Simon Benninga. Financial Modelling 3-rd edition

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698 Chapter 26

Thus, if the yield falls, bond 1 will no longer fund our obligation,

whereas bond 3 will overfund it. Bond 2’s ability to fund the obliga-

tion—not surprisingly, in view of the fact that its duration is exactly 10

years—hardly changes. We can repeat this calculation for any new yield

to maturity. The results are shown in the following ﬁ gure, which was

produced by running a Data|Table (see Chapter 31):

Immunization Properties of the Three Bonds

1,550

1,750

1,950

2,150

2,350

2,550

2,750

2,950

02468

New interest rate (%)

Terminal value ($)

Bond 1

Bond 2

Bond 3

16141210

Clearly, if you want an immunized strategy, you should buy bond 2!

26.4 Convexity: A Continuation of Our Immunization Experiment

The duration of a portfolio is the weighted average duration of the assets

in the portfolio. Therefore, there is another way to get a bond investment

with a duration of 10: If we invest $665.09 in bond 1 and $344.91 in bond

3, the resulting portfolio also has a duration of 10. These weights are

calculated as follows:

λλ∗+−∗=+−=Duration Duration

Bond Bond12

1 7 6655 14 6361 1 0() . .()

λλ

..6551

699 Immunization Strategies

Suppose we repeat our experiment with this portfolio of bonds. Starting

in row 15 of the following spreadsheet, we repeat the experiment of the

previous section (varying the YTM), but add in the portfolio of bond 1

and bond 3. The results show that the future value in row 23 does not

vary for the portfolio.

BACDEFG

Yield to maturity (YTM) 6%

bond 1 bond 2 bond 3

Coupon rate 6.70% 6.988% 5.90%

Maturity 10 15 30

Face value 1,000 1,000 1,000

Bond price $1,051.52 $1,095.96 $986.24 <-- =-PV($B$2,D6,D5*D7)+D7/(1+$B$2)^D6

Face value equal to $1,000 of market value 951.00$ 912.44$ 1,013.96$ <-- =D7/D9*D7

Duration 7.6655 10.0000 14.6361 <-- =dduration(D6,D5,$B$2,1)

New YTM 7%

Bond 1 Bond 2 Bond 3

Bond 1 & 3

portfolio

Bond price $1,000.00 $999.51 $883.47

Reinvested coupons $925.70 $965.49 $815.17

Total $1,925.70 $1,965.00 $1,698.64

Multiply by percent of face value bought 95.10% 91.24% 101.40%

Product 1,831.35$ 1,792.95$ 1,722.34$ 1,794.84$ <-- =B26*B23+(1-B26)*D23

Portfolio of bonds 1 and 3

Proportion of bond 1 0.6651 <-- =(10-D12)/(B12-D12)

Proportion of bond 3 0.3349 <-- =1-B26

EXPERIMENTING WITH BOND PORTFOLIOS AND CONVEXITY

Performance of Bond 2 versus Bond Portfolio

1,750

1,800

1,850

1,900

1,950

2,000

2,050

2,100

0246810121416

Interest rate (%)

Terminal value ($)

Bond 2

Bond portfolio

Building a data table based on this experiment and graphing the results

shows that the portfolio’s performance is better than that of bond 2 by

itself:

700 Chapter 26

Look again at the graph: Notice that, while for both bond 2 and the

bond portfolio, the terminal value is somewhat convex in the yield to

maturity, the terminal value of the portfolio is more convex than that of

the single bond. Redington (1952), one of the inﬂ uential propagators of

the concept of duration and immunization, thought this convexity very

desirable, and we can see why: No matter what the change in the yield

to maturity, the portfolio of bonds provides more overfunding of the

future obligation than the single bond. This is obviously a desirable

property for an immunized portfolio, and it leads us to formulate the

following rule:

In a comparison between two immunized portfolios, both of which are

to fund a known future obligation, the portfolio whose terminal value

is more convex with respect to changes in the yield to maturity is

preferable.

