Simon Benninga. Financial Modelling 3-rd edition

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708 Chapter 27

Another possibly mispriced bond is the Kinder Morgan bond matur-

ing 1 September 2012, whose yield appears to be signiﬁ cantly above the

regression curve, indicating that the bond may be underpriced:

327

328

329

330

331

332

333

334

BACDE GF

CONTINENTAL CORP 109.71 8.38% 15-Aug-12 6.40% 6.02

PEPCO HOLDINGS INC 104.2 6.45% 15-Aug-12 5.62% 6.02

VIACOM INC 100.47 5.63% 15-Aug-12 5.53% 6.02

KINDER MORGAN INC 98.52 6.50% 1-Sep-12 6.80% 6.06 <

Yield is significantly above curve. Bond underpriced?

SAFECO CORP 107 7.25% 1-Sep-12 5.86% 6.06

ALCAN INC 97.28 4.88% 15-Sep-12 5.41% 6.10

AMERISOURCEBERGEN CORP 99.08 5.63% 15-Sep-12 5.81% 6.10

CINCINNATI GAS & ELEC CO 101.45 5.70% 15-Sep-12 5.42% 6.10

27.2.1 Very Short-Term Bonds May Not Fit the Model

The preceding is a valuable example of when the yield curve can be

described well by a polynomial regression. Unfortunately this technique

does not always work if we include very short-term bonds. Here’s a nega-

tive example drawn from the same data source but a different date. This

time we have included data for all corporate bond maturities:

BACD GFE

Issuer Price Coupon

Maturity

date

Maturity in

years

Yield

HOUSEHOLD FIN CORP INT NT BE 102.10 5.75% 15-Mar-07 0.05 -30.82% <

=YIELD(DATE(2007,2,23),D3,C3,B3,100,2,1)

MERRILL LYNCH CO INC MTN BE 101.62 3.00% 30-Apr-07 0.18 -5.74% <

=YIELD(DATE(2007,2,23),D4,C4,B4,100,2,1)

