An application of duality theory

AN APl’LlCATION OF DUALlTY THEORY

.

A Portfdio CompositGonof the West-German Private Non-Bank Sector, 1968. 1975

Rccci\cd January The

1979, final \crGou rccciwd

objcctikc of this paper ih to apply the dualit)

approach

for dcri\ inp demand systc~m IO analyac the allocation hank sector. A commodity liabilities

rntcr

the direct

capital

cntcr

the

apply

functions

indirect for

approach utility utilit>

~qimnl

function

and

the

function.

The

approach

stocks

in the thcorv

of ;Isvzts

is chosen where wtice

rcsult~

tmplcmcntati~w

m werall

firht.

wbmodclb

arc con\lrucIcd

IU~

the liahilit>

for a11optimal

aggregate to its \aricw\ components

for time wwc\ datu of the
that

pri\ ate non-barih

bch;i\ lrll

of the pi-i\&

non-

of assets. physictiE capital ,md

corresponding

con&cnl

of con~amer

and liabiiitich

ilow

kvith ulilit? model in ;III aswt aggrcpatc. capital

Nat.

Augt~~t 1974

intcreut

rates

in a s\strm

niaximiAon. and ;I liability

allocution

price of and

F-or econometriC agrrpatc i4 fitted

of the aswt

rn;dc up the apprqatc.

and

of dcmand

apprc;rntc a~tf

Fmpirlcai

realty

~2ctor .lrc presen(etl.

1. Introduction The objccti\e of this paper is to apply the duality approach in rho thecrr! of consumer bcha\,ior tct fhc allocation of assets rtnd liabilities in the pri\;iIc non-bank sector. The model 11~1sto be seen as an alterna~i\ c apprc~ch to the problem of portfolio selection or to the theory of rclatiw prices. According to this theory the objective of an in\cstor 01 ~1 pri\ate household is to find ate optimal structure of net wealth by allocating assets and liabilities kvith respect to revenue and cost proportions. The theoretical foundation for the specification of supply and donland functions is usually provided by the theory of portfolio selection. dc\.cloped by Marko\vitL ( 1952) and, Tobin (19%). in this theory the trade-off between expected rwenue (mean) nncl corresponding risk (variance) is determined by utility maximization under conditions of uncertainty. The o~~tcome is a structure of assets and liabilities which satisfies in an optimal way under gi\,cn risk behavior the individual preferences for revenue and risk. The reaction of the structure of the portfolio to changes in intcrcst rata dcpcnds on the rish ii\ersion of the *The their

author

would

like

to acknnwlcdgc

with

valuable comments.

I h?

~ratitudc

hi*

Indehtcdnc~~

(0 the rcfcrcc~ 1;~

K. Conrad.

Ih-l

An

applicationefdualitp

rhror)

investor and, similarly to the theory of consumer behavior, on the assumed preferences: that is on the position of the indifference curves. FBm~m.u&g e.conometric nmd,els of the financial sector it seems that the mode) b.&&r has tf& theory in mind when he arg43es that the stmcture of assets a& lk&Sities depends on the interest r&m and net weahh, However. as it is assumed for estimation that the anticipa&d risks remain constant and that the anticipated future iuterest rates and capital prices depend cm their present and past values, there is no linkage between the portfolio approach of forming expectations and variances of returns (to incorporate the aspect of risk and uncertainty) to the finally specified system of demand equations. Although empirical studies about the process determining the structure of asset prices and returns are highly sophisticated on the micro-level - but not as yet satisfactory [see Jensen (1972) and Roll (1977)] - econometric models on the macro-level are usually based on ad hoc hypothesising and enormous data mining. It is known that the theoretical underpinning of econometric models of the financial sector is rather weak.’ Although economic theory does not specify either what variables determine the behavior of the demanders and suppliers of assets nor the precise form of asset demand or supply functions, a list of facts is put ahead of the model construction: (a) The sectors are assumed to have a ‘desired’ or ‘optimal’ composition of their asset portfolios which depends on the yields of all assets. (b) The desired composition is to be viewed as a set of long-run preferences. (c) These preferences for individual assets are assumed to be consistent with rational maximizing behavior although it is not said what it is that is being maximized. (d) Partial stock adjustment processes take care of delays in adjustment from actual to desired positions. To justify some of these assumptions a portfolio model based on the theory of consumer behavior seems to be a reasonable alternative for characterizing the structure of financial preferences. For empirical investicivation we assume that the anticipated risks remain constant and incorporate the aspect of risk implicitly in the slopes of the indifference curves for alternative combinations of services from assets and liabilities. The advantage oI‘such a model is that it consists of a complete system of supply functions of assets and demand functions for liabilities which under the assumption of utility maximization subject to a balance identity obeys certain properties such as the adding-up property, homogeneity of the functions in ‘normalized’ intcn:st rates, symmetry of the Slutzky matrix and the property of a negative ‘SW. for instance.

Parkin.

Gray and Barrct (1970)

and Christ

(1971 i,

semi-definite substitution matrix.’ Such a model will also satisfy the Yale principles of financial model building specified by Brainard and Tobin ( 1968 ).

