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Ex post efficiency and mutual fund evaluation
OMEGA Int. J. of Mgmt Sci., Vol. 20, No. 2, pp. 249-255, 1992 Printed in Great Britain. All rights reserved
0305-0483/92 $5.00 + 0.00 Copyright © 1992 Pergamon Press pie
Ex Post Efficiency and Mutual Fund
Evaluation LP J E N N E R G R E N Stockholm School of Economics, Sweden (Received March 1991; in revised form May 1991) The ex post efficient frontier is the tradeoff between portfolio average return and standard deviation which could have been attained in some capital market over a set of consecutive historical subperiods. This article formulates three different problems of deriving the ex post efficient frontier. These problems differ in the extent to which portfolio revision is permitted between subperiods. "l'nc ex post efficient frontier may be useful as a reference point in evaluating the performance of mutual funds and other institutional portfolios. This is mustrated for a set of Swedish mutual funds. Key words--portfolio selection, finance, mutual funds, quadratic programming
1. INTRODUCTION IN ASSESSMENTS of mutual fund performance, the funds are sometimes plotted in a twodimensional diagram with average subperiod return on one axis and standard deviation of subperiod return on the other. We are here talking about the ex post, actually attained average returns and standard deviations over a set of historical subperiods. For instance, Sharpe [3] plotted attained average annual returns for 34 mutual funds against their attained standard deviations (cf. also [1, pp. 571-584]). In Sharpe's investigation, the input data were historical, actually attained rates of return for the 34 funds for the years 1954-1963. To take another example, a French financial information services company (Associ6s en Finance) regularly publishes a survey of French mutual fund performance which includes plots of actually attained average returns vs standard deviations over historical subperiods, In addition to the dots representing individual mutual funds, it may be interesting also to display the ex post efficient frontier, i.e. the tradeoff between portfolio average return and standard deviation that could actually have been attained over the same subperiods using
some relevant underlying universe of stocks. This leads to the problem of determining that ex post frontier. More precisely, there are several such problems which can be defined, depending on the extent to which portfolio revision is allowed between the subperiods. The following section formulates three different problems of deriving the ex post efficient frontier. After that, results are displayed in the context of performance evaluation of Swedish mutual funds ("aktiefonder")(based on monthly return observations between June 30, 1985 and April 30, 1987). The portfolio selection problem, as presented in standard texts like [1] and [2], is an ex ante one. That is, the object is to derive the ex ante tradeoff between expected rate of return and standard deviation (or variance) of rate of return, f o r one future subperiod only. Perhaps surprisingly, the ex post efficient frontier does not seem to have been discussed all that much in the literature.
249
2. THREE DIFFERENT EX P O S T EFFICIENT
FRONTIERS
This paper considers three different problems of deriving the e x p o s t efficient frontier, referred
250
Jennergren--Ex Post Efficiency and Mutual Fund Evaluation
to as the buy-and-hold problem, the rebalancing problem, and the non-buy-and-hold problem, Suppose there are n stocks in some capital market, indexed by i. In each of T previous subperiods (for instance, months), stock i has provided a rate of return rit (i = 1, 2 . . . . . . n; t = 1, 2 . . . . . T). The final portfolio wealth which could have been attained in subperiod t is denoted by IV,, and the portfolio rate of return in the same subperiod by R, (t = l, 2 . . . . . T). The final portfolio wealth in any subperiod t depends on the individual stock returns rit in that subperiod, but it is the nature of that dependence which differentiates the three ex post efficient frontier problems. All three start out as follows (all summations over t go from 1 to T): Minimize[(1/r).y.~&-((l/r).~R,~2T/2 L T ( \ TI) J
(1)
subject to: RI = [(W~- 1)/1], R2= [(W2- W1)/W~],
[ I
(2)
•~r = [(Wr- Wr_t)t Wr_t ], (l/T) . ~ R , > K .
Systematic risk is the only component of risk which is rewarded in the market and hence the only one which is relevant in performance evaluation. As a matter of fact, the standard deviation of actually attained subperiod returns is a total risk measure. However, mutual funds are diversified and hence involve only systematic risk (i.e. the systematic risk of such a fund equals the total risk). In Sections 4 and 5 below, the performance of a set of mutual funds will be compared to the ex post efficient frontier, and that frontier will then also pertain to diversified portfolios (i.e. it will be generated with an individual upper bound on each stock, to assure the inclusion of at least l0 different stocks in each portfolio on the frontier). See also [1, pp. 571-608] for a discussion of various other risk measures in connection with mutual fund performance evaluation. The above formulation (1)-(3) is evidently incomplete. What is needed in addition is a specification of how the subperiod final portfolio wealths W~ depend on the subperiod portfolio choices, as already indicated. Those specifications are stated formally in Appendix 1 and may be summarized as follows:
(3)
t
As seen in the equation defining RI, it is assumed without loss of generality that the initial amount to invest, at the beginning of subperiod 1, was unity. The objective function (1) is seen to be minimization of the ex post standard deviation, but in computations it is actually more convenient to minimize the variance. By solving the problem indicated by (1)-(3) in a parametric fashion, i.e. varying the required average portfolio return K, one can generate a pointwise approximation of the ex post efficient frontier, The objective function (1) hence implies that the ex post risk measure under consideration is the standard deviation of actually attained subperiod returns (or, equivalently, the variance of such returns). This risk measure is evidently the ex post analogue of that in the ordinary Markowitz ex ante portfolio selection problem, Apart from that, the standard deviation of actually attained subperiod returns is an appropriate risk measure for evaluating mutual fund performance, according to the following line of argument:
(i) In the buy-and-hold problem, all portfolio choices are assumed to have been made at the beginning of the first subperiod, and to have been held without any revision at all until the end of the final subperiod. (ii) In the rebalancingproblem, the portfolio choices are assumed to have been made at the beginning of subperiod 1, but the portfolio is rebalanced at the beginning of each subsequent subperiod, so that the portfolio proportions are reset to the original ones, like at the outset of subperiod 1. (iii) In the non-buy-and-hoM problem, it is assumed that arbitrary portfolio revisions could have been undertaken between subperiods. These three different specifications lead to different ex post efficient frontiers, as will be seen presently.
