Electronic copy available at: http://ssrn.com/abstract=1096182

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! Term Structure Estimation Sanjay K. Nawalkha Gloria M. Soto !

IRR

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Electronic copy available at: http://ssrn.com/abstract=1096182

1

Term Structure Estimation

Sanjay Nawalkha*

Gloria M. Soto**

http://www.fixedincomerisk.com/

Last updated: February 19, 2009

Abstract

The term structure of interest rates gives the relationship between the yield on an

investment and the term to maturity of the investment. Since the term structure is typically

measured using default-free, continuously-compounded, annualized zero-coupon yields, it is

not directly observable from the published coupon bond prices and yields. This paper focuses

on how to estimate the default-free term structure of interest rates from bond data using three

methods: the bootstrapping method, the McCulloch cubic-spline method, and the Nelson and

Siegel method. Nelson and Siegel method is shown to be more robust than the other two

methods. The results of this paper can be implemented using user-friendly Excel spreadsheets.

Practical Guide Series in Fixed Income

The aim of the Practical Guide Series in Fixed Income is to provide a simple introduction to the basics of fixed

income valuation, interest rate modelling, and credit risk modelling. These guides are based upon selected

chapters in the Trilogy on the Fixed Income Valuation Course by Wiley Finance. The guides are accompanied with

free excel spreadsheets that allow readers with basic excel skills to instantly play with the various interest rate

models and credit risk models. A more detailed analysis of the topics herein and other subjects related to fixed

income analysis, is provided in the Trilogy on the Fixed Income Valuation Course. A description of the Trilogy can

be found at http://www.fixedincomerisk.com/

*Department of Finance and Operations Management, Isenberg School of Management, University of

Massachusetts, Amherst, MA 01003. E-mail: nawalkha@som.umass.edu

** Departamento de Economia Aplicada, Universidad de Murcia, Murcia, Spain.

2

1. Introduction

The term structure of interest rates, or the TSIR, can be defined as the relationship

between the yield on an investment and the term to maturity of the investment. The TSIR is

typically measured using continuously-compounded, annualized zero-coupon yields. The TSIR

is not directly observable due to the non-existence of zero-coupon bonds over a continuum of

maturity dates. As a consequence, the TSIR is generally estimated by applying some term

structure estimation method to a set of coupon bearing bonds with different maturities.

As noted by Bliss [1997], the TSIR estimation requires making three important

decisions. First, one must consider the assumptions related to taxes and liquidity premiums in

the pricing function that relates bond prices to interest rates or discount factors. Second, one

must choose a specific functional form to approximate the interest rates or the discount factors.

And third, one must choose an empirical method for estimating the parameters of the chosen

functional form. This practitioner guide focuses on the three popular TSIR estimation methods

given as the boostrapping method, the McCulloch cubic spline method and the Nelson and

Siegel method.

The structure of the guide is as follows. First, we review the notation and some

important concepts. Next, we describe each of the three tern structure estimation methods and

illustrated them using numerical examples. Finally, we describe the excel spreadsheet file for

applying the three term structure estimation methods, using a few snapshots of the

spreadsheet file.

2. The Building Blocks: Bond Prices, Spot Rates, and Forward Rates

The TSIR can be expressed in terms of spot rates, forward rates, or prices of discount

bonds. This section shows the mathematical relationship between these concepts.

2.1. The Discount Function

Under continuous compounding, the price (or present value) of a zero-coupon bond

with a face value of $100 and a term to maturity of t years can be written as:

3

!" " "( )( )

100( ) 100 100 ( )y t t

y t tP t e d t

e (1)

where y(t) is the continuously-compounded rate corresponding to the maturity term t. The

function y(t) defines the continuously-compounded term structure based upon zero-coupon

rates. The expression e-y(t)t is referred to as the discount function d(t). The typical shape of the

discount function is shown in Figure 1. This function starts at 1, since the current value of a $1

payable today is $1, and it decreases with increasing maturity due to the time value of money.

0

1

Term

Figure 1 The discount function

If a series of default-free zero-coupon bonds exist for differing maturities, then it is

possible to extract the term structure by simply inverting equation (1) to obtain y(t). However,

due to the lack of liquidity and unavailability of zero-coupon bonds for all maturities, the term

structure cannot be simply obtained by using zero-coupon bonds such as U.S. Treasury

STRIPS.

2.2. Bond Price and Accrued Interest

A coupon bond can be viewed as a portfolio of zero-coupon bonds. Consequently, the

4

price of a coupon-bearing bond that makes a total of N coupon payments, k times a year at

times t1, t2,..., tN, and a face value F, be given as:1

"

" #$ ( ) ( )1

j j N N

N

y t t y t tj

C FP

ee (2)

where C is the periodic coupon paid k times a year. Equivalently, we have:

"

" % # %$1

( ) ( )N

j Nj

P C d t F d t (3)

These formulas give what is called the cash price of a bond. This is the price that

purchaser pays when buying the bond. However, bond prices are not quoted as cash prices.

The quoted prices are clean prices, which exclude the accrued interest. Accrued interest is the

interest accumulated between the most recent interest payment and the present time. If t0

denotes the current time, tp denotes the date of the previous coupon payment, and tq denotes

the date of the next coupon payment, then the formula for accrued interest is given as:

& '!

" ( )( )!* +

0 p

q p

t tAI C

t t (4)

and the bond’s quoted price is then expressed as:

1 When compounding is discrete, each ey(t)t is replaced by (1+APR(t)/k)tk, where APR(t) is the Annual Percentage

Rate for term t compounded k times a year.

5

"

& '!" ! " # ! ( )( )!* +

$ 0( ) ( )

1

Quoted Pricej j N N

Np

y t t y t tj q p

t tC FP AI C

e t te (5)

Computation of accrued interest requires the day count basis used in the market. The

day count basis defines how to measure the number of days:

i) between the current date and the date of the previous coupon payment, or t0 – tp, and

ii) between the coupon payment dates before and after the current date, or tq – tp.

The most widely used day count basis are given as follows:

, Actual/Actual: both t0 – tp and tq – tp are measured using the actual number of

days between the dates.

, Actual/360: t0 – tp is measured using the actual number of days between t0 and tp,

and tq – tp equals 360/k, where k is the number of coupon payments made in one

year.

, 30/360: t0 – tp is measured as 30 % number of remaining and complete months +

actual number of days remaining between the dates t0 and tp, and tq – tp equals

360/k, where k is the number of coupon payments made in one year.

The Actual/Actual basis is used for Treasury bonds, the Actual/360 basis is used for

U.S. Treasury bills and other money market instruments, and the 30/360 basis is used for U.S.

corporate and municipal bonds.

Example 1

Consider a semi-annual coupon bond with a $1,000 face value and a 5% annualized

coupon rate with coupon payments on June 1 and December 1. Suppose we wish to calculate

the accrued interest earned from the date of the last coupon payment to the current date,

November 3.

