J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 89
that is, the sum of the quantities invested in the fund and in the riskless asset. Note that β
0j
− c · δ
0j
is a fraction, as δ
0j
= β
0j
, and for the same reason, β
0j
− c · δ
0j
= β
0j
· 1 − c. If α
0j
is the desired investment, then the following ought to take place α
0j
= A
· β
0j
− c · δ
0j
A · 1 − c · δ
0j
, 9
from where we obtain β
0j
= α
0j
1 − c + c · α
0j
. 10
Up to now we have referred to the initial moment. In order to consider dates within the investment period which we have normalized to be 0
≤ t ≤ 1, let us agree on dividing this in n days and on using the subindex τ = 1, 2, . . . , n to date the variables on a daily basis. Let us also assume that after one more day τ
= 1 the fund has grown by a factor of x
1
, and the riskless asset according to e
r ·1t
. The wealth at the beginning of the following day would be A
· [α
0j
· x
1
+ 1 − α
0j
· e
r ·1t
] · 1 − c · δ
0j
≡ A
1
. 11
With this new initial wealth and the current price of the share, the corresponding proportion α
1j
is determined. Be β
1j
the new purchase order under transaction costs; we then have δ
1j
≡ A
· α
0j
· x
1
· 1 − c · δ
0j
A
1
− β
1j
≡ |D| . 12
Thus c ·δ
1j
will be paid and the net wealth will be A
1
1 − c · δ
1j
. As in the previous stage, the question is to calculate β
1j
from the expression α
1j
= A
1
· β
1j
− c · δ
1j
A
1
· 1 − c · δ
1j
. 13
This equation has two solutions: if D 0,
β
1j
= α
1j
1 + c · 1 − α
1j
+ 1
− α
1j
1 + c · 1 − α
1j
· c
· α
0j
· x
1
· 1 − c · δ
0j
· A A
1
, 14
if D 0, β
1j
= α
1j
1 − c · 1 − α
1j
− 1
− α
1j
1 − c · 1 − α
1j
· c
· α
0j
· x
1
· 1 − c · δ
0j
· A A
1
. 15
Similarly for τ = 2, 3, . . . , n.
3. Empirical results
Previously to the analysis of the results obtained, certain characteristics of the data used ought to be commented upon.
3.1. Data The data consists of daily closing prices of 35 Spanish mutual funds for the period between 1 July 1991 and 30
June 1995, funds that are detailed and described in Tables 1 and 2. They were all constituted between 1986 and 1988, thus proving sufficient seniority. The funds are both of fixed 12 and variable income 8 and mixed 15.
90 J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 Table 1
Some characteristics of the funds in the sample Funds
Year Type
Financial group Wealth 30-VI-95
AHORROFONDO 1986
MFI Ahorro Corporaci´on
5975 million pesetas BEX BOLSA
1987 VI
Argentaria 4371
BEX RENTA 1988
MFI Argentaria
61941 FONDPOSTAL
1987 MFI
Argentaria 70580
FONDOATLANTICO 1987
FI Banco Atlantico
51711 BK FONDO
1987 VI
Bankinter 12017
FONDBARCLAYS 3 1988
MFI Barclays Bank
3689 BBV AHORRO
1987 MFI
BBV 99820
BBV INDICE 1987
VI BBV
7986 BBV RENDIMIENTO
1987 VI
BBV 5189
BBV RENTA 1987
FI BBV
262598 FONBANIF
1987 VI
BCH 15393
FONDHISPANO 1987
MVI BCH
20026 HISPADINER
1987 FI
BCH 61508
HISPAFONDO 1987
MVI BCH
7468 RENTFONDO
1987 FI
BCH 45581
PLUSMADRID 1988
MVI Caja Madrid
7408 IBERCAJA AHORRO
1988 FI
Ibercaja 62173
FONCAIXA 2 1986
FI La Caixa
182966 FONCAIXA 5
1986 MVI
La Caixa 7701
FONCAIXA 7 1988
FI La Caixa
236807 FONCAIXA 9
1988 FI
La Caixa 51800
FONCAIXA 10 1988
FI La Caixa
83351 FONCAIXA 11
1988 FI
La Caixa 122423
MERCHFONDO 1987
IVI Merch Bank
7086 EUROVALOR 1
1987 MVI
Banco Popular 5574
EUROVALOR 2 1987
FI Banco Popular
50718 FONBANESTO
1988 FI
Banesto 48614
FONTISA 1987
MVI Banesto
4441 BSN ACCIONES
1988 VI
Banco Santander 16158
SANTANDER ACCIONES 1986
VI Banco Santander
13378 SANTANDER AHORRO
1987 MFI
Banco Santander 69458
SANTANDER 8020 1986
MFI Banco Santander
12994 SANTANDER PATRIMONIO
1988 MFI
Banco Santander 58003
SUPERFONDO ST II 1986
MFI Banco Santander
109369 Table 2
Composition of the sample of funds Funds
Number Wealth
Total 35
1886275 million pesetas Type
Fixed income 12
67.3 Mixed fixed income
9 26.1
Mixed variable income 6
2.3 Variable income
8 4.3
Group Argentaria
3 7.25
Banco Bilbao Vizcaya 4
20.0 Banco Central Hispano
5 8.0
Banesto 2
2.8 Banco Popular
2 3.0
Banco Santander 6
14.8 La Caixa
6 36.3
Others 7
7.75
J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 91
As there is certain freedom of composition, some of the funds have changed category, going from one modality to another, during the sample period.