Omega, Vol. 20, No. 2
3. ANEXAMPLE:E X P O S T
EFFICIENT FRONTIERS IN THE SWEDISH STOCK MARKET
Figure 1 shows one example of the three efficient frontiers, for the Swedish stock market. E x post efficient portfolios have been computed based on a universe of 69 stocks quoted on the Stockholm Stock Exchange. The names of the 69 stocks are given in Appendix 2. The stocks in question
a Standard
251 deviation
7
~z~• ~ /
e
ex post
6 4
D•
z
_/
Ex poet portfolioa ~ Efficient frontier
2
have been selected because they were among the most actively traded ones during the relevant period of time, which is from June 30, 1985 to April 30, 1987, under the restriction that they must have been listed during all of that period. Taken together, these 69 stocks constitute a very substantial part of the Swedish stock market. For each stock in the list of 69, monthly returns between June 30, 1985 and April 30, 1987, have been collected. This means that there are 22 consecutive monthly return figures for each stock. The efficient frontiers have been generated as pointwise approximations, by solving the three different problems mentioned in the previous section. In all cases, individual upper bounds have been set to 10% of the amount to invest, which means that all three frontiers pertain to diversified stock portfolios composed of at least l0 stocks in any one subperiod (month). The scales on both axes in Fig. 1 (as well as Figs 2 and 3) are in per cent per month,
,r
A Randomportfolioa [] Mutual funds
t o
0
,
1
,
2
,
3
, , , 4 6 6 Average return
,
7
,
8
,
9
Fig. 2. Mutual funds in relation to random portfolios and ex post frontier.
It is seen from Fig. 1 that the buy-and-hold and rebalancing ex post efficient frontiers are very close, so close, in fact, that they are difficult to distinguish from each other in the figure. This indicates that the rebalancing frontier is a good approximation of the buy-and-hold frontier (a similar result has also been obtained in investigations using different data sets). The non-buy-and-hold ex post efficient frontier, on the other hand, extends considerably below and to the right of the other two frontiers. That is what one would expect, given no restrictions on portfolio revisions between consecutive months. Standard deviation
Standard deviation 8 16 7 14
Ex post frontiers
12
--P~- Rebalancing
10
~
Buy-and-hold
6
Non-buy-and-hold
~
5
~[
/
~.
~,~
;
4 8 Ex post portfolios 2
~
4 1
Efficient frontier
~.
Random portfolioa
[2
Mutual funds
2 0 0 0
2
4
6
8
10
12
14
Average return Fig. 1. E x post efficient frontiers,
16
i
i
i
t
2
3
i
i
i
4 5 6 Average return
i
i
7
8
9
Fig, 3. T-bills included in random portfolios and e x post frontier.
252
Jennergren--Ex Post E~ciency and Mutual Fund Evaluation
4. AN EVALUATION OF SWEDISH MUTUAL FUNDS
been held without revision over the 22-month period). One may compute what the monthly return would have been on each random portfoFigure 2 is an example of a simple way lio in the 22 months between June 30, 1985 and of visualizing the performance of a group April 30, 1987, and then calculate an average of mutual funds. The funds in question are return and a standard deviation. The resulting nine Swedish mutual funds. For those funds, 10 pairs of random portfolio average returns monthly returns have been collected between and standard deviations are also plotted in June 30, 1985 and April 30, 1987. In other Fig. 2. It should be mentioned that taxes have words, 22 consecutive monthly returns have been disregarded in producing Fig. 2 (and Figs 1 been obtained for each fund, the subperiods and 3, as well). This means that the mutual fund (months) being exactly the same as for the evaluation is from the point of view of a taxuniverse of 69 Swedish stocks. According to exempt investor (e.g. a foundation). The same Ohmans Biirsguide 1985/86 (an information analysis could have been carried out after taxes, source about the Swedish stock market), there by making specific tax bracket assumptions. were 19 mutual funds in Sweden then. Of There are thus three categories of portfolios those 19, nine were investing mainly in Swedish plotted in Fig. 2: the nine mutual funds, 10 stocks (although they did also include money random portfolios, and the ex post efficient market instruments, bonds and foreign stocks; frontier. These three categories are comparable the money market instrument holdings will be to each other in several respects: All three further discussed below). Those nine mutual contain diversified portfolios. They all draw on funds are the ones which have been included in the same underlying universe of 69 Swedish this investigation. It may be mentioned that stocks (since the mutual funds were specialized the number of mutual funds in Sweden has in Swedish stocks, they would largely have been since increased greatly. For instance, Ohmans composed of the 69 stocks in question). All Bfrsguide 1988/89 lists 76 mutual funds. For three categories represent buy-and-hold investeach of the nine funds, the average return and ment strategies (this is so by construction for the standard deviation of return has been computed efficient frontier and the random portfolios; the over the 22 months. The resulting pairs of points representing the mutual funds indicate average returns and standard deviations have combinations of average monthly return and been plotted in Fig. 2. The names of the nine standard deviation which could have been obmutual funds are given in Appendix 3. tained by buying and holding each fund over the The mutual funds may be assumed to have 22-month period). Finally, the average returns been diversified (since that is one of the sales and standard deviations pertain to monthly arguments of such funds: to provide diversifica- returns for the same 22 months for all three tion for small investors). In Fig. 2, the perform- categories. ance of the mutual funds is compared to that From an investor's point