Estimating the Term Structure of Interest Rates

by

Mark Deacon

&

Alldrew Derry"

·Bank of England, ThreadnecdJe Street, London, EC2R RAI-f, T h e views expressed are those of the authors and not necessari ly t hose of the Bank of England. We are extreme l y grateful to our coll eagues N i cola Anderson, lan Bond, Francis Breedon, Martin Brookes, Kater ina Mastronikola and David Miles for many helpful comments and suggestions. We would also like to thank Dr Nick Webber of the Un i ve r s i t y o f W a r w i c k w h o m a d e e x tensive c o m m e n t s on a n e a r l i e r d r a f t. Rosemary Denney and Jim Lewin provided excel lent research assistance. T h e usual disclaimer applies. ©Bank of England 1994 ISBN 1 85730 082 3 ISSN 0142-6753

Contents

Abstract

3

1

Introduction

5

2

Nota tion a nd some definitions

6

3

Estima ti ng yield curves

4

Modelling the effect of tax

17

(the "Coupon Effect")

36

5

A comparison of the three models

52

6

Conclusion

56

A ppend ix A

58

Append ix B

64

A ppendix C

66

References

69

Abstract This pa per examines va riou s tec h n i q u es u sed to estima te the term s tructu re of i n terest rates from the prices of go vernment bond s; i n partic u l a r compa ring the cu rre n t B a n k of Engla nd mod e l w i th t wo approaches suggested in the academic litera ture. There a re two main a spec ts of this problem : estimating the rela tionsh i p bet ween bo nd yiel d s and ma turi ty, a nd the rel a tionship be tween bond y ield s a nd coupon. TI,e paper outlines how these problems are approached by the three models, a nd co mpa res them on both theoretical a nd practical gro u nd s . I t concl u des tha t t here i s a trad e-off bet ween theore tical rigour and practical considerations.

3

1

Introduction

When pricing fi nancial i n s truments, agents throughou t the financial markets a re (either explici tly or implici tly) revealing information on the i n terest rates that they rega rd as bei ng appropriate for the pa rticular transactions they a re making; but these prices or yields may also reflect o ther factors such as the effect of taxa tion ru les and the perceived risk of defa u l t by the issuer. Isolating the implied interest rates is therefore a fa r from tri vial task. I t can reasonably be a ss u med tha t a u nique ( theore tica l ) u nderlying ra te e x i s ts for each m a t u ri ty, and so when trying to reco ver these w e a re a i ming to con s t ru c t a f u nction that describes a single i n terest rate for each maturity

interest rates.

-

the term structure of

This i s u sed for a nu mber o f purposes. For exa mple, the

Bank of England advises HM Treasury on appropria te i nterest ra tes to charge local au thori ties and some na tiona lised indu stries who borrow money through the Na tiona l Loa ns F u nd (NLF ) or the Public Works Loan Board (PWLB). Insti tu tions or individuals u nderta king financial t ra nsactions ma y w a n t to k n o w how their o w n o pi n i o n s rel a te to ' ma rket' o p i n i o n s . I t is a l so u sefu l fo r f i n a n c i a l eco n o m i s t s; for example, s u ch d a ta a re / can be u sed to es tima te the para me ters o f general equ ilibrium term structure models, and to test their stabili ty (eg Cox, I ngersoll and Ross 1985, Longsta ff and Schwartz 1992 ). Government securi ties a re generally u sed in the estima tion of the term s tru cture of i n teres t ra tes, si nce they a re free of defa u l t ri sk . I f there were a 'su i table' go vernment bond (ie si ngle payment, l i q u i d , etc) ma tu ring a t every fu ture date we cou ld simply take the interest rate on tha t bond as the u nderlying i n terest ra te for tha t ma turity. I n the U K, however, gove rnment bon d s

-

gilt-edged seCllrities

-

a re no t e q u a l l y

spaced through the maturi ty spectru m : there a re 'ga ps' for w hich w e n eed s o m e fo rm o f i n te r po l a t i o n to i d e n t i fy a co n t i n u o u s t e r m s tru cture. Moreover, there a re n o single pa ymen t ( zero cou po n ) UK

5

governme n t bonds, m so the problem i s fu rther complica ted by the existence of semi-annual {2 } interest or 'coupon' payments. This pa pe r exa m i nes v a r i o u s techniques u sed to recover the term structu re o f i n t eres t rates from U K government bond prices. Some f u n d a mental concepts are defi ned in Section

2,

while the res t of the

p a per c o m p a re s the B a n k ' s c u rren t y i eld c u r ve m o d e l w i th two commonly u sed term structure mod e l s i n the academic l i tera ture. Section

3 describes how the various models estimate the fundamental

term structure (or yield -maturity rela tionship), and Section 4 outlines how each model acco u n ts for the complica tions cau sed by cou p o n ( i n terest) payments. Section 5 presents examples o f curves produced by the various methods and Section 6 concludes.

2

Notation and some definitions

Before d iscussing the i ssues invol ved in estima ting yield cu rves, i t is u seful to set ou t the nota tion and terminology u sed i n the rest of the paper. Whilst some of the analysis is specific to the gilt-edged market}3> the ma i n i ssues are rel e v a n t when estima ting the term stru c t u re of i n terest ra tes for any govemment bond market.

(1)

Other than short-term Treasury bills.

(2)

2 112 % Consols pays interest quarterly. but is the .:x ception rath.:r than the rule.

(3)

In particular the treat ment of taxation (Section 3) is specific to the UK case. the details of which can be found in "British Government Securities: The Market i n Gilt-Edged Securities" (pages 24-5) pu blished by the Bank of England.

6

2 .1

The bond price e q uation

A bond is simply the obl igation on the bond' s issuer to provide one o r more fu ture cashflow(s). For a conventional (4) UK government bond, the stream of cashflows consists of regular (semi-annual) fixed in terest, or ' coupon' payments a nd a redemption payment w hich is paid with the final coupon payment on the gil t' s maturity date. The market price o f a co n v e n ti o n a l bond i s t h e ma rket v a l u a t i o n o f the s t rea m o f cashflows associa ted with that bond. A

the rate at wh ich an individ ual cashflow (either a

spot interest rate is

c o u pon o r a redemption paymen t ) i s d i sco u n ted . I f s p o t ra tes for payments at a l l da tes in the fu ture a re known, then the price (S) of a bond ma turing in

m period s can be equa ted

to the pre sent va l ue o f

futu re cashflows:

c

Price

(1+ r

where:

C R

r. ,

(4)

(5)

= = =

+ 1

)

c

(l+r ) 2

2

+

•

•

•

•

•

•

•

+

R

+ C

(1+r )

(1)

m

m

coupon redemption payment the spot ra te applicable for a payment in period (i=l, ... , m)

i

There are other kinds of UK govern ment bonds: i ndex - l i nked (with payments linked to t h e R e tail Price Inde x), irredeemable ( w i t h no con t r a c t u a l rede m p t io n d a te ), double-dated (with a period, usual l y of several years, in w h i ch the gove r n me n t can repay the bond) and convertihle (which give the holder the option to convert into other (conventional) bonds at particular dates). Where 'price' is the sum of the quoted ('clean') price and accrued interest - see section 2. 3 below.

