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Model of Connection Between Inflation and Interest Rate Based on the Polish Financial Market

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A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 152, 2000

S t e f a n i a G i n a l s k a * , B e a t a S k o w r o n - G r a b o w s k a *

M O D E L O F C O N N E C T IO N B E T W E E N IN F L A T IO N A N D IN T E R E S T R A T E B A SED O N T H E P O L IS H F IN A N C IA L M A R K E T

Abstract. The paper presents some empirical models of financial markets, which describe international interest rates and exchange rates. The main emphasis is placed on the model based on J. A. Frenkel’s model of exchange rates, which presents the theory in detail and gives some practical applications.

The paper includes Fisher-Chow test оГ the stability of financial markets, shown for a large number of observations and concerning simulation models for small changes in interest rates and exchange rates, which can be used to estimate future interest rates.

1. ECONOMETRIC MODEL FOR FORECASTING INTEREST RATES AND EXCHANGE RATES

T h e m odels o f financial m a rk e ts analysed in th e p a p e r d eal w ith in tern a tio n al interest rates and exchange rates. T h e m odel in trod uced below is a version o f F ra n k e l’s m odel. On the basis o f this m odel, w hich applies p arity o f interest rates, it is possible to p ro po se the follow ing relatio n betw een the spot and forw ard exchange rates ( M i l o , G o n t a r , 1994):

lo g St = a + b1 logF , - 1 + h 2 lo g F, _ 2 + b3 lo g Ff_ 3 + fc4lo g F, _ 4 + e„ log S, = b t l ogS, + Cj lo g F t _! + с 2 logF, _ 2 + c3 log F,_3 + c4lo g Ft _4 + lo g e „ where:

S, - the sp o t ra te in period t,

F t- ! ~ the forw ard ra te in period t — 1, F , -2~ the forw ard ra te in period t — 2,

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F , - 3~ th e forw ard ra te in period í —3, F (_ 4- th e forw ard ra te in period t —4,

a, ht , b2, fe3, c lt c 2, c 3, c4 - are p aram eters, e, - is the erro r in period t.

T h is m odel holds good for such currencies as th e U.S. d o llar, C an ad ia n d o lla r, p o u n d sterling, and D e u tsc h m a rk . T h a t is th e m ain cau se o f forecasting sp o t and forw ard rates.

U sing F ra n k e l’s m odel as a basis, it is possible to establish a m odel for forecasting fu tu re interest rates. T h e forecasted in terest ra te is d ep en d en t o n th e interest rate from previous period and th e change in b o th the inflation ra te and the exchange rate. T h e above - m en tio n ed p aram eters are described w ith the use o f the follow ing symbols:

S, - interest ra te in period t, St - t - interest ra te in period t — 1, W, - exchange rate in period t, I, - inflation ra te in period t,

A I, = I, — I , - i - difference between inflation ra te in period t and in period í — 1,

A W , = Wt — W , - i difference betw een exchange ra te in period t and in period t — 1.

In the sim plest case, assum ing th a t It = 0, we get:

S'^S'-ál+AVQ

(

1

)

and assum ing th a t A W t = 0, then

S t = S , - l ( l + A I t) (2)

M o re com plicated and m ore real - life situations can be expressed as follows:

S, = St - i ( l + a • A (3)

if it is assum ed th a t Al t = 0 and

Ą - S ^ O + b - A / , ) (4)

if it is assum ed th a t A W t = Q.

O n th e basis o f (3) and (4), th e follow ing dependence is derived:

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where: coefficients a, b m ak e the above dependence (5) m o re elastic th a n firm eq u a tio n s (1), (2). Som e elem ents o f financial policy o f th e cen tral b an k o r th e governm ent can be included in coefficients a, h. I f it is assum ed th a t AI, and AW, arc insignificant, it is possible to reject co m p o n en t ab AW, and get equation:

S, = V t ( l + a A W , + bAI,) (6) T h is eq u atio n has been chosen as a tool o f calcu lation because it best suits the Polish econom y. It is elastic and includes th ree m a jo r factors. A n e rro r facto r can be added, but this w ould m ak e calculations, m o re difficult and less clear.

By using the values in T ab . 1 (S0, S„ AW„ AI,) fo r t = 1, 2, 3, m to calculate coefficients a, b and applying the least sq uare m e th o d , the follow ing function can be generated:

m

F(a, b) = £ [St_ j ( l + a A W , + b A I t) - S,]2. f = l

C oefficients a, b should be chosen in such a way th a t th e values o f fu nction F(a, b) are sm allest ( G a j d a , 1994). M ethods o f m ath em atical analysis are used to calculate a and b and the m inim um o f differential fu nction F(a, b).