of view, Fig. 2 of two other categories of diversified stock gives an impression of the potential benefit of portfolios: The ex post efficient frontier for investing in mutual funds rather than putting buy-and-hold portfolios, and 10 randomly together one's own portfolio: The random portselected buy-and-hold portfolios. The efficient folios represent performance which the investor frontier in Fig. 2 is the same as the ex post could hope to attain on his/her own, whereas buy-and-hold efficient frontier in Fig. 1 and the efficient frontier would require perfect forehence indicates the efficient tradeoffwhich could sight. By investing in a mutual fund and hence have been obtained for diversified buy-and-hold obtaining the benefit of professional portfolio portfolios of Swedish stocks in the time period management, the investor would like to move of consideration, from random performance to a higher degree of The 10 random buy-and-hold portfolios in efficiency represented by the ex post efficient Fig. 2 have been generated by picking sets of 10 frontier. In the case of the nine Swedish mutual stocks out of the universe of 69 Swedish stocks funds, the evaluation implied by Fig. 2 is actu(i.e. each random portfolio is composed of 10 ally not entirely positive: It would appear that stocks with an equal amount invested in each the mutual funds, as a group, have obtained stock; each such portfolio is assumed to have lower average returns and standard deviations
253
Omega, Vol. 20, No. 2
than the random portfolios. However, it is not clear that the mutual funds are any closer to the e x p o s t efficient frontier than the random portfolios,
5. INCLUDING MONEY MARKET INSTRUMENTS IN THE E X P O S T EFFICIENT FRONTIER AND IN THE RANDOM PORTFOLIOS The comparison in Fig. 2 may be considered slightly unfair to the mutual funds, in that they include money market instruments, as already indicated above. In fact, their annual reports for the years 1985 and 1986 indicate that the portfolio holdings of such instruments were between 5 and 25% at the ends of those years. One reason why a mutual fund may want to include money market instruments is to provide liquidity to facilitate market transactions. To the extent that mutual funds consider themselves obliged to do so, this could represent a handicap in relation to the random portfolios and the ones on the e x p o s t efficient frontier, which do not include any such instruments, Against this reasoning it should be pointed out, though, that the evaluation discussed here is from the point of view of individual investors, not the management of the mutual funds. If a mutual fund must necessarily include money market instruments, then that is merely another aspect, possibly negative, for the investor to take into account when choosing between investing in a mutual fund and putting together his/her own portfolio, Nevertheless, it is interesting to see how the inclusion of money market instruments in the random portfolios and the ones on the e x p o s t efficient frontier affects the relative performance of the mutual funds. For that reason, Swedish government treasure bills (T-bills) with one month to expiration have been included in the random and e x p o s t efficient frontier portfolios plotted in Fig. 3. More precisely, the return on T-bills with one month left to expiration has been collected for each month between June 30, 1985 and April 30, 1987, i.e. the same 22 subperiods as before. These 22 monthly T-bill returns have been taken as proxies for the money market in general. An initial money market instrument holding of 15% of the available capital (which is then maintained in a buy-and-hold fashion over the 22 subperiods)
has been forced into all random and e x p o s t efficient frontier portfolios in Fig. 3. The random portfolios in Fig. 3 thus contain an initial holding of 15% in money market instruments and l0 stocks, each with a fraction of 8.5%. The stock parts of the random portfolios in Fig. 3 have been generated using the same random number seed as the random portfolios in Fig. 2. The e x p o s t efficient frontier portfolios in Fig. 3 have been generated with the same 15% initial holding of money market instruments and individual upper bounds on the stocks of 8.5% of the amount to invest. Hence, all random and e x p o s t efficient frontier portfolios in Fig. 3 include at least 10 stocks and may still be regarded as diversified. Comparing Fig. 3 with Fig. 2, it is seen that the mutual funds (the positions of which are exactly the same in the two figures) are now somewhat more favorably located, relative to the random portfolios and the e x p o s t efficient frontier. The effect of forcing money market instruments into the random portfolios and the efficient frontier is to reduce their average returns as well as standard deviations. However, it is still not clear that the mutual funds are any closer than the random portfolios to the e x p o s t efficient frontier. 6. CONCLUSION This paper has formulated three problems of deriving the e x p o s t efficient frontier. The latter may be useful as a reference point in assessing the performance of institutional portfolios. This article has provided an example of that type of evaluation, relating to Swedish mutual funds. ACKNOWLEDGEMENTS The author is indebted to Bertil N/islund for discussions and comments. The data on the Swedish mutual funds have been taken from a diploma thesis at the Stockholm Schoolof Economics written by C Boustedt, P H~kansson and B Jansson. REFERENCES 1. Elton EJ and Gruber MJ (1987) Modern Portfolio Theory and Investment Analysis, 3rd edn. Wiley, New York. 2. Markowitz HM (1987) Mean-Variance Analysis in Portfolio Choice and Capital Markets. Blackwell, Oxford. 3. Sharpe WF (1966) Mutual fund performance. J. Bus. 39, 119-138. 4. Zangwill WI (1969)Nonlinear erogramming:A Unified Approach. Prentice-Hall, Englewood Cliffs, NJ.