Instead of deriving supply and demand equations for assets and liabilities based on the theory of portfolio selection we suggest a theoretical foundation of these equations based on the theory of consumer behavior. In this case we can employ the elegant theoretical tool of the duality approach to derive a complete system of demand and supply functions which is consistent wirh the assumed utility maximizing behavior. For deriving behavioral equations we suppose that the financial preferences of an agent or private household can be characterized by a utility function, the argument of which are the expected interest payments in terms of revenues and costs implied b! the stocks of assets and liabilities under the expected interest rates. We considtz1 the balance sheet shown in table 1. Table 1 Balance sheet of private non-banks. _ -----____-____ --.._.___

_-__ AS%% A,

Currency and demand deposit5

,q2

Time deposits

A3

Savmps deposits

J,

Government bonds

A5

bearer bonds of banks

K

Physical capital at replacement cost .-.__--------

LiabilitieQ t,

Short-term commercial bills

L2

Medium- and long-term commercial bills

L.3

Private bonds (net position)

Lb

Foreign liabilities met position)

U’

Private net wealth _____

__.--.--_-_.

.._-.--

-..

“At the beginning of the period.

We introduce the following sign convention: A&O,

i=l

Lj~O,

j=

. . . .’ I?,

K 20.

l,..., m,

CV>O.

lSee Fischer (1972) for a derivation of Slutzky equations for a continpnt and an asset model.

commodit)

model

Under

our sign convention

is as li~llo\~s:

the balance sheet identity

(1)

The

of the pri\,atc household

objectil\x

private net \vcalth in financial allocation transaction

costs

borro~ving

and

optimal

ahdon

aspects

with

respect

to

the prefcrcncc structure of wets

the

in\estmcnt

and

arc the reason

and liabilities

single asset with the highest interest rate and a single liability provided by the traditional

ho\vever. is to quantify

relationship

iid

Gf the flow

[SW

to wnsumcr 3

utility

l~~wn~~late demand

tlrc

t3y

~incl

assuming

of ;kwtx

charxtcristics

can

Another

ivrittcn

bc

flwvs

as ;1 utility

:II-ptncnts

of a utility

payments as the clLlantificution

intwst

on

dcpck

istic pcculiaritics of the assic~~and liabilities. .A further

nloiictary

results

from

Ii~llding

rc\cnucs must cntcr rhc utilit> with

alid

dtmand

deposits

the rate of inflation.

tlcmand deposits lhc

rate

medium

iis m

of inflation. to

stocks.

payments

A

on

third

and prclfit

and

WC consider returns

on the diffcrcnt charac.tcr-

I-~I~L’~

walth.

type of asset is currency and demand deposits. sa> .-I,. M.hich the

IL-.!:tlllch

currency

function

The prices arc the intcrc:st

hou~~:k~-iil want5 to hold a5 acti\‘c or iiiactiw !!‘IL1i’Lf:.l

II’ tllc

of scr\ kc flows and their w;\lu;ition

in the prcfcrcncc s! s~cm of 111~‘household for altcrnati\c forms elf inicsting

to

liabililics

ad

the utility

function

function.

is

w:ij

[.C’hctty (l!W)J.

~tpproacli cniplo~~~~ in this paper i5 to choose intcrcst from stcxks as the

measure

approach to consumer

the number of charactcrislics

qi~;ds

as ~1prosy

their problem in terms of

( 1966)

and scr\kc

;t

point out the analogs

~vcalth constraint.

ii

of the

rendered by the

that the consimicr comhincs ;1sscts

10 prcx~uct’ certain charxtcristics number

formulate

in terms of Lancaster’s

problem

tlw~r~

flow

of the stock quantity

theory hut do not explicitly priw3

utilization

the scrvicc flo\vs. One can ;wumc

Fcigc ( l364)]. In this case economists

function.

instead of u

theory of consumer beha\,ior.

bctv,ccn the stock and the scr\kc

stock \vhich permits the utiliratioii

01

with the lowest

enc. The emphasis on flo~vs, rather than on stocks, permits problem.

the

safety. solxmzy. time of repay~icnt.

Iii;;: liquidity,

other

~11~ the household holds ;i combination

The

For

service flows rendered b\- the assets play ;I

of money determine

analytic framavork

of

assets. physical capital and liabilities.

certain non-pecuni;ir!

crucial role Churacteristics

is to find an

rcconcilc

By

argument utility

cash.

function.

in the as

;iss~iiiic

that the wvicc

\olu~ncs

utility

lljilil

01

ilitct.c\t

flo\i of

to the stock and increases ser~%x

a

an

in tc’rm~

argnj-iiclit

Wc

including

iikwmc

no

~ill~~ti~~r

is proportional

of cash

changing

cash. .Is

Ilow

function

as?;et for

of currency

bvhich transactions

of transxtionh

changes and

is cxpresscd.

and m ith ;I<

;I

To

%.;I, = Expected %..Ii

m,

%K

Expected asset Ai = Expected liability = Expected =

service flow from cash (currency and demilnd deposits A, ). interest revenue at the end of the period from the stock of at the beginning of the period (i= 2.. . . . II ). interest payments at the end of the period from the stock of Lj izt the beginning of the period (,j = 1.. . .. III 1.. prol’it at tlrc end of the period from physical capital K at the

beginning

of the period.

