Insurance: Mathematics and Economics 27 2000 83–104
Mutual fund evaluation: a portfolio insurance approach A heuristic application in Spain
José M. Chamorro
a, ∗
, José M-
a
. Pérez de Villarreal
b,1
a
Dpt. Fundamentos del Análisis Económico, Instituto de Econom´ıa Pública, Fac. CC. Económicas y Empresariales, Univ. Basque Country, Avda. Lehendakari Aguirre 83, 48015 Bilbao, Spain
b
Dpt. Econom´ıa, Fac. CC. Económicas y Empresariales U. Cantabria, Avda. de los Castros sn, 39005 Santander, Spain Received 1 October 1998; received in revised form 1 August 1999; accepted 19 November 1999
Abstract
Usual techniques for evaluating mutual funds are based on asset pricing models CAPM and APT that are related to the mean–variance analysis, where risk aversion is assumed. Nonetheless we think that, at least in Spain and perhaps in continental
Europe, a significant group of investors can be further portrayed by skewness preference. With this kind of investors in mind, we adopt a different approach based on option pricing theory. In particular, we show how to apply the portfolio insurance
dynamics to the evaluation of mutual funds. We estimate insurance premia for 35 Spanish funds of diverse composition, and we also compute their net-of-downside-risk returns. We then rank funds according to both criteria. We also analyze the effect
of transactions costs on these variables. © 2000 Elsevier Science B.V. All rights reserved.
JEL classification: G22; G12; D81 Keywords: Fund evaluation; Skewness preference; Portfolio insurance; Transactions costs
1. Introduction
Mutual funds are financial institutions that specialize in gathering family savings. The advantages they offer are well known. Savers can take part, through monetary contributions suitable for them, in financial asset portfolios
managed by professionals who can diversify them in the right way, as they operate at great scale, incurring at the same time in low transactions costs. Thus, “market portfolios”, which enjoy efficient combinations of return
and risk, become divisible and for the same reason accessible to less skilled savers. Besides, the liquidity of the shares is very high. Finally, favorable fiscal treatments also benefit participants. Due to all these advantages, it is
not surprising that the savings gathered by funds in Spain be close to 33 billion Spanish pesetas by the end of 1998 39 of GDP.
Not only have mutual funds multiplied their portfolio holdings but also their range has become wider. The offer is no longer limited to the traditional funds in monetary assets, fixed income assets with longer maturity,
∗
Corresponding author. Tel.: +34-9460-13769.
E-mail addresses: jepchgombs.ehu.es J.M. Chamorro, perezjmccaix3.unican.es J.M-
a
. P´erez de Villarreal.
1
Tel.: +34-9422-01649.
0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 9 9 0 0 0 6 0 - 8
84 J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104
risky assets, and corresponding hybrids.
2
The financial internationalization of the economy is mirrored in the appearance of funds with foreign asset portfolios as well as the increasing sensitivity of domestic portfolios to
worldwide environment. Therefore, the crisis of international debt markets, which took place at the beginning of 1994, affected a great number of depositor-minded shareholders in Spanish funds. The feeling that taking part
in mutual funds was not a decision without risk is what made them less attractive as a means of saving; the financial sector then reacted offering the so-called “guaranteed funds”.
3
Financial upheavals during the sum- mer of 1997 and 1998 due to the worsening in Asian and Latin American economies have surely reinforced this
feeling. More sophisticated funds will make rating more necessary. The protection of the less skilled savers also calls
for an improvement in the information on the financial health of the funds. Though some funds be organized as “guaranteed”, there will be others probably most of them entailing risks. Taking part in the small and medium size
firms, together with the opacity arising from operations with derivatives, will lead to shareholders undergoing a greater uncertainty. In this context, one might expect an increase in the demand for evaluation
services.
The improvement in the “rating” activities favors low scale savers as it allows them to place their savings more efficiently. It also benefits the fund managers themselves as they are reported on their relative positions in a ranking
of results. Finally, it also helps the supervising authorities to identify the most troublesome ones and to take action in safeguarding the interests of small scale investors.
The usual techniques for evaluating funds are based on asset pricing models CAPM and APT that are related to the mean–variance analysis, where risk aversion is assumed. In general, the use of these techniques is focused
on equity or variable income funds. Besides, it is necessary to set a “benchmark portfolio” which in addition must be mean–variance efficient; otherwise the comparisons among different funds may be biased. Sharpe et al. 1995
provides a good overview of current practice in Chapter 25.
The starting point of this paper is that investors are further supposed to show strong skewness preference. We focus on those who are so sensitive to the risk of losing that they reject probabilistic distributions with negative
skewness.
4
In our opinion, there is a significant group of investors who can be portrayed in this way. Let us think, for example, about Spanish families and many others in continental Europe who have been traditionally
savers rather than investors, most of whom have become new shareholders in the fund industry recently in spite of their depositor-minded profile. Perhaps by inertia, they are still used to safety, dreaming that “capital never
vanishes”.
Skewness preference is in a sense similar to the asymmetrical risk perception behind the so-called “prospect theory” Kahneman and Tversky, 1979; Tversky and Kahneman, 1992. They demonstrate that subjects’ choices
of lotteries or “prospects” exhibit a wide range of anomalies that violate expected utility theory. In particu- lar, they show that losses loom much much larger than gains, an asymmetry of such magnitude that it can-
not be explained by income effects or curvature in the classical utility function; they call this preference loss
2
In Spain, fixed income funds FIFs are 100 composed of fixed income securities, mixed fixed income funds MFIFs are at least 75 fixed income and at most 25 risky, mixed variable income funds MVIFs are at least 30 fixed income up to 75, and variable income funds
VIFs are more than 30 risky.