254
Jennergren--Ex Post E~cieney and Mutual Fund Evaluation
ADDRESSFORCOmIESPOISDENC~ Professor LP Jennergren, T h e Vet and Rt can be eliminated from the Stockholm School of Economics, Box 6501, S-II3 83 rebalancing problem. If so, one obtains an Stockholm, Sweden. ordinary Markowitz portfolio selection problem, with expected returns estimated as APPENDIX 1 (I/T).~.r,,=~
(i= 1,2 ..... n),
t
Ex Post Efficient Frontier Problem Specifications
variances estimated as (l/r).~.(r,t--ei) 2 (i = 1,2 . . . . . n),
Let x; be the amount of money invested in stock i at the beginning of subperiod 1. The buy-and-hold problem consists of the objective function (1) plus restrictions (2)--(3) plus the following restrictions (all summations over i go from 1 to n): x, = 1, (4) , 0 < x~ < U (i = 1, 2 . . . . . n), (5) Wt=~x,(l+r,~), 7 ' I W 2 = ~ x ' ( l ~ + r ' l ) ( 1 +r'2)' (6)
and covariances estimated as (l/T).~(r,,-~,)(rj,-fj)t
( i , j = l , 2 . . . . . n;i#j).
In a certain sense, the ordinary Markowitz portfolio selection problem may hence be interpreted as an ex post efficient frontier problem. Consider finally the non-buy-and-hold problem. Let x~, be the amount of money invested in stock i at the beginning of subperiod t. The non-buy-and-hold problem consists of (1)-(3) plus the following restrictions: '~X~I
=
l,
i
Wr=~'xi(l+ril)(l+ra)'''"(l+rir)'~
As seen from (5), short positions are not allowed. Also, an upper bound U less than unity (e.g. 0.1) is imposed on each portfolio weight, if one is interested in deriving the ex post efficient frontier for diversified portfolios. The buy-and-hold problem (1)-(3), (4)-(6) is not a convex problem. In fact, the efficient frontier may have local "bumps" (non-convexities), unlike the ordinary Markowitz portfolio selection problem. Nevertheless, it can be solved (although slowly on a personal computer) by putting the restriction (3) into the objective function (i.e. minimizing a weighted sum of variance and minus average return) and then applying a constrained optimum seeking procedure; see, e.g. [4, pp. 158-162]. The rebalancing problem consists of (1)--(3) plus the restrictions:
~,i x, = 1, O
O~xtl
( i = !,2, .. . , n ) ,
wtf~,x,~(l+r~), Y~x~ = w~, O < x a < U . W I (i = 1,2 . . . . . n),
w~= Y~xa (I + r~), . '
Y~x,r= Wr_j, 0
1 ( i f f i 1 , 2 . . . . . n),
+r,r ).
The non-buy-and-hold problem can be refermulated as an equivalent one without the x~, and I4", variables. The reformulated problem is a small one, consisting only of the objective function (1), the restriction (3), plus lower and upper bounds on each individual R,. Those bounds are easy to calculate in advance. It is thus easy to solve the non-buy-and-hold problem as a quadratic programming problem. In the previous discussion of the rebalancing and non-buy-and-hold problems, it was assumed that funds could not be withdrawn or added between subperiods (in addition, transaction cOStS were ignored as well). In other words, the amount available to invest at the
Omega, Vol. 20, No. 2
outset of subperiod t was equal to the final portfolio wealth at the end of subperiod t - 1. This assumption is not necessary: More general formulations, permitting the initial amount to invest in each subperiod to be selected freely, but still resulting in the same calculated efficient frontiers, could have been presented.