The aggregate wealth of the funds in the sample is quite significant, having doubled during the period under con- sideration from 955 044 million Spanish pesetas at the end of 1991 up to 1 886 275 million by mid 1995. In 1995 it
represented about 40 of the patrimonial value of the whole of the funds. On the other hand, the distribution accord- ing to types is biased towards fixed income, which represents 67 of the aggregate wealth, whereas variable income
hardly surpasses 4. The rest is all in mixed income, although in these funds fixed income securities also prevail.
Most of the funds analyzed are managed by a few companies linked to bank groups the numbers in brackets refer to year 1995: Banco de Santander 6 and 14.8 of aggregate wealth, La Caixa 6 and 36.3, Banco Central Hispano
5 and 8, Banco Bilbao Vizcaya 4 and 20, Argentaria 3 and 7.25, Banco Popular 2 and 3, and Banesto 2 and 2.8. This concentration would allow us to study both risk and performance behaviors by bank groups.
Transforming the closing prices of the different funds into daily logarithmic returns has been a first treatment of data series. Afterwards their quarterly standard deviations have been calculated so as to use them as measures of
volatility in the B–S formula. In particular, σ
j
has been estimated from daily returns for the concerned quarter.
7
Finally, the quarterly interest rates of the riskless asset are those of the Treasury Bills TB. 3.2. Results
First, we look at the insurance premia estimated as indicators of funds’ risk; then we put some emphasis on the “covered” or hedged returns and compare them with the raw returns on funds. In both parts, we carry out a sensitivity
analysis towards transactions costs and rescue thresholds assuming three different levels for c 0, 0.1 and 1 and z 0.90, 0.95, and 1. Nevertheless, the pair of values c
= 1, z = 1 stands as the base case, since we think they reflect more accurately the actual costs
8
and the strong loss aversion hypothesis. As it is shown in Figs. 1 and 2 below, both insurance premia and covered returns vary throughout the 16 quarterly
periods so any ranking based on the last estimates may seem rather myopic. In consequence, we deem it sensible to rely on some average values in order to rank the funds more consistently. For this purpose we apply the arithmetic
mean to the premia estimates whereas we use the geometric mean in the case of the covered returns. This is computed here as
1 + R
G cj
=
16
Y
i =1
1 + R
i cj
116
. 16
We propose the geometric mean of the RARs as a global indicator of performance since it sounds well in a loss aversion context. This measure penalizes extreme values, specially the negative ones. In fact, a portfolio with
any ruinous event would have a zero geometric mean and consequently would be ranked in the last position. In other words, note that both 1
+ R
G cj
and ln1 + R
G cj
lead to the same ranking since the latter is a monothonic transformation of the former, and that ln1
+ R
G cj
=
1 16
P
16 i
=1
ln1 + R
i cj
. According to the right-hand side, it is easily shown that the negative values of R
c
are more and more weighted than the positive ones.
9
Note that covered
7
We thus measure historical risk and performance, which are not necessarily indicative about future. Historical results are nonetheless useful and interesting information. Before deciding how to invest going forward, investors are likely to want to know how various funds performed in
the past, and whether they were adequately compensated for the risks they faced.
8
Keep in mind that it is 1 due both to an entry operation purchase of shares in the fund as well as an exit operation sale. Therefore, we are not dealing only with a standard refunding fee but, in these costs, mutual fund entry charges are also included. For example, if the adjustment
in the replicating portfolio involved selling today the same sum that it is to be purchased tomorrow, the transaction cost incurred would be 2 of such sum. Capital gains or losses implied by these readjustments are taxable but we do not analyze this effect.
9
As discussed in Footnote 5, it is possible to represent the intuition of a no-lose behavior in a utilityreturn space by means of a kink in the utility function when the fund return is zero. This feature of concavity is akin to the one showed by the 16th root which determines our geometric
mean.
92 J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104
Fig. 1. Volatility and insurance premia: temporal evolution. Quarters: from 1 July 1991 until 30 June 1995.