6

If we use the Actual/Actual day count basis to measure the number of days between

dates, then,

t0 – tp = the actual number of days between June 1 and November 3 = 155, and

tq – tp = the actual number of days between June 1 and December 1 = 183.

The accrued interest is given as:

! & '" " "( )! * +

0 15525 $21.17

183p

q p

t tAI C

t t

where C = 50/2 = $25, is the semi-annual coupon payment.

If we used the Actual/360 convention, then,

t0 – tp = the actual number of days between June 1 and November 3 = 155, and

tq – tp = equals 360/k = 360/2 = 180.

The accrued interest is given as:

! & '" " "( )! * +0 155

25 $21.53180

p

q p

t tAI C

t t

Finally, if we used the 30/360 convention, then,

t0 – tp = 30 %5 (five complete months times 30 days each month) plus 2 (number of actual days

from November 1 to November 3) = 152, and

tq – tp = equals 360/k = 360/2 = 180.

The accrued interest is given as:

7

! & '" " "( )! * +0 152

25 $21.11180

p

q p

t tAI C

t t

Now, assume that the bond in the above example is a U.S. Treasury bond quoted at

the price of $95-08 on November 3. Since Treasury bonds are quoted with the accuracy of

thirty-two seconds to a dollar, a quoted price of 95-08 corresponds to a price of $95.25 on a

$100 face value, and hence a price of $952.50 for the $1,000 face value. Using an

Actual/Actual day count basis, the accrued interest on November 3 is $21.17, giving a cash

price equal to:

" # "Quoted Price + = 952.5 21.17 $973.67P AI

It is not the cash price, but the quoted price that depends on the specific day count

convention being applied. Any increase (decrease) in the accrued interest due to a specific

day count convention used is exactly offset by a corresponding decrease (increase) in the

quoted price, so that the cash price remains unchanged. Since the TSIR is computed using

cash prices, it is also independent of the day count convention used. Of course, it is

necessary to know the day count convention in order to obtain the cash price using the quoted

price and the accrued interest.

2.3. Yield to Maturity

The yield to maturity is given as that discount rate which makes the sum of the

discounted values of all future cash flows (either of coupons or principal) from the bond equal

to the cash price of the bond, that is:2

2 When compounding is discrete, each eyt is replaced by (1+y/k)tk. Since cash price is used in equation (6), sometimes the discount rate is also called the “adjusted” yield to maturity.

8

% %"

" #$1

j N

N

y t y tj

C FP

ee (6)

A comparison of equations (6) and (2) reveals that the yield to maturity is a complex

weighted average of zero-coupon rates. The size and timing of the coupon payments influence

the yield to maturity, and this effect is called the coupon effect. In general, the coupon effect will

make two bonds with identical maturities but with different coupon rates or payment

frequencies have different yield to maturities if the zero-coupon yield curve is non-flat. The

coupon effect makes the term structure of yields on coupon bonds lower (higher) than the term

structure of zero-coupon rates, when the latter is sloping upward (downward).

2.4. Spot Rates, Forward Rates and Future Rates

Zero-coupon rates as defined above are spot rates because they are interest rates for

immediate investments at different maturities. The forward rate between the future dates t1 and

t2 is the annualized interest rate that can be contractually locked in today on an investment to

be made at time t1 that matures at time t2. The forward rate is different from the future rate in

that the forward rate is known with certainty today, while the future rate can be known only in

future.

Consider two investment strategies. The first strategy requires making a riskless

investment of $1 at a future date t1, which is redeemed at future date t2 for an amount equal to:

!% 1 2 2 1( , )( )1 f t t t te (7)

The variable f(t1, t2) which is known today is defined as the continuously-compounded

annualized forward rate, between dates t1 and t2.

Now consider a second investment strategy that requires shorting today (which is the

same as borrowing and immediately selling) a $1 face value riskless zero-coupon bond that

matures at time t1 and investing the proceeds from the short sale in a two year riskless

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investment maturing at time t2. The proceeds of the short sale equal P(t1), the current price of

$1 face value riskless zero-coupon bond that matures at time t1. This investment costs nothing

today, requires covering the short position at time t1 by paying $1, and receiving the future

value of the proceeds from the short sale, which at time t2 equals:

%

%%"

2 2

2 2

1 1

( )( )

1 ( )( )y t t

y t ty t t

eP t e

e (8)

where y(t1) and y(t2) are zero-coupon rates for terms t1 and t2. Since both riskless investment

strategies require $1 investment at time t1, and cost nothing today, the value of these

investment strategies at time t2 must be identical. This implies that the compounded value in

expression (7) must equal the compounded value in equation (8), or:

%

!%"

2 2

1 2 2 1

1 1

( )( , )( )

( )

y t tf t t t t

y t t

ee

e (9)

Taking logarithms on both sides of the above equation and further simplification we

obtain:

!

"!

2 2 1 11 2

2 1

( ) ( )( , )

y t t y t tf t t

t t (10)

Rearranging the terms:

!

" #!

2 11 2 2 1

2 1

( ) ( )( , ) ( )

y t y tf t t y t t

t t (11)

The above equation implies that if the term structure of zero-coupon rates is upward

(downward) sloping, then forward rates will be higher (lower) than zero-coupon rates. For a flat

term structure, zero-coupon rates and forward rates are identical and equal to a constant.

10

Example 2

Table 1 illustrates the calculation of forward rates. The second column of the table

shows the continuously-compounded zero-coupon rates for terms ranging from 1 to 10 years.

The third column gives the one-year forward rates implied by the zero-coupon rates.

Table 1 Computing implied forward rates

Maturity

t

Spot rates

y(t)

Forward rates

f(t-1,t)

1 5.444

2 5.762 6.080

3 5.994 6.457

4 6.165 6.679

5 6.294 6.811

6 6.393 6.888

7 6.471 6.934

8 6.532 6.961

9 6.581 6.977

10 6.622 6.987

For example the forward rate f(2,3) is computed using equation (10) as follows:

% ! %

" " "!

0.05994 3 0.05762 2(2,3) 0.06457 6.457%

3 2f (12)

All forward rates derived in the above example apply over the discrete time interval of

one year. In general, forward rates can be computed for any arbitrary interval length, and each

length implies a different term structure of forward rates. To avoid this indeterminacy, the term

structure of forward rates is usually defined using instantaneous forward rates. Instantaneous

forward rates are obtained when the interval length becomes infinitesimally small.