INTERNATIONAL LEASE FIN MTN BE 101.62 3.13% 3-May-07 0.19 -5.24%

HSBC FIN CORP HSBC FIN 101.94 5.20% 15-May-07 0.22 -3.36%

WELLS FARGO FINL INC 101.88 4.88% 12-Jun-07 0.30 -1.36%

NORWEST CORP MEDTERM NTS BK EN 102.62 6.75% 15-Jun-07 0.31 -1.70%

PROCTER & GAMBLE CO 101.94 4.75% 15-Jun-07 0.31 -1.51%

CITICORP MTN SUB BE 102.53 7.00% 1-Jul-07 0.35 -0.15%

INTERNATIONAL LEASE FIN CORP 101.34 3.75% 1-Aug-07 0.44 0.69%

GOLDEN WEST FINL CORP DEL 101.46 4.13% 15-Aug-07 0.47 1.05%

INTERNATIONAL LEASE FIN MTN BE 102.20 5.80% 15-Aug-07 0.47 1.17%

U S BANCORP MTNS BK ENT 101.36 3.95% 23-Aug-07 0.50 1.21%

BANK NEW YORK MTN BK ENT 101.28 3.90% 1-Sep-07 0.52 1.40%

BANK ONE CORP 101.41 4.13% 1-Sep-07 0.52 1.38%

SUNAMERICA INC 102.79 6.75% 1-Oct-07 0.60 2.06%

BANKAMERICA CORP 102.77 6.63% 15-Oct-07 0.64 2.24%

HOUSEHOLD FIN CORP INT NT BE 103.65 8.00% 15-Oct-07 0.64 2.22%

INTERNATIONAL LEASE FIN CORP 101.04 4.00% 15-Nov-07 0.73 2.54%

MERRILL LYNCH CO INC MTN BE 101.10 4.00% 15-Nov-07 0.73 2.45%

MERRILL LYNCH & CO INC 103.25 6.56% 16-Dec-07 0.81 2.48%

BANKBOSTON N A 102.91 6.50% 19-Dec-07 0.82 2.87%

BANK ONE CHICAGO ILL MTN BE 100.69 3.70% 15-Jan-08 0.89 2.91%

GOLDMAN SACHS GROUP INC 101.09 4.13% 15-Jan-08 0.89 2.88%

BANK NEW YORK NY MEDTERM NT BE 100.81 3.80% 1-Feb-08 0.94 2.92%

CITIGROUP INC 100.50 3.50% 1-Feb-08 0.94 2.95%

CHEVRONTEXACO CAP CO 100.47 3.38% 15-Feb-08 0.98 2.88%

WELLS FARGO & CO NEW 100.94 4.13% 10-Mar-08 1.04 3.20%

ABBOTT LABS 102.96 6.00% 15-Mar-08 1.06 3.12%

GILLETTE CO 99.72 2.88% 15-Mar-08 1.06 3.15%

LILLY ELI & CO 99.78 2.90% 15-Mar-08 1.06 3.11%

YAHOO AA BOND DATA, 23 FEBRUARY 2007

y = 2E-05x

– 0.0008x

+ 0.0109x + 0.012

= 0.262

-35

-30

-25

-20

-15

Yield (%)

Maturity

-10

-5

0 2 4 6 8 1012141618202224262830

709 Modeling the Term Structure

The data include some odd points, such as the highly negative yields

on the Household Finance bonds (row 3) and even the −5.74 percent on

the Merrill Lynch bonds. If we are willing to make some judgment calls

and throw out some of the data as inexplicable, we can improve our

prediction of the term structure.

BACD GFE

Issuer Price Coupon

Maturity

date

Maturity in

years

Yield

U S BANCORP MTNS BK ENT 101.36 3.95% 23-Aug-07 0.50 1.21% <

=YIELD(DATE(2007,2,23),D3,C3,B3,100,2,1)

BANK NEW YORK MTN BK ENT 101.28 3.90% 1-Sep-07 0.52 1.40%

BANK ONE CORP 101.41 4.13% 1-Sep-07 0.52 1.38%

SUNAMERICA INC 102.79 6.75% 1-Oct-07 0.60 2.06%

BANKAMERICA CORP 102.77 6.63% 15-Oct-07 0.64 2.24%

HOUSEHOLD FIN CORP INT NT BE 103.65 8.00% 15-Oct-07 0.64 2.22%

INTERNATIONAL LEASE FIN CORP 101.04 4.00% 15-Nov-07 0.73 2.54%

MERRILL LYNCH CO INC MTN BE 101.10 4.00% 15-Nov-07 0.73 2.45%

MERRILL LYNCH & CO INC 103.25 6.56% 16-Dec-07 0.81 2.48%

BANKBOSTON N A 102.91 6.50% 19-Dec-07 0.82 2.87%

BANK ONE CHICAGO ILL MTN BE 100.69 3.70% 15-Jan-08 0.89 2.91%

GOLDMAN SACHS GROUP INC 101.09 4.13% 15-Jan-08 0.89 2.88%

BANK NEW YORK NY MEDTERM NT BE 100.81 3.80% 1-Feb-08 0.94 2.92%

CITIGROUP INC 100.50 3.50% 1-Feb-08 0.94 2.95%

CHEVRONTEXACO CAP CO 100.47 3.38% 15-Feb-08 0.98 2.88%

WELLS FARGO & CO NEW 100.94 4.13% 10-Mar-08 1.04 3.20%

ABBOTT LABS 102.96 6.00% 15-Mar-08 1.06 3.12%

GILLETTE CO 99.72 2.88% 15-Mar-08 1.06 3.15%

LILLY ELI & CO 99.78 2.90% 15-Mar-08 1.06 3.11%

MBNA AMER BK NA SUB MTN 144A 103.54 6.75% 15-Mar-08 1.06 3.31%

U S BANCORP MTNS BK ENT 99.85 3.13% 15-Mar-08 1.06 3.27%

MERRILL LYNCH & CO INC MERRILL 99.77 3.05% 17-Mar-08 1.06 3.27%

US BK NATL ASSN MINNEAPOLISMTN 100.94 4.13% 17-Mar-08 1.06 3.21%

BANKBOSTON NA SUB BK MTN 103.20 6.38% 25-Mar-08 1.08 3.34%

BANKBOSTON NA SUB BK MTN 103.18 6.38% 15-Apr-08 1.14 3.50%

INTERNATIONAL LEASE FIN CORP 101.16 4.50% 1-May-08 1.19 3.49%

FLEET FINL GROUP INC NEW 103.33 6.38% 15-May-08 1.22 3.56%

HOUSEHOLD FIN CORP INT NT BE 100.19 3.75% 15-May-08 1.22 3.59%

YAHOO AA BOND DATA, 23 FEBRUARY 2007

Yields on bonds with maturity > 1/2 year

y = 6E-06x

– 0.0003x

+ 0.005x + 0.0292

= 0.8154

Maturity

Yield (%)