J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 93
Fig. 2. The returns of the funds: temporal evolution Quarters: from 1 July 1991 until 30 June 1995.
94 J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104
returns can be negative because of imperfect or incomplete hedging due both to discontinuous trading, transactions costs andor rescue values below unity.
Thus, and for the sake of briefness, we only present the average estimates of the both variables we have just mentioned. They are shown at individual level so as to outline classifications according to risk and return, as well as at
group level FI, MFI, MVI, VI and total of the sample so as to make comparisons among the different types of funds.
3.2.1. Insurance premia and classifications by risk As explained above, Eq. 5 of the previous section can be solved for one unknown, K
j
, by a numerical routine
10
for each observed F
0j
and σ
j
; when transactions costs are considered, the standard deviation is transformed according to Eq. 7. Then, given the solution value K
j
, an estimate of the insurance premium P
0j
can be computed using Eq. 2 and hence the ratio P
0j
F
0j
. We have followed this procedure for each of the sample funds on a quarterly basis over the study period 16
quarters, from 1 July 1991 until 30 June 1995 with the different transaction costs 0, 0.1, and 1 and threshold values 0.90, 0.95, and 1. We set the time to expiration at 0.25, implicitly assuming that, in purchasing insurance,
funds buy a net put every quarter with a maturity of 14 year. Aggregate results. Fig. 1 shows the time evolution of the insurance premia of the whole sample, as well as the
specific ones of the VIFs and FIFs, computed under the base assumptions of c = 1 and z = 1. Note the great
correlation between premia and volatility in the case of VIFs, and also in all of the sample; not so in the FIFs group, where the size of both variables is significantly more reduced. On the graphs mainly in the variable income one it
is the peaks of the quarterly periods of July–September 1992, and April–June 1994, that stand out, periods during which financial markets run into trouble. The relatively high premia of VIFs are also explained by the increase in
volatility derived from transactions costs.
All the aggregate results are shown, by means of simple averages, at the bottom of Table 3. We can here observe that the whole average premium is 7.62
‰
if at least a 100 of the initial investment is guaranteed and no transaction costs are considered. Nevertheless, in the case of including these costs 0.1 and 1, premia increase at 9.89
‰
and at 2.94, respectively. Thus we are aware that insurance premia are effectively sensitive to transactions costs. At
greater cost there is more volatility, and for the same reason the price of the insurance increases. The same sensitivity is observed between insurance premia and rescue levels. Focusing on the case c
= 1, the results show that the higher the values of z demanded, i.e., 0.90, 0.95 and 1, the greater the premia required, 2.40
‰
7.65
‰
and 2.94. As expected, the price of the insurance differs significantly among the different types of funds. From 7.03 with
c = 1, as an average of the 16 quarters, in those of variable income, to much lower levels, about 3.77
‰
, in those of fixed income. Therefore, the figures are consistent with the supposition that funds involving higher risks ought
to pay more for insuring their wealth. Note that this relation takes place for the different values of c and z. Individualized results and classifications. Table 3 also details our individualized calculations under the different
assumptions about transactions costs and rescue levels. The central column remakes the base case c = 1, z = 1.
On the other hand, Table 4 displays the five classifications of funds based on these figures. The ranking is from greater to less risk as gauged by premium. This table worths more commenting.
First, as it is logical, FIFs appear at the lowest positions, whereas VIFs are at the head of the ranking. Among these, FONBANIF has the first position with 8.83. On the other hand, among those of fixed income, there are
funds FONCAIXA, mainly with very small premia. In general, FIFs are positioned behind MFIFs, these follow MVIFs, which, in turn, appear behind VIFs. Nevertheless, FONCAIXA 5 especially and EUROVALOR 1, both
in the MVI segment, appear as involving higher risk than some of VI.
10
We used subroutine BRENTM algorithm no. 504, ACM. In a trial run, we found that the convergence is not sensitive to the initial estimates of the solution. The initial estimate that we used for the value, K
j
, was the actual price at the beginning of the investment period, that is, the closing price on the day before. As for the values of N., they were obtained using a polynomial approximation Cox and Rubinstein 1985, p. 261.