26.5 Building a Better Mousetrap

Despite what was said in the preceding section, there is some interest in

deriving the characteristics of a bond portfolio whose terminal value is

as insensitive to changes in the yield as possible. One way of improving

the performance (when so deﬁ ned) of the bond portfolio is not only to

match the ﬁ rst derivatives of the change in value (which, as we saw in

section 26.2, leads to the duration concept), but also to match the second

derivatives.

A direct extension of the analysis of section 26.2 leads us to the con-

clusion that matching the second derivatives requires

tt P

()

∑

The following example illustrates the kind of improvement that can

be made in a portfolio where the second derivatives are also matched.

Consider four bonds, one of which, bond 2, is our old friend from the

1. There is another interpretation of the convexity shown in this example: It shows the

impossibility of parallel changes in the term structure! If such changes describe the

uncertainty relating to the term structure, a bond position can be chosen that always

beneﬁ ts from changes in the term structure. This is an arbitrage, and therefore impos-

sible. I thank Zvi Wiener for pointing this fact out to me.

701 Immunization Strategies

previous example, whose duration is exactly 10. The bonds are described

in the following table:

BACDFE

Yield to maturity 6%

Bond 1 Bond 2 Bond 3 Bond 4

Coupon rate 4.50% 6.988% 3.50% 11.00%

Maturity 20 15 14 10

Face value 1,000 1,000 1,000 1,000

Bond price $827.95 $1,095.96 $767.63 $1,368.00 <-- =-PV($B$2,E6,E5*E7)+E7/(1+$B$2)^E6

Face value equal to $1,000 of market value 1,207.80$ 912.44$ 1,302.72$ 730.99$ <-- =E7/E9*E7

Duration 12.8964 10.0000 10.8484 7.0539 <-- =dduration(E6,E5,$B$2,1)

Second derivative of duration 229.0873 136.4996 148.7023 67.5980 <-- =secondDur(E6,E5,$B$2)/bondprice(E6,E5,$B$2)

BOND CONVEXITY

Here secondDur(numberPayments, couponRate, YTM) is a

VBA function we have deﬁ ned to calculate the second derivative of

the duration:

Function secondDur(numberPayments, _

couponRate, YTM)

For Index = 1 To numberPayments

If Index < numberPayments Then

secondDur = couponRate * Index * _

(Index + 1) / (1 + YTM) ^ Index + _

secondDur

Else

secondDur = (couponRate + 1) * Index * _

(Index + 1) / (1 + YTM) ^ Index + _

secondDur

End If

secondDur = secondDur

Next Index

End Function

We need three bonds in order to calculate a portfolio of bonds whose

duration and whose second duration derivative are exactly equal to those

of the liability. The proportions of a portfolio that sets both the duration

and its second derivative equal to those of the liability are as follows:

702 Chapter 26

bond 1 = −0.5619, bond 3 = 1.6415, bond 4 = −0.0797.

As the following

ﬁ gure shows, this portfolio provides a better hedge against the terminal

value than even bond 2:

Immunization Using Second Derivatives

1,750

1,800

1,850

1,900

1,950

2,000

New interest rate (%)

Terminal value ($)

Bond 2 Bond portfolio

161412108642

26.5.1 Computing the Bond Portfolio

We want to invest proportions x

, x

in bonds 1, 3, and 4 so that

•

The portfolio is totally invested: x

+ x

= 1.

•

The portfolio duration is matched to that of bond 2: x

+ x

= D

, where D

is the duration of bond i.

•

The second derivative of the portfolio duration is matched to that of

bond 2: x

+ x

= D

, where D

is the duration derivative.

Writing this problem in matrix form, we get

111 1

134

DDD

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥⎥

⎥

2. See next subsection for the details of this computation.

703 Immunization Strategies

whose solution is given by

DDD

134

111 1

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

−

⎤⎤

⎦

⎥

This matrix can easily be set up in Excel:

IJKLMN

Calculating the bond portfolio:

Vector of

Matrix of coefficients constants

111 1

12.8964 10.8484 7.0539 10.0000

229.0873 148.7023 67.5980 110.0000

Solution

-0.5619

1.6415 <

{=MMULT(MINVERSE(I18:K20),M18:M20)}

-0.0797

Explanation: We want to invest proportions

, x

, and x

in bonds 1, 3, and 4, respectively, in order that

(a) The total investment is $1,000; this means x

=1.