Mathematically, the instantaneous forward rate f(t), is the annualized rate of return

11

locked-in today, on money to be invested at a future time t, for an infinitesimally small interval

dt - 0, and can be derived using equation (11), by substituting t2 = t + dt and t1 = t:

-

." # " #

.0

( )( ) lim ( , ) ( )

dt

y tf t f t t dt y t t

t (13)

The instantaneous forward rates can be interpreted as the marginal cost of borrowing

for an infinitesimal period of time beginning at time t. Using equation (13), the term structure of

instantaneous forward rates (or simply forward rates, from here on) can be derived from the

term structure of zero-coupon rates.

Equation (13) can be expressed in an integral form as follows:3

/ " 00( ) ( )t

y t t f s ds (14)

or,

" 00( )

( )

tf s ds

y tt

(15)

The above equation gives a relationship between zero-coupon rates and forward

rates. It implies that the zero-coupon rate for term t is an average of the instantaneous forward

rates beginning from term 0 to term t. Since averaging reduces volatility, this relationship

suggests that forward rates should be in general more volatile than zero-coupon rates,

especially at the longer end. An excellent visual exposition of the difference in the volatilities of

3 Equation (14) is the solution to the first order differential equation given in (13). This can be verified by taking the

partial derivative of both sides of equation (14), with respect to t.

12

the zero-coupon yields and those of the instantaneous forward rates is given in the excel file

guide01movie.xls explained at the end of this guide.

3. Other Practical Topics

In this section we discuss two important topics in the estimation of the TSIR: the

shape of the term structure and the selection of market data. For other advanced issues, such

as the use error-weighting schemes or penalty parameters for the estimation, we refer the

reader to Nawalkha, Soto and Beliaeva [2005], chapter 3.

3.1. The Shape of the Term Structure of Interest Rates

Estimation of the term structure involves obtaining zero-coupon rates, or forward

rates, or discount functions from a set of coupon bond prices. Generally, this requires fitting a

functional form that is flexible in capturing stylized facts regarding the shape of the term

structure. The TSIR typically takes four different shapes given as the normal shape, the steep

shape, the humped shape and the inverted shape. Figure 2 shows these four typical shapes.

Term

Zero

-cou

pon

rate

s (%

)

inverted

humped

normal

steep

Figure 2 Basic shapes of the term structure

13

The normal shape is generally indicative of an economy that is expanding normally.

That is, the term structure tends to be sloping upwards, reflecting the fact that longer-term

investments are riskier. A higher risk implies a higher risk premium and hence, a higher interest

rate. The steep shape of the term structure typically occurs at the trough of a business cycle,

when after many interest rate reductions by the central bank, the economy seems poised for a

recovery in the future. The inverted shape of the term structure typically occurs at the peak of a

business cycle, when after many interest rate increases by the central bank, the economic

boom or a bubble may be followed by a recession or a depression. Finally, the humped shape

typically occurs when the market participants expect a short economic recovery followed by

another recession so that there are different expectations at different terms. It could also occur

when moving from a normal curve to an inverted curved or vice versa4.

It is also worthy to highlight that whatever the shape, the TSIR tend to be horizontal at

longest maturities. The reason for this is twofold. First, although investors can hold different

expectations about the future of interest rates for the short, medium, and long terms, their long

term their expectations are more diffused, which makes it difficult to establish differences

between different long rates. Second, risk premiums tend to be more stable for longer terms.

3.2. Selection of Market Data

Usually, not all the bonds that trade in the market at a given time are used for the

estimation of the TSIR. The bond selected must cover a wide spectrum of maturities, should

have an enough degree of liquidity and their prices shouldn’t incorporate high distortions due to

tax effects or other market frictions. Usually, these requirements are fulfilled by the

establishment of filtering criteria for determining the bonds that qualify for inclusion in the

sample. For example, liquidity is related to both the daily trading volume and the outstanding

4 The shape of the term structure is also explained by other variables not related to expectations such as liquidity

premium, market segmentation, etc. Alternative term structure hypothesis have assigned different roles to these

variables. For a brief discussion about the main hypothesis, see Nawalkha, Soto and Beliaeva [2005], pp. 52-55.

14

amount of a bond.

The definition of filtering criteria, however, is not a simple task. A loose criterion might

improve the robustness in estimation but at the cost of introducing unrepresentative data that

might distort the resulting term structure. On the contrary, a tight filtering criteria reduces the

number of data points - and hence, detrimental to the estimation - but ensures higher quality of

data. Moreover, the choice of a threshold value for any filtering criteria might be somewhat

arbitrary. Consequently, the empirical analysis and good judgment of the researcher must

determine the filtering criteria.

4. Basic Methods for Term Structure Estimation

First attempts to estimate the term structure relied on fitting smooth functions to the

yields to maturity of bonds using regression analysis. However, this approach was

unsatisfactory due to its limitation in identifying the zero-coupon yields, and in dealing with the

coupon effect. The seminal work of J. Huston McCulloch in 1971 suggested a new method

based on quadratic splines, which focused directly on estimating zero-coupon yields and

discount factors. Much research has extended the work of McCulloch in the past three

decades. Methods for TSIR estimation must find a way to approximate the spot rates, or the

forward rates, or the discount function. Generally, this requires fitting a parsimonious functional

form that is flexible in capturing stylized facts regarding the shape of the term structure. A good

term structure estimation method should satisfy the following requirements:

, The method ensures a suitable fitting of the data.

, The estimated zero-coupon rates and the forward rates remain positive over the

entire maturity spectrum.

, The estimated discount functions, and the term structures of zero-coupon rates

and forward rates are continuous and smooth.

, The method allows asymptotic shapes for the term structures of zero-coupon

rates and forward rates at the long end of the maturity spectrum.

15

The commonly used term structure estimation methods are given as the bootstrapping

method, the polynomial/exponential spline methods of McCulloch [1971,1975] and Vasicek and

Fong [1982], and the exponential functional form methods of Nelson and Siegel [1987] and

Svensson [1994]. Extensions of the above methods are given as the heteroscedastic error

correction based model of Chambers, Carleton, and Waldman [1984], and the error weighing

models such as the B-spline method of Steely [1991] and the penalized spline methods of

Fisher, Nychka and Zervos [1995] and Jarrow, Ruppert, and Yu [2004]. In this section we

focus on the three most commonly used term structure estimation methods: the bootstrapping

method, the McCulloch polynomial cubic-spline method, and the Nelson and Siegel

exponential-form method. Some applications of the Nelson and Siegel method for term

structure modeling are analyzed in Nawalkha, Beliaeva and Soto [2007].

4.1. Bootstrapping

The bootstrapping method consists of iteratively extracting zero-coupon yields using a

sequence of increasing maturity coupon bond prices. This method requires the existence of at

least one bond that matures at each bootstrapping date.

To illustrate this method, consider a set of K bonds that pay semi-annual coupons.