7

2.2

Discount factors and the discount function

The bond price equa tion (1) describes how the price of a bond can be calculated if all the spot rates rj (i=1, ...,m) are known. This equation is often w ri tten i n terms of discoun t factors, so tha t the present value of each cashflow i s written a s the prod uct of i ts nominal value and its discount factor:

or:

m Price

=

C E d

i

+

d

i =l

w here

dj i s

m

(l)

R

the d iscoun t factor for period

i (i=l,. . ,m) and .

i s si mply a

transformation of the ilh period spot rate:

d

i

1

=

i =

(l + r ) i

i

1,

..

.

,m

(3)

I t i s o ften u se ful to t h i n k o f the co n t i n u o u s a nalogue to the set o f

discollnt fllHctioH 5(t), a s a con tinuous function that maps time t to a discount factor. Equi valently 5(t) is the present value of £1 recei vable at time t, and so gi ven a con tinuous discou nt fu nction d iscount factors, the

the present value of a cashflow at any point in the fu ture can easily be calculated. A set of d i scou n t factors

dj (i=l, ,m) ...

ca n therefo re be

thought of as discrete points on the continuous discount function

d . = 5(t.) ,

,

8

5(t):

where

tj is the

time to the end of the

ith period .

In terms of the discount

function, the bond price equation becomes:

m Price = C E 6 ( t ) + i i=l

2.3

(4)

6 ( t )R m

Accrued interest and continuous compounding

The bond price equation (1) is over-simplified since i t assu mes that the next cashflow is due in exactly one period's time. In fact, while coupon pa yments on individual bonds are made at fi xed d a tes, bonds can be traded on any working day. Whenever a bond is traded on a day that is not a coupon payment d a te, the valua tion of the bond will reflect the proximity of the next cou pon paymen t d a te. In the UK, for exa mple, the buyer pays

accmed i1lferest

to compensa te the seller for the period

since the last coupon payment during which the seller has held the gilt b u t for which they w i l l recei ve no coupon pa yment. (61 The accrued interest is by market convention calcu la ted si mply as the proportion of t he c o u p o n forego n e by t h e selle r, e x p re s s e d a l ge b r a i ca l ly i n equa tion (5): ai

where:

=

QI

t1

(6)

Cl

x

= =

C

(5 )

accrued interest time to the next receivable dividend payment (as the actual nu mber of d ays d i vided by the n u m ber o f days in a "standard" year. (7) )

There is a period (usually 37 days) bdore each coupon date when the bond is traded

ex-divideud, ie without the right to the next coupon paymen t, and in this period

(between the ex-dividend date and the coupon payment date) accrued in terest

is negative since it is the buyer who is givi n g up part of the n ext coupon payment. (7)

For the Un ited Kingdom. the market conven tion is to assume that a "standard" year consists of 365 da ys. In some other cou ntries. such as the United S t ates. accrued i nterest is instead calculated on a 360 day basis.

9

A bond's p rice can therefore be decomposed i nto two components: the accrued interest a nd the bond's

clean price.

I t is the clean price of a gilt

t h a t i s u s u a l l y q u o te d, s i n c e m o v e m e n t s in t h e c l e a n p r i ce a re i ndependent of the (exactly predic table) cha nges i n accrued interest. The

dirty price is

the actual market valuation of the bond as given by

equa tion (1), at which transactions take place; and is simply the clean (quoted ) price plu s any accrued interest. Between coupon payment dates, the bond price equation (1) needs to be m od ified to a l l o w for the fact tha t the ne x t cou pon paymen t is not exactly o ne period in the fu tu re. (S) This is straightforward with either discrete discount factors or a continuous discount function; the latter c a se ( fo r a bond w i t h

m

rema i ning coupon payments) is sh o w n i n

equa tion (6) below:

P

+ ai

w here:

::

C 6 (t I)

P

=

al

=

t1

=

C

=

R

=

+

C 6 ( tI+ I)

+

•

•

•

+

( C +R )

6 ( t I + (m-I ) )

(6)

clean price accrued interest (equation (5» t i m e to fi r s t cou pon p a y m e n t ( a s a f r a c t i o n o f a period) coupon redemption payment

A l t h o u g h a c c r u ed i n t e res t ca l c u l a t i o n s a re c o n c e p t u a l l y s traigh tforward, in prac tice they can be a n a wkward compl ica tion to empirical work . To avoid this, McCu lloch

(8)

(1971 , 1975) approxima tes

There are further complications when considering a bond that has recently been issued. If (as i s usua\1 y the case ) i t was not issued on a coupon payment date. the first-ever rece i vable d i vidend wil\ be less than t he usual coupon payment. reduced to reflect the fact that the holder will not hold the bond for the ful1 coupon period. Furthermore. gilts are often issued part l y paid. which reduces the first coupon payment sti\1 further and i n troduces negative cashflows i nto the right- hand side of the price equation (amounts pa y a b l e by t h e h o l d e r ) . The r e q u i re d ( a l gebra i c ) a l t e r a t i o n s are r e a s o n a b l y straightforward but are not give n here.

10

the bond price equ a tion (1) by a ssu ming that cou pon paymen ts are made con tinuously ra ther than at d iscrete poi n ts i n time, so i n terest does not accrue. This assumption of C011ti11lWHS

compounding means that

the price equation can be slightly simpli fied :

p

=

rn

C f cS (�) o

where:

P

d� + R cS (rn)

(7)

=

clean price ma turity of the bond (c, R and 6 as defined before)

m

=

The con tinuous compounding approximation ca n significa n tl y a l ter estima tes of the discoun t function (and of the deri ved yield cu rves),

so

this a pprox imation error sho u l d be weighed against the percei ved benefit from simpli fying the ca lcula tions if continuous compou nding is to be considered. The following sections describe the methodology for both the con tinuous and discrete compou nding cases, but all resu l ts i n Section 5 were produced using only the (more precise) discrete method.