0 2 4 6 8 1012141618202224262830

It is clear that the regression provides an even better explanation for

yields of bonds with maturities greater than two years. Examining the

regression shows that the contribution of the two nonlinear terms is quite

marginal: The coefﬁ cient of Time

is 0.000006 and the coefﬁ cient of Time

is −0.0003. The term structure is, in fact, more or less linear.

27.2.2 Doing the Regressions with Linest

Excel’s Linest function (discussed in Chapter 34) can be used to estimate

the regression, as follows:

710 Chapter 27

Note that we have added two additional columns to our data—in columns

F and G we added the maturity squared and the maturity to the power

three, both factors in our regression.

By using the function Index, also discussed in Chapter 34, we can pick

out individual items from the Linest array. In the following cells we show

the regression intercept and coefﬁ cients, and also the t-statistics for the

coefﬁ cients:

JKLMNO P

Coefficients

Time 0.000006 <

=INDEX(LINEST(H3:H238,E3:G238,,TRUE),1,1)

Time

-0.000320 <

=INDEX(LINEST(H3:H238,E3:G238,,TRUE),1,2)

Time

0.004991 <

=INDEX(LINEST(H3:H238,E3:G238,,TRUE),1,3)

Intercept 0.02918 <

=INDEX(LINEST(H3:H238,E3:G238,,TRUE),1,4)

R-squared 81.54% <

=INDEX(LINEST(H3:H238,E3:G238,,TRUE),3,1)

Coefficient t-statistic

Time E),1,1)/INDEX(LINEST(H3:H238,E3:G238,,TRUE),2,1)

Time

-13.250878 <

=INDEX(LINEST(H3:H238,E3:G238,,TRUE),1,2)/INDEX(LINEST(H3:H238,E3:G238,,TRUE),2,2)

Time

20.763478 <

=INDEX(LINEST(H3:H238,E3:G238,,TRUE),1,3)/INDEX(LINEST(H3:H238,E3:G238,,TRUE),2,3)

10.351011 <

=INDEX(LINEST(H3:H238,E3:G238,,TRU

27.3 Description of the Data

Bond data are much more difﬁ cult to access than stock price data. The

data for the regression illustrated in the previous section come from

Yahoo’s bond screener (http://screen.yahoo.com/bonds.html), the data

for which are supplied by ValuBond. The following screen ﬁ nds corpo-

rate bonds with a Fitch AA rating:

ABCDEFGHIJKLM

Issuer Price Coupon

Maturity

date

Maturity in

years

Maturity

Yield

U S BANCORP MTNS BK E 101.36 3.95% 23-Aug-07 0.50 0.25 0.12 1.21%

time

time intercept

BANK NEW YORK MTN BK 101.28 3.90% 1-Sep-07 0.52 0.27 0.14 1.40% Slope 0.000006 -0.00032 0.005 0.02918

BANK ONE CORP 101.41 4.13% 1-Sep-07 0.52 0.27 0.14 1.38% Standard error 0.00000 0.00002 0.00024 0.00059

SUNAMERICA INC 102.79 6.75% 1-Oct-07 0.60 0.36 0.22 2.06%

0.81539 0.00347 #N/A #N/A

BANKAMERICA CORP 102.77 6.63% 15-Oct-07 0.64 0. 41 0.26 2.24% F statistic 341.57800 232.00000 #N/A #N/A

HOUSEHOLD FIN CORP IN 103.65 8.00% 15-Oct-07 0.64 0.41 0.26 2.22%

0.01234 0.00279 #N/A #N/A

USING LINEST TO ESTIMATE THE REGRESSION

The cells K4:N8 below contain the array equation

{=LINEST(H3:H238,E3:G238,,TRUE)}

711 Modeling the Term Structure

A typical screen of data looks like this:

712 Chapter 27

We can copy/paste this screen into Excel. Cleaning up the page some-

what gives the following Excel spreadsheet:

BACDEF HG

Issue Price Coupon Maturity YTM

Excel

Yield

function

BANKAMERICA CORP 102.77 6.63% 15-Oct-07 2.12% 2.24% <

=YIELD(DATE(2007,2,23),D3,C3,B3,100,2,1)

HOUSEHOLD FIN CORP INT NT BE 103.65 8.00% 15-Oct-07 2.09% 2.22% <

=YIELD(DATE(2007,2,23),D4,C4,B4,100,2,1)

INTERNATIONAL LEASE FIN CORP 101.04 4.00% 15-Nov-07 2.51% 2.54% <

=YIELD(DATE(2007,2,23),D5,C5,B5,100,2,1)

MERRILL LYNCH CO INC MTN BE 101.10 4.00% 15-Nov-07 2.42% 2.45% <

=YIELD(DATE(2007,2,23),D6,C6,B6,100,2,1)

MERRILL LYNCH & CO INC 103.25 6.56% 16-Dec-07 2.39% 2.48%

BANKBOSTON N A 102.91 6.50% 19-Dec-07 2.79% 2.87%

BANK ONE CHICAGO ILL MTN BE 100.69 3.70% 15-Jan-08 2.89% 2.91%

GOLDMAN SACHS GROUP INC 101.09 4.13% 15-Jan-08 2.85% 2.88%

BANK NEW YORK NY MEDTERM NT BE 100.81 3.80% 1-Feb-08 2.89% 2.92%

CITIGROUP INC 100.50 3.50% 1-Feb-08 2.94% 2.95%

CHEVRONTEXACO CAP CO 100.47 3.38% 15-Feb-08 2.87% 2.88%

WELLS FARGO & CO NEW 100.94 4.13% 10-Mar-08 3.19% 3.20%

ABBOTT LABS 102.96 6.00% 15-Mar-08 3.08% 3.12%

GILLETTE CO 99.72 2.88% 15-Mar-08 3.15% 3.15%

LILLY ELI & CO 99.78 2.90% 15-Mar-08 3.12% 3.11%

YAHOO'S BOND SCREENER--TYPICAL PAGE

Data from 23 February 2007

Comparing the Yahoo YTM with Excel's Yield function

In this spreadsheet we have used Excel’s Yield function to recompute

the Yahoo reported yield (column E). The differences between the

reported Yahoo yields and those computed by Excel’s Yield function are

not large.

The Excel Yield function looks like this:

Two Screens from Excel’s Yield Function

713 Modeling the Term Structure

Some notes on the Yield function’s inputs:

•

The Settlement date is the transaction date. The data were recorded on

23 February 2007, and we have used Excel’s Date function to enter this

date in the serial number form required by the function.

•

Rate refers to the bond’s coupon. Yahoo reports this coupon as a whole

number; we have divided by 100 to make it into a percentage.

•

Pr refers to the bond’s price.

•

Redemption is the bond’s face value.

•

Frequency is set to 2; this assumes the bond pays interest

semiannually.

•

Basis is set to 1, indicating that the computation is based on actual days

to maturity and a 365-day year.

27.4 The Treasury Yield Curve

The discussion in this chapter has thus far concentrated on corporate

bonds. The largest and most efﬁ cient bond market in the world is,

of course, the market for United States Treasury securities. A won-

derful paper by Gürkaynak, Sack, and Wright computes the daily

term structure for U.S. Treasuries from 1961 to 2006.

Here is a snapshot

from the data; note that in the early part of the period covered by

this paper, not all currently available long-term Treasury securities

existed, so that it is impossible to compute a longer-term term structure.

Notice also that the interest rates are given in whole numbers, not

percentages.

2. “The U.S. Treasury Yield Curve: 1961 to the Present,” by Refet S. Gürkaynak, Brian

Sack, and Jonathan H. Wright. http://www.federalreserve.gov/Pubs/feds/2006/ (this site

also has the data ﬁ le for this paper). The version of this data set used for this chapter

is on the disk that comes with this book.

714 Chapter 27

A few term structures from this data set are shown in the following

graph:

Four Zero-Coupon Term Structures

Maturity (years)

Coupon rate (%)

31-Oct-00

3-Apr-01

18-Nov-03

21-Jun-06

29282726252423222120191817161514131211109876 5432

715 Modeling the Term Structure

27.5 Computing Par Yields from a Zero-Coupon Yield Curve

One of the interesting details on the Federal Reserve data set is the

computation of the par yield. This is deﬁ ned as the yield X on a coupon

bond computed so that the price of the bond on a given date is 1.