As mentioned

itbo\ e. the folhvil;g

%.-li = Ai . 1.“. rate

of inflation. ,j =

mkrc

I r-f. . , ., r;,’ ) is

holds:

i = 1. . . ,. II.

’ the expected /*II IS

where

relationship

the

and

1.. . .. 111.

iector

of

I

expected

interest

rates

relc~~ant

to

3I

111~

pri\2tc asset compositinn wctor

of expected

(I,;2 t0 implies

interest

%L,~O

%h’ =k’

Accordiiig re\wi

iit’s.

h~~iischold stocks. liabilities.

for

This

in our

implies

to the ‘budget

ser~ke

the

interest

pa) mcnts.

coniposition

the

pri\.atc

Max I’(%.4 subject

to the

sign con\ention).

itpprOXl1.

illld

a certain

gi\,en

relo~~ant

prilate Finally.

liitbilit> \\‘L’ li;t\e

. ,a&.

to our

profit

rates

follow-ing

net \veillth

I..

,

I.

%

of model

flows

stacks.

that

iS iiiwut

the

;lSSetS

IiabilitiCs. illId

ilIld

for

the

prckrcnws

alloc’;llion

of

1~1‘ the 1101

aSsetS

thcZ

and

It’:

-I,,. %fi. %I., . . . *. %I

constraint’

of

determine

.),,.

f

1.

1-1

(6) is equivalent

budget constraint

to the balance sheet identity

( I ). The

\ilriable f is time and represents omitted varirrbles like the change in the distribution of wealth and a change in the preferences for some components

of the balance sheet. In analogy to the theory of consumer behavior are 1:r ihe ‘prices’ in our portfolio approach. If the household expects a higher interest rate r’;:”of an asset. the ‘price’ I it -Adrops and he will re-allocate in the 11ormal case at the beginning of the period his assets and liabilities in such a ~v;ly. that he receives higher interest revenues %Ai from the stock Ai which exceed the increase in interest revenues due to the higher interest rate: that is. IX will increase his stock .4,. Similarly. if the household expects a higher interest

rate I$. it will normally

rr-allocate

his portfolio

in such a way as to

avoid the full impact of higher interest payments %Lj. that is. it will reduce the liability Li by reshuffling his liabilities (and assets). To obtain the optimal structure

of the portfolio

maximizes

the utility

assmne that strongly

at the beginning

function

quasi-coccave in interest

substitution

differentiable.

revenues.

implies that the marginal

diminishes. to

The

stitution

serves

explain

portfolio.

Gi\.en the structure

ment in an asset ivith

strictly

profits

the

desire

of

the

(6). We

increasing

and interest

rate of interest

law of the diminishing

and

payments

revenue (payment)

marginal

household

rate of sub-

to

dib-ersify

its

of expected interest ,rates, an enlarged invest-

it higher

asset \iith a Iowcr interest

the household

(5) subject to the balance constraint

L.’ is twice continuously

(negati\c). This

of the period

interest

rate results

marginal rate of interest substitution

rate by reducing

in\,estment in an

indeed in more re\cnues. howe\zr. the diminishes

\vith a continuing

investment

in .i.iht this asset. because interest I’L’WIIWS as a result of an intensified increased stock of an asset are conntxtrd witI1 risk factors like a falling quotation. illId

insol\,ency or wrong espectations.

negative

service

tlows

seems to

Our way of considering

be a consequent

extension

positive of

the

treatment of stocks in the theory of consumer behavior. For illtcrest intcrcst

a correspondence of dimensions rate5 wc rclvritc

in the dual spaces of quantities

the bal;mce identity

(6) in terms

and

of normalized

I’iltCS.

(7)

The ratio (( i q’ )‘W’) interest raenues

is the share of net wealth necessary to obtain one DM

frt?m the in\zstment

the share of net wealth liability

L;.

From

implying

the duality

in the asset Ai.J Similarly,

one DM

approach’

interest

payment

we obtain

‘If % I, - I. \\c C~l~l,~lllI r,’ -I(. tllai i5, .I: := 1 r(’ or $1: Ii’=(I %cc c.g. Lau t I WJ. I’),7 11;br ;I &Wild dixx~ssion on duality.

((l/r: from

the maximum

I.,‘) It:

j/M/) is holding Ie~4

of

=

mm L: (Z.4 , , . . .. Z,4,,. ZK. ZL. . . .. ZL ,,,.f ) dil ,“‘” %I.,),

subject to (7 ).

(XI

I’ is strictly decreasing in the normalized interest rates of the asset3 and strictly increasing in the normalized intcrcst rates of the liabilities. end quasiconvex in all normalized \ ariables. Using Roy’s identity. LVC can dcri\e a complctc system of dcm:lild functions for assets and liubilitics. For empirical ilnpi~mcnt~htion. hc?\\.c\cr. \\‘c haw to reduce the number of wkblcs. We rhtxfc~re ccwstruct sub-utilit\ that the direct utility function LT is grc)up\\ie functions by wuming separable in the Mowing sense:

r*[r., IZ.4

1.‘.

. .

%.4,,. t ). ZK. L’, (ZL,.

. .. ZL )),’I 1.r-j.