2.4

Yields

S i nce the c o u p o n a n d the r e d e m p t i o n p a y m e n t a re k n o w n , i t i s s traightforwa rd to measure the return o n a gilt trading a t a particular p ri c e . There a re two mea s u r e s c o m m o n l y u se d : the flat ( so m e t i m e s referred to a s t h e

ClIrre71t

or

rll1l1li1lg

yield

yiel d ) a n d the

redemptioll yield. The flat yield is a nalogous to the 'di vidend yield' on an equ i ty, and is defined as:

Flat Yield

=

Coupon

(8)

Clean Pric e

11

The fla t yield i s essen tially u sed to v a l u e the ret u rn from hol d i ng bond for a short period - and is often thought of as the

a

income from the

bon d . Common market practice is to compare the fla t yield on a bond w i th a short-term interest rate - if the flat yield is below the short-term i nterest rate, the holder is

ceteris paribus

incu rring a short-term cost by

holding the bond . The

redemption yield

(or

yield to maturity) correspond s to the internal rate

of retu rn on the bond. As such, it can be seen that the redemption yield i s d er i v e d fro m the b o n d p r i ce e q u a t i o n (1 ) w i th a l l c a s h flo w s discounted a t the same rate:

P

c

+ ai

+

( l+y)

where

y

=

c

--

(1+ y)

2

+

R •

•

•

•

•

•

•

+

+

C

--

(1+y)

m

(9)

(gross) redemption yield

(P, ai, C and R are defined as before) Gi ven a price, equ a tion (9) is sol ved for the red emption yield

y u sing

some form of non-linear itera tion technique (eg Newton-Raphson). If the bond is to be held to redemption, the redemption yield is clearly a better measure of return than the fla t yiel d . Ho wever, i t rarely equ a ls the reali sed return since it assumes that all fu ture cou pon payments can on average be reinvested at the internal rate of return. Of the two mea sures, the red emption yield is the more widely u sed . For the rest of the paper the term ' y i eld' will speci fica lly refer to the redemption yield.

12

2.5

Yield curves

The d i scou n t fu nction 6(t) can be u niquely tra n s fo rmed i n to other useful functions, such as the spot ra te (or zero coupon) cu rve, par yield curve a n d i mpl ied forwa rd ra te cu rve. Simi l a rly, a set of regularly spaced

discrete

d iscou n t factors

dj ( i = l, ... ,m )

can be transformed into

correspond ing discrete spot rates, par yields and implied forward rates w hich, if su fficiently closely spaced, can be plotted as a con tinuo u s curve. This section describes how, gi ven a discoun t function or set of d isco u n t fac tors, the other cu rves can be deri ved . I t i s i mporta n t to note tha t all these transforma tions a re u nique, so gi ven any one of the four curves the other three can be derived .

Implied forward rates In equ a tion (3) the discou n t factor for period i (in d iscrete ti me), dj, is given in terms of the corresponding spot rate, rj, by the rela tionship:

d = 0 + rJi I

I

The spot rate rj can be thought of as an average(9) of all the implied one period forward rates [1' [2' ..., [j

so

that:

(10)

lid

III

(9)

=

O+r )111 = 0+[1)0+/2) III

. • .

0+'J ) III

From equat ion (10) it is clear that ( 1 H,) is the geometric mellt1 of (1 +[1)' (1 +[2)'

(1 +1;).

13

. , ..

The

implied forward rate fi for any period can lid

( 1+[ ) ( 1+[ ) 1 2

i =

lid

..

d

d

i

...

( 1 +[ ) i -l

1

i - d

i-1

i

=

-

[

( 1 +[ ) (1d ) i i-1

( 1 +['>

=

d [ i

( 1 +[ ) ( 1 +[ ) 1 2

i -l

i-1

...

therefore be isolated using:

d

i

lld

i

where

=

d

lld

i

=

d

i

d

( 11)

i -1

i

The above is the d i screte compou nd i ng case. Using the continuous d i scou n t f u n c t i o n

6 (t )

a n d a ss u m i n g tha t i n terest i s compou nded

il1stalltal1eOIlS forward rate curve p(t) by considering equa tion (11) with period s i and (i-1) infini tesimally

continuou sly we can therefore derive an close:

p (t)

=

-

("

(t)

(1�)

(, (t)

The i n s t a n t a n eo u s fo r w a rd r a t e cu rve i s a t heore ti c a l con s t r u c t , providing the interest rate appl icable on a fu ture loa n that i s repaid a n i n s t a n t l a te r . A m o r e u s e f u l mea s u re to co n s i d e r ( w he n u s i n g c o n t i n u o u s compo u n d ing) i s the a v erage o f interval

[ tl, t2].

This

p(t)

over a pa r t i c u l a r

mean forward rate fal, t2 ) is given by:

14

{(c

1

,

C

1

C ) 2

C - C 1 2

The for ward ra te

I

2 p ( jl)

C

( 13 )

djl

1

f(tj_l,t,>

i n equa tion ( 1 3 ) there fo re represen t s the

continuous compounding approximation to the d iscrete forward rate fj i n equation (11).

Spot (or zero coupon) curve The s p o t r a te

rj

i s some t i me s c a l led t he

zero coupon yield

s i nce i t

represents the y ield to maturi ty o n a (hypo thetical) pure d i scoun t or zero coupon bon d , and can be easily deri ved from t he a ppropria te d i scou n t factor u s i ng equ a t i o n (3). The con t i n u o u s compo u n d i ng approxi mation

curve,

ll(t)

to the term

structu re of spot rates, or zero coupon yield

can be derived from equation (13) since the spot rate for payment

a t t i me

t

i n t he fu t u re is the a verage i ns ta n ta ne o u s for w a rd r a te

between now

'I1 ( C )

=

(t1=O) and

time t

(t2=t).

So:

f(O,c)

and hence from equation (13):

t

••

7J (t)

- In 6(t) t

(assu ming

(10)

(14)

=

15(0) 1 ). (10) =

The assumption that t he discou nt fu nction equals u ni t y at t i me t=O is a sensible restriction, Implying that an amount receivable now is not discoun ted. This,

and other, restrictions are discussed in more detail in Section 3.

15

The equivalent of (14) for the case of discrete compounding is:

-

1

( 15)

The zero cou pon y ield c urve i s the con s t r u c t to which economis ts u sually refer when talking about the tenn structure of interest rates.

The par yield curve A (cou pon-paying) bond is said to be priced

at par if i ts curren t market

price is R , its face (or par) value. F rom equ a tion (9) i t can be shown that for a bon d to be trading at par, i ts redemption yield must equal i ts c o u po n . U s i n g t h i s fa c t , t h e

par y ield Y m

e q u a ti o n s ( 2 ) a n d (9) for a n y period discount factors P

=

In

c a n be d e r i v ed f r o m

(gi ven a series o f d i screte

d1, ... ,dlll ) by setting the coupon C

R:

=

Ym a nd the price

m R

=

y

m

R

:. y

m

=

E d i =l

i

+

d R m

( l -d ) m

(16 )

m E d i=1

i

Similarly, the continuous compounding approxima tion to the par yield curve ye t".> can be estima ted u sing a rearranged version of equa tion (7), setting C

=

y( tn/

16

y (e

m

)

( where m

R (l-6(e )) m

t; is

the time to the

( 17)

jth

regular coupon payment on the notional

period bond .)