AP P END IX 2
List of 69 Swedisk Stocks AGA A Alfa-Laval B fr. ASEA A Asken Astra A Atlas Copco fr. BGB Beijer Investment A Bergrnan & Beving B Bilspedition B Boliden A Catena A Custos A Dacke B ESAB A Electrolux B fr. Ericsson B fr. Esselte A Euroc A Export-Invest A Fabege B F1/ikt Gambro B fr. Gullsp~ng B G6tabanken Hasseblad A Holmens Bruk A Hufvudstaden A Iggesunds Bruk A Incentive B fr. Industriv~iriden A Investor A JM Bygg. och Fasth. B Kebo B Korsn~is B
255
Lundbergf6retagen B MoDo B fr. Nisses B PKBanken PLM A Papyrus Perstorp B Pharmacia B Protorp Proventus B Providentia A Ratos A Reinhold B SEBanken A SCAB SIAB A SKF B fr. Saab-Scania A Sandvik A Skandia fr. Skanska B Skaraborgsbanken Sk~ne-Gripen B Stockholms Badhus B Stora A Svenska Handelsbanken st. Swedish Match B fr. Sydkraft C Transatlantic Trelleborg B Volvo B Wermlandsbanken st. Akermans Verkstad A ~stg6tabanken APPENDIX
3
List of Nine Swedish Mutual Funds SEB Delta SEB Placeringsfonden Banco Fond Concita Fond SHB B PK-Spar SPB Contura SPB Sparinvest SPB Kapitalinvest
0305-0483/92 $5.00 + 0.00 Copyright © 1992 Pergamon Press pie
Ex Post Efficiency and Mutual Fund
Evaluation LP J E N N E R G R E N Stockholm School of Economics, Sweden (Received March 1991; in revised form May 1991) The ex post efficient frontier is the tradeoff between portfolio average return and standard deviation which could have been attained in some capital market over a set of consecutive historical subperiods. This article formulates three different problems of deriving the ex post efficient frontier. These problems differ in the extent to which portfolio revision is permitted between subperiods. "l'nc ex post efficient frontier may be useful as a reference point in evaluating the performance of mutual funds and other institutional portfolios. This is mustrated for a set of Swedish mutual funds. Key words--portfolio selection, finance, mutual funds, quadratic programming
1. INTRODUCTION IN ASSESSMENTS of mutual fund performance, the funds are sometimes plotted in a twodimensional diagram with average subperiod return on one axis and standard deviation of subperiod return on the other. We are here talking about the ex post, actually attained average returns and standard deviations over a set of historical subperiods. For instance, Sharpe [3] plotted attained average annual returns for 34 mutual funds against their attained standard deviations (cf. also [1, pp. 571-584]). In Sharpe's investigation, the input data were historical, actually attained rates of return for the 34 funds for the years 1954-1963. To take another example, a French financial information services company (Associ6s en Finance) regularly publishes a survey of French mutual fund performance which includes plots of actually attained average returns vs standard deviations over historical subperiods, In addition to the dots representing individual mutual funds, it may be interesting also to display the ex post efficient frontier, i.e. the tradeoff between portfolio average return and standard deviation that could actually have been attained over the same subperiods using
some relevant underlying universe of stocks. This leads to the problem of determining that ex post frontier. More precisely, there are several such problems which can be defined, depending on the extent to which portfolio revision is allowed between the subperiods. The following section formulates three different problems of deriving the ex post efficient frontier. After that, results are displayed in the context of performance evaluation of Swedish mutual funds ("aktiefonder")(based on monthly return observations between June 30, 1985 and April 30, 1987). The portfolio selection problem, as presented in standard texts like [1] and [2], is an ex ante one. That is, the object is to derive the ex ante tradeoff between expected rate of return and standard deviation (or variance) of rate of return, f o r one future subperiod only. Perhaps surprisingly, the ex post efficient frontier does not seem to have been discussed all that much in the literature.
249
2. THREE DIFFERENT EX P O S T EFFICIENT
FRONTIERS
This paper considers three different problems of deriving the e x p o s t efficient frontier, referred
250
Jennergren--Ex Post Efficiency and Mutual Fund Evaluation
to as the buy-and-hold problem, the rebalancing problem, and the non-buy-and-hold problem, Suppose there are n stocks in some capital market, indexed by i. In each of T previous subperiods (for instance, months), stock i has provided a rate of return rit (i = 1, 2 . . . . . . n; t = 1, 2 . . . . . T). The final portfolio wealth which could have been attained in subperiod t is denoted by IV,, and the portfolio rate of return in the same subperiod by R, (t = l, 2 . . . . . T). The final portfolio wealth in any subperiod t depends on the individual stock returns rit in that subperiod, but it is the nature of that dependence which differentiates the three ex post efficient frontier problems. All three start out as follows (all summations over t go from 1 to T): Minimize[(1/r).y.~&-((l/r).~R,~2T/2 L T ( \ TI) J
(1)
subject to: RI = [(W~- 1)/1], R2= [(W2- W1)/W~],
[ I
(2)
•~r = [(Wr- Wr_t)t Wr_t ], (l/T) . ~ R , > K .
Systematic risk is the only component of risk which is rewarded in the market and hence the only one which is relevant in performance evaluation. As a matter of fact, the standard deviation of actually attained subperiod returns is a total risk measure. However, mutual funds are diversified and hence involve only systematic risk (i.e. the systematic risk of such a fund equals the total risk). In Sections 4 and 5 below, the performance of a set of mutual funds will be compared to the ex post efficient frontier, and that frontier will then also pertain to diversified portfolios (i.e. it will be generated with an individual upper bound on each stock, to assure the inclusion of at least l0 different stocks in each portfolio on the frontier). See also [1, pp. 571-608] for a discussion of various other risk measures in connection with mutual fund performance evaluation. The above formulation (1)-(3) is evidently incomplete. What is needed in addition is a specification of how the subperiod final portfolio wealths W~ depend on the subperiod portfolio choices, as already indicated. Those specifications are stated formally in Appendix 1 and may be summarized as follows:
(3)
t
As seen in the equation defining RI, it is assumed without loss of generality that the initial amount to invest, at the beginning of subperiod 1, was unity. The objective function (1) is seen to be minimization of the ex post standard deviation, but in computations it is actually more convenient to minimize the variance. By solving the problem indicated by (1)-(3) in a parametric fashion, i.e. varying the required average portfolio return K, one can generate a pointwise approximation of the ex post efficient frontier, The objective function (1) hence implies that the ex post risk measure under consideration is the standard deviation of actually attained subperiod returns (or, equivalently, the variance of such returns). This risk measure is evidently the ex post analogue of that in the ordinary Markowitz ex ante portfolio selection problem, Apart from that, the standard deviation of actually attained subperiod returns is an appropriate risk measure for evaluating mutual fund performance, according to the following line of argument:
(i) In the buy-and-hold problem, all portfolio choices are assumed to have been made at the beginning of the first subperiod, and to have been held without any revision at all until the end of the final subperiod. (ii) In the rebalancingproblem, the portfolio choices are assumed to have been made at the beginning of subperiod 1, but the portfolio is rebalanced at the beginning of each subsequent subperiod, so that the portfolio proportions are reset to the original ones, like at the outset of subperiod 1. (iii) In the non-buy-and-hoM problem, it is assumed that arbitrary portfolio revisions could have been undertaken between subperiods. These three different specifications lead to different ex post efficient frontiers, as will be seen presently.