I‘unctionk. Ihis where Ix*, and L’, art‘ homothetic sub-utility c>r :I,,~~w3ytor assumption implies that the in\xNor first all~watcs his net uulrh tc) tcJtal assets. capital. and total liabilities and in n wcond stage hc carritis out an optimal allocation within the t\vo aggrcg,atc+assets and liabilitic3. repc’cdem;and funcriws lit‘ emplov t;x tiw1y. For da-i\ ing utility-masimizitl~ cc~rrcsponding indirect utilit! function.:’

with the homc~thctic ~135utility functions

Groupwisc separability implic< that within the aggregate IS indcpcndent t

i;

;intl

Ii.

Homolheticity of the sub-utility functions implies that the asset (liability) share within the aggregate is independent of total assets (liabilities) depending only on interest rates of assets (liabilities) that make up the group. For

the study of the effsct of changes in normalized interest rates. in private net we&h IUB& &aqge~ in pr&r~nces over time OR the portfolio composition we begin the first sage of our analysis with the overall model

subject to the aggregate balance equation

or equivalently.

We next transform the indirect utility function emplo:.* the logarithmic version of Roy’s identity.

( 1 1)

logarithmically

and

For a specification of demand functions this system may be representt’d as ratios of first-order polynomial functions. S~uch demand functions may be interpreted as first-order approximations to arbitrary systems of demand functions. To obtain a demand system of this type with the adding--up property, homogeneity, qmmetry of the SJutsky matrix and a negative semidefinite Slutzky matrix, four different types of specification of an indirect uti!ity function can be chosen.’ One of them is the translog utility function introduced by Christensen, Jorgenson and Lau (1975). Using this representation of an indirect utility function, whlere In I/ is quadratic - in the logarithms of normalized interest rates and of time r, the system of asset. capital and liability share functions to be estimated follows from ( 15).

Denominator~~as in

( 16)

with the simplified notation

and the normalization x,r. = - 1 which does not affect the shares. We estimate this system of demand functions. To ensure consisten+ c~fthis system with maximization of utility we have to impose ii set of paramstcr rcstrictions.~ If the parameters in lhc denominator arc fhc sitm4! for ;A relations in (19) are necessary and sutkient equations, the parameter conditions for the adding-up property. that is. the sum of the shares is equal

to unity. Furthermore. we have to observe integrability conditions to ensure consistency of our system of demand functions with utility maximization. For the translog demand system symmetry of the parameters Pij ensures this consistency.

These conditions result from the symmetr! of the matrix of compcnsatcd o\+n- and cross-interest substitution effects and prolidc a link betwcn the equations for the shares and the utility function. For econometric estimation ne add to each of the three equations an error term. As the shares sum to unity. these error terms are not distributed independently but sum to zero in each year. This implies that the variance cowriance matrix is singular. We therefore drop the last equation and compute the parameters of this equation from the parameter restrictions (19) and the normalization x1, = - I. For the estimation we use quarterly t-i‘me series data for the aggregates assets .4. liabilities L and capital K and normalized interest rates (1 r’) 11: (1 I*‘) If and ( 1 I*‘- I ItI computed from ( 13) and ( 14). This terminates the first stage in our model for the allocation of pri\ute net walrh. We nest turn to the sub-models for the aggresatcs.

3. Jlodels of aggregation

and sub-utility

functions

construction of sub-models for the ;&xation of the aggregated wets and liabilities we 11a1.e1;) derilc demand functions for the components of the ;lpgrcgate~ from the sub-utility functions. We first consider the h~~nlothe~\ic direct utility function in the interest rc\cnucs and cash scr\icc Ilr~u I)!’ tllc‘ t4jt;il iiitcrcht iwc’ntlc ;~ggrcg;~Ic. For

the

%:I = I,. , /Z.-l , . . . .. %A,,.r I= w’T(%,4p. \\.llCI.CI'* , -F

. ..%.I,,.

I I).

' (Z.4 )

is homogeneous of degree ow. The corresponding indirect utility function IIT is homogeneous of degree minus one.” where WC omit the subscript -I in the notation for 1’;‘.

F‘- ’ (i-,.1)= H,

*(,l,.. ...,].I). ,I

/

(21)

To show that 1he homothetic aggregat”r reciprocal indirec1 utility function” of

function

NT nlLlltiply

introduced

in (9) is the

both sides of (21 ) by IIT and obtain because of 111~homu~eneit~ of degree one for F _ 1 and degree minus one for 1-J:.

~,liicli is lh,c honiotheric

sub-utility

function

in (‘)I.

Next. we obtain by using Roy’s identity in the logarithmic maximizing asset shares of the portfolio composition,

version the utility-

i=l .

Ill v,

, . . .. Il. (25)

As t; is !>omogeneous of degree one. the denominator of (25) sums to unity and the asset share functions are independent of total assets: (25) reduces to

i=

l.....~.

(26)

The desired shares arc homogeneous of degree zero in 1 I'i(i= I.. . .” II 1. Doubling 1 I’~.or equi\.alcntly. halving the inflation rate r, and the interest rates dots not change under the assumed preferences the optimal structure of assets. The next step for deriving a system of desired shares of assets consists in specifying an indirect utility function V,. Again. we choose the translop specification as a quadratic function in In(l;ri) and in f. Then, according to (26). the optimal shares are lop-linear.

