ 1,781.79
$ 1,786.37
$ 1,789.02
$ 1,790.32
$ 1,790.78
$ 1,790.85
$ 1,790.91
$ 1,791.31
$ 1,792.38
$ 1,794.38
$ 1,797.58
$ 1,802.21
$ 1,808.46
$ 1,816.55
$ 1,826.65
69

Immunization Using Second Derivatives
$2,000
$1,950
$1,900
$1,850
$1,800
$1,750
0% 2% 4%
Bond 2
6% 8%
New interest rate
Bond portfolio
10% 12% 14% 16%
70