ć) F dF

In the extrem um p oint, the partial differentials — , —r are equal to 0, as da ob in eq u atio n (7): ’ m £ [5 ,- ,(1 + a AW, + bl,) - SJS,_ yAW, = 0, r= 1

(

7

)

£ [S,_ i d + a AW, + bl,) - S J S ,_ , AI, = 0. ^r=i

and from the above we get:

( m rn fl I (5(_ , A ^ ) 2 + b X (S,2- , A 1 В Д ) = У 5 , _ , А В Д - S,_i), »=x t = t , = i m m m ( Ю fl £ (Sf- , A W A I , ) + b £ ( S t - y A I , ) 2 = £ S , - ! A I,(S, - St _ t ). r=t t= l I=1 m m / rn \ 2

O n the basis o f inequality £ X ? • £ Y f > ( fo r d isp ro p o rtio n a l »=i t = i \ t= i J

sequences ( X v X m) and (У,, . . . t Ym), it is possible to say, th a t system o f eq u a tio n s (8) is C ra m e r’s system o f eq u atio n s therefore, and has only one solution ( if there is no such p ro p o rtio n = —— = =

(4)

W hen we solve this system o f eq u atio n s we get the following:

m m m m

a =

(9)

D e n o m in a to rs in th ese e q u a tio n s a re p o sitive, b ecau se o f in dex es p, q e { l , 2, m } for which A W pA I q Ф AWqA I p.

D u e to econom ic and financial reasons it is possible to have such assum ption s th a t d en o m in ato rs in system o f eq u a tio n are positive. A n o th er m ath em atica l problem is w hether a, b calculated a, h fo r eq u a tio n s (9) for fun ction F(a, b) achieve m inim um .

F ro m m athem atical analysis it follows th a t it is eno ug h to fulfil such an inequ atio n

using a and b from (9) we can write:

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T a b l e 1 T s, ДI, Д w, 0 1 0.316 0.317 -0 .0 0 3 - 0 .0 1 0 2 0.319 0.003 - 0 .0 2 2 3 0.286 - 0 .0 2 2 - 0 .0 1 0 4 0.267 -0 .0 0 9 0.018 5 0.268 -0 .0 0 3 - 0 .0 1 2 6 0.247 -0 .0 1 6 - 0 .0 1 0 7 0.232 - 0 .0 2 -0 .0 0 7 8 0.228 -0 .0 0 3 - 0 .0 0 2 9 0.190 -0 .0 3 6 0 .0 2 0 10 0.185 -0 .0 0 9 -0 .0 1 5 11 0.165 -0 .0 1 6 - 0 .0 1 0 12 0.151 -0 .0 1 6 - 0.008

S o u r c e : Author’s calculations based on empirical data from: Report on inflation rale published by and Report about monetary policy National Bank of Poland Warsaw.

F ro m the system o f eq u atio n (9), the follow ing values are derived: a = 0.268965, b = 4.6359.

T his m eans th a t for

S, = St—i (1 + 0,26AW; + 4,63Д /,) (10) function F(a, b) has the sm allest value.

A fter com p arin g o n S t given in T ab . 1 with values S', calculated o n th e basis o f eq u atio n (10) we have:

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T a b l e 2

S, and square difference of S, and SJ needed for receiving value of error, based on data given in Tab. 1

T s, s , - s ; ( s . - s , ) 1 1 0.317 0.319 0 .0 0 2 0.00004 2 0.319 0.323 -0 .0 0 4 0.000018 3 0.286 0.287 - 0 .0 0 1 0.0000017 4 0.267 0.273 -0 .0 0 6 0.000032 5 0.268 0.264 0.004 0.000015 6 0.247 0.249 - 0 .0 0 2 0.0000034 7 0.232 0.225 0.007 0.000055 8 0.228 0.229 -0.0009 0.0000008 9 0.19 0.189 0 .0 0 1 0.0000016 10 0.185 0.183 0 .0 0 2 0.0000047 11 0.165 0.172 -0 .0 0 7 0.000046 12 0.151 0.153 - 0 .0 0 2 0.0000044 S o u r c e : Authors’ calculations. F ro m T a b . 2 we get: m £ ( s ( - s ; ) 2 = 0.0002226 (11) t = 2

T h e result given in (11) shows th a t m odel (6) is a good m odel for searching dependencies betw een S t from S , _ b A Wt and AI,. C han ges in the inflatio n rate I, are o f greater im portance and have a bigger influence on S, th a n changes in the exchange ra te Wt. T herefore, the difference betw een m odel (1), (2) and m odel (6) is obvious.

