On the Average Return Rate for a Group of Investment Funds

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

L e s ł a w G a j e k * , M a r e k K a l u s z k a *

O N T H E A V E R A G E R E T U R N R A T E F O R A G R O U P O F IN V E S T M E N T F U N D S1

Abstract. In the paper a new definition of the average return rate for a group of investment (or pension) funds is proposed. The definition is derived via integration of the financial results of the group of funds during a given period of time. It satisfies a set of postulates which every coherent definition is supposed to fulfil contrary to the definition which is used in the Polish law of August 1997 on Organisation and Operation of Pension Funds. A very simple formula for the average return rate is available provided that the fund’s shares are stable in time.

1. INTRODUCTION

C onsider a g ro u p o f n pension (or investm ent) fu n d s w hich sta rt th eir activity selling accounting (or p artic ip a tio n ) units a t the sam e price. D en ote by k t(t), / = 1, 2, n, the n u m b er o f all units possessed by the clients o f the i-th fund at the m o m en t t and by w,(i) - th e value o f i-th fund unit a t the m o m en t t. T h e value w ,(0 is established by dividing the to tal assets o f the i-th fund, say A t(t), by the n u m ber o f th e units k t(t). T h e assets A t(t) can change due to the change o f k t(t) or d u e to the chan ge o f th e u n it’s value w f a) according to the form ula

/1 , ( 0 = kf a)wf a).

F o r the individual investor the change o f w ,(0 is o f m ain interest because it results in his ow n re tu rn rate. So define the re tu rn ra te a t th e i-th fund d u rin g the tim e period (ŕ, i - f ДО, by [vv/i + Д 0 — wfa)]/wfa). A ssum e,

* Technical University of Łódź, Institute of Mathematics. 1 Supported by the KBN. G rant No. 11102B 018 14.

fo r m athem atical sim plicity, th a t there exists a lim it o f this re tu rn ra te divided by At, as A r—>0, and d en o te it by ô fa). H ence

F o rm u la (1) m ay be derived also in a n o th e r way. A ssum e th a t b o th £ ,(•) an d w,( ) are differentiable functions. T h e infinitesim al relative change o f the assets o f the i-th fund d u rin g the tim e interval (t, í + At ) is

T h e first sum m and co rrespo nds to the allocation o f u nits as well as to ap p e arin g new clients o r disap p earin g old ones and so on. T h e second sum m an d describes a p ure investm ent effect at the i-th fu nd , an d is equal to öt(t)dt. H ence á j( t) á í has tw o in terpretations: it is the infinitesim al re tu rn ra te for the accounting unit in the i-th fund and , sim ultaneously, it is the in finitesim al re tu rn ra te fo r the assets o f th is fu n d , d u e to th e p u re investm ent effects (we shall use this duality in Section 2 to define the average re tu rn ra te for the whole group).

Let r, d en o te the re tu rn ra te o f the i-th fund d u rin g a given tim e period [Tj, T2]. C learly

T h e ra te r, inform s the client w hat would be his re tu rn a t tim e T2 if he b o u g h t one accounting unit o f the i-th fund at tim e T v

N ow the problem arises how to define an average w eighted re tu rn rate r ( T j, T2) fo r the whole g roup o f n investm ent funds. T h e average re tu rn rate r should reflect the investm ent results o f all the funds. In the Polish pension fund law it is also used in ord er to verify if a given pension fund achieves the so called m inim um required return rate (compare: Security through..., 1997). If the retu rn rate r, is smaller than the m inim um required one, a deficiency arises w hich should be covered by the com pany m an ag in g th e fund. Since the definition has severe financial consequences, it should be very carefully fo rm u lated tak in g into acco u n t th e follow ing “ coherency p o stu la te s” .

( 1)

dA,(t) _ d kt(t) dw£t) Т Т Г — i “Г ----7~r~ A ,(t) k,(t) w,(t)

(2)

P ostu late 1. In case th e g ro u p consists o f one fund (n = 1) r ( T x, T 2 ) should reduce to (2).

P ostulate 2. I f all funds have the sam e values o f th eir ac co u n tin g u n its all the tim e, i.e. w,(i) = . .. wn(t) for all i6|T1, T 2 ], then

T \ — w^ 2 ) ~ )

1 , 2 v v ^ T J

It m eans th a t if the u n it’s value changes in tim e in the sam e w ay in all funds th en it does n o t m a tte r if the clients alocate from a fund to a n o th e r one o r where the new com ers place them selves; th eir individual re tu rn rates will alw ays be the same.