J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 95
Table 3 Quarterly Insurance Premia period: July 1991 to June 1995
Fund c
= 0, z = 1 c
= 01, z = 1 c
= 1, z = 1 c
= 1, z = 0.95 c
= 1, z = 0.90 AHORROFONDO
3.13E −03
5.19E −03
2.44E −02
4.09E −03
7.27E −04
BBV AHORRO 2.65E
−05 2.65E
−04 6.02E
−03 1.89E
−04 2.51E
−06 BBV INDICE
2.40E −02
2.95E −02
7.32E −02
2.23E −02
7.70E −03
BBV RENDIMIENTO 1.24E
−02 1.68E
−02 5.22E
−02 1.33E
−02 3.61E
−03 BBV RENTA
1.58E −08
2.38E −05
2.17E −03
1.29E −05
6.51E −09
BEXBOLSA 2.10E
−02 2.60E
−02 6.61E
−02 1.95E
−02 6.63E
−03 BEXRENTA
8.77E −05
4.89E −04
7.06E −03
4.20E −04
1.38E −05
BK FONDO 2.37E
−02 2.94E
−02 7.38E
−02 2.26E
−02 7.68E
−03 BSN ACCIONES
2.29E −02
2.85E −02
7.23E −02
2.19E −02
7.44E −03
EUROVALOR 1 1.39E
−02 1.84E
−02 5.41E
−02 1.47E
−02 4.48E
−03 EUROVALOR 2
1.59E −04
5.53E −04
6.63E −03
4.03E −04
2.09E −05
FONBANESTO 5.72E
−05 2.80E
−04 5.13E
−03 1.91E
−04 6.54E
−06 FONBANIF
3.27E −02
3.90E −02
8.83E −02
2.95E −02
1.13E −02
FONCAIXA 10 1.54E
−17 1.37E
−07 5.39E
−04 3.77E
−07 1.93E
−11 FONCAIXA 11
1.32E −17
1.41E −07
5.83E −04
3.72E −07
1.84E −11
FONCAIXA 2 7.16E
−22 7.79E
−08 4.72E
−04 1.41E
−07 1.24E
−12 FONCAIXA 5
2.93E −02
3.53E −02
8.26E −02
2.68E −02
9.92E −03
FONCAIXA 7 2.55E
−20 2.60E
−07 6.89E
−04 2.38E
−07 3.21E
−12 FONCAIXA 9
4.39E −18
1.07E −07
5.04E −04
3.33E −07
1.35E −11
FONBARCLAYS 3 4.55E
−03 6.92 E
−03 2.81E
−02 5.34E
−03 1.10E
−03 FONDHISPANO
6.60E −03
1.01E −02
3.92E −02
8.31E −03
1.73E −03
FONDOATLANTICO 2.18E
−05 2.77E
−04 6.56E
−03 2.29E
−04 3.44E
−06 FONDPOSTAL
4.27E −05
3.18E −04
5.49E −03
2.44E −04
5.02E −06
FONTISA 7.73E
−03 1.11E
−02 3.92E
−02 8.82E
−03 2.14E
−03 HISPADINER
3.37E −05
3.54E −04
7.28E −03
2.87E −04
4.65E −06
HISPAFONDO 1.21E
−02 1.65E
−02 5.15E
−02 1.33E
−02 3.68E
−03 IBERCAJA AHORRO
1.19E −04
5.00E −04
7.18E −03
3.82E −04
1.41E −05
MERCHFONDO 1.47E
−02 1.95E
−02 5.72E
−02 1.53E
−02 4.41E
−03 PLUSMADRID
6.17E −03
9.39E −03
3.65E −02
7.52E −03
1.57E −03
RENTFONDO 3.22E
−05 3.63E
−04 7.45E
−03 2.94E
−04 4.22E
−06 SANTANDER ACCIONES
2.72E −02
3.31E −02
7.94E −02
2.51E −02
8.98E −03
SANTANDER AHORRO 2.87E
−07 2.92E
−05 2.69E
−03 3.77E
−05 2.86E
−07 SANTANDER PATRIMONIO
1.03E −03
2.45E −03
1.76E −02
2.15E −03
2.22E −04
SANTANDER 8020 3.01E
−03 5.25E
−03 2.59E
−02 4.26E
−03 6.91E
−04 SUPERFONDO ST II
6.12E −05
1.69E −04
2.55E −03
1.79E −04
1.69E −05
Sample 7.62E
−03 9.89E
−03 2.94E
−02 7.65E
−03 2.40E
−03 Fixed income
3.52E −05
1.96E −04
3.77E −03
1.50E −04
4.49E −06
Mixed fixed income 1.33E
−03 2.34E
−03 1.33E
−02 1.88E
−03 3.09E
−04 Mixed variable income
1.26E −02
1.68E −02
5.05E −02
1.32E −02
3.92E −03
Variable income 2.23E
−02 2.77E
−02 7.03E
−02 2.12E
−02 7.21E
−03
Second, the classifications hardly differ due to the transaction costs and the rescue thresholds. As it is summarized in Table 7, Kendall’s concordance coefficient
11
among rankings with c = 0, 0.1 and 1 under the assumption z = 1
the first three columns in Table 4 is 98.94, which suggests the classifications are not sensitive to homogeneous costs. On the other hand, a similar result is obtained when the comparison is made among rankings with z
= 0.90, 0.95 and 1 under c
= 1 the last three columns, since correlation also approaches 99.