The shortest maturity bond is a six-month bond, which by definition does not have any

intermediate coupon payments between now and six months, since coupons are paid semi-

annually. Using equation (2), the price of this bond is given as:

#

" 0.5 0.5(0.5)0.5

(0.5)y

C FP

e (16)

where F0.5 is the face value of the bond payable at the maturity of 0.5 years, C0.5 is the semi-

annual coupon payment at the maturity, and y(0.5) is the annualized six-month zero-coupon

yield (under continuously-compounding). The six-month zero-coupon yield can be calculated

by taking logarithms of both sides of equation (16), and simplifying as follows:

16

1 2#

" 3 45 6

0.5 0.51(0.5) ln

0.5 (0.5)

F Cy

P (17)

In order to compute the 1-year zero-coupon yield, we can use the price of a 1- year

coupon bond as follows:

#

" #1 1 1(0.5)0.5 (1)

(1)y y

C F CP

e e (18)

where F1 is the face value of the bond payable at the bond’s 1-year maturity, C1 is the semi-

annual coupon, which is paid at the end of 0.5 years and 1 year, and y(1) is the annualized 1-

year zero-coupon yield. By rearranging the terms in equation (18) and taking logarithms, we

get the 1-year zero-coupon yield as follows:

1 23 4#

" 3 43 4!5 6

1 1

1(0.5)0.5

(1) ln(1) y

F Cy

CP

e

(19)

Since we already know the six-month yield, y(0.5) from equation (17), this can be

substituted in equation (19) to solve for the 1-year yield. Now, continuing in this manner, the six-

month yield, y(0.5), and the 1-year yield, y(1), can be both used to obtain the 1.5-year yield,

y(1.5), given the price of a 1.5-year maturity coupon bond.

Following the same approach, the zero-coupon yields of all of the K maturities

(corresponding to the maturities of the bonds in the sample) are computed iteratively using the

zero-coupon yields of the previous maturities.

The zero-coupon yields corresponding to the maturities that lie between these K dates

can be computed by using linear or quadratic interpolation. Generally, about 15 to 30

bootstrapping maturities are sufficient in producing the whole term structure of zero-coupon

17

yields.

Instead of solving the zero-coupon yields sequentially using an iterative approach as

shown above, the following matrix approach can be used for obtaining a direct solution.

Consider K bonds maturing at dates t1, t2, …, tK, and let CFit be the total cash flow payments of

the ith (for i= 1,2,3,…,K) bond on the date t (for t = t1, t2, …, tK). Then the prices of the K bonds

are given by the following system of K simultaneous equations:

& '& ' & '( )( ) ( )( )( ) ( )" ( )( ) ( )( )( ) ( )( )* + * +* +

!

!" "" " # "

!

1

1 2

1 2

11 1

2 22 2

0 0( ) ( )

0( ) ( )

( ) ( )K

t

t t

K KKt Kt Kt

CFP t d t

CF CFP t d t

P t d tCF CF CF

(20)

Note that the upper triangle of the cash flow matrix on the right hand side of equation

(20) has zero values. By multiplying both sides of equation (20) by the inverse of the cash flow

matrix, the discount functions corresponding to maturities t1, t2,...,tK can be computed as

follows:

!& '& ' & '( )( ) ( )( )( ) ( )" ( )( ) ( )( )( ) ( )( )* + * +* +

!

!" "" " # "

!

1

1 2

1 2

1

11 1

2 22 2

0 0( ) ( )

0( ) ( )

( ) ( )K

t

t t

K KKt Kt Kt

CFd t P t

CF CFd t P t

d t P tCF CF CF

(21)

The above solution requires that the number of bonds equal the number of cash flow

maturity dates. The zero-coupon rates can be computed from the corresponding discount

functions using equation (1).

Example 3

This example demonstrates the bootstrapping method using the ten bonds given in

18

Table 2. For expositional simplicity all bonds are assumed to make annual coupon payments.

Table 2 Bond data for bootstrapping method

Bond # Price Maturity (years)

Annual coupon rate

(%)

1 $96.60 1 2

2 $93.71 2 2.5

3 $91.56 3 3

4 $90.24 4 3.5

5 $89.74 5 4

6 $90.04 6 4.5

7 $91.09 7 5

8 $92.82 8 5.5

9 $95.19 9 6

10 $98.14 10 6.5

Bond 1’s price is given as follows:

! %" # (1) 196.60 (2 100) ye

which gives the 1-year zero-coupon yield as:

#1 2" " "3 45 6

100 2(1) ln 0.05439 5.439%

96.60y

Using the 1-year yield of 5.439% to discount the first coupon payment from the 2-year bond,

the price of the 2-year bond is given as:

! % ! %" % # #0.05439 1 (2) 293.71 2.5 (2.5 100) ye e

19

Solving for y(2), we get,

! %

#1 2" " "3 4! %5 60.05439 1

1 2.5 100(2) 0.05762 5.762%

2 93.71 2.5y

e (22)

Following this iterative procedure, the zero-coupon rates for all ten maturities can be

computed. More directly, we can use the matrix solution given by equation (21), as follows:

& '( )( )( )( )( )( )

"( )( )( )( )( )( )( )( )* +

(1) 102 0 0 0 0 0 0 0 0 0

(2) 2.5 102.5 0 0 0 0 0 0 0 0

(3) 3 3 103 0 0 0 0 0 0 0

(4) 3.5 3.5 3.5 103.5 0 0 0 0 0 0

(5) 4 4 4 4 104 0 0 0 0 0

(6) 4.5 4.5 4.5 4.5 4.5 104.5 0 0 0 0

(7) 5 5 5 5 5 5 105 0 0 0

(8) 5.5 5.5 5.5 5.5 5.5 5.5

(9)

(10)

d

d

d

d

d

d

d

d

d

d

!& ' & '( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )* + * +

196.60

93.71

91.56

90.24

89.74

90.04

91.09

5.5 105.5 0 0 92.82

6 6 6 6 6 6 6 6 106 0 95.19

6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 106.5 98.14

Multiplying the two matrices gives the solution as:

20

& ' & '( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )

"( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )* + * +

(1) 0.947

(2) 0.891

(3) 0.835

(4) 0.781

(5) 0.730

(6) 0.681

(7) 0.636

(8) 0.593

(9) 0.553

(10) 0.516

d

d

d

d

d

d

d

d

d

d

The zero-coupon rates are obtained from the corresponding discount functions by the

following relationship derived from equation(1):

!

"ln ( )

( )d t

y tt

(23)

The zero-coupon rates are displayed in Figure 3. The points between the estimated

zero-coupon yields are obtained by simple linear or quadratic interpolation, and thus, the whole

term structure of zero-coupon rates is obtained.