P ostulate 3. If the n u m b er o f units is c o n sta n t at every fund d u rin g the tim e interval [Tl t T2], then

£ л , ( Г 2 ) - Í ^ ( T i )

r c r l f T2) - i = i — - — i= i--- (3) 1 4 ( 7 1 )

/= 1

Indeed, w hen none o f the clients change the fund o r com e in to o r o u t o f the business, then any change o f the assets A t reflects only the investm ent results in the i-th fund. T re atin g all the unds as a solid one leads to the fo rm u la (3) then. U sing the n o ta tio n k t = /с,(г), we o b ta in from (3) th a t

t r M i M ) г ( Т ц T2) = ^ --- (4) I W T . ) i= i where _ wi(T2) - w i(T1) ' “ w , m ) (5>

is the re tu rn ra te o f the i-th fund d urin g the tim e period [T i,T 2 ]. C learly r, satisfies (1).

Postulate 3’. If /с j (í) = ... = kn(ť) = k fo r every t e [ T u T2 ], then

П т 1гт2 ) = ^ --- (6)

1= 1

Indeed, (4) im plies (6). P o stu la te 3 implies also

P ostulate 3” . A ssum e th a t the nu m b er o f units is c o n sta n t a t every fund d u rin g th e tim e interval [Tlt T2], the initial assets (at t = T{ ) o f every fund h av e th e sam e values an d fo r som e / с < п /2, r t = — rk + l ,

r 2 = ~ rk-¥ 2. rk = ~ r 2kt r 2k +l = 0| •••> r „ ~ 0. 1 hen

r ( T 1, T 2 ) = 0.

Indeed, under the assum ptions o f P o stu late 3” , the to tal assets o f the g ro u p are c o n sta n t and since the nu m b er o f units does n o t change at any fu n d , the average re tu rn should be 0.

Postulate 4 (Multiplication Rule). F o r every Г е ^ . Т ^ ] it should hold

1 + r ( T v T2) = [ 1+ г ( Т „ T)][ 1 + r ( T , T2 )] (7) It m eans th a t the average re tu rn since T | until T2 should equal the average re tu rn since T until T2 given th e average re tu rn since T, to T. C learly, the individual re tu rn ra te r, defined by (5) satisfies (7).

Postulate 5. I f th e re are n u m b e rs n lt n 2 e { l , 2, n} su ch th a t ^ n i ( 0 < ^ ( 0 < ^ ( i j ( 0 fo r all t e [ T u T2] and every * = 1 , 2 , n, th en

m in r, ^ r( T j, T2 ) ^ m a x r ,.

i i

C learly, m i n r ; = rni and т а х г ; = гП2.

i i

P o stu la te 5 describes tw o extrem e situations: all clients have chosen the best fund , or all have chosen the w orst one. In both cases n o n e o f them alocate d u rin g th e considered tim e period.

I h e next p o stu late takes into account th a t clients m ay ch an ge the fund w hen its re tu rn ra te has changed com paring with o th er funds.

P ostulate 6. It should hold -T

exp|^ J m in — 1 < r ( T t , T2 ) < exp|^ J m ax Ö, (t)dt J - I.

P o stu late 6 m eans th a t the average re tu rn ra te r is n o t g reater th a n the ra te corresp o n d in g to the case all clients alocate a t each t e [ T l t T2 ] to the fund o b tain in g the highest return rate, and n o t sm aller th a n the ra te co rresp o n d in g to the case all clients alocate to th e fund o b tain in g the sm allest return ra te , respectively.

P ostulate 7. A ssum e th a t n js 2 and f c ^ r ^ O , kt(t) = 0 fo r i = 2, n, r e [ T i , T2 — Ai], w here A f > 0 is such th a t T2 - A t > T v T h en

lim f ( T u T2 ) = r,. Д1

-Sim ilarly, if k t(t) = 0 for i = 2, ..., n and i e / T , + Ai, T2], then lim r ( T lt T2 ) = rv

Af- 0

It m eans th a t if all the clients were m em bers o f a one fund d u rin g alm ost all tim e then the average re tu rn rate w ould be ap p ro xim ately equal to the re tu rn ra te o f th is fund.