()( )1

∑

A little ﬁ ddling shows that the solution for X is

−

∑

()

Computing these yields is simple if we recognize that the denominator

needs to be computed using an Excel array function. (This speciﬁ c array

function is described in the following subsection; for a general descrip-

tion of array functions, see Chapter 35.)

ABCDEF G H I JK

Maturity

(years)

Zero

coupon

yield

coupons

terminal

Par

yield

Contents of column C

Contents of

column D

Contents of

column E

1 4.98% 0.9526 0.9526 4.98% {=SUM(1/(1+$B$3:B3)^$A$3:A3)} =1/(1+B3)^A3 =(1-D3)/C3

2 4.94% 1.8606 0.9080 4.94% {=SUM(1/(1+$B$3:B4)^$A$3:A4)} =1/(1+B4)^A4 =(1-D4)/C4

3 4.92% 2.7263 0.8657 4.92% {=SUM(1/(1+$B$3:B5)^$A$3:A5)} =1/(1+B5)^A5 =(1-D5)/C5

4 4.92% 3.5515 0.8251 4.92%

5 4.94% 4.3373 0.7858 4.94%

6 4.97% 5.0847 0.7474 4.97%

7 5.01% 5.7947 0.7100 5.00%

8 5.06% 6.4683 0.6736 5.05%

9 5.11% 7.1067 0.6384 5.09%

10 5.16% 7.7112 0.6045 5.13%

11 5.21% 8.2832 0.5720 5.17%

12 5.25% 8.8243 0.5411 5.20%

13 5.29% 9.3361 0.5118 5.23%

14 5.32% 9.8203 0.4842 5.25%

15 5.34% 10.2785 0.4582 5.27%

16 5.36% 10.7124 0.4339 5.28%

17 5.37% 11.1237 0.4113 5.29%

18 5.37% 11.5138 0.3902 5.30%

19 5.36% 11.8844 0.3706 5.30%

20 5.35% 12.2367 0.3524 5.29%

21 5.34% 12.5722 0.3355 5.29%

22 5.32% 12.8921 0.3199 5.28%

23 5.29% 13.1975 0.3054 5.26%

24 5.26% 13.4896 0.2921 5.25%

25 5.23% 13.7693 0.2797 5.23%

26 5.19% 14.0376 0.2683 5.21%

27 5.15% 14.2953 0.2577 5.19%

28 5.11% 14.5433 0.2480 5.17%

29 5.06% 14.7822 0.2389 5.15%

COMPUTING THE PAR YIELD ON 9 MAY 2006

Zero versus Par Yields on 9 May 2006

4.90

4.95

5.00

5.05

5.10

5.15

Yield (%)

5.20

5.25

5.30

5.35

5.40

0 2 4 6 8 1012141618202224262830

Maturity

Zero

coupon yield

Par yield

716 Chapter 27

27.5.1 Excel Sidebar: Computing

()++

∑

Using an Array Function

The expression

()+

∑

requires that each interest rate r

in the zero-

coupon term structure be raised to the power t. There is no built-in

Excel ﬁ nancial function that performs this task. However, if we write

=SUM(1/(1+$B$3:B3)^$A$3:A3), then Excel knows that each term in

column B should be raised to the power given in the corresponding cell

of column A. This is an array function: It has to be inserted by pressing

[Ctrl]+[Shift]+[Enter].

Because we are suspicious, we always check to see if the array function

indeed works. Here’s a test; this time we use a recursive function in

column C. The results are the same as before—the Excel array function

is correct.