11
Kendall 1963, Chapter 6 defines the coefficient of concordance W = 12 · Sm
2
· n
3
− n, where m is the number of rankings, n is the number of funds, and S is the sum of squares of the actual deviations between the sum of ranks for each fund and the mean value of the sums,
1 2
mn + 1. The coefficient W ranges from 0 to 1; as it increases there is a greater measure of agreement in the rankings.
96 J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 Table 4
Rankings by mean insurance premia period: July 1991 to June 1995 c
= 0, z = 1 c
= 0.1, z = 1 c
= 1, z = 1 c
= 1, z = 0.95 c
= 1, z = 0.90 FONBANIF
FONBANIF FONBANIF
FONBANIF FONBANIF
FONCAIXA 5 FONCAIXA 5
FONCAIXA 5 FONCAIXA 5
FONCAIXA 5 SANTANDER
ACCIONES SANTANDER
ACCIONES SANTANDER
ACCIONES SANTANDER
ACCIONES SANTANDER
ACCIONES BBV INDICE
BBV INDICE BK FONDO
BK FONDO BBV INDICE
BK FONDO BK FONDO
BBV INDICE BBV INDICE
BK FONDO BSN ACCIONES
BSN ACCIONES BSN ACCIONES
BSN ACCIONES BSN ACCIONES
BEXBOLSA BEXBOLSA
BEXBOLSA BEXBOLSA
BEXBOLSA MERCHFONDO
MERCHFONDO MERCHFONDO
MERCHFONDO EUROVALOR 1
EUROVALOR 1 EUROVALOR 1
EUROVALOR 1 EUROVALOR 1
MERCHFONDO BBV RENDIMIENTO
BBV RENDIMIENTO BBV RENDIMIENTO
BBV RENDIMIENTO HISPAFONDO
HISPAFONDO HISPAFONDO
HISPAFONDO HISPAFONDO
BBV RENDIMIENTO FONTISA
FONTISA FONDHISPANO
FONTISA Mean
Mean FONDHISPANO
FONTISA FONDHISPANO
FONTISA FONDHISPANO
Mean PLUSMADRID
Mean FONDHISPANO
PLUSMADRID PLUSMADRID
Mean PLUSMADRID
PLUSMADRID FONBARCLAYS 3
FONBARCLAYS 3 FONBARCLAYS 3
FONBARCLAYS 3 FONBARCLAYS 3
AHORROFONDO SANTANDER 8020
SANTANDER 8020 SANTANDER 8020
AHORROFONDO SANTANDER 8020
AHORROFONDO AHORROFONDO
AHORROFONDO SANTANDER 8020
SANTANDER PATRIMO-
NIO SANTANDER
PATRIMO- NIO
SANTANDER PATRIMO-
NIO SANTANDER
PATRIMO- NIO
SANTANDER PATRIMO-
NIO EUROVALOR 2
EUROVALOR 2 RENTFONDO
BEXRENTA EUROVALOR 2
IBERCAJA AHORRO IBERCAJA AHORRO
HISPADINER EUROVALOR 2
SUPERFONDO ST II BEXRENTA
BEXRENTA IBERCAJA AHORRO
IBERCAJA AHORRO IBERCAJA AHORRO
SUPERFONDO ST II RENTFONDO
BEXRENTA RENTFONDO
BEXRENTA FONBANESTO
HISPADINER EUROVALOR 2
HISPADINER FONBANESTO
FONDPOSTAL FONDPOSTAL
FONDOATLANTICO FONDPOSTAL
FONDPOSTAL HISPADINER
FONBANESTO BBV AHORRO
FONDOATLANTICO HISPADINER
RENTFONDO FONDOATLANTICO
FONDPOSTAL FONBANESTO
RENTFONDO BBV AHORRO
BBV AHORRO FONBANESTO
BBV AHORRO FONDOATLANTICO
FONDOATLANTICO SUPERFONDO ST II
SANTANDER AHORRO SUPERFONDO ST II
BBV AHORRO SANTANDER AHORRO
SANTANDER AHORRO SUPERFONDO ST II
SANTANDER AHORRO SANTANDER AHORRO
BBV RENTA BBV RENTA
BBV RENTA BBV RENTA
BBV RENTA FONCAIXA 10
FONCAIXA 7 FONCAIXA 7
FONCAIXA 10 FONCAIXA 10
FONCAIXA 11 FONCAIXA 11
FONCAIXA 11 FONCAIXA 11
FONCAIXA 11 FONCAIXA 9
FONCAIXA 10 FONCAIXA 10
FONCAIXA 9 FONCAIXA 9
FONCAIXA 7 FONCAIXA 9
FONCAIXA 9 FONCAIXA 7
FONCAIXA 7 FONCAIXA 2
FONCAIXA 2 FONCAIXA 2
FONCAIXA 2 FONCAIXA 2
3.2.2. Covered returns and corresponding classifications In the previous section we have estimated the insurance premium for each fund on each quarterly period. It is then
straightforward to compute the initial proportions of the underlying asset and the riskless asset in the replicating portfolio by means of Eq. 6. Henceforth the pattern of readjustments goes along the lines described by Eqs. 8–15
with c adopting the value stated in each case. We follow the same order of presentation as before. We only add a section where classifications of funds according
to “covered” or hedged returns are compared to other more traditional ones based on raw returns. Aggregate results. Fig. 2 shows the evolution of the different types of return throughout the 4-year period in the
base case. In the first of them, we outline the profiles of the quarterly covered and raw returns of the whole sample. In the following ones, details about the variable income and fixed income groups of funds are enclosed. In each case
J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 97
the profiles of their returns are compared with the return on a substitute asset, the Madrid Stock Exchange General Index IGBM in the case of variable income, and TB in that of fixed income.