The above p o stu lates describe partly a kind o f econom ical in tu itio n and p artly m athem atical self consistency o f any good definition o f a w eighted average re tu rn ra te o f a g roup o f investm ent funds. In the P olish law regulatio ns (The L a w on Organisation and Operation o f Pension Funds, “ D ziennik U staw ” n r 139 poz. 934, A rt. 173; for the English tra n sla tio n , see: Polish Pension..., 1997) the follow ing definition o f the average re tu rn ra te ap pears ^ o ( t ; , t 2 ) = i \ r t i=i* ^ л т . A t(T2 ) \ t w o Í 4 T 2 ) \i = i 1 = 1 /

(

)

U n fo rtu n ately , r ( T u T2 ) defined by (6) does n o t satisfy P ostu lates 3, 3’, 3” , 4 and 7. In Section 2 we derive a definition o f th e average re tu rn ra te basing o n the in teg ratio n o f th e financial results o f th e w hole g ro u p o f funds. I he definition satisfies all the P ostulates 1—7. In Section 3 we derive a sim ple form ula

П

r ( T l t T2) = _n

Y * i wi ( Ti )

fo r th e av e rag e re tu rn ra te w hich is valid w hen th e re la tiv e sh ares k |( 0 / £ k t(t) a a, are co n stan t in tim e fo r i = 1, n. W e show th a t (8)

i= i

alw ays overestim ates T ( T „ T 2 ) in th a t case.

Let A(t) denote the total assets o f the group a t the m om ent £ е [Г р T2], i.e.

A ssum e th a t b o th k t( ) and w(( ) are differentiable functions. T hen

1 he first sum on the right side o f (10) correspo nd s to the influence on the to tal assets value o f fluctuations o f the n u m b er o f u n its at each fund; the second sum co rresponds to the influence o f flu ctu atio n s o f the u n it’s values. T he second sum is corresponding only to the effects o f investing the assets, n o t to alocating the clients between the funds o r so. T h is is exactly w hat we are interested in when defining the average re tu rn rate. T h e second sum on the right side o f (10) m ay be w ritten as

П

2. DEFINITION OF H IE AVERAGE RETURN RATE

/I

A (t) = £ fci(0w,(0

-(9)

A fte r rescaling (9) by the to tal assets, we get

Z

(0

£ fciWw,-(i)

L n 1 = 1

Z fci(0 wi(0

w here <5,(i) = -7- [log w,(t)]> for i = l , n. Sim ilarly as in Section 1, the at

infinitesim al re tu rn ra te for the g roup o f funds d u rin g th e tim e ( t , t + dt ) is equal to

1=1 i* i

and the weighted average re tu rn ra te o f the g roup, d u rin g a given tim e period [Tl t T2], is r (Tu T2 ) = exp ( Í Ä L S M J l T j 1 = 1 Z k,(t)wj(r) i= i

(

)

(com pare (1)). F ro m the econom ical p o in t o f view this is the m ain ca n d id ate to be used as the average re tu rn ra te o f th e group.

3. BASIC PROPERTIES

P roposition 1. T h e average re tu rn ra te r ( T j, T2 ) defined by (11) satisfies all P ostu lates 1-7. A dditionally, if k ^ w ^ t ) = ... = k n(t)wn(t), then:

l + F ( T 1, r 2) = ( l + r 1) . . . ( l + r „ ) , where r, are defined by (2).

P ro o f. O m itted.

F ro m P o stu late 3 we get a very sim ple and useful fo rm u la fo r the average re tu rn ra te г ( Т и Т 2 ) if the n u m b er o f units /c;(i) o f the i-th fund does n o t change in tim e (i.e. kt(t ) = kt):

T rik iwi(Ti)

П т 1,т 2) = 1- ^

---Z W T i ) i= i

T h e follow ing p rop o sitio n show s the relationship betw een r ( T x, T 2 ) and г0( Г , ,Г 2 ), defined by (8), in the case к fa) = const.