Omega, Vol. 20, No. 2
3. ANEXAMPLE:E X P O S T
EFFICIENT FRONTIERS IN THE SWEDISH STOCK MARKET
Figure 1 shows one example of the three efficient frontiers, for the Swedish stock market. E x post efficient portfolios have been computed based on a universe of 69 stocks quoted on the Stockholm Stock Exchange. The names of the 69 stocks are given in Appendix 2. The stocks in question
a Standard
251 deviation
7
~z~• ~ /
e
ex post
6 4
D•
z
_/
Ex poet portfolioa ~ Efficient frontier
2
have been selected because they were among the most actively traded ones during the relevant period of time, which is from June 30, 1985 to April 30, 1987, under the restriction that they must have been listed during all of that period. Taken together, these 69 stocks constitute a very substantial part of the Swedish stock market. For each stock in the list of 69, monthly returns between June 30, 1985 and April 30, 1987, have been collected. This means that there are 22 consecutive monthly return figures for each stock. The efficient frontiers have been generated as pointwise approximations, by solving the three different problems mentioned in the previous section. In all cases, individual upper bounds have been set to 10% of the amount to invest, which means that all three frontiers pertain to diversified stock portfolios composed of at least l0 stocks in any one subperiod (month). The scales on both axes in Fig. 1 (as well as Figs 2 and 3) are in per cent per month,
,r
A Randomportfolioa [] Mutual funds
t o
0
,
1
,
2
,
3
, , , 4 6 6 Average return
,
7
,
8
,
9
Fig. 2. Mutual funds in relation to random portfolios and ex post frontier.
It is seen from Fig. 1 that the buy-and-hold and rebalancing ex post efficient frontiers are very close, so close, in fact, that they are difficult to distinguish from each other in the figure. This indicates that the rebalancing frontier is a good approximation of the buy-and-hold frontier (a similar result has also been obtained in investigations using different data sets). The non-buy-and-hold ex post efficient frontier, on the other hand, extends considerably below and to the right of the other two frontiers. That is what one would expect, given no restrictions on portfolio revisions between consecutive months. Standard deviation
Standard deviation 8 16 7 14
Ex post frontiers
12
--P~- Rebalancing
10
~
Buy-and-hold
6
Non-buy-and-hold
~
5
~[
/
~.
~,~
;
4 8 Ex post portfolios 2
~
4 1
Efficient frontier
~.
Random portfolioa
[2
Mutual funds
2 0 0 0
2
4
6
8
10
12
14
Average return Fig. 1. E x post efficient frontiers,
16
i
i
i
t
2
3
i
i
i
4 5 6 Average return
i
i
7
8
9
Fig, 3. T-bills included in random portfolios and e x post frontier.
252
Jennergren--Ex Post E~ciency and Mutual Fund Evaluation
4. AN EVALUATION OF SWEDISH MUTUAL FUNDS
been held without revision over the 22-month period). One may compute what the monthly return would have been on each random portfoFigure 2 is an example of a simple way lio in the 22 months between June 30, 1985 and of visualizing the performance of a group April 30, 1987, and then calculate an average of mutual funds. The funds in question are return and a standard deviation. The resulting nine Swedish mutual funds. For those funds, 10 pairs of random portfolio average returns monthly returns have been collected between and standard deviations are also plotted in June 30, 1985 and April 30, 1987. In other Fig. 2. It should be mentioned that taxes have words, 22 consecutive monthly returns have been disregarded in producing Fig. 2 (and Figs 1 been obtained for each fund, the subperiods and 3, as well). This means that the mutual fund (months) being exactly the same as for the evaluation is from the point of view of a taxuniverse of 69 Swedish stocks. According to exempt investor (e.g. a foundation). The same Ohmans Biirsguide 1985/86 (an information analysis could have been carried out after taxes, source about the Swedish stock market), there by making specific tax bracket assumptions. were 19 mutual funds in Sweden then. Of There are thus three categories of portfolios those 19, nine were investing mainly in Swedish plotted in Fig. 2: the nine mutual funds, 10 stocks (although they did also include money random portfolios, and the ex post efficient market instruments, bonds and foreign stocks; frontier. These three categories are comparable the money market instrument holdings will be to each other in several respects: All three further discussed below). Those nine mutual contain diversified portfolios. They all draw on funds are the ones which have been included in the same underlying universe of 69 Swedish this investigation. It may be mentioned that stocks (since the mutual funds were specialized the number of mutual funds in Sweden has in Swedish stocks, they would largely have been since increased greatly. For instance, Ohmans composed of the 69 stocks in question). All Bfrsguide 1988/89 lists 76 mutual funds. For three categories represent buy-and-hold investeach of the nine funds, the average return and ment strategies (this is so by construction for the standard deviation of return has been computed efficient frontier and the random portfolios; the over the 22 months. The resulting pairs of points representing the mutual funds indicate average returns and standard deviations have combinations of average monthly return and been plotted in Fig. 2. The names of the nine standard deviation which could have been obmutual funds are given in Appendix 3. tained by buying and holding each fund over the The mutual funds may be assumed to have 22-month period). Finally, the average returns been diversified (since that is one of the sales and standard deviations pertain to monthly arguments of such funds: to provide diversifica- returns for the same 22 months for all three tion for small investors). In Fig. 2, the perform- categories. ance of the mutual funds is compared to that From an investor's point