Note that the covered return does not significantly go into the area of negative values, as is logical, whereas the raw return and that of the IGBM do. The value of R
c
approaches zero in the quaterly periods in which the financial markets plummet. On the other hand, this return, not because of being insured, is fixed or invariant; in fact, it varies
more than the return on TB. The latter dominates evolves above R
c
, but not so the raw return. Finally, the global correlation between R
c
and R is 93, 91 in the case of VIFs and 98 in that of FIFs. From Fig. 1 it is also very easy to test that the quarterly frequency distribution does not show negative skewness in the R
c
case while it is much more symmetrical when R is considered. The former is just the profile which would logically be appreciated by
investors with strong skewness preference. Table 5 at the bottom collects our estimates of R
cj
for 16 quarters, both at the aggregate and segment level, under the different costs and thresholds considered. As can be seen, there are no surprises in the relation between return
and transaction costs: the quarterly covered return is reduced as costs are greater. This reduction affects especially VIFs and MVIFs which, due to their composition, are more exposed to the portfolio dynamic revision, though they
are not clearly dominated by FIFs nor MFIFs.
The relation between return and rescue value is less clear. For each set of funds, the mean covered return increases as the threshold increases from z
= 0.90 to z = 1, but except for VIFs it decreases at z = 0.95. The setting of a floor becomes particularly convenient when a loss takes place; otherwise it represents just a
sunk, worth avoiding cost. Although not reported here,
12
it is possible to show that whenever a fund loses more than 5 on some period so that the rescue requirement turns binding, the inequality R
i cj
z = 0.95
R
i cj
z = 0.90 holds; it can then be said that the “threshold-effect” more than offsets the “premium-effect”.
Conversely, when the fund’s loss is less significant the latter dominates the former and R
i cj
z = 0.90 R
i cj
z = 0.95.
Most of the losses, both in frequency and size, refer to the MVI and VI segments, especially to the latter. Concerning VIFs, the number of losses is almost evenly distributed above and below 5 so on average it is possible
to observe that the higher the value of the floor, the higher the covered return. However, only one in three of the losses in the MVIFs is greater than 5; the relation between z and R
c
is thus blurred since the a priori riskier strategy z
= 0.90 is cheaper than the more conservative one z = 0.95. In other words, as the chance of a heavy loss decreases the premium-effect overshadows the threshold’s role. The same fact may be operating in the MFI and
FI segments R
c
decreases as z increases from 0.90 to 0.95 though less strongly since their insurance premia are noticeably lower.
Finally, the interest rates of TB almost always surpass the average covered returns on funds, with few exceptions variable income with low costs; look at the bottom line in Table 5. As TB also lack the risk of losing, even investors
with strong skewness preference should have preferred them to insured funds. How can then be explained that they invested in funds? First of all, shall we clear it, return dominance of the TB and other riskless assets was really
evident in the fact that the greatest slice of household savings had flowed to them over the years 1991–1995; only 5200 billion pesetas less than 5 of households and non-financial firms assets were poured in funds by the end of
1995. Note further that, as mentioned before, fixed income funds represented about two thirds of that figure. On the other hand, fiscal advantages, and perhaps investments by people less sensitive to risk, can explain this remainder
of wealth placed in funds. In any case, we think that the return dominance of TB has been vanishing along with the fall in interest rates during the last years.
Individualized results and classifications. Once again we are more interested in the ranking than in remarking the quantitative individualized estimates reported in Table 5. The only one detail we insist on is the negative values
of the geometric mean returns that emerge for some funds when the rescue threshold is lower than unity. There are
12
The reader interested in these results or in the individualized estimates of premia and returns under the different assumptions and periods can ask the authors for them.