The par yield curve ya,n> describes the coupon requ ired on a (notional) coupon-paying bond with time to matu ri ty par. Ol)

3

tIll

for tha t bond to trade a t

Estimating yield curves

The p re v i o u s sec t i o n d e ta i l ed the rel a t i o n s h i ps between d i f fe rent variables and curves o n the basis tha t ei ther a set of d iscrete d i scount factors o r a con tinuou s disco u n t function is known. A l so, since the d i scou n t f u nc t i o n, par yield c u rve, zero co u po n yield c u r v e a n d i mplied forward rate curves are all algebraically rela ted, knowing any one of these four means tha t we can readily compute the other three. In reality, however, none o f the four curves is d i rectly observable; t hey must instead be deri ved from bond prices. Two fu n d a me n t a l problems need to be a d d ressed by a n y m o d e l a t tempting t o identify the term structure of interest ra tes i mpl ied b y prices o f go vernmen t bonds . The first is the problem of 'ga ps' i n the ma turi ty spectrum - there is not always a sui table bond, or any bond a t all, ma turing at a date o f interest. Second, the term structure is defined in terms of zero coupon bon d s - but all UK go vern ment bon d s pa y

(1 1)

The par yield curve is essentially the same as a swap rate curve ( i n the absence of default risk). si nce a par yield represents the f i xed i n terest payments required by the market 10 match the same number of future (unknown) Ooating pa yments.

there are a number of practical d iffere nces in estimating the two curves.

17

Howevet".

coupons,

SO

a zero cou po n yield cannot be i n ferred directly from the

price of a coupon-paying bond. These two problems lead to further practical estimation problems. First, the problem of filling the gaps - what shapes should the term structure be a l l o we d to t a ke? To a n s w e r t h i s q u e s t i o n , a deci s i o n o n the appropriate trade-off between ' smoothness' (removing 'noise' from the d a t a ) a n d ' responsi veness' ( flexib i l i t y to a ccommod a te a genuine movement in the term structure) is required . For example, it might be fel t tha t the estimated term structure should be smooth, bu t not to the e x te n t that it is seriously misreprese n ted . Second, is i t preferable to esti mate the term structure v ia the d isco u n t fu nction or via the par y i e l d cu rve? There a re o ther practical h u rd le s to o verco me: for e x a m p l e , i n t h e U K m a n y i n ve s to r s pa y i nc o me t a x o n c o u pon payments whereas any capital gai n is tax-free. This differential taxation of cou pon payments and capita l gains resul ts in taxpayers preferring, and hence paying a premium for, low coupon bonds; so the size of the coupon on a bond will affect its yiel d . Such coupon effects, along w i th any other tax effects, need to be removed from any esti ma te of the term structure. The rest of this paper describes three model s used to estimate the term structure of i n terest ra tes : the model currently u sed by the Bank o f England (Mastronikola 1991) and two from the academic l i terature, d ue to M cCu l loch

(1971, 1975) and

Schaefer

(1981).

The many problems

i nherent in any estima tion of the term structure can be nea tly split i nto three ca tegories: which curve to estimate (Section cu rve should be esti mated (Section

3.2 ), and

3.1 ), how

the chosen

how to deal w i th other

factors w hich might infl uence rela tive bond prices, such as ta x effects (Section 4).

18

3 .1

Yield curve or discount function?

M odels used to estimate the term structu re of interest rates fall into two d isti nct categories: those that fi t the par yield curve and those tha t fi t a discount function. The Bank's current model ( Mastronikola op

cit) is a n

exa mple o f the former, whereas most o f the la tter a re based o n fit ti ng discount functions, pioneered by McCulloch (1 971 ).

Fitting a curve through redemption y ields The B a n k ' s y i e l d c u r ve m o d e l e s se n t i a l1y f i t s a c u r ve t h r o u g h red e m p t i o n y ie l d s, deri ved d i re c t l y from observed p r i ce s u s i n g equ a tion (9). Thi s methodology, w hile simple to u ndersta nd, has the theoretical d rawback that it does not explicitly restrict payments due o n the sa me date t o be d iscounted at the same rate. T o see w h y this is the case consider two bon ds; the fi rst, bond A, ma tu ring in o ne period s' time and the second, bond B, in two periods: R

Price o f

Bond A

a

C a

+

=

(l+y )

a

Price o f

Bond B

( 18)

C =

b

(lty ) b

R +

b

+

C

b

(l +y ) b

2

E stima ting the yield cu rve by fi tting a cu rve through the redemption yield s on these two bond s does not restrict the first coupon payment on bond B to be discounted at the same rate as the redemption payment on bond A even though both payments are d ue at exactly the sa me time. I nstead, w hen esti ma ting a yield cu rve in this manner the assumption must be made that the first coupon on bond B i s d iscoun ted u si ng the rate indicated by the yield on bond A, the yiel d on bond B reflecting the d i fference in ra tes between period 1 and period

2.

I n o ther w ord s,

bond A i s a ss u med to pro v i d e all the i n fo r m a t i o n req u i red fo r

19

in feren ce s abo u t h o w the ea rl ier coupo n payme n t s o n bond B a re d iscounted. G i ven a specifica tion of the functio nal form for the y ield cu rve (see Section 3.2), the estimatio n procedure i s simply to f i t a curve o f the gi ven f u n ct i o n a l form to m i n i mise t he s u m of squared d i fferences be tween t h e o b served a n d f i t ted yiel d s . The e s t i ma ted c u rve is i mplici t l y a par yield c u rv e . Th i s a pproach is rea sonable i f other a spects o f the model define this cu rve explici tly as the par yield curve (eg as in M astronikola 1991 see Section 4). However, whether or not a -

regres s i o n o f red e m p t i o n y ie l d a g a i n s t m a t u r i t y is a re a l i s tic approximation to the par y ield curve depends on market condi tions. If b o n d s a re t ra d i n g so t h a t t h e a verage red e m p t i o n y i e l d a t e a c h ma turity - the rate derived from a yield against ma turity regression - is c l o se to t h e p a r y i e l d a t t h a t m a t u ri t y , t he n t he a ss u m p ti o n is reasonable. However, the less well this assumption ma tches the reality, the worse the approxima tion.

Fitting a discount function Most of the academic li tera ture follows McCulloch

(1971 )

in explicitly

constraining cashflows from d ifferent bond s due at the sa me ti me to be d i scoun ted a t the same ra te, and estima tes a d iscount function from which the term structure can be derived . ( 1 2 ) McCulloch u ses the form of the bond price equa tion w i th a con tinuous d isco u n t function and makes the assumption of continuous compounding - tha t the coupon payments are made continuously through time ra ther than at regular d iscrete intervals. (]3 ) Under this assumption interest does not accrue, and equation (7) is used to give the price on bond

p

i

=

(12)

(13)

C .! 1

m i

6 (�)

d�

+

R

i

i (i=1 ,...,n) :

(19)

6(m) i

o

In particular. Schaefer

(198 1 ) follows this approach.

As stated in Section

2 this is mere l y a si mplifying assu mpt ion. and the results

presented later in this section were derived using McCulloch' s technique with discrete compounding. For clarity the description here follows the origi nal.