of view, Fig. 2 of two other categories of diversified stock gives an impression of the potential benefit of portfolios: The ex post efficient frontier for investing in mutual funds rather than putting buy-and-hold portfolios, and 10 randomly together one's own portfolio: The random portselected buy-and-hold portfolios. The efficient folios represent performance which the investor frontier in Fig. 2 is the same as the ex post could hope to attain on his/her own, whereas buy-and-hold efficient frontier in Fig. 1 and the efficient frontier would require perfect forehence indicates the efficient tradeoffwhich could sight. By investing in a mutual fund and hence have been obtained for diversified buy-and-hold obtaining the benefit of professional portfolio portfolios of Swedish stocks in the time period management, the investor would like to move of consideration, from random performance to a higher degree of The 10 random buy-and-hold portfolios in efficiency represented by the ex post efficient Fig. 2 have been generated by picking sets of 10 frontier. In the case of the nine Swedish mutual stocks out of the universe of 69 Swedish stocks funds, the evaluation implied by Fig. 2 is actu(i.e. each random portfolio is composed of 10 ally not entirely positive: It would appear that stocks with an equal amount invested in each the mutual funds, as a group, have obtained stock; each such portfolio is assumed to have lower average returns and standard deviations
253
Omega, Vol. 20, No. 2
than the random portfolios. However, it is not clear that the mutual funds are any closer to the e x p o s t efficient frontier than the random portfolios,
5. INCLUDING MONEY MARKET INSTRUMENTS IN THE E X P O S T EFFICIENT FRONTIER AND IN THE RANDOM PORTFOLIOS The comparison in Fig. 2 may be considered slightly unfair to the mutual funds, in that they include money market instruments, as already indicated above. In fact, their annual reports for the years 1985 and 1986 indicate that the portfolio holdings of such instruments were between 5 and 25% at the ends of those years. One reason why a mutual fund may want to include money market instruments is to provide liquidity to facilitate market transactions. To the extent that mutual funds consider themselves obliged to do so, this could represent a handicap in relation to the random portfolios and the ones on the e x p o s t efficient frontier, which do not include any such instruments, Against this reasoning it should be pointed out, though, that the evaluation discussed here is from the point of view of individual investors, not the management of the mutual funds. If a mutual fund must necessarily include money market instruments, then that is merely another aspect, possibly negative, for the investor to take into account when choosing between investing in a mutual fund and putting together his/her own portfolio, Nevertheless, it is interesting to see how the inclusion of money market instruments in the random portfolios and the ones on the e x p o s t efficient frontier affects the relative performance of the mutual funds. For that reason, Swedish government treasure bills (T-bills) with one month to expiration have been included in the random and e x p o s t efficient frontier portfolios plotted in Fig. 3. More precisely, the return on T-bills with one month left to expiration has been collected for each month between June 30, 1985 and April 30, 1987, i.e. the same 22 subperiods as before. These 22 monthly T-bill returns have been taken as proxies for the money market in general. An initial money market instrument holding of 15% of the available capital (which is then maintained in a buy-and-hold fashion over the 22 subperiods)
has been forced into all random and e x p o s t efficient frontier portfolios in Fig. 3. The random portfolios in Fig. 3 thus contain an initial holding of 15% in money market instruments and l0 stocks, each with a fraction of 8.5%. The stock parts of the random portfolios in Fig. 3 have been generated using the same random number seed as the random portfolios in Fig. 2. The e x p o s t efficient frontier portfolios in Fig. 3 have been generated with the same 15% initial holding of money market instruments and individual upper bounds on the stocks of 8.5% of the amount to invest. Hence, all random and e x p o s t efficient frontier portfolios in Fig. 3 include at least 10 stocks and may still be regarded as diversified. Comparing Fig. 3 with Fig. 2, it is seen that the mutual funds (the positions of which are exactly the same in the two figures) are now somewhat more favorably located, relative to the random portfolios and the e x p o s t efficient frontier. The effect of forcing money market instruments into the random portfolios and the efficient frontier is to reduce their average returns as well as standard deviations. However, it is still not clear that the mutual funds are any closer than the random portfolios to the e x p o s t efficient frontier. 6. CONCLUSION This paper has formulated three problems of deriving the e x p o s t efficient frontier. The latter may be useful as a reference point in assessing the performance of institutional portfolios. This article has provided an example of that type of evaluation, relating to Swedish mutual funds. ACKNOWLEDGEMENTS The author is indebted to Bertil N/islund for discussions and comments. The data on the Swedish mutual funds have been taken from a diploma thesis at the Stockholm Schoolof Economics written by C Boustedt, P H~kansson and B Jansson. REFERENCES 1. Elton EJ and Gruber MJ (1987) Modern Portfolio Theory and Investment Analysis, 3rd edn. Wiley, New York. 2. Markowitz HM (1987) Mean-Variance Analysis in Portfolio Choice and Capital Markets. Blackwell, Oxford. 3. Sharpe WF (1966) Mutual fund performance. J. Bus. 39, 119-138. 4. Zangwill WI (1969)Nonlinear erogramming:A Unified Approach. Prentice-Hall, Englewood Cliffs, NJ.