98 J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 Table 5
Quarterly covered and raw returns period: July 1991 to June 1995 Fund
c = 0, z = 1 c = 0.1, z = 1 c = 1, z = 1 c = 1, z = 0.95 c = 1, z = 0.90 Meandev raw
AHORROFONDO 0.01902
0.01762 0.00582
0.00034 0.00267
0.45800 BBV AHORRO
0.02109 0.02000
0.01033 0.01019
0.01025 1.29708
BBV INDICE 0.02587
0.02364 0.00740
−0.0067 −0.0108
0.10921 BBV RENDIMIENTO
0.01765 0.01595
0.00285 −0.0128
−0.0125 0.01777
BBV RENTA 0.02091
0.01986 0.01019
0.01071 0.01071
2.33170 BEXBOLSA
0.01973 0.01780
0.00342 −0.0055
−0.0076 0.15968
BEXRENTA 0.02074
0.01972 0.00999
0.00861 0.00878
1.04198 BK FONDO
0.03753 0.03520
0.01742 0.00681
0.00696 0.33119
BSN ACCIONES 0.04148
0.03922 0.02154
0.01305 0.01629
0.42949 EUROVALOR 1
0.02384 0.02203
0.00953 −0.0032
−0.0018 0.22422
EUROVALOR 2 0.02096
0.01991 0.01016
0.00878 0.00893
1.05686 FONBANESTO
0.01974 0.01870
0.00932 0.00835
0.00845 1.14217
FONBANIF 0.03614
0.03366 0.01558
0.00414 0.00081
0.19703 FONCAIXA 10
0.02058 0.01956
0.01032 0.01037
0.01037 3.24219
FONCAIXA 11 0.02234
0.02132 0.01203
0.01212 0.01212
3.52545 FONCAIXA 2
0.02065 0.01963
0.01039 0.01044
0.01044 3.25610
FONCAIXA 5 0.02964
0.02738 0.01089
−0.0010 0.00033
0.28998 FONCAIXA 7
0.02390 0.02287
0.01350 0.01366
0.01366 3.74130
FONCAIXA 9 0.01941
0.01839 0.00918
0.00921 0.00921
3.05911 FONBARCLAYS 3
0.01924 0.01808
0.00772 −0.0020
0.00082 0.36836
FONDHISPANO 0.02961
0.02807 0.01459
0.00388 0.00635
0.37254 FONDOATLANTICO
0.02114 0.02006
0.00975 0.01051
0.01058 1.42828
FONDPOSTAL 0.02154
0.02053 0.01132
0.00964 0.00973
1.07146 FONTISA
0.02396 0.02245
0.00926 0.00150
0.00097 0.27737
HISPADINER 0.02130
0.02009 0.00958
0.01049 0.01059
1.22438 HISPAFONDO
0.02347 0.02161
0.01017 −0.0052
−0.0091 0.06102
IBERCAJA AHORRO 0.02162
0.02066 0.01104
0.00984 0.01005
1.17235 MERCHFONDO
0.02285 0.02080
0.00895 0.00212
0.01015 0.50394
PLUSMADRID 0.02124
0.01958 0.00609
−0.0002 0.00362
0.44650 RENTFONDO
0.01947 0.01827
0.00792 0.00836
0.00847 1.16197
SANTANDER ACCIONES 0.03737
0.03460 0.01565
0.00575 0.00045
0.17051 SANTANDER AHORRO
0.01974 0.01869
0.00869 0.00953
0.00954 2.54881
SANTANDER PATRIMONIO 0.02450
0.02310 0.01159
0.01036 0.01157
0.80884 SANTANDER 8020
0.02504 0.02385
0.01284 0.00406
0.00563 0.43865
SUPERFONDO ST II 0.01967
0.01860 0.00875
0.00948 0.00958
2.95945 Sample
2.38E −02
2.23E −02
1.03E −02
0.53E −02
0.56E −02
1.17E +00
Fixed income 2.10E
−02 1.99E
−02 1.03E
−02 1.02E
−02 1.03E
−02 2.20E
+00 Mixed fixed income
2.12E −02
2.00E −02
9.68E −03
6.69E −03
7.62E −03
1.22E +00
Mixed variable income 2.53E
−02 2.35E
−02 1.01E
−02 −7.32E−04
5.88E −05
2.79E −01
Variable income 2.98E
−02 2.76E
−02 1.16E
−02 8.45E
−04 4.48E
−04 2.40E
−01 Treasury Bills
2.62E −02
2.62E −02
2.62E −02
2.62E −02
2.62E −02
4.89E +00
five cases when z = 0.90 but eight cases with z = 0.95; one of them is a MFIF and the others are either MVIFs or
VIFs so the above discussion about premium- and threshold-effect probably applies. The first comment on Table 6, whose backing is the former one, has to do with the degree of coherence among the
classifications shown in it. Concerning the first three ones with different values of c but sharing a common z = 1, we
can conclude that they are rather consistent since Kendall’s coefficient amounts to 89 for the whole sample, and gets to 95 and 97 when the comparison is carried out separately for the fixed and variable incomes see Table 7.