A;

.~ =3Ii + ~

pi, Ill ~

t

/sjlr

“j .;

:

,

i=l ,*

= %i -

C /jij In I’;f /l;,f. j=l

The parameter restrictions arc

. . . .. 11.

(27)

Under these parameter restrictions the shares sum to unity and I’, is homogeneous of degree one. For an interpretation of the parameters of the system (27) we computr .31e share elasticities with respect to an interest rate,

?(;ji.;,4).._.I _ =_ -.__Bij._._.

[:ij = ....__- .. (’ 111 “j

Aj

.4

Ail,4

,j,i=l.....

n. .

If PijO, the share elasticity is negative. In the previous section we did not specify 311 ari-iustment process to ‘explain’ the derivation of actual shares from optimal shares as the introduction of such a process makes the equations to be estimated too complex. Here we can introduce such a process. Eq. (27) determines the desired share of asset i. Ii‘ general it is assumed that time is needed to adapt the actual portfolio to the desired one. It is therefore assumed that the‘ difference between the desired and the actual share is reducted by a certain fraction in each period. We assume that the obscrvcd changes in the balance shares can partly be explained by the adjustment to the desired b;lPan~z shares.

As the asset shares sum to unity and the changes of the asset shares to zero. the parameter restriction on the ;‘ir is ~i;‘ir. =;’ for all k. If the adjustment of any asset share _1(.4i(C) .-t(f )I does not depend on the discrcpanq 4 -iklr B .A1 (t ) - .-I ,, (I - I ) A (f - 1 )) for other shares Ii - i. then ;lir, = 0 for I, = i. ;nid ltic model is equivalent to L1(.3i(t) .4(r))=;*(.ji(~) .-llf)-.-li(rt t .-lit1 ~II for iill i, implying equal fractional adjustment of each pre\ious asset share toward\ the desired share.‘” To include the possibility of full adjustment of each previous share to the desired share so that .-tit/ 1 .-l(! I= .riirt) .-I(1 1 for a11 i. we impose the stronger constraint ri;‘;r = 1. that is. ;*= I. Rewriting (PO) NC obtain .;li(t) .itt)

=

x li

i’ib

A,crI .-j(f)

y

-hyk

., .-l,U-

..&l)

1) +tt

-;‘I,)

.-li(ta_l(r_

t) t

)‘

I...1I

Substituting

the desired shares in (27) into (31) we obtain

(32)

Lvhere, because of (28). (33)

note that under the parameter restrictions (28) and (33) to (35) all parameters cm be identified. F’rom the estimation of ;sir,and ii \~t’ obtain the uriginitl xb. from ;.nlc and Pi,, the ori$nal /~w and from ;qir, and /ri, the parameters /IA,. As NY are not interested in the original parameters IVC estimate the system of eq. (32) under the parameter restrictions (33) (35). WC next turn to the allocation of the liability aggregate and deri1.e demand functions for the components of the aggregate using again an indirect sub-utility funcfion. The homothetic direct utility function in the interest payments is We

%l!_= I:, (ZL,.. . .. ZL,,. r J= F ( L’; IZL,. . . .. ZL,,, r ) ).

(36)

where L*z= F * (ZL ) is homogeneous of degree one and (ZL, ( is the expected interest payment at the end of the period from the stock of the ith liability at the beginning of the period (ZL, 5 0. Li S-0 J. Under similar reasoning as before. we can show that the corresponding reciprocal indirect utility function is the subutility function C> in (10). With I > homogeneous of degree one NT consider the indirect sub-utility function’ * 1 /ye ,.I*

‘1

1, ~- L .,... ;1(, 1’*

1

-~ ‘L,r

I’,,,

).’ *

(37)

subject to the liability constraint

To interpret (36), we 0bserF.e that higher interest payments accrue hm an increase in the interest rate of a liability. To kveaken this effect the hou~ch~ld will reshuffle its liabilities at the beginning of the period according to his preference structure. The analogy to the theory of consumer beha\~iol becomes evident. if the utility function Lyz is associated uith a nepatiie number expressing the disutility of interest payments. Higher intewt pa!ments augment the disutility and the household’s 0bjectii.e is to keep thib disutility as low as possible by re-allocating its liabilities. The detrrmination of the optimal structure of liabilities at the beginning of the period re4tb from the objective to maximize the utility function (36) (from more negatks to less negative) subject to the constraint (38). As before. the utility function is strictly increasing and quasi-concave. An increase of ZP, tfrom more negative to less negative) results in an increase in utility. because an incrcasc in ZP, is equivalent to a reduction of interest payments I%/‘!. The cIu;tsiconcavity implies the diminishing marginal rate of interest sub~titutinn. The indirect utility function 1; is strictly monotone and quasi-con\Ct\. / ( 1 I’~) Li is the share of total liability nhich implies one I)31 intcrc\t payments from holding the ith liability. If the interest rate I*,incrtxstx ihat i> I( 1‘ri) L) increases IL-CO). the indirect utility decreases. To deri\,e a system of desired shares of liabilities \f’c’ chocw &Ltrawlers specification for V:, where the inLerest rates are no\t the interest rattx rclc‘\;int to the liability side of the balance sheet. The optimai shares are

n

t.