(b) Portfolio duration is matched to that of bond 2; this means

that x

= D

, where D

is the duration

of bond I.

to that of bond 2.

These three conditions give us the matrix system in

cells I18:K20 and the corresponding solution in

cells I23:I25.

Given this solution, the last chart is produced by the following data

table:

704 Chapter 26

BACDFE

Data table: Sensitivity of Bond 2 and bond

portfolio terminal values to interest rate

Bond 2

Bond

portfolio

=I23*B23+I24*D23+I25*E23 , data table header (hidden)

0% 1,868.87$ 1,774.63$

1% 1,844.71$ 1,781.79$

2% 1,825.14$ 1,786.37$

3% 1,810.05$ 1,789.02$

4% 1,799.35$ 1,790.32$

5% 1,792.97$ 1,790.78$

6% 1,790.85$ 1,790.85$

7% 1,792.95$ 1,790.91$

8% 1,799.26$ 1,791.31$

9% 1,809.76$ 1,792.38$

10% 1,824.46$ 1,794.38$

11% 1,843.37$ 1,797.58$

12% 1,866.53$ 1,802.21$

13% 1,893.98$ 1,808.46$

14% 1,925.77$ 1,816.55$

15% 1,961.98$ 1,826.65$

26.6 Summary

The value of an immunized portfolio of bonds is insensitive to small

changes in the underlying yield to maturity of the bonds. Immunization

involves setting the bond portfolio’s duration equal to the duration of

the underlying liability against which the portfolio is held. This chapter

shows how to effect the immunization of the portfolio. Needless to say,

Excel is an excellent tool for immunization calculations.

Exercises

1. Prove that the duration of a portfolio is the weighted average duration of the port-

folio assets.

2. Set up a spreadsheet that enables you to duplicate the calculations of section 26.5

of this chapter.

3. Follow the example of section 26.3 for this exercise.

a. Find a combination of bonds 1 and 3 with a duration of 8.

b. Find a combination of bonds 1 and 2 with a duration of 8.

4. In exercise 3, which portfolio (a or b) would you prefer to immunize an obligation

with a duration of 8?

5. In exercise 3, recalculate the portfolio proportions assuming that you need a target

duration of 12. Which portfolio would you prefer now?

Modeling the Term Structure

27.1 Overview

In this chapter we model the term structure using polynomial and other

regression models. The object of this exercise is to ﬁ t a curve to an exist-

ing data set of bond yields. In many cases this technique works remark-

ably well. Its use has already been illustrated in section 2.8, where we

used a ﬁ tted yield curve to estimate the cost of capital for Kraft.

27.2 An Initial Example

The following example shows data for BBB corporate bonds down-

loaded from Yahoo on 11 August 2006.

The data have been graphed

using an Excel XY Scatter Chart.

1. A description of how to download and massage the data is given in section 27.3.

706 Chapter 27

The polynomial regression shows that 85 percent of the variability of

the yields to maturity is explained by the curve

YTM Time Time Time=+∗+∗+∗0 0221 0 0152 0 0023 0 0001

.. . .