6.006.16

6.296.39 6.47 6.53 6.58 6.62

5.76

5.44

5.0

5.5

6.0

6.5

7.0

0 2 4 6 8 10

Term (years)

Zero

-cou

pon

rate

s (%

)

21

Figure 3 Zero-coupon rate estimated using the bootstrapping method

The bootstrapping method has two main limitations. First, since this method does not

perform optimization, it computes zero-coupon yields that exactly fit the bond prices. This

leads to over-fitting since bond prices often contain idiosyncratic errors due to lack of liquidity,

bid-ask spreads, special tax effects, etc., and hence, the term structure will not be necessarily

smooth as shown in Figure 3. Second, the bootstrapping method requires ad-hoc adjustments

when the number of bonds is not the same as the bootstrapping maturities, and when cash

flows of different bonds do not fall on the same bootstrapping dates. For example, when two or

more bonds mature on the same bootstrapping maturity, the estimated spot rates resulting from

using each of these bonds are usually averaged. In the opposite case, when no bond exists at

a required bootstrapping maturity, a common practice is to estimate a par yield curve (that is,

the yield to maturities of bond priced at par) using simple regression models that make the

yields to maturity on current bonds depend on a series of bond characteristics including the

coupon rate and the time to maturity. Then, the yields on par bonds are estimated by assuming

that the coupon rate of each bond equals its yield to maturity. The next two methods overcome

these difficulties by imposing specific functional forms on the term structure.

Cubic-spline method

Consider the relationship between the observed price of a coupon bond maturing at

time tm, and the discount function. Using equations (1) and (3), the price of this bond can be

expressed as:

7"

" / #$1

( ) ( )m

m j jj

P t CF d t (24)

where CFj is the total cash flow from the bond (i.e., coupon, face value, or both) on date tj (j =

1,2,…,m). Since bond prices are observed with idiosyncratic errors, we need to estimate some

functional form for the discount function that minimizes these errors. We face two problems in

22

doing this. First, the discount functions may be highly non-linear, such that we may need a

high-dimensional function to make the approximation work. Second, the error terms in

equation (24) may increase with the maturity of the bonds, since longer maturity bonds have

higher bid-ask spreads, lower liquidity, etc. Due to this heteroscedasticity of errors, estimation

of the discount function using approaches such as least squares minimization, generally fits

well at long maturities, but provides a very poor fit at short maturities (see McCulloch [1971]

and Chambers Carleton and Waldman [1984]).

The cubic-spline method addresses the first issue by dividing the term structure in

many segments using a series of points that are called knotpoints. Different functions of the

same class (polynomial, exponential, etc.) are then used to fit the term structure over these

segments. The family of functions is constrained to be continuous and smooth around each

knot point to ensure the continuity and smoothness of the fitted curves, using spline methods.

McCulloch pioneered the application of splines to term structure estimation by using quadratic

polynomial splines in 1971 and cubic polynomial splines in 1975. The cubic spline method

remains popular among practitioners and is explained below.

Consider a set of K bonds with maturities of t1, t2,..., tK. years. The range of maturities

is divided into s-2 intervals defined by s-1 knot points T1, T2,..., Ts-1, where T1=0 and Ts-1=tK. A

cubic polynomial spline of the discount function d(t) is defined by the following equation:

8"

" #$1

( ) 1 ( )s

i ii

d t g t (25)

where g1(t), g2(t),..., gs(t) define a set of s basis piecewise cubic functions and 81,..., 8s are

unknown parameters that must be estimated.

Since the discount factor for time 0 is 1 by definition, we have:

" " $(0) 0 1,2, ,ig i s (26)

The continuity and smoothness of the discount function within each interval is ensured

23

by the polynomial functional form of each gi(t). The continuity and smoothness at the knotpoints

is ensured by the requirement that the polynomial functions defined over adjacent intervals (Ti-

1,Ti) and (Ti,Ti+1) have a common value and common first and second derivatives at Ti. The

above constraints lead to the following definitions for the set of basis functions g1(t), g2(t),...,

gs(t):

!

!!

!

! !#

#

# ! ## ! #

9

9:;;; !

< 9;!;

;;" =! ! ! ! !; # # ! < 9; !

;;; ! ! !& '; ! # >( ); * +?

""

1

31

11

2 2 31 1

11

1 1 11 1 1

Case 1:

0

( )6( )

( )( ) ( )( ) ( ) ( )

6 2 2 6( )

2( )

6 2

Case 2:

( )

i

ii i

i i

i

i i i i i i ii i

i i

i i i ii i i

i

i s

t T

t TT t T

T T

g tT T T T t T t T t T

T t TT T

T T T t TT T t T

i s

g t t

(27)

Substituting equation (25) into equation (24), we can rewrite the price of the bond

maturing at date tm as follows:

8 7" "

& '" # #( )* +

$ $1 1

( ) 1 ( )m s

m j i i jj i

P t CF g t (28)

By rearranging the terms, we obtain:

24

8 7" " "

! " #$ $ $1 1 1

( ) ( )m s m

m j i j i jj i j

P t CF CF g t (29)

The estimation of the discount function requires searching of the unknown

parameters, 81, 82,…,8s, that minimizes the sum of squared errors across all bonds. Since

equation (29) is linear with respect to the parameters 81, 82,…,8s, this can be achieved by an

ordinary least squares (OLS) regression.5

The above approach uses s – 2 number of maturity segments, s-1 number of

knotpoints, and s number of cubic polynomial functions. An intuitive choice for the maturity

segments may be short-term, intermediate-term, and long-term, which gives three maturity

segments of 0 to 1 years, 1 to 5 years, and 5 to 10 years, four knot points given as, 0, 1, 5, and

10 years, and five cubic polynomial functions.

McCulloch recommends choosing knotpoints such that there are approximately equal

number of data points (number of bonds’ maturities) within each maturity segment. Using this

approach, if the bonds are arranged in ascending order of maturity, i.e., t1< t2 < t3... < tK, then

the knot points are given as follows:

@ #

":;" # ! < < !=; " !?

1

0 1

( ) 2 2

1i h h h

K

i

T t t t i s

t i s

(30)

where h is an integer defined as:

5 Other functions can be optimized. McCulloch, indeed, proposed to weight the errors by the inverse of the buy-

ask spread. This, however, precludes the use of OLS techniques.

25

A B!1 2

" 3 4!5 6

1

2

i Kh INT

s (31)

and the parameter @ is given as:

A B@!

" !!1

2

i Kh

s (32)

McCulloch also suggests that the number of basis functions may be set to the integer

nearest to the square root of the number of observations, that is:

1 2" 5 6s Round K (33)

This choice of s has two desired properties. First, as the number of observations

(bonds) increases, the number of basis functions increases. Second, as the number of

observations increases, the number of observations within each interval increases, too.

Example 4

Consider the fifteen bonds shown in Table 3. This set includes five more bonds with maturities

ranging from 11 to 15 years in addition to the ten bonds given in Example 3.