20

where Pi' C;,

m

and R; are the price, coupon, maturity and redemption payment of the ith bond . ;

6(m), i t

To estimate t he d i scou n t function,

k

combi n a ti o n o f a se t o f

is d e fi ned to be a li nea r

(l inearly i ndepend e n t ) u n derl y i n g

basis

functio11S :

6 (m)

1

k

t

E

( l O)

a f ( m) j j

j =l

ph b a s i s f u n c t i o n , a n d aj i s t he corre s p on d i n g coefficient (j=l, . . ,k). There are a number o f functional forms tha t the

w h e re f/ m) i s t he .

basis functions �(m) can take to produce a sensible d i scou n t function, and this choice is discussed in detail in Section 3.2. A s y s tem o f

l i ne a r equa t i o n s c a n be deri ved (l4 ) b y combi n i n g

7l

equa tions (19) and (20), with the fu nction weights aj as the coefficien ts in each equation:

k

Y,

1

where:

a

E

j= l

Y.

P

=

1

x

j

C

-

i

i

(n)

ij m

i

-

R

i

m x

ij

=

c

.i

i

f

f

0

j

(")

d"

+

R

i

f

j

(m .J 1

The coefficients aj (j= 1 , ...,k) can be estima ted from equation (21 ) u sing ordinary least squares, and the estimated discoun t function can then be calculated using equation (19).

( 1 4)

Se!! Appendi x A for the full derivation.

21

H a ving estima ted the discoun t function, equ a tions (13), (14) a nd (17) can be u sed to esti ma te the implied forward ra te, zero coupon and par yield curves respec ti vely. G iven the assumption made by the Bank's model, the same inferences abo u t the term s tructure of i nterest rates can be drawn from the estima ted curves regardless of methodology. The a d v a n tage of M cCulloch' s technique is tha t it makes explicit the a ss u m p t i o n o f a n efficient ma rke t, ie one in eq u i l ibriu m . F i t t i n g through redemption yields c a n b e regard ed as simply fi tting a curve t h rough d a ta a nd as such requires no assumption about the state of the market; however, the assumption is implicit as soon as such a curve is i n terpreted as a par yield curve.

3.2

Estimating functions

As described so far, both the McCul loch and Bank method s require a speci fica t i o n of one or more estima ting f u n c tion(s):

w hen fi t t i ng

through redemption yields, t he functiona l form needs to be specified; w h e re a s e s t i m a t i n g a d i s c o u n t f u n c t i o n u s i n g M c C u l l o c h ' s methodology req u i res the speci fica tion o f basis functions

(I/m) i n

equa tion (20». The choice o f fu nctions i n both cases i s crucial since it ul tima tely determines the trade-off between smoothness and flexibility discussed earlier, and therefore reflects prior beliefs abou t the shapes a y i e l d c u rve s ho u l d be a b l e to t a ke . Th i s cho i ce i s u n a v o i d a b l y s u bjec t i ve bu t cer ta i n properties a re essen t i a l ;

i n particu lar an

estima ted d isco u n t function shou l d be both posi tive and mono tonic non-i ncrea si ng ( to a void nega tive forwa rd ra tes) and should equal unity a t time

t=O (the present value of £1 receivable now is £1).

The Si mplest approach to fi tting the discount fu nction is tha t u sed by Carleton and Cooper

(1976), who estimate the term structure of i nterest

rates for the US government cou pon secu ri ties (ie n o tes and bond s) market. They u tilise the fact tha t the semi-annual i nterest paymen ts made by nearly all securities in this market a re made on only four days

22

o f each yea r. (15)

This even-spa cing o f d a ta poi n t s mea n s tha t the

d i scou n t factors ca n be estima ted d i rectly from equ a tion (2) u si n g ord i n ary lea s t sq u a res f o r ma t u r i ties u p to seven yea rs,
expo"e"tial splint

(see below), since the time to maturity on each bon d is transformed using

equation (22) before estimation. However, since the motivation for using the

transformation is different from that for using an exponential spline, it seems mo re useful to desc r ibe the Bank's model as usi ng a cubi c spline (in transformed time).

26

Bernstein polynomials Schaefer

(1981) u ses approxima ting

fu nctions to es tima te the d iscount

function i n the same ma n ner as M cCulloch, but instead o f cubics he u ses

Bernstein polynomials .

I t can be shown u sing the Weierstrass

approximation theorem (eg Williams 1991, page 74) that combi na tions of Bernstein functions will approxima te any continuous func tion with a rbitrary accuracy. A n ad vantage of these functions over conventional polynomial appro ximating fu nctions is tha t they gi ve considerably better approxima tions to the deri va tives; impo rtant since the forward curve depends on the first deri vative of the discou n t function. ( 2 3) By imposing constrain ts, Schaefer ensures tha t the a/ s a re non-nega tive, tha t the estima ted discou n t function is non-negative a nd tha t 6 (0) = 1 . With these conditions, negative forward rates are a voided . (2 4 )

( 23 )

For a more d e tailed account of the use of B e rn s tcin functions in this con text see S chaefer (1982).

(24)

If p(m ) is t h e forward r ate curve and 6(m ) t h e discount func t ion it c an be ' shown that p(m) = -6 (m)/ 6(m) (equation ( 1 2». Clea rly, p(m ) will be n ega ti ve , - if either 6 (m) is posit ive or 6(",) is negati ve - Schaefer's constra i n ts ensure tha t neither of t hese con ditions arise. Since Schaefer's d iscount fun c t ion is a

l in ear combination of monotonic non-increasing approxim ating func tions he ensures that it is monoton ic by constraining the

27

n;.s to be non-negative.

Exponential s p lines O ne o f the m a i n cri t i c i s m s l e v e l l e d a t bo t h c u b i c a n d Ber n s te i n polynom i a l functions a s a choice o f approxi mating functions is that these can lead to forward ra te cu rves which exhibi t u ndesirable (and u nrealistic) properties for long maturities ie rise or fall steeply. Vasicek a n d F o n g ( 1 982) d e t a i l a m e t h o d t h a t ca n be u sed to p ro d u ce a symptotically fla t forward curves. Central to their a pproach is the characterisa tion of the discoun t function as essentially exponential in s h a pe . They a rgue tha t spl i n es, as piecewise polynomials, h a ve a d i fferent curva ture from exponentials a nd so will not provide a good local fit to the discoun t function. ( 2 S ) Vasicek and Fon g claim tha t this poor local fi t will res u l t in the spline "wea vi ng" a round the d i scoun t f u n c t i o n , t h u s p ro d u c i n g h i g h l y u n s t a b l e fo rwa rd r a tes . A l so, polynomial splines cannot be forced to tail off in a n exponential form as maturi ty increases. Va sicek and Fong suggest applying a transform to the a rgu ment the discount function

m = -

( l /a) ln ( l - x) ,

5(m).