-.f = Yi I

z

pi j In I’; + pir . I.

i=l

. . . .. ti.

where the parameter restrictions adjustment mechanism, specified liabilities. Liit 1

-----=3ii+

~

Lit)

_

jJ-

Piilll~i+/~ir’t 1

-.,

L,!!..-

h- ri ‘I’ Lit-

Li(t - 1 ) + ( 1 -yii) ---L(t .- I I’ 1I * )

Ikrr

4. Empirical results of the structure

For thr: analysis the utility

maximizing

of the portfolio

composition

wc‘ estimate

shares of the overall model in the aggrqptes

wscts

-l.

maximizing shares within the capital K and liabilities 1,. and the utility aggregates .4 and L. WC first add ;tdditivc error terms to the ecluations for the shares

in

the

aggregates

and

to

of the aggregates. Due

components

model and the summability

the

eqwtions

for

the

shilres

to the b;\hnce identity

of tllc

iI1 thC

in the overall

aggregates in the t\vo submodcls

the

stochastic terms for the shares add to .wro SO that only Ii - I

corresponding

of k equations are required for analysing portfolio behavior. We therefore estimate the first k- 1 equations and compute the parameters of the kth We have fitted the two equations equation from the parameter restrictions. for

the

asset

aggregate

(20). This

test the hypotheses

rather

FOI- a presentation

than

difficulties

of aggregation

and

unrealistic

implications

aggrcgatc

prixute

\\.ith

components prilatc

respect time

non-bank

intcrcst

rates

rate.

dqmsits.

Incwtgugcs.

j-h.

pXWlS

and and

t~~al~~:~ti~n 5tOck The intaut inlaw I’ .:, h4~lldS.

KSUltb

nominal

rate

of return

tie subtract

cjf the capital

on r:itt’

stock. rillC

1~ intcrcst short-term on

the

on 011

IX\

iding returns

wxritici.

listed

the

commorc~~l

in tahlc

mwths

1968. I rcporth

cjf

tllis

nominal I’,

intcrcst

IS5

intt’rcst

raw

on

rate

on

Cilpitill.

uagc

data on

income itdd

and

I arc

of I1 qiwtcr: I’,\

cltlilrterl}r

\illllc

income

of

frtwl

sckmployed

capital

gaina

dtlc

by the

nominal

to

capit:ll

Iqh.

for the liabilities

I*,

bilscd in the

has been ctilculiltcd

deposits.

physical

tas

a11c1 lonpterm

arc

diffcrcnt

c:lpitiil

the m~mthly

securities.

imputed

(,I'

rates

thL’

arc thcrcforc

ph>~sical

stock

three

time

011

inconw

in the nc)minal

capital

frc~nl

an

corpowte

rate‘ on mtx~iltm itltetwt

intt’rest

entreprcncurship the

o\cr rate

to

diltil

o\cr the period

frcw

t;km

for the ilssctx

interest rate

profit.

corrcspimcliii~ rate

1.:

WC C\XII ha\~

c\aluatcd at replacement costs.

and

rates

;I5 il\‘CrilgC

I*~

F’ot- calculating prvpcrty

method

results and

ccc~11cw1~~

t-4

to disregard

all its restrictive

function.

empirical

hccn

we ha\,e with

rCiiI CStiltC!. iilld

liabilities

qwrterly

intcrcst

calculated

GDP-inflation *ia\ insr*

and

in\xntory

Our assets.

haw

in the

research is to

aa IIO disaggregutcd

and

G~rm;~n

of the

Bundesbank

The corresponding

Glpitill

on2

model utility

rind firms

and liabilities

by the perpettul

of our

many households

liabilities. data

sector

1973. IV. Assets the Ccrman

and

series

way for further

for the underlying

to ph~sicii

to the pi~ritm~tC!

of parameters

them,

results

o\w

households

of assets

on quarterly

imposing

of empirical

the

a\3ilablc

(17). subject

cilpit:\l

reduces the number

system to eleven. An altcrnati\v

simultaneous

to

alld for

(16)

( 19) and

restrictions

bills commercial J.icld

on

listed

in table

I are:

(Wcchscldiskontsatz). bills long-term

I*) /a1

(Kontohorrentsat/I. US

gowrntncnt

Although expected interest rates enter the analysis. NX did not perform autorcgressive extrapolations to take the aspect of cxpcctation into account. For estimation we assume that the expected interest rates arc equal to the actual ones. Instead of including dummy \a-iables we 1lal.e seasonally ad.iusted the quarterly time series using the method of ratio of the series to a rno\Glg average. Finally. for the overall model the normalized interest rata (1 I*“) Ml (I I-“)’ Wand (I I*‘,) It’ha\e been scaled to unity in the first period of 1972 (I =O in 1972.1). The estimated paramctcrs of the o\.crall m~xicl (161 of the estimated parameters (18) are gi\w in table 2.” For an interpretation \ve calculate the elasticity of the shares kvith respect to an interest rate. f-‘cw deriving the formulas for these direct and cross-interest rate elasticirics N’L denote the nominator in ( 16) by :2’ and the denominator bl- D.Y. For the Aare .-I It: for instance. ~‘e obtain

elasticities vary over time. For the aggregate models the formulas for the elasticities can be simplified as all flwj are zero because of homogeneity. To get an idea of the size of the elasticities we have computed them in the b@S@ &x&@&A WI&J&WhW fkke to CM% n~~~~~~~~~ the ho&Wkhnaitvalues of th ~~~~~~~~ ia&mst m&mam ~qlaal tol zero. For the overaN mode&in the

aggregatesthe fimwEm redwe Eij

=

-flijlxi-PWp

to $F=

-PWi~ri-(PIC'A+BIVk'+PrYL)r

i.j=A,K.L.