26

Table 3 Bond data for cubic-spline method

Bond # Price Maturity (years)

Annual coupon rate

(%)

1 $96.60 1 2

2 $93.71 2 2.5

3 $91.56 3 3

4 $90.24 4 3.5

5 $89.74 5 4

6 $90.04 6 4.5

7 $91.09 7 5

8 $92.82 8 5.5

9 $95.19 9 6

10 $98.14 10 6.5

11 $101.60 11 7

12 $105.54 12 7.5

13 $109.90 13 8

14 $114.64 14 8.5

15 $119.73 15 9

According to the McCulloch criterion, the number of basis function is given as:

1 2" "5 615 4s Round

which implies that the number of knot points is s – 1= 3, and the number of intervals for the

maturity range that extends from 0 to 15 years is s –2 = 2.

According to equation (30), the first knot point, T1 = 0, and the last knot point T3=15

years. The second knot point, T2, is obtained as follows:

@ #" # !2 1( )h h hT t t t (34)

where using equations (31) and (32), h and @ are given as:

27

A B A B! ! %1 2 1 2

" " " "3 4 3 4! !5 6 5 6

1 2 1 15[7.5] 7

2 4 2

i Kh INT INT INT

s, and

A B A B@! ! %

" ! " ! "! !1 2 1 15

7 0.52 4 2

i Kh

s

Also, using Table 3, t7 = 7 and t8 = 8. Substituting the values of h, @, t7, and t8, given above in

equation (34), we get,

" # % ! "2 7 0.5 (8 7) 7.5T

Hence, the three knot points are T1 = 0, T2 = 7.5, and T3=15. The three knot points

divide the maturity spectrum into two segments, 0 to 7.5 year and 7.5 years to 15 years. The

number of basis functions is given as s = 4. Using equation (27), these functions are given as

follows6:

6 For calculation, note that the basis functions are continuous at the knotpoints and T1=0.

28

:! < 9;

;;" =; !& '; # >( ); * +?

:< 9;

;;" =; ! ! !; # # ! < <; !?

9:;;" =

!; < <; !?

"

2 3

1 22

1

2 22 2

3

1 22

2

2 2 32 2 2 2 2

2 33 2

2

3 32

2 33 2

4

2 6

( )

3 2

6

( )

( ) ( ) ( )6 2 2 6( )

0

( )( )

6( )

( ) for all

t tT t T

T

g t

T t TT t T

tT t T

T

g t

T T t T t T t TT t T

T T

t T

g tt T

T t TT T

g t t t

The shapes of these basis functions are shown in Figure 4. The vertical line divides

the maturity range into two segments. As can be seen these basis functions ensure continuity

and smoothness at the knot points. The discount function is given as a linear weighted

average of the basis functions as follows:

8 8 8 8" # # # #1 1 2 2 3 4 4 4( ) 1 ( ) ( ) ( ) ( )d t g t g t g t g t

29

0

10

20

30

40

50

60

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

g1(t) g2(t) g3(t) g4(t)

Figure 4 Basis functions of the discount function

The parameters 81, 82, 83, and 84 are estimated using an OLS linear regression

model given as:

8 8

8 8 7

" " "

" "

1 2 1 2! " #3 4 3 4

5 6 5 61 2 1 2

# #3 4 3 45 6 5 6

$ $ $

$ $

1 1 2 21 1 1

3 3 4 41 1

( ) ( ) ( )

+ ( ) ( )

m m m

m j j j j jj j j

m m

j j j jj j

P t CF CF g t CF g t

CF g t CF g t

Using the bond data in Table 3, the estimated values of the parameters are:

8 8

8 8

" ! "

" " !

1 2

3 4

0.00035 0.00347

0.00095 0.05501

30

The discount function is obtained using the above four parameter values and the four

basis functions. The zero-coupon rates are obtained from the discount function using equation

(23), and are displayed in Figure 5. The points in the curve correspond to the maturities of the

bonds used in the estimation and can be compared with those of the bootstrapping method in

Example 3.

5.986.13

6.266.38

6.476.55 6.60 6.63 6.65 6.68 6.70 6.72 6.76

5.675.83

5.0

5.5

6.0

6.5

7.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Term (years)

Zero

-cou

pon

rate

s (%

)

Figure 5 Estimated zero-coupon rates using cubic-spline method

A potential criticism of the cubic-spline method is the sensitivity of the discount

function to the location of the knot points. Different knot points result in variations in the

discount function, which can be sometimes significant. Also, too many knot points may lead to

overfitting of the discount function. So, one must be careful in the selection of both the number

and the placing of the knot points.

Another shortcoming of cubic-splines is that they give unreasonably curved shapes for

the term structure at the long end of the maturity spectrum, a region where the term structure

must have very little curvature.

Additionally, the OLS regression used for the estimation of the parameters in equation

(29), gives the same weights to the price errors of the bonds with heterogeneous

characteristics, such as liquidity, bid-ask spreads, maturity, etc. Other functions can be used

31

for optimization to overcome this limitation but at the cost of precluding the use of OLS

techniques7.

Finally, the choice of polynomials as basis functions is also controversial. It is argued

that the shape of the discount function estimated using cubic splines is usually reasonable up

to the maturity of the longest bond in the dataset but tend to be positive or negative infinity

when extrapolated to longer terms. This implies that it is possible to generate unbounded

positive or negative interest rates. Moreover, although the use of polynomial splines moderates

the wavy shape of simple polynomials around the curve to be fitted, this shape might not

disappear completely and hence, the fitted discount function might wave around the real

discount function introducing a significant variability in both spot and forward rates. Despite

these shortcomings, the use of polynomial splines to estimate the TSIR is widespread in the

financial industry. In fact, the superiority of exponential splines over polynomial splines has not

been clearly established.

Nelson and Siegel Model

An alternative approach that overcomes many of the shortcomings of spline

techniques is the methodology of Nelson and Siegel. The Nelson and Siegel [1987] model uses

a single exponential functional form over the entire maturity range. Nelson and Siegel suggest

a parsimonious parameterization of the instantaneous forward rate curve given as follows:

7 In fact, there are many alternative error-weighing schemes which might lead to more robust estimates of the term

structure. For example, Bliss [1997] suggests weighting each bond price error by the inverse of the bond’s

duration as a way to improve the fitting of long interest rates, which might be poor. This is due to the fact that in

absence of a weighting scheme for pricing errors, the quality of the fit of the term structure decreases with

maturity. To understand this, consider the relationship between prices, yields and maturities. A same change in

price implies a much greater change in yield in short-term bonds compared to long-term bonds. Therefore,

following a price error minimization criterion in the estimation will make interest rates corresponding to long-term

bonds to be over-fitted at the expense of shorter-term interest rates.