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2. SIMULATION MODEL FOR STABILITY OF INTEREST RATES

O ne o f the m eth o d s com m only used as a sim ulation m odel is the F ish er-C h o w test. T his test enables to predict the fo rw ard ra te fo r a given series o f dates.

F isher-C ho w test takes the form ( M i l o , G o n t a r , 1994):

S SS S ,

n — k where:

ss - is th e sum o f squares for the entire series o f th e in terest rate, ss, - is the sum o f squares for the first h alf o f series o f interest rate, m - is the n u m b er o f observations excluded w hen calcu latin g s.v,, n - is the n u m b er o f observations included w hen calcu latin g ss,, к - is the n u m b er o f param eters.

I f 0< F” - * < 1 the interest ra te has a desirable (decreasing) tendency. T his m odel helps to sim ulate stability o f interest rate. T h e result fo r this exam ple shows F “-* = 0.679.

3. SOM E ASPECTS OF RISK PREMIUMS: CAPITAL ASSET PRICING M ODEL AND ARBITRAGE PRICING THEORY

R ecognition th a t greater system atic risk m akes an asset less desirable can be used to understan d the capital asset pricing m odel (C A P M ). T h e C A P M is useful because it provides an explan atio n fo r the m a g n itu d e o f an asset’s risk prem ium ; the difference betw een th e assets expected re tu rn an d the risk-free interest rate.

A n asset con tributes risk to a well-diversified p o rtfo lio in th e a m o u n t o f its system atic risk as m easured by beta. W hen an asset h as a high beta, m eaning th a t it has a large a m o u n t o f system atic risk, an d is th ere fo re less desirable, we would expect th a t investors w ould be willing to hold this asset only if it yielded a higher expected retu rn . T his is exactly w h a t the C A P M show s in the eq u a tio n ( M i s h k i n , 1995).

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where:

R e - expected re tu rn on the assets, R f - risk-free interest rate,

ß - beta o f the asset,

R* - expected re tu rn for the m ark e t portfolio.

T h e C A P M eq u a tio n provides th e com m on sense result th a t w hen an assets b eta is zero, w ith m eans th a t it has n o system atic risk, its risk prem ium will be zero. I f its beta is 1.0, i.e. it has the sam e system atic risk as the entire m ark e t, it will have the sam e risk prem ium as the m a rk e t, R ‘m — R f . If the asset has an even higher beta, e.g. 2.0, its risk prem ium will be greater th a n th a t o f the m ark et. F o r instance, if the expected re tu rn on the m a rk e t is 7% and th e risk-free rate is Z, the risk prem ium fo r the m a rk e t is 5% . T h e asset w ith the beta o f 1.5 w ould th en be expected to have a risk prem ium o f 7.5% ( = 1.5 x 5% ).

A lth o u g h the capital assets pricing m odel has proved to be useful in the real life applications, it assum es th a t there is only one source o f system atic risk th a t is found in the m a rk e t p ortfolio. H ow ever, an altern ativ e theory, the arbitrage pricing theory (A PT), takes the view th at there are several sources o f risk in the econom y th a t cannot be eliminated by diversification. These sources o f risk can be considered as related to econom y wide factors such as inflation and aggregate o u tp u t. Instead o f calculating a single beta, like the C A P M , arbitrage-pricing theory calculates m any betas by estim ating the sensitivity o f an asset’s re tu rn to changes in each factor. T h e arb itra g e pricing theo ry e q u a tio n is ( M i s h k i n , 1995) as follows:

Risk premium = Re - R , = р ^ % с1ог - R f ) + ß2(R}ac,or2 ~ R f) + ... + РЩасшъ - R f )-T h u s the arb itrag e pricing th eo ry thus indicates th a t th e risk prem ium for an asset is related to the risk prem ium for each factor, and th a t as th e asset’s sensitivity to each fa cto r increases, its risk prem ium will increase as well.

It is still uncertain w hich o f these theories provides a better ex p lan a tio n o f risk prem ium s. B oth agree th a t an asset has a higher risk prem ium w hen it has a higher system atic risk, and b o th are considered to be valuable tools fo r explaining risk prem ium s.

REFERENCES

M i s h k i n F. (1995), The Economics o f Money, Banking and Financial Markets, Harper Collins College Publishers, New York.

M i l o W., G o n t a r Z. (1994), On Stability o f Econometric Models o f Financial Markets, Conference Materials, Absolwent, Łódź.

G a j d a J. (1994), Ekonometria (Econometrics), Absolwent, Łódź. Raport o inflacji NBP (1997), Warsaw.

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