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Jennergren--Ex Post E~cieney and Mutual Fund Evaluation
ADDRESSFORCOmIESPOISDENC~ Professor LP Jennergren, T h e Vet and Rt can be eliminated from the Stockholm School of Economics, Box 6501, S-II3 83 rebalancing problem. If so, one obtains an Stockholm, Sweden. ordinary Markowitz portfolio selection problem, with expected returns estimated as APPENDIX 1 (I/T).~.r,,=~
(i= 1,2 ..... n),
t
Ex Post Efficient Frontier Problem Specifications
variances estimated as (l/r).~.(r,t--ei) 2 (i = 1,2 . . . . . n),
Let x; be the amount of money invested in stock i at the beginning of subperiod 1. The buy-and-hold problem consists of the objective function (1) plus restrictions (2)--(3) plus the following restrictions (all summations over i go from 1 to n): x, = 1, (4) , 0 < x~ < U (i = 1, 2 . . . . . n), (5) Wt=~x,(l+r,~), 7 ' I W 2 = ~ x ' ( l ~ + r ' l ) ( 1 +r'2)' (6)
and covariances estimated as (l/T).~(r,,-~,)(rj,-fj)t
( i , j = l , 2 . . . . . n;i#j).
In a certain sense, the ordinary Markowitz portfolio selection problem may hence be interpreted as an ex post efficient frontier problem. Consider finally the non-buy-and-hold problem. Let x~, be the amount of money invested in stock i at the beginning of subperiod t. The non-buy-and-hold problem consists of (1)-(3) plus the following restrictions: '~X~I
=
l,
i
Wr=~'xi(l+ril)(l+ra)'''"(l+rir)'~
As seen from (5), short positions are not allowed. Also, an upper bound U less than unity (e.g. 0.1) is imposed on each portfolio weight, if one is interested in deriving the ex post efficient frontier for diversified portfolios. The buy-and-hold problem (1)-(3), (4)-(6) is not a convex problem. In fact, the efficient frontier may have local "bumps" (non-convexities), unlike the ordinary Markowitz portfolio selection problem. Nevertheless, it can be solved (although slowly on a personal computer) by putting the restriction (3) into the objective function (i.e. minimizing a weighted sum of variance and minus average return) and then applying a constrained optimum seeking procedure; see, e.g. [4, pp. 158-162]. The rebalancing problem consists of (1)--(3) plus the restrictions:
~,i x, = 1, O
O~xtl
( i = !,2, .. . , n ) ,
wtf~,x,~(l+r~), Y~x~ = w~, O < x a < U . W I (i = 1,2 . . . . . n),
w~= Y~xa (I + r~), . '
Y~x,r= Wr_j, 0
1 ( i f f i 1 , 2 . . . . . n),
+r,r ).
The non-buy-and-hold problem can be refermulated as an equivalent one without the x~, and I4", variables. The reformulated problem is a small one, consisting only of the objective function (1), the restriction (3), plus lower and upper bounds on each individual R,. Those bounds are easy to calculate in advance. It is thus easy to solve the non-buy-and-hold problem as a quadratic programming problem. In the previous discussion of the rebalancing and non-buy-and-hold problems, it was assumed that funds could not be withdrawn or added between subperiods (in addition, transaction cOStS were ignored as well). In other words, the amount available to invest at the
Omega, Vol. 20, No. 2
outset of subperiod t was equal to the final portfolio wealth at the end of subperiod t - 1. This assumption is not necessary: More general formulations, permitting the initial amount to invest in each subperiod to be selected freely, but still resulting in the same calculated efficient frontiers, could have been presented.
AP P END IX 2
List of 69 Swedisk Stocks AGA A Alfa-Laval B fr. ASEA A Asken Astra A Atlas Copco fr. BGB Beijer Investment A Bergrnan & Beving B Bilspedition B Boliden A Catena A Custos A Dacke B ESAB A Electrolux B fr. Ericsson B fr. Esselte A Euroc A Export-Invest A Fabege B F1/ikt Gambro B fr. Gullsp~ng B G6tabanken Hasseblad A Holmens Bruk A Hufvudstaden A Iggesunds Bruk A Incentive B fr. Industriv~iriden A Investor A JM Bygg. och Fasth. B Kebo B Korsn~is B
255
Lundbergf6retagen B MoDo B fr. Nisses B PKBanken PLM A Papyrus Perstorp B Pharmacia B Protorp Proventus B Providentia A Ratos A Reinhold B SEBanken A SCAB SIAB A SKF B fr. Saab-Scania A Sandvik A Skandia fr. Skanska B Skaraborgsbanken Sk~ne-Gripen B Stockholms Badhus B Stora A Svenska Handelsbanken st. Swedish Match B fr. Sydkraft C Transatlantic Trelleborg B Volvo B Wermlandsbanken st. Akermans Verkstad A ~stg6tabanken APPENDIX
3
List of Nine Swedish Mutual Funds SEB Delta SEB Placeringsfonden Banco Fond Concita Fond SHB B PK-Spar SPB Contura SPB Sparinvest SPB Kapitalinvest