BACDE

Issuer Price Coupon Maturity YTM

Years to

maturity

PANHANDLE EASTN PIPE LINE CO 100.29 2.75% 15-Mar-07 2.24% 0.59

SAFEWAY INC 104.75 9.88% 15-Mar-07 1.62% 0.59

WEYERHAEUSER CO 102.35 6.13% 15-Mar-07 2.03% 0.59

CONSTELLATION ENERGY GROUP INC 102.35 6.35% 1-Apr-07 2.53% 0.64

PENNEY J C INC 103.10 7.60% 1-Apr-07 2.56% 0.64

MOHAWK INDS INC 102.61 6.50% 15-Apr-07 2.50% 0.68

PROGRESS ENERGY INC 102.26 6.05% 15-Apr-07 2.59% 0.68

AOL TIME WARNER INC 102.34 6.15% 1-May-07 2.78% 0.72

ARAMARK SVCS INC 102.77 7.00% 1-May-07 3.01% 0.72

BANK UTD CORP 104.32 8.88% 1-May-07 2.67% 0.72

COMCAST CABLE COMMUNICATIONS 103.96 8.38% 1-May-07 2.68% 0.72

STARWOOD HOTELS&RESORTS WRLDWD 103.00 7.38% 1-May-07 3.05% 0.72

VIACOM INC 101.90 5.63% 1-May-07 2.89% 0.72

REALTY INCOME CORP 103.41 7.75% 6-May-07 2.94% 0.73

CIGNA CORP 103.31 7.40% 15-May-07 2.88% 0.76

TELUS CORP 103.58 7.50% 1-Jun-07 2.87% 0.81

EOP OPER LTD PARTNERSHIP 102.98 6.76% 15-Jun-07 3.08% 0.84

POTASH CORP SASK INC 103.25 7.13% 15-Jun-07 3.11% 0.84

VORNADO RLTY TR 102.04 5.63% 15-Jun-07 3.10% 0.84

KENTUCKY POWER CO 102.06 5.50% 1-Jul-07 3.08% 0.89

AVALONBAY CMNTYS INC MTN BE 101.45 5.00% 1-Aug-07 3.44% 0.97

ARCHSTONE SMITH TR 101.62 5.00% 15-Aug-07 3.33% 1.01

BANK OKLA N A TULSA 103.54 7.13% 15-Aug-07 3.48% 1.01

LIBERTY PPTY LTD PARTNERSHIP 103.56 7.25% 15-Aug-07 3.57% 1.01

MASCO CORP 101.15 4.63% 15-Aug-07 3.44% 1.01

RAYTHEON CO 103.09 6.75% 15-Aug-07 3.56% 1.01

TIME WARNER COMPANIES INC 104.45 8.18% 15-Aug-07 3.59% 1.01

TANDY CORP 103.33 6.95% 1-Sep-07 3.66% 1.06

ICI WILMINGTON INC 103.27 7.05% 15-Sep-07 3.92% 1.10

PHH CORP INTERNOTES BOOK ENTRY 101.90 5.60% 15-Sep-07 3.78% 1.10

OCEAN ENERGY INC DEL 100.82 4.38% 1-Oct-07 3.62% 1.14

USA WASTE SVCS INC 103.58 7.13% 1-Oct-07 3.83% 1.14

CALENERGY INC 104.21 7.63% 15-Oct-07 3.88% 1.18

ENTERPRISE PRODS OPER LP 100.15 4.00% 15-Oct-07 3.87% 1.18

POTOMAC ELEC PWR CO 102.73 6.25% 15-Oct-07 3.82% 1.18

ROYAL CARIBBEAN CRUISES LTD 103.13 7.00% 15-Oct-07 4.20% 1.18

UNION PAC CORP 103.99 5.75% 15-Oct-07 2.24% 1.18

CVS CORP 100.01 3.88% 1-Nov-07 3.87% 1.22

YAHOO: NONCALLABLE BBB CORPORATE BONDS WITH

MATURITY 0.5–10 YEARS

11 August 2006

BBB Bond Yields, 11Aug06

y = 0.0001x

– 0.0023x

+ 0.0152x + 0.0221

= 0.8479

012345678910

Maturity

Yield (%)

707 Modeling the Term Structure

This simple model for the yield curve can be very useful. In Chapter 2,

for example, we used this model to estimate the cost of debt for Kraft.

Assuming that Kraft’s average debt maturity is ﬁ ve years, its cost of debt

is estimated to be 5.31 percent:

CBA

Average time to maturity (years) 5

Yield 5.31% <

=0.0001*B2^3-0.0023*B2^2+0.0152*B2+0.0221

COMPUTING KRAFT'S r

FROM A YIELD CURVE

YTM=0.0001*time

-0.0023*time

+0.0152*time+0.0221

The same regression relation can be used to attempt to spot mispriced

bonds. For example, the Union Paciﬁ c bond in row 39 appears to be

overpriced (its yield is signiﬁ cantly below the regression line).