ABCDEF G H I

Maturity

(years)

Zero

coupon

yield in %

coupons

terminal

Par

yield

Contents of column C

Contents of

column D

Contents of

column E

1 4.98% 0.9526 0.9526 4.98% =1/(1+B3)^A3 =1/(1+B3)^A3 =(1-D3)/C3

2 4.94% 1.8606 0.9080 4.94% =C3+1/(1+B4)^A4 =1/(1+B4)^A4 =(1-D4)/C4

3 4.92% 2.7263 0.8657 4.92% =C4+1/(1+B5)^A5 =1/(1+B5)^A5 =(1-D5)/C5

4 4.92% 3.5515 0.8251 4.92% =C5+1/(1+B6)^A6

5 4.94% 4.3373 0.7858 4.94%

6 4.97% 5.0847 0.7474 4.97%

7 5.01% 5.7947 0.7100 5.00%

8 5.06% 6.4683 0.6736 5.05%

9 5.11% 7.1067 0.6384 5.09%

10 5.16% 7.7112 0.6045 5.13%

CHECKING THE ARRAY FUNCTION COMPUTATION OF THE PAR YIELD

Instead of an array function in column C, we've used a recursive function:

Cell C3 = 1/(1+r

)

; Cell C2 = C3 + 1/(1+r

)

, etc.

27.6 Summary

We can use Excel efﬁ ciently to compute a polynomial curve to ﬁ t the

term structure. This chapter has illustrated this technique using both

corporate bond yields and using a U.S. Federal Reserve data set of zero-

coupon bond rates. As illustrated in Chapter 2, curve-ﬁ tting technique is

useful for computing the ﬁ rm’s cost of debt; it may also be helpful in

spotting mispriced bonds.

717 Modeling the Term Structure

Exercises

1. The Excel notebook fm3_problems27.xls includes data for noncallable CCC-rated

bonds. The data were reported on Yahoo on 20 July 2007.

a. Graph the bond yield to maturity (YTM) on the x-axis of an XY-scatter plot,

with the bond maturity in years on the x-axis.

b. Use a polynomial regression to determine the relation between the YTM and

the bond maturity.

FEDCBA

Type Price Coupon(%) Maturity YTM(%)

Fitch

Ratin

LEVEL 3 COMMUNICATIONS INC 105.50 11.00 15-Mar-08 3.134 CCC

CCC361.580-tcO-152.705.201LED OC RTM DROF

CCC165.590-naJ-0257.657.101CNI CESU

XM SATELLITE RADIO HLDGS INC 88.75 1.75 1-Dec-09 6.864 CCC

LEVEL 3 COMMUNICATIONS INC 98.75 6.00 15-Mar-10 6.505 CCC

CCC732.701-guA-152.857.201PROC NOETSIV

CCC020.7211-beF-105.800.95CNI SDNI CEBMET

CCC505.811-guA-136.700.79CNI SU R SYOT

TENET HEALTHCARE CORP 94.00 6.38 1-Dec-11 8.010 CCC

CCC696.2221-raM-5157.700.85CNI SDNI CEBMET

TENET HEALTHCARE CORP 93.50 6.50 1-Jun-12 8.126 CCC

CCC655.721-guA-100.900.601PROC RMA

TENET HEALTHCARE CORP 93.50 7.38 1-Feb-13 8.872 CCC

CCC843.931-rpA-5188.705.39CNI SU R SYOT

CCC417.1131-nuJ-183.800.68CNI TOBLAT & EPOP

CCC308.831-guA-5188.600.19PROC DIA ETIR

CCC300.841-naJ-0211.705.59CLL oC cificaP aitocS

CCC081.941-raM-0100.752.98PROC NOETSIV

TENET HEALTHCARE CORP 102.00 9.88 1-Jul-14 9.478 CCC

TENET HEALTHCARE CORP 99.00 9.25 1-Feb-15 9.434 CCC

CCC006.861-peS-5100.994.201PROC RMA

CCC687.881-tcO-5183.700.09CNI SU R SYOT

BORDEN CHEMICAL INC 102.50 9.20 15-Mar-21 8.878 CCC

BORDEN CHEMICAL INC 92.62 7.88 15-Feb-23 8.748 CCC

CCC398.872-beF-5107.700.98PROC DIA ETIR

TENET HEALTHCARE CORP 83.00 6.88 15-Nov-31 8.543 CCC

CCC473.0174-beF-5189.952.69LED OC RTM DROF

CCC CORPORATE BONDS, NONCALLABLE, 20 JULY 2007

2. Clean up the data in exercise 1 by eliminating obviously problematic points. Repeat

the regression exercise.

3. The Excel notebook fm3_problems27.xls includes data for 249 noncallable BB-

rated bonds. The data were reported on Yahoo on 20 July 2007.

a. Graph the bond yield to maturity (YTM) on the x-axis of an XY-scatter plot,

with the bond maturity in years on the x-axis.