This transform has the form:

where

OSx< l

m

of

(ll)

a n d h a s the a ffec t o f t r a n s formi n g the d i scou n t f u n ction from a n a p proximately exponen tial function o f m to a n approximately l i near function o f x. ( 26) Polynomial splines can then be employed to estimate this tra nsfo rmed discoun t function . Using this transform i t is easy to i mpose a d d i ti o n a l cons t ra i n ts on the d i scou n t f u n c t i on . ( 2 7 ) The

( 1 98 5 ) who insist s t h at a piecewise polynomial function should be able to mimic well a piecewise exponential function.

( 25 )

This is refuted by Shea

(26)

Here.

(27)

One such condition that they impose is the non-negative condition.

x

is referred to as transformed time.

28

parameter a constitutes the limiting value of the forward rates, and can be fi tted to the data as part of the estimation. V a sicek a n d Fong u se a c u b i c s p l i ne to e s t i ma te the t ra n s fo rmed discoun t function . I n terms of the ori gi nal variable

m

this is equivalent

to estima ti ng the discoun t function by a third order exponential spline ie between each pair of knot points

6(m) takes the form: (23)

A l though Vasicek and Fong claim to ha ve tes ted exponential s plines su ccessfully, they provide no evidence. Con sequen tly, Shea ( 1 985) presents some empirical results on the su itabili ty of exponential splines for yield curve modelling. He concludes tha t there is no evidence to s u pp o r t the c l a i m th a t e x po nen t i a l s p l i ne s p ro d u ce mo re s table estima tes of the term structure than polynomial spl ines - the disco u n t fu nction often devia ting from the expected exponential deca y . Shea found that the asymptotic property only constrained the forward cu rve to fl a t ten a t m a t u ri ties beyond the l on ges t obse r v a b le bond a nd exhibi ted li ttle influence over i ts shape or level a t other ma tu ri ties. A n addi tional observa tion was tha t one of the factors d riving the instability of the Vasicek and Fong model was the da ta-cond itioning properties of the expone n ti a l t r a n s form, observed

x

x=l-e-a", .

For sma l l �, this c a u se d the

to become b u n ched so tha t s u b s ta n tial portions o f t he

estima tion i n terval

[ 0,1] con ta i ned

no d a ta , lea d i n g to pa rticula rly

u n s t a b l e a n d u n re a l i s t i c a s y m p t o t i c f o rw a r d r a t e s .

In such

circu mstances Shea had to coa x the nonlinea r estima tion p rogra m to converge to a solution. It is possible tha t this problem was ca used by S hea ' s c h o i ce of k no t poi n t s , w h i c h a p pe a r s t o be in l i n e w i t h M cCulloch ' s con ven tion o f placing equal numbers o f observa tions ( i f possible) between knots. (28)

(28)

This was certainly the rule used in S hea

29

(1 984).

C h a mbers, C a r l eto n a nd W a l d m a n

(1984 )

ha ve i nco rpora ted t h e

exponential characteristic i n a d i fferent manner. Here, a polynomial functional form i s applied directly to the spot cu rve. The spot c urve can then be related to observable bond prices by exponentiation of this functional form.

B-splines A n important observa tion made by Shea

(1984) concerns the choice of

basis functions w hen defi ni ng a spl i ne function. He reports that some spline bases, such as that chosen by McCulloch

(1971,1975) can generate

a regressor ma trix w i th col u m n s tha t are nea rly perfectly col l inear, resul ti ng in possible i naccuracies arising from the subtraction of l arge n u mbers . A s a solu tion he ad vocates the u se of a basis of "B-splines" . These a re functions which are identically zero over a large portion of the a pproxi ma tion space ( u n l i ke those u sed by M cCulloch) and so prevent the loss of accu racy d ue to cancell a tion . By using a B-spl i ne basis i t is also easier to impose constraints on the spline function. Steeley

(199 1 )

a l so recommen d s the u se o f B-spli nes for the s a me

reason . H e provides comprehensi ve details o f how B-splines can be u sed to fi t a d i scou nt function, a nd concludes tha t by their u se spline f u n c t i o n s ca n be viewed as a rob u s t a l terna t i ve to both cubic a n d Bemstein polynomials.

Prob lems u sing spline functions as estimation functions Shea

(1984)

considers some of the pi tfa l l s enco u n tered w hen u si n g

spli nes to model t h e term s tru ctu re. First, he demons tra tes t h a t the constra i n ts implicit in the M cCul loch cubic spl ine do not restri c t the d i scoun t fu nction to i ts desired nega tive slope, and ca n consequently prod uce an esti ma te for the d i sco u n t function which starts to slope u pward a t the longest ma turities . ll1e forward rate curve genera ted by such a discou n t fu nction will fea ture nega tive interest-rate estima tes . W i thou t the im pos i tion of constraints (d iscussed ea rlier) t h e Schaefer

30

polynomial wou ld d i splay simi l a r characteristics. Shea a rgues tha t Scha e fer' s c o n s t ra i n t o n the s l o pe o f the d i scou n t fu n c t i o n t o be everywhere negati ve, though serving to prevent nega ti ve forward rates does nothing for the general stability of the forward curve. One a l ternative "fix" suggested by Shea on such occasions is the use o f

ad hoc constra i n t speci fication.

I n its more obviou s form t h i s might

c o n s i s t of c h a n g i n g the n u mbe r o r l oca t i o n of the k n o t p o i n t s . However, Shea goes o n to suggest the use o f l oca l i sed con straints to deal w ith specific problem areas. One such constraint suggested was a simple restriction of fixed proportions between the first deri v a ti ves o f the discoun t function a t di fferen t maturi ties. lll i s i s of particular u se a t the l o n g end where i t c a n b e a p p l ied t o e n s u re t h a t the d i sc o u n t f u n c t i o n rem a i n s n e g a t i v e l y s l o pe d . A l t h o u g h t h e se m a n u a l a d j u stments to the term structure a re acceptable i n a resea rch a n d d e v e l o p me n t c o n te x t , t h e y w i l l c l e a r l y b e o f l i m i te d u se f o r practi tioners in an operational environment, where yield curve u pd a tes may be required on a rea l ti me basis. Also, changes in the cu rve may be w rongl y a ttribu ted to even ts i n the ma rket when in fact they a re solely due to a change in the constraint specifica tion . Knot points Another decision that need s to be made when using a ny ki nd of spl i ne fu nction is the appropriate number of knot poi n ts . I f the nu mber o f k nots i s too low then the model will not fi t the d a ta closel y when the term s t r u c t u re ta kes on d i fficu l t sha pes, w hile i f i t is too high the estimated curve may conform too readily to unrepresentative outliers. The Bank yield curve model currently u ses si x kno ts, which a re spaced evenly i n transfo rmed time (see above ) . The a pproach ado pted by M cC u l loch

(1975)

a nd several subsequent resea rchers i s to set the

number of kno ts to be equal to the square root of the number of bonds to be u sed in the estima tion process . These knots a re then s paced e venly amongst the number o f observat ions (ma turi ties). G i ven the