32

C C8 8 8! !" # #1 2/ /1 2 3( ) t tf t e e (35)

Finding the above model to be over-parameterized, Nelson and Siegel consider a

special case of this model given as:8

C C8 8 8C

! !" # #/ /1 2 3( ) t tt

f t e e (36)

The zero-coupon rates consistent the forward rates given by the above equation can

be solved using equation (15), as follows:

A B A BC CC8 8 8 8! !" # # ! !/ /1 2 3 3( ) 1 t ty t e e

t (37)

The Nelson and Siegel model is based upon four parameters. These parameters can

be interpreted as follows:

, 81+82 is the instantaneous short rate, i.e., 81+82 = y(0) = f(0).

, 81 is the consol rate. It gives the asymptotic value of the term structure of both the zero-

coupon rates and the instantaneous forward rates, i.e., 81 = y(D) = f(D).

, The spread between the consol rate and the instantaneous short rate is –82, which can be

interpreted as the slope of the term structure of zero-coupon rates as well as the term

structure of forward rates.

8 The new function resembles a constant plus a Laguerre function, which consists of polynomials multiplied by

exponentially decaying terms. This indicates a method for generalization to higher-order models.

33

, 83 affects the curvature of the term structure over the intermediate terms. When 83 > 0,

the term structure attains a maximum value leading to a concave shape, and when 83 < 0,

the term structure attains minimum value leading to a convex shape.

, C > 0, is the speed of convergence of the term structure towards the consol rate. A lower

C value accelerates the convergence of the term structure towards the consol rate, while a

higher C value moves the hump in the term structure closer to longer maturities.

Figure 6 illustrates how the parameters 81, 82, and 83, affect the shape of the term

structure of zero-coupon rates (given a constant C = 1). A change in 81 can be interpreted as

the height or parallel change, a change in 82 can be interpreted as the slope change (though

this parameter also affects the curvature change slightly), and a change in 83 can be

interpreted as the curvature change in the term structure of zero-coupon rates.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Term

Zero

-cou

pon

rate

cha

nges

E8F

E8G

E8H

Figure 6 Influence of the alpha parameters of Nelson and Siegel on the term structure of

zero-coupon rates

Figure 7 demonstrates that Nelson and Siegel method is consistent with a variety of

term structure shapes, including monotonic and humped, and allows asymptotic behavior of

34

forward and spot rates at the long end. For illustrative purposes, the consol and instantaneous

rates have been set at the same level.

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

Term

Zero

-cou

pon

rate

s

ECI

Term

Zero

-cou

pon

rate

s

E8HI

Figure 7 Influence of the curvature and hump positioning parameters of Nelson and

Siegel on the spot curve

The discount function associated with the term structures in (36) and (37) is given as:

A BA BC C8 C 8 8 8! !! ! # ! #

"/ /

1 2 3 31( )

t tt e ted t e (38)

35

Substituting this functional form into the pricing formula for a coupon-bearing bond, we

have:

A BA BC C8 C 8 8 8! !! ! # ! #

"

"$/ /

1 2 3 31

1

( )t tj j

j jm t e t e

m jj

P t CF e (39)

where tm is the bond’s maturity and CFj is the cash flow of the bond at time tj.

The parameters in equation (39) can be estimated by minimizing the sum of squared

errors, that is:

8 8 8 C

7"$

1 2 3

2

, , ,1

K

ii

Min (40)

subject to the following constraints:

88 8C

J# JJ

1

1 2

0

0

0

(41)

where 7i is the difference between the ith bond’s market price and its theoretical price given by

equation (39). The first constraint in equation (41) requires that the consol rate remain positive;

the second constraint requires that the instantaneous short rate remain positive; finally, the

third constraint ensures the convergence of the term structure to the consol rate.

Since the bond pricing equation (39) is a non-linear function, the four parameters are

estimated using a non-linear optimization technique. As non-linear optimization techniques are

usually sensitive to the starting values of the parameters, these values must be carefully

chosen.

Despite this computational difficulty, the Nelson and Siegel model, and its extended

version given by Svensson [1994], have a prominent position among term structure estimation

36

methods. The smoothness of the estimated curves for both spot rates and forward rates, the

asymptotic behavior of the term structure over the long end, and their robustness to outliers

and errors in market data are the main advantages these methods compared to spline

methods. In fact, as reported in BIS [2005], most Central Banks use these methods for term

structure estimation. Also, in recent years, these models are attracting the interest of

researchers in the area of interest modelling and portfolio risk management. Matzner-Løber

and Villa [2004] and Diebold and Li [2006], for example, reinterpret them as modern three-

factor models of level, slope and curvature factors in the most pure tradition of Litterman and

Scheinkman [1991] and Bliss [1997] and obtain empirical evidence in favor of them.

Example 3.5

Reconsider the bond data in Table 3. The initial values of the parameters may be

guessed using some logical approximations. For example, the starting value for the parameter

81, which indicates the asymptotic value of the term structure of zero-coupon rates, may be set

as the yield to maturity of the longest bond in the sample. In Table 3, the longest bond is Bond

15, whose continuously-compounded yield-to-maturity can be calculated using the “solver”

function in excel and is given as 6.629%.9

The starting value for the parameter 82, which is the difference between the

instantaneous short rate and the consol rate, may be set as the difference between the yields

to maturity of the shortest maturity bond and the longest maturity bond in the sample. In Table

3, this corresponds to the difference between the yields to maturity of Bond 1 and Bond 15,

given as 5.439-6.629 = –1.19%.

The starting values of the other two parameters, 83 and C, which are associated with

the curvature of the term structure and the speed of convergence towards the consol rate,

9 Generally, the yield to maturity is reported using discrete compounding. This yield may be used as the starting

value too, since it is quite close in value to its continuously-compounded counterpart.

37

respectively, are difficult to guess. Therefore, it might be necessary to consider a grid of

different starting values. The feasible range for these parameters for realistic term structure

data are –10 < 83 < 10, and C ranging from the shortest maturity and the longest maturity in

the sample of bonds. Hence, using Table 3, 1 < C < 15.

Using the data in Table 3, we obtain the following values of the parameters using non-

linear optimization:

8 8

8 C

" " !

" "

1 2

3

0.07000 0.01999

0.00129 2.02881

The corresponding term structure of zero-coupon rates is shown in Figure 8.

5.00

6.176.29

6.396.47

6.53 6.58 6.62 6.66 6.68 6.71 6.73 6.75

5.76

5.44

5.99

5.0

5.5

6.0

6.5

7.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Term (years)

Zero

-cou

pon

rate

s (%

)

Figure 8 Estimated zero-coupon rates using Nelson and Siegel method

5. Software description and download

In this section we briefly describe the Excel files that accompany this guide. They

allow readers with basic Excel skills to instantly play with term structures and the methods

described in this guide.

38

There are two files associated to this guide, named TSIRmovie.xls and TSIR.xls. They

are fully operative and can be downloaded from the website: www.fixedincomerisk.com. See

the Guides tab.