P roposition 2. A ssum e th a t kfa) = k t, i = 1, n. T hen

D2 r 0( Tl , T 2 ) = r ( T i , T 2 ) + 2(1 + r ) where

»’ =

is the variance o f re tu rn rates, corresp onding to r ( T i t T2 ). Proof. D enote r ( T t , T 2 ) by f and w,(7]) by w,. T hen

k,Wj _ fc;W,(l + r,) ľ fc,w, Z f y v j O + r , ) 1=1 i=l W, i= 1 Z fciwi ( l + r<) —

Z

+ Z

[(1+r- > r M =

Z

r

, Z

= T + I 2(1 + Г ) £ •> k,w, _{---(Г ) 2}„ , 1 1 Ё rtk iwt (■= t

= r +

C orollary 1. U nder the assum ptions o f P ro p o sitio n 2, r 0(Tu T2) > f ( T lt T2 )

unless r, = . . . = r„. H ence the fo rm u la (8), used in th e P olish pension fund law, overestim ates the real average re tu rn rate.

Exam ple 1. A ssum e th a t the g ro u p consists o f n = 10 funds fo r which the re tu rn rates and the initial un it’s values are as follows

i _{" , T1} 1 10 30% 2 10 25% 3 10 17% 4 10 23% 5 10 28% 6 10 1 1% 7 10 24% 8 10 26% 9 10 27% 10 10 24%

A ssum e th a t the nu m b er o f units is c o n sta n t d u rin g th e considered period o f tim e and k t = ... = kn. T h en the average re tu rn ra te r = 23.500% while the re tu rn ra te defined in law is r0 = 23.614% . T h o u g h the difference seems to be relatively sm all, it w ould result in a large a m o u n t o f deficiency. A fter 5 -6 years the assets in a typical pension fund in P o land will be larger th a n 4 bln P L N . T hen a fund w here the re tu rn fo r th e last 24 m o n th s is low er th a n the m inim um required re tu rn (11.0807% ) is obliged to cover the resulting deficiency. D ue to o verestim ating th e financial results o f the funds, the fund № 6 w ould have to cover an ad d itio n al 2 .2 8 - 106 P L N o f deficiency. All th a t concerns a very typical situ atio n b u t w h at w ould h ap p e n if som e o f the funds had very bad financial results. S uppose fo r

in sta n c e th a t r3 = - 50% r 4 = - 7 0 % . T h e n Г = 7 .50 00 % w hile r о = 12.99977% , hence the overestim ating the financial results o f th e g ro u p is r0 —r = 5.49977% . In th a t case the ad d itio n al deficiency to be covered by a fu nd, du e to a w rong definition, would be 1 1 0 .0 0 -106 P L N . T he largest differences between rh the m ore stran ge values r0 produces. If, for instance, five o f the funds have th e retu rn rates equal to 50% an d the rest five equal to —50 % , then the real average re tu rn ra te is 0% , because the to ta l assets after tw o years are th e sam e. H ow ever, the definition used in law gives r0 = 12.5%.

T h e next p ro p o sitio n provides a sim ple form u la fo r the average re tu rn ra te in the case &,(() are n o t co n stan t in tim e bu t the relative share o f each fund is co nstant.

Proposition 3. A ssum e th a t there are a function q>: [Tly T2] —►R f and

Л

reals a ( > 0 such th a t a, = 1 and i = i

fo r (alm ost) all e [ 7 i, T2 ], i = 1, ..., n. T hen

( 0 ' Zr Pi Wt i Ti ) П r ( T u T2 ) = i =l ₍₁₂₎

00

Proof, (i) O bserve th a t

M D w .iO r‘ , =1 i > , (0w, ( 0

- Í Í *

Using (11) and the equality r, = w,(T2)/w ,(7 ;) - 1, we get (i). T o p ro ve (ii), observe th a t 2| = i 1 " - i l r . z l= 1 В Д ) « < . + Q ft ' и Z ^ ( T i ) « , £ « ^ ( 7 1 X 1 + r , ) < =1 i= 1

T h e rest o f the p ro o f is sim ilar to the p ro o f o f P ro p o sitio n 2.

O bserve th a t a, from P ro p o sitio n 3 satisfy the eq u a tio n s

□

“ i = W I 1 K ( t ) , i = l ... И. i=i

H ence a very sim ple form ula (12) m ay be used as a definitio n o f the average re tu rn ra te even w hen fe,(t) are no t c o n sta n t in tim e, provided the relative share o f each fund in the to tal n u m b er o f u n its is co n stan t.

REFERENCES

Security through Diversity (1997), Office of the Government Plenipotentiary for Social Security

Reform.

Polish Pension Reform Package (1997), Office of the Government Plenipotentiary for Social