BACDE GF

Issuer Price Coupon Maturity YTM

Years to

maturity

PANHANDLE EASTN PIPE LINE CO 100.29 2.75% 15-Mar-07 2.24% 0.59

SAFEWAY INC 104.75 9.88% 15-Mar-07 1.62% 0.59

WEYERHAEUSER CO 102.35 6.13% 15-Mar-07 2.03% 0.59

CONSTELLATION ENERGY GROUP INC 102.35 6.35% 1-Apr-07 2.53% 0.64

PENNEY J C INC 103.1 7.60% 1-Apr-07 2.56% 0.64

MOHAWK INDS INC 102.61 6.50% 15-Apr-07 2.50% 0.68

PROGRESS ENERGY INC 102.26 6.05% 15-Apr-07 2.59% 0.68

AOL TIME WARNER INC 102.34 6.15% 1-May-07 2.78% 0.72

ARAMARK SVCS INC 102.77 7.00% 1-May-07 3.01% 0.72

BANK UTD CORP 104.32 8.88% 1-May-07 2.67% 0.72

COMCAST CABLE COMMUNICATIONS 103.96 8.38% 1-May-07 2.68% 0.72

STARWOOD HOTELS&RESORTS WRLDWD 103 7.38% 1-May-07 3.05% 0.72

VIACOM INC 101.9 5.63% 1-May-07 2.89% 0.72

REALTY INCOME CORP 103.41 7.75% 6-May-07 2.94% 0.73

CIGNA CORP 103.31 7.40% 15-May-07 2.88% 0.76

TELUS CORP 103.58 7.50% 1-Jun-07 2.87% 0.81

EOP OPER LTD PARTNERSHIP 102.98 6.76% 15-Jun-07 3.08% 0.84

POTASH CORP SASK INC 103.25 7.13% 15-Jun-07 3.11% 0.84

VORNADO RLTY TR 102.04 5.63% 15-Jun-07 3.10% 0.84

KENTUCKY POWER CO 102.06 5.50% 1-Jul-07 3.08% 0.89

AVALONBAY CMNTYS INC MTN BE 101.45 5.00% 1-Aug-07 3.44% 0.97

ARCHSTONE SMITH TR 101.62 5.00% 15-Aug-07 3.33% 1.01

BANK OKLA N A TULSA 103.54 7.13% 15-Aug-07 3.48% 1.01

LIBERTY PPTY LTD PARTNERSHIP 103.56 7.25% 15-Aug-07 3.57% 1.01

MASCO CORP 101.15 4.63% 15-Aug-07 3.44% 1.01

RAYTHEON CO 103.09 6.75% 15-Aug-07 3.56% 1.01

TIME WARNER COMPANIES INC 104.45 8.18% 15-Aug-07 3.59% 1.01

TANDY CORP 103.33 6.95% 1-Sep-07 3.66% 1.06

ICI WILMINGTON INC 103.27 7.05% 15-Sep-07 3.92% 1.10

PHH CORP INTERNOTES BOOK ENTRY 101.9 5.60% 15-Sep-07 3.78% 1.10

OCEAN ENERGY INC DEL 100.82 4.38% 1-Oct-07 3.62% 1.14

USA WASTE SVCS INC 103.58 7.13% 1-Oct-07 3.83% 1.14

CALENERGY INC 104.21 7.63% 15-Oct-07 3.88% 1.18

ENTERPRISE PRODS OPER LP 100.15 4.00% 15-Oct-07 3.87% 1.18

POTOMAC ELEC PWR CO 102.73 6.25% 15-Oct-07 3.82% 1.18

ROYAL CARIBBEAN CRUISES LTD 103.13 7.00% 15-Oct-07 4.20% 1.18

UNION PAC CORP 103.99 5.75% 15-Oct-07 2.24% 1.18 <

Yield appears to be unusually low; bond overpriced?

CVS CORP 100.01 3.88% 1-Nov-07 3.87% 1.22

ENSCO INTL INC 103.33 6.75% 15-Nov-07 3.97% 1.26

EOP OPER LTD PARTNERSHIP 104.56 7.75% 15-Nov-07 3.96% 1.26

CAN WE SPOT MISPRICED BONDS?

Mispriced Bonds?

y = 0.0001x

– 0.0023x

+ 0.0152x + 0.0221

= 0.8479

012345678910

Maturity

Yield (%)