5.1. TSIRmovie.xls

This file gives a visual “movie” illustration of the evolution of the U.S. term structure of

interest rates between December 1946 and February 1991 based upon McCulloch and Kwon

[1993] term structure data.

The above picture gives a snapshot of the excel file. As can be seen, there are three

boxes with options and controls and a graph showing different term structures. The first box in

grey area in the top-left corner allows the user to choose one or more term structures among

the zero-coupon yield curve (labeled as TSIR), the par-bond yield curve (PAR) and the

instantaneous forward rate curve (FWD). The second box in grey area to the top-right corner

gives the option of including in the graph an additional curve for comparative purposes. This

curve is represented by a dashed line in the graph and will not change during the movie.

Choosing the option ‘None’ imply that no comparative curve will be shown. The option ‘A

39

specific month’ shows the curve of the month entered by the user. The option ‘A specific year’

shows the average curve of the year chosen. Finally, the yellow area in the bottom-left is the

box of controls for the movie. The user must first enter the starting month for the movie. Then,

the user can move forward or backward in time to see how the term structure changes

dynamically in the graph using the spinner control or view the entire movie from the starting

month to the last month in the dataset by pressing the button ‘View Movie’. In this latter case,

when the box ‘Slow motion’ is checked, the movie goes at a slower speed.

5.2. TSIR.xls

This file allows estimating the term structure of interest rates from a set of bond data

and then using the estimated curves to price bonds. The file has two sheets, one for each

purpose, titled as ‘TSIR Estimation’ and ‘Bond Pricing,’ respectively.

The picture below gives a snapshot of the sheet ‘TSIR Estimation’. The sheet includes

a graph showing the estimated term structures and four areas of data labelled as ‘Input bond

data’, ‘Input terms for the graph’, ‘Cubic spline’ and ‘Exponential form’. In the top of the sheet,

the button ‘Clear all’ when pressed deletes all the data in the sheet.

40

In the first two areas, whose titles are in a yellow font, the user must enter the input

data for term structure estimation. The blue arrows in the different columns of these two areas

allow sorting the data either in ascending or descending order. The area ‘Input bond data’ must

include the list of bonds and their characteristics. The maximum number of bonds is 100. For

each bond, the user must enter an alphanumeric code or name, the cash (or dirty) price in

dollars, the face value, the coupon rate in annual terms, the number of coupons the bond pays

a year and the bond’s maturity in years. If bond characteristics are incomplete or wrong, the

user will receive an error message during the estimation. Also, bonds lacking a code are

discarded from the estimation if they are placed at the end of the list. The column ‘Input terms

for the graph’ must be filled with the terms that will constitute the estimated term structure of

interest rates.

Once the required data has been entered, the user can press the buttons ‘Estimate’

41

under ‘Cubic spline’ or/and ‘Exponential form’ to obtain the estimated interest rates for the

terms specified by the user using McCulloch’s cubic splines or Nelson and Siegel’s method,

respectively. The estimation could take some time, especially in the case of Nelson and

Siegel’s method due to the non-linear optimization. The original data in the sheet allows the

user to obtain the term structures shown in this guide for both methods.

The snapshot of the second sheet labelled ‘Bond Pricing’ is shown bellow. The sheet

consists on a graph showing the bond prices obtained from the estimated term structures, an

input area titled ‘Input bond data’ and an output area titled ‘Calculate bond price’.

The bond data to be filled in the first area is the same described above with the only

42

exception of bond prices, not needed in this case. The estimated prices of the bonds with the

specified characteristic are shown in the area ‘Calculate bond price’ once the button ‘Spline’ or

‘Exponential’ is pressed. The bond prices under the button ‘Spline’ use the term structure

obtained using McCulloch’s cubic splines and those under ‘Exponential’ use the term structure

estimated using Nelson and Siegel’s method. The button ‘Clear all’ at the top of the sheet

delete all the data in the sheet.

43

References

BIS, 2005, Zero-coupon yield curves: Technical documentation, Monetary and Economic Department, BIS Papers 25, October 2005, Bank for International Settlements.

Bliss, R.R., 1997, “Movements in the Term Structure of Interest Rates”, Economic Review, FRB of Atlanta, fourth quarter, 16-33.

Chambers, D.R., W.T. Carleton and D.W. Waldman, 1984, “A New Approach to Estimation of the Term Structure of Interest Rates”, Journal of Financial and Quantitative Analysis 19(3), 233-252.

Diebold, F.X., Ji, L. and Li, C., 2006, A Three-Factor Yield Curve Model: Non-Affine Structure, Systematic Risk Sources, and Generalized Duration, in L.R. Klein (ed.), Long-Run Growth and Short-Run Stabilization: Essays in Memory of Albert Ando. Cheltenham, U.K.:Edward Elgar, 240,274.

Fisher, M., D. Nychka and D. Zervos, 1995, Fitting the Term Structure of Interest Rates with Smoothing Splines, Working Paper 95-1, Finance and Economic Discussion Series, Federal Reserve Board.

Jarrow, R., D. Ruppert and Y. Yu, 2004, “Estimating the Term Structure of Corporate Debt with a Semiparametric Penalized Spline Model”, Journal of the American Statistical Association 99, 57-66.

Litterman, R., Scheinkman J., 1991. Common factors affecting bond returns, Journal of Fixed Income, June, 54-61.

Matzner-Løber, E. and Villa, C., 2004, Functional Principal Component Analysis of the Yield Curve, International Conference AFFI 2004, France.

McCulloch, J. H. and H.C. Kwon, 1993, U.S. Term Structure Data 1947-1991, Ohio State University, Working Paper, 93-96.

McCulloch, J.H., 1971, “Measuring the Term Structure of Interest Rates”, Journal of Business 44, 19–31.

McCulloch, J.H., 1975, “The Tax Adjusted Yield Curve”, Journal of Finance 30, 811–830.

Nawalkha, Sanjay K., Gloria M. Soto, and Natalia A. Beliaeva, 2005, Interest Rate Risk Modeling: The Fixed Income Valuation Course, Wiley Finance, John Wiley and Sons, NJ.

Nawalkha, Sanjay K., Natalia A. Beliaeva, and Gloria M. Soto, 2007, Dynamic Term Structure Modeling: The Fixed Income Valuation Course, Wiley Finance, John Wiley and Sons, NJ.

Nelson, C.R. and A.F. Siegel, 1987, “Parsimonious Modeling of Yield Curves”, Journal of Business 60(4), 473-489.

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Steeley, J. M., 1991, “Estimating the Gilt-Edged Term Structure: Basis Splines and Confidence Intervals”, Journal of Banking, Finance and Accounting 18(4), 513-529.

Svensson, L.E.O., 1994, Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994, Institute for International Economic Studies.

Vasicek, O.A. and H.G. Fong, 1982, “Term Structure Modeling Using Exponential Splines”, Journal of Finance 37(2), 339-348.