To get an idea of the change of the elasticities, we have also computed them for the years 1969.1 and 1975.1. For analysing the change in preferences over time. we calculate the elasticities of tlnz shares with respect to time: L’i~ =

Pir~Ni - B,,IDN.

i-AK.

L.

For the base period 1972.1 we obtain

With the parameters given in table 2 the elasticities of the shares with respect to interest rates and time for the first quarters of 1969. 1972 and 1975 are

- 0.045 0.20

- 0.035 =:

0.24

E-?=

0.11

-0.02

0.03

-0.002

0.27

-0.08

0.23

-0.008

-0.10

-0.01

- 0.06

0.13

-

0.03

0.25

-0.07

0.31

- O.oO:S.

-0.22

0.03

0.03

0.13

0.002

0.04

-0.003 >

0.21

-0.006

0.12

0.009

-5 E

,

.

=

- 0.03 0.09

0.22 0.33

0.05 - 0.09

0.08 0.27

-0.006 >

The signs of the elasticities are economically reasonable. Own interest elasticities of the two assets are positive, while the own intcrcst elasticity of

the liability is negative. The elasticity of the asset A with respect to the rate of return on capital has increased over time (in absolute terms) showing a faster reshuming of the asset side. This fits into the picture of the weak investment activity in Germany since 1970. Even in 1977 real private investment had not yet reached the level of 19701and no one expected a substantial boost in capital spending. The rate of return on capital decreased and investors considered less risky financial assets more attractive. The substitution relationship appears to have strengthened over time as capital under-utilization and high wage contracts prevented yield-orientated capacity expansion and as the public mood during these years was against grokvth and business. This explanation is underlined by the elasticity of the shares with respect to time which shows an increasing preference for fmancilal assets. Furthermore. assets and liabilities appear to be independent of one another. This underlines our separability assumption that private non-banks reallocate their asset side independently from the structure of the liability side. A change in the interest rate of an asset does not result in a re-allocation of the total balance sheet: it will only change the structure of assets but not of liabilities. In contrary to the behavior of commercial banks private nonbanks do not borrow money to increase its stock of assets if the interest rate on assets rises. This will be only the case if the interest rate for liabilities is lower than for assets, what rarely happens. The elasticity of the liability share with respect to the rate of return on capital is the highest one and indicates that an increase in the rate of profit implies borrowing of funds to tinancr investment in physical capital. However. a low interest rate for liabilities do not stimulate investment in physical capital. Finally. share elasti&tics with respect to net wealth are positive, that is, the wealth elasticitics of assets and liabilities are above unity but for physical capital near unity. We next turn to the empirical results for the asst-lt aggregate. In order to avoid the problem of multicollinearity we note that portfolio selection equations for different assets need not include identical groups of yield variables [Helliwell et al. (1971). Friedman (1977)]. We therefore set a priori some of the parameters equal to zero and it seems plausible to include in the demand functions for time deposits or savings deposits only one long-term interest rate (r4 or r4 ) or in the demand function for government bonds or bearer bonds of banks only one medium-term interest rate (I-? or 1.~1which take up the influence of omitted variables. Similarly. the ad_iustment of an! asset share need not depend on discrepancies for all other asset sh&cs. We include only the own lagged share of an asset and lagged shares of those assets whose interest rates ‘lave been included among the explaining variables . I3 Subject to all parameter restrictions the following system has been

estimated:

0.01 x7 (1.X) 0.335 (5.9)

0.3Y5 (17.9) - O.OOC)7 (5.-+)

~-0.55 I6.3 ) 0. 16 (1.9)

- 0. I h

-- 0.31 I?..3 1

K!t, (2.i)

(1.Y)

In table 3 we present the estimated paraimters of the equations for the iis3ct sh;ircs and in table 4 the short-run elasticities Lvith respect to the inflation and interest wtes for the first quarters of 1972 and 1975. Tticse elasticities are rather small and even long-run elasticities are not substantialI! higher as the speed of adjustment is not low. We observe that the &u-a increase if the own interest rates and the rate of inflation. respecti\.ely. increase (/jji ~0). A higher rate of inflation I’,. for instance. fGrces the pri\ Utc: sector to keep a higher share of liquidity assets for transrictions in the portfolio. Furthermore. an expected inflation motivates a household to purchase prior as planned consumer durables and residential structures MCI