31

current number of bonds in the U K market, this approach also suggests the u se of six knot points. One advantage of the McCulloch convention i s tha t the positioning of the knots w i l l au toma tica l l y change w ith a shift i n the structure of government debt - unlike the knot poi nts i n the Bank's model, which will remain fixed . On the negative side, allowing the knots to move on a day to day basis may gi ve the false impression that the term structu re has changed . Figures

3.2 -3.5 illustrate the kinds of effect tha t changing the number or

location of the knots i n the Bank model can have on the forward ra te curve.( 2 9)

Figure 3.3

Figure 3.2 Forward curve for d ifferent numbers of knot points cob 30/3192

'

'-

Forward curve for different n umbers of knot points Per Cleal - 10.5

cob

- 10.0

-

'- 9. 5 .... - � - :::":' -- -:::; -- ...::. ' '.. ... 4 1m" --,,: ""","-

-

3019192

- 11

Per Cleal

10

""

-

-

',,,I,,I!d11!I"11dII'II!I"I"I1111"I"I11 '11d o

2

5

Figures

10

3.2

and

Yean

15

3.3 sho w

20

25

9, 0

-

" 1.1'11'"11'11'"d11111111d11111111"!11'1111al o

2

5

10

Ycara

15

20

25

8

the effect of reducing the number of knots to

fou r or five, bu t still spacing these points evenly in transformed time. In the case of 30 September 1992 , red ucing the number of knots from six to four raises the forward curve by over 30 basis points in places. This smoothing also removes the point of inflection at the 3 year horizon .

(29 )

S uch effects a l so occur when considering a par or zero coupon y i e ld curve. but are less significant.

32

Figure 3.5

Figure 3.4 Forward curve for d i fferent positioning of knots cob 3 013/92

Per 00 01 - 10.$

-

-

',.,11111.1,1'11111" 1" 11111,11111111111" 1111" .1 o

2

10

�

Figu res

Yean

U

20

2j

Forward curv� for d i fferent positioning of knots

cob 3019192

-

11..11111.tI1111I"II"!,"It!I..'111111I...1111"I o

2

3.4 and 3 .5 compa re the effec t of

10

Y.,.

20

swi tching from kno ts spaced

e ven l y i n tra nsformed time to knots spaced evenl y by n u mber o f observa tions. I n the example o f 30 September this prod uces a shift i n the forward curve o f up to 13 basis points. Surprisi ngly, aside from Steeley (op dO, there seems to have been little effort in the l i tera ture devo ted to testi ng sophistica ted techniques for speci fi ng the optimal nu mber a nd loca tion of knot poin ts. Tha t such techni ques a lread y exist (eg de Boor, 19 78 ) makes this a l l the more surprising.

33

11

-

-

9.0

fltrcea!

I

Nelson and Siege I (1987) A very different approach is tha t due to Nelson a nd Siegel (1 987), w ho explici tly a t tempt to model the implied forward rate curve (ra ther than the term structure of i nterest rates). They choose a functional form for the forward rate curve tha t allows it to take a number of shapes that the authors feel are "sensible". The functional form that they suggest is: [(m)

=

tJo + tJ 1 exp( m/ d + tJl(m/ dexp( - m/dJ -

(24)

w here [(m) is the forward rate at maturity m, and 13 ' 13 ' 13 and r are the 0 1 2 pa rameters to be est i m a ted . This fu nction can be transformed to a d i scoun t function (using the rela tionships i n Section 2) from which the parameters are estimated . (30) By considering the three components tha t make up this function ( see Figure 3.6) i t is clear how, with appropriate choices of weights, it can be u sed to generate forw ard rate curves of a va riety of shapes, including monotonic a nd "humped". An importan t property of this model is that

tJ o spec i f i e s t h e l o n g ra te t o w h i c h the fo r w a r d ra te a s y m p t o tes horizontally . Furthermore, this approach avoid s the problem in spline­ based models of choosing the "best" knot point specification.

(30)

Equa tion

(24)

can also be tran sformed to a spot rate curve. to which Nelson and

S iegel fit U S Treasury Bill data (because Treasury Dills are zero coupon i nstruments).

34

Figure 3.6 Com ponents of the forward rate curve Model curv� (Nelson & Sicgcl, 1 987)

_

"?�(

. _ . _ . _ . _ . _ . _ . _ . _

_

1 .0

- O.S

, ,

... ....;::: ... .., ::;: -..;:.�_..,...,... .. .... ... ...,. .. . 0.0 +-,--.-....--.o

25

S Years

7.S

10

From the Nelson a nd Si egel forw a rd ra te eq u a tion it i s possible to d e r i v e a l gebra i c e x p ressions for the spo t cu rve and the d i sco u nt function, though not unfortuna tely for the par yield curve. (3 1 ) I t is interesti ng to note tha t Svensson (1 993) esti ma tes spot and forward r a te cu rves u s i ng M cC u l l och' s ( 1 971 , 1 975) a pproach o f fi t t i n g a d iscount function to bond price data, but uses the Nelson a nd Siegel fu nctional form instead of a spli ne fu nction . Svensson a rgues that for monetary policy applica tions a simpl istic functional form of this nature i s perfectly acceptable. I n his paper on estima ting Swedi sh for ward ra tes (Svensson 1 994) he increases the flexibil i ty of the original Nelson a nd Siegel model by add ing a fourth tenn to the forward rate equation (eq u a ti o n (24». This term takes the form 1l 3 (m / T 2 )exp(-m / T 2 ) a n d provides t w o extra pa ra meters for estima tion . Ho wever, S vensson con c l u d es tha t the origi nal Nel son and Siegel mod el prod uced sa tisfactory fi t on most occasions.

(31 )

To obtain a par yield curve numerical methods must be applied.

35

a

4

Modelling the effect of tax (the "Coupon Effect")

The techniques o u t l i ned i n the previous sec tion can be thought of as method s to esti ma te the y ield -matu rity structure of a bond ma rke t . However, the existence of coupon paying bonds complicates estimation of the term s tru cture . I n particular, ta x rules can grea tly a ffect the prices of bon d s a nd , if thei r effe c t s a re ignored i n the mod e l l i n g process, ca n d i stort a n y estima te of the term structure of interest rates . This is what is commonly known as the COUpOll

effect and

is particularly

i mportant in the UK becau se of the w ide range of coupon s on bonds c u rren t l y trad i n g in the market ( the curre n t ran ge o f coupons o n conventionals being 3 % to 15 1 / 2%). (32 ) A substa n t i a l propo rtion o f i nvestors i n the UK governme n t bond market a re taxed at their margi nal ra te of ta x on any cou pon income they recei ve, but are exempt from taxa tion on capi ta l gain . Bonds w i th h i g h c o u p o n s c l e a r l y p r o v i d e more of thei r re t u rn i n the form o f coupon i ncome than do bonds with low cou pons. Therefore investors facing a non-zero ma rgi nal income tax ra te bu t no ta x on capi tal gai n will

ceteris paribus

prefer l o w cou pon to high coupon bonds, whereas

those paying no income or capital gains tax will be indifferent between the two types . The preference of tax-paying i nvestors for low coupon b on d s w i l l i n cre a s e their price rel a t i v e to h i gh co u po n bo n d s , a d istortion that need s to be removed when a ttempting to measure the u nderlyi ng term structure. Thi s section outl i nes and compares three methodologies for taking account of the coupon effect.