3
At maturity, these funds reimburse as a minimum the initial capital plus a known rate and as a maximum the initial investment plus some proportion of the gain in a stock index. The time to maturity is typically several years so they look like a term deposit; no surprise then that many
investors disregard these products and turn to traditional funds.
4
As is well known in the expected utility framework, risk aversion is characterized by the second derivative of the utility function u
′′
. 0. Skewness preference can also be considered by requiring the third derivative to be positive u
′′′
. 0. To see it we simply apply Taylor’s expansion to the expected utility function: E[uR]
= uµ +
1 2
u
′′
µσ
2
+
1 3
u
′′′
µκ + remainder, where E denotes expected value, R is the
portfolio return, and µ, σ
2
and κ are the first three moments of its distribution. Now, provided u
′′′
. 0, the higher negative skewness the lower expected utility. Note that if u
′′′
. takes a very high value then skewness effect may be strong enough to define a “loss rejection” behavior or “no-lose” investor.
J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 85
aversion. Our loss rejection hypothesis could then also be interpreted as an extreme case of loss averse behavior.
5
Although skewness preference can be accounted for in the expected utility context see Footnote 4, the analysis becomes rather complex since we need to consider an additional parameter besides the usual mean and variance. On
the other hand, “safety-first” models, which are also related to skewness preference while keeping mean–variance framework, deviate from the expected utility paradigm in a rather ad hoc way. This is why we propose a new
approach to deal with “no-lose” investors.
The approach is based on option pricing theory. Particularly we focus on the “portfolio insurance” technique, and among others, we refer to previous important works such as those of Gatto et al. 1980, Leland 1980, 1985,
Bookstaber and Clarke 1984, Rubinstein 1984, Benninga and Blume 1985, Benninga 1989, Bird et al. 1990 and Basak 1995.
We deem the portfolio insurance approach to be a quite natural way to grapple with the strong skewness prefer- ence.
6
In short, the basic idea is as follows: if we purchase a stock and simultaneously purchase a put option on that stock, we know that the dollar return from the purchase will never be lower than the exercise price on the put. Let
us now consider the possibility of insuring mutual funds in this way; then, for our investors, the best fund would be the one with the highest payoff provided they were insured against any potential loss, i.e., the one with the highest
net-of-downside-risk return. In our case, there are not put options on the mutual funds in which we want to invest. However, under certain assumptions, their payoffs can be replicated by means of dynamic strategies involving the
underlying portfolio and the riskless asset.
In this paper, we propose a measure of evaluation which is applicable, in principle, to all funds, either fixed or variable income funds. By making use of the portfolio insurance technique, we shall show how it is possible to build
a ranking of funds according to their quality of risk and to their net-of-downside-risk return. Besides, it may be worth stressing that, concerning the information required, our method is hardly very demanding. We only take daily
closing prices for each fund during the sample period together with the interest rate on the riskless asset. Given that a “benchmark portfolio” is not constructed, we are not able to spot which funds if any outperform the benchmark.
In spite of this, performance is evaluated on a relative basis since we place the different mutual funds at the same starting point, at least to guarantee the investor’s initial wealth, and we rank them from best to worst according to
the net-of-downside-risk or hedged returns they afford at the end of the investment period.
We show some results, such as the ranking by risk, which are not too surprising. Variable income funds are placed at the top positions, whereas fixed income ones are placed at the bottom. More surprises arise from the classifications
by hedged return. There are equity funds which head the classifications and suggest that fixed or mixed fixed income are not always the most attractive shelter for those investors especially sensitive to the “risk of losing”. On the other
hand, these rankings differ significantly from other more conventional ones based on raw return.
5
Consider an investor with a utility function of the form: a · R
j
if R
j
0, R
j
if R
j
≥ 0, where R
j
is the return on fund j, and a is a constant ≥1. Such a function exhibits risk aversion in the large, because the loss in utility associated
with a return below zero is greater than the gain in utility associated with a return equally far above zero. Within return ranges that lie wholly above or wholly below zero, however, the function is linear and thus shows risk neutrality. The parameter a reflects the fact that, when considering the
probability distribution of return, people weight possible losses more heavily than possible gains. A “no-lose” investor could then be described by an extremely high coefficient of loss aversion a.
6
Leland 1980 shows that investors who have average expectations on the return of the underlying stock, but whose risk tolerance increases with wealth more rapidly than average, will wish to obtain portfolio insurance. Institutional investors falling in this class might include pension
or endowment funds which at all costs must exceed a minimum value, but thereafter can accept reasonable risks. “Safety-first” investors would find portfolio insurance attractive on this basis. Note that u
′′′
. 0 is a necessary condition to an increasing risk tolerance, as can be seen by differentiating the Arrow–Pratt measure of absolute risk aversion A
= −u
′′
.u
′
.; so the link between skewness preference and portfolio insurance becomes clearer see Footnote 4.
86 J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104
The paper is organized as follows: in Section 2, we explain the key theoretical elements of “portfolio insurance”; we then show how to work out indicators of risk and return, without and with homogenous transactions costs,
that allow us to compare the different funds. In Section 3, we apply these measures to a heterogeneous group of 35 Spanish funds and build some classifications. We dedicate Section 4 to summarize the main results, make a
wary comment, and suggest some further extensions. Finally, Appendix A reports some statistics concerning the goodness of the dynamic strategy as it impinges on our reliance on the results.
2. Portfolio insurance