Table 1 Share elasticities with respect to inflation

and interest rates: awt

Interest rate Share Currency and demand deposits

I972 1975

0.010 0.01 I

Time deposits

1972 1975

- 0.020 -0.01x

Savings deposits

1972 1975

Government bonds

1972 1975

Bearer bonds

1972 1975

1-P

“Not significantly different from zero.

r.1

---m_-

r4

rs .__-._._i_n__ __ .__--___

-0A110 - 0.011 O.W6 O.O88

--O.IVU -0.174

- 0.03 I - 0.03 I

0.040 0.041 0.320 0.27 I

- 0.036 - O.03 I 0.019 0.018 - _-_-

-

w-/-I_ rz

r1

side,

0.mi” 0SMt6”

- 0.0 i -0.01 0.076 0,066 O.c~lY OIlI’” .I _ ____ _-------

- 0.34X .- tw9 - o.ow -- O.Of7” _I..--

1

therefore it wilt increase its stock of money at the beginning of the period by withdrawing time deposits and selling government bonds (the coefficient of savings deposits turned out to be zero: one would expect1 a withdrawal of savings deposits by private households and of time deposits’by firms). However, all these effects are not very significant and the elasticities arc tow. The negative effect of the interest rate on time deposit+ on the demand for currency and demand deposits reflects the opportunit! cost of holding rnonq~. How,c\-et-. the low standard error and the small elasticity indicate that currency plus demand deposits and time deposits are wc:ak substitutes. The own price coeficients of time deposits and of saviniis deposits imply that a11 increase in the own rate of return will increase the demand for these deposits: both regression equations imply that satrings and ‘time deposits are substitutes. Time deposits and bearer bonds appear to be weak complements but if we look as a control on the symmetry of the sign of the coefficients we realize that the share of bearer bonds is rather independent of the interest rates. Only an expected higher rate of inflation has a positive effect on the share of bearer bonds: the run into physical assets (e.g. residential structures) increases the demand for bank loans and banks will issue higher-yielded bonds on favorable terms. This effect is however not very important and the share of bearer bonds appears to be independent of the other shares with a .4~nitlcant trend in favor of this asset. We also observe increasing preferences for government bonds and decreasing preferences for wsh and savings deposits. Finally. savings deposits and government bonds are neither substitutes nor complements as the sign of the corresponding coefficients in the two regression equations are not symmetric. A possible ex$anation of such

an inconclusi\.e result is that the estimated parameters fiii are sums of products ;*ik(-Pki) of the adjustment coefficients in the ith equation and the parameters of the ,jth interest rate in all equations for the desired shares. The size of the own and cross adjustment coefficients influences the sign of the estimated parameter /lij and may lead to an observed complementary relationship even if the two assets are substitutes in the desired portfolio. This shows the need for substituting explicitly the expression (34) for fiij in the regression equations. In table 5 we present the estimated parameters of the equations ‘for .the liability shares of the liability aggregate. The parameters on the main diagonal show the expected negative sign. An increase in the own interest rate reduces the corresponding share although. except for long-term commercial bills. these effects arc not significant. The substitution relationship T’able 5 Edimatcd

- 0.0 15 (1.X) O.I3X (2.1)

- 0.O(HP5 12.8I 0.8X (9.0) -0.10 11.X)

paramctcrs: liabilily

iigprC@tC.

0.0’4 (2.X) 0.7 (34.1) 0.0006 (6.1 1 - 0.88 (9.‘) 1 iJ.10 I1.X)

0.X9 lY.0)

-0.77 18.1)

0.82

0.76

1.Y

2.5

- O.SY 19.0b 0.77 (8.1 L

I Sh

S. C’unclusion Our nbjcctiw portl’olio

has btxn to hnd 0111 lvithin

bchaGor

the

substitution

physical capital and liabilities.

a ccrmpl~tf systcni ol il model of

relationship principal

FOlli’

bctwccn financial

conclusions

assets.

cmcrgc from

the

empirical work presented in this pqxr. First.

siib3titution

relationships

;irc

empiricall!,

less

significant

is

t hati

ust~:rll> assumed. Second. physical suhstitutcs

although

capital

and financial

the substitiiticw

assets

relationship

‘The stC;1d! decline of ,thc rate of return outlook

since

continuously awts ;I

raises

mwhanism

marlit

1972

made financi;tl

low elasticity

doubts about a stimulation

has strengthcncd

c?n capital

assets

ol’ capital with

appear to bc ~wy

;ln

purchases by

n7oneIar~~

authorities

wcr

time.

and thu pcwmistic

attractive

substitute.

respect to the yield

The

variable

on

of economic activity on the basis of

that lo\\a interest rates OII assets \‘iil discount

as3cts and into physical capital.

wwk

policy or opcn-

will Icnd in\.cstors to shift

out of

Third. justifying ascts

the separability

indcpcndcntly

Fourth, deposits. ha~~ks

assets and liabilities

cm

nwrncss

apptxr

assumption

be indcpendcnt

that pri\ ate non-banks

of OIIC :moth~r I-C-allo~atc liicir

from the liabilities.

the weak substitution and time deposits partially

to

dots

relationship not

permit

between currency

and demand

to conclude that

pi-i\ate’ non-

offset the cffccti\cncss of monuta-y

of money of these deposits.

policy bccrruw of the