(32)

A nother possible effect caused by bonds paying coupons is what might be termed a ·duration effect · . si nce t wo bonds of the same maturilY but with di fferent coupons w i l l have differe nt durations and different exposure to i nterest rate risk. Such e ffects are not considered by a n y of the three models described here. presumabl y si nce the tax e ffect is consiJer.:J to dominale.

36

McCul l och (1975) In his original work, McCulloch ( 1 971 ) overlooked the possible effect o f taxation rules o n bond prices, bu t developed the model to take account of such effects in his second paper ( 1 975). In this paper, M cCulloch sets up a number of equ a tions for various types of bonds not all of which are relevant to this study as they reflect US tax laws i n the early 1 970s. Instead, when a pplied to the U K , only o ne equ a tion i s req u i red - a n a m e n d ed v e r s i o n o f t h e p r i ce e q u a t i o n a s s u m i n g c o n t i n u o u s compounding l /C I ; i e the cou pon C l must be larger than the cou pon C2 . Thi s illu stra tes the general property of such d iagrams that

consta n t cou po n lines representing high coupon bonds a re less s teep than those representing low coupon bonds .

The i n tersections o f C 1 a n d C 2 w i th C C ' (deno ted ( P I ( C ) a nd P 2 (G» i n d ica te how gross in vestors will value the strea m of cashflows from bon ds

1

a nd

2

respectivel y . Li kewise, P 1 (H) a nd P 2 (H) represen t the v a l u a tions of the same two bonds made by taxpayers. I t ha s alread y

been noted tha t an increase in a bond's price moves i t along i ts constant coupon line towards the origin o n a c a p i t a l - i ncome d i agram ( see Figu re 4 . l (b», and so i t is clear from Figure 3.3 tha t the gross investors w i l l value bond 1 higher than the ta x-paying investors ( s i nce P I (C) i s closer to the origi n than P (H» , whereas bond 2 will be priced higher by 1 the taxpayers. So, if investors a re ra tional, the higher coupon bond's price will be set by the ta x-exempt investor, wherea s the lower coupon bon d ' s p rice w i l l be determi ned by the tax-paying i nvestor . Such a 46

speci fica tion mod el s a market a ssumed to be in "equi libri u m u nder s wi tching" ( M a s tro nikola 1 99 1 , page 1 0), i n which no i n vestor can swi tch from one s tock to a n y combi nation o f o ther s tocks i f such a switch results in: - higher capital gain and maintained income, or - higher income and maintained capital gain, or - higher income and higher capital gain. These cond itions define an equ ilibrium equivalen t to the "no arbi trage" equilibrium in Schaefer's model (4 1 ) - for each bond it is the ca tegory of taxpayer who values it the highest who detennines i ts price. Figure 4 .4(a) shows the two extreme indi fference l i nes, for the gross investors and 1 00% taxpayers. The line for an investor facing a 1 00% i ncome tax rate is horizontal, since such an investor will only invest i n bond s providing a pu re capital gain.

Figure 4.4(a)

Figure 4.4(b) Cupitul Guin v Income : I n termcd iute Cuscs

Capital Gain v I ncome : Extreme Cases

�

Capital

Capotal

Gain

HtOO--..l.,-., -----

�

Gain

�oo

lneon..

-)

(4 1 )

-)

Therefore this model also depends on

47

an

assumption that short sales are restricted.

This model can easily be generali sed to any n umber o f ca tegories of ta x payers, as illustra ted in Figu re 4 .4(b) . All the intersections between indi fference c urves are a ssumed to occur abo ve the par line, since all bonds tra d ing bel o w the par line are priced above par and therefore c a u se a capital /oss if held to redemption ( this loss being bala nced by above par cou po n payments). Such bond s should t herefore be held only by gross investors, and only bonds lying above the par line will be held by tax-paying investors of any kind.(4 2 } (43 ) I f the market is in equilibrium u nder swi tching, the prices of bonds i n this diagram will be set along the heavy boundary (corresponding to a n "efficient frontier") . However, a s mentioned earlier (with reference to Sch a e fe r ' s m o d el ) , i t i s d i fficu l t to speci fy a n u mber o f d i s tinct categories of taxpayers si nce, apart from the four current personal rates in the UK (0%, 20%, 25% and 40%), t here are a number of institutions tha t h a v e e x e m p tions ( i ncl u d i ng pension fu nd s and some forei g n investors) whilst others pay at their corpora tion tax rate (currentl y 25% or 33% in the U K) and may be able to offset some income against o ther l os ses for ta x pu rposes. For this rea son, therefore, the Bank model a l l o w s for a con tinuous spec trum of income ta xpayers between the g r o s s i n v e s t o r a n d 1 00 % t a x r a t e p a y e r , a n d t h e b o u n d a ry i n F i g u re 4 . 4 ( b ) becomes the

capital-illcome ClI rve

i n Figu re 4 . 5 . So,

al though t he theory behind Schaefer's model and the Bank's model is the same, there is an importan t di fference in implementation . Schaefer

a priori d etermines

the specific ta x ra tes for which term structures a re

req u i red , w he rea s t h e B a n k ' s m o d e l d e f i n e s h o w ca tego r i e s o f

(42 )

This assumpt ion may be too restrict i ve since there may be other reasons why some ta xpayers might want to hold bonds that w i l l provide them with

a

capital loss. The

model could be amended to rela x this assumption if it was felt unreasonable by (for e x a mpl e ) r e s t r i ct i n g a l l i n tersec t i o n s to occur above t h e constant coupon l i ne representing the highest coupon bond in the market.

(43 )

The Bank model does not constrai n the gross in vestor' s tax rate to be 0% but i ns tead

a l l ow s it to v a r y w i t h maturi t y . The e s t i mated v a l u e of t h i s pa rameter at each mat uri t y perhaps gi ves an i ndication of whether or not bonds with that maturity and prices above par are i n fact held by

0% taxpayers.

48

ta x pa yers i n te ra c t a n d thereby e s t i ma tes a si ngle term s t r u c t u re representative of the market as a whole.

Figure 4.5 Capital Income Curve CapiUlI Gain

Income

-1

The ca p i ta l - i n come cu rve i s d e fi ned i n t h e B a n k ' s model b y the following equa tion :

cc

=

at

{

at

( y (m) - r )

(y (m) - r )

+

}.

(m)

(y ( m ) - r )

6

r

r