On the other hand, the ranking is a bit more sensitive to the rescue threshold. According to Kendall’s measure the agreement among classifications with different values of z and a common c
= 1 gets to 73, 80 and 83 when the whole, FI and VI samples are regarded.
J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 99
100 J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104
Secondly, we look at the base case z = 1, c = 1 in order to remark the feature ranking. Unlike the
one based on insurance premia, this is no longer a staggered juxtaposition or the different partial, or by seg- ments, classifications we could consider. All of them appear rather mixed. So there are some VIFs and MV-
IFs at the top BSN ACCIONES, BK FONDO, SANTANDER ACCIONES, FONBANIF, FONDHISPANO and others at the bottom BBV RENDIMIENTO, BEXBOLSA, PLUSMADRID, BBV INDICE. The same is ob-
served amongst the FIFs or MFIFs RENTFONDO, FONDBARCLAYS 3, AHORROFONDO are at lower posi- tions, but FONCAIXA 7, FONCAIXA 11, SANTANDER 8020 and SANTANDER PATRIMONIO are at higher
ones.
It is worth recalling that two elements have an influence on R
cj
: the revaluation of the fund and the risk of losing. Both factors explain the good behavior of fixed income. Not only does it involve less risk the discount is very
small but the base return was substantial. It seems that in some funds the risk discount has contributed to their low covered returns; for example, the two that
head the ranking by risk FONBANIF and FONCAIXA 5 do not head the classification by return. Nevertheless, it seems that the last positions of others, like BEXBOLSA and BBV RENDIMIENTO, cannot be explained exactly
due to their degree of risk but to their poor revaluation. Conversely, the positions held by some of VIFs BSN ACCIONES, BK FONDO seem to obey to their great revaluation and not to their low discount for risk, as their
insurance premia are substantial and their position in the ranking by risks relatively high.
The fact that several VIFs appear in the highest part of the ranking shows that not always those of FI, let alone those of MFI, are the ones that offer higher insured return. In this case, even wise investors, whose preferences aim
at “not to lose”, should find them attractive. Finally, following our calculations it is also possible to outline, although not clearly enough, a ranking by bank
groups. In this sense, it seems that the one led by Banco de Santander because of its funds in VI and MFI is at the top, and that of BBV at the bottom mainly because of its VIFs; La Caixa would be in the middle of the ranking.
“Covered” versus “raw” returns. Since a conventional or traditional criterion to evaluate the different funds has been the Sharpe’s ratio, which relates mean to standard deviation, we show in Table 5 the individualized estimates
of this ratio, and in Table 6 the ranking based on them. As it is easily seen FIFs and MFIFs appear at the top while MVIFs and VIFs are at the bottom.
Table 7 Correlation coefficients
Mean quarterly insurance premia Insurance premia z
= 1, c = 0 Insurance premia z
= 1, c = 0.1 98.94
Insurance premia z = 1, c = 1
Insurance premia c = 1, z = 1
Insurance premia c = 1, z = 0.95
98.71 Insurance premia c
= 1, z = 0.90 Sample
FIF + MFIF
MVIF + VIF
Mean quarterly covered returns Covered returns z
= 1, c = 0 Covered returns z
= 1, c = 0.1 88.95
95.32 97.06
Covered returns z = 1, c = 1
Covered returns c = 1, z = 1
Covered returns c = 1; z = 0.95
73.81 80.89
83.49 Covered returns c
= 1, z = 0.90 Covered c
= 1, z = 0.90 vs. meandev raw 82.40
65.71 90.76
Covered c = 1; z = 0.95 vs. meandev raw
82.49 72.98
63.07 Covered c
= 1; z = 1 vs. meandev raw 10.72
22.98 30.98
J.M. Chamorro, J.M-
a
. P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 101
Comparing these classifications to the other ones, particularly the referred to the base case c = 1, z = 1,
some great differences can be observed. In fact Spearman’s correlation coefficient
13
amounts to 10.72 when all the funds are accountd for 23 in the case of fixed income and 31 in variable income.
As it is collected in Table 7, the correlation is much higher when lower values for z are considered. In particular for variable income funds, as the size of the allowable loss increases the value of the Spearman’s coefficient also
increases reaching 90.76 when z = 0.90.
4. Summary, cautions and extensions
