Our calculations involved the cash flow from payrolls to pension funds, the cost of commissions, and the parameters that define the investment options. They start in Septem-ber of 1997, when the first contribution to the AFORE was made. When applicable, we used the national consumer price index to index for inflation.

Data on SIEFORE share prices (CONSAR, 2019a), interest rates, and exchange rates (Banco de Mexico, 2019) are reported daily. Other data are reported monthly. To match daily data with monthly data, we used the last working day of the month. SIEFOREs have four primary funds, which are indexed from SB1 to SB4 according to the target age group. SB1 is used for workers 60 years and older; SB2 is for those between 46 and 59 years old; SB3 is for those between 37 and 45; and SB4 is for those 36 and young-er. In the first years, there were only SB1 and SB2 funds, and the SB2 funds still held 45% of the assets by 2019. We concentrated on the SB2 funds and on the five firms that have operated since 1997.

The measure of a risk-free asset is the 28-day federal government bill (CETES), which is the more liquid paper in the Mexican market. We also used data for CETES at ma-turities of 91, 182 and 364 days; there were four missing values relatively early in the period for data at 182 and 364 days of maturity, which we filled linearly. The latter issue reflects the hardship that the government faced selling bills at extremely short maturities during the 1990s.

To measure foreign investment opportunities, we used the SP500 index, for which highly liquid investment opportunities are available, including funds traded in the Mexican financial markets.

To model payroll, we used data on monthly taxable salaries reported by the Social Security Institute (IMSS, 2019). Hourly wages are uncommon in the Mexican labor market, as the more usual arrangement is the payment of a salary each day. Thus, taxable salaries are daily salaries adjusted by mandatory benefits, and employers usually pay for weekends and holidays (proportionally for non-full-time workers). Therefore, we multiplied daily taxable salaries by 30.4 to obtain monthly salaries. Taxable salaries are capped at the equivalent of 25 times the minimum wage. The average taxable salary during the researched period was 4.08 times the minimum.

Data on commissions is available from CONSAR (2019b), which also publishes an index of net return (IRN). Data on returns to housing sub-accounts were taken from IN-FONAVIT (2018, 2019a) and Villagomez (2001). There are 24 AFORE in the CONSAR records. Five have data for the entire period of 21 years and 5 months. Others were short-lived, and four have data for more than 10 years; however, these merged with other firms by 2019. Thus, we had an unbalanced panel, with some entities having 257 observations and one entity having only 172 observations. We had no full data for some of the defunct corporations. The AFORE XXI, the largest one, is the result of a merger; the pre-merger correlation of prices was 0.997, and we consider this as a single corporation for our purpos-es.

We are interested in measuring the impact of public debt on the real returns of pension funds. The general strategy is to measure the deviation of returns from a standard and investigate whether it is affected by public debt. We present careful calculations of contributions and net returns. We also needed to purge the returns that SIEFOREs pay from the yields on government debt and the fixed effects of the return in each firm and investigate whether the residual correlates with government debt. We interpreted an affirmative answer as evidence of the public debt causing variations in the real return of the SIEFORE.

From the standpoint of individuals, before the individual becomes eligible for retirement, SIEFOREs’ shares are bonds that mature at retirement. Investor-workers have different ages and expected ages of retirement, making it difficult to pin down an average expected maturity for a fund. We model the monthly rate of return of each SIEFORE ( r i t ) as the return on one-year government bills ( r t g ) plus a SIEFORE specific risk premium ( L i t ), as seen in equation 1. We use the one-month bill as the risk-free asset because it is the most liquid in the Mexican market. On the other hand, the return on a government bond of maturity k is formed by the expected return of short-term bonds and a term-specific risk premium L t g , as seen in equation 2.

r i t = r t g k + L i t ( k ) (1)

r t g k = 1 k ∑ j = 1 k ‍ E t r t + j - 1 g 1 + L t g ( k ) (2)

Merging equations 1 and 2, we obtain a relation between the SIEFOREs rates and public debt rates:

r i t = 1 k ∑ j = 1 k ‍ E t r t + j - 1 g 1 + L t g ( k ) + L i t ( k ) (3)

We obtained short-term rate forecasts and used them to construct the first term on the right side of equation 4, which comes from equation 3 for the terms k and k=1. To forecast, we used two specifications; one is a dynamic model of the 28-day CETES rate, the other is the actual (future) rates. The public debt premium is, thus, estimated as the observed difference L ^ t g ( k ) :

L ^ t g k = r t g k - r t g 1 - 1 k i = 1 k - 1 j = 1 k E t ? r t + j g (4)

Putting the pieces together, we reach equation 5. The firm-specific variable, L i t ( k ) , is not observable, but it should not be correlated with government debt. It could be different across firms for a variety of reasons, such as lack of competition, differences in the profile of clients that lead to different investment strategies, slow response of worker-investors to relative returns. Thus, a finding of a significant negative effect of government debt when we add it as a variable in a regression model would support the hypothesis that there is a negative effect of debt on the rate of return on pension funds.

r i t = 1 k ∑ j = 1 k ‍ E t r t + j - 1 g 1 + L ^ t g ( k ) + L i t ( k ) (5)

In summary, the rate of return in a SIEFORE is a function of short-term rates in government bonds plus the risk premium of the k-period bond, and the SIEFORE fixed effect.

To evaluate equation 5, we use SUR and fixed effects models, adding the total public debt as a variable, under the hypothesis that public debt should not affect the results (i.e., should be uncorrelated with returns). The main difference between the two models is that while SUR allows the firm-wise calculation of coefficients and the same-period covariation of errors, the fixed-effect model allows the use of firm-clustered time-related errors but assumes the independence of the errors across entities. The nature of the unobservable L i t ( k ) gives some support to each approach: it makes sense to expect contemporary covariation in unobservable factors affecting returns across firms (which support the use of SUR), but within-firm returns may also have a serial correlation due to slowly changing portfolios. We perform calculations using both approaches, which support the use of a fixed-effects model that allows for a calculation of standard errors that is robust to heteroskedastic mistakes (i.e., within-entity serially correlated errors).

The heuristics are that funds have as floor the return on government bonds and should not do worse than that.

This section describes the evolution of the value of pension funds, their real returns, and the way that public debt has interacted with them. We produced detailed estimates of the value of funds for workers who contributed continuously since 1997; future research will address the impact of incomplete labor careers on the value of funds.

The total value of the fund at time t for a worker with a wage and participation his-tory w(t),I(t), in the absence of commissions, is shown in equation 6:

V T t + 1 = τ w t + τ s w M t I t + V p t 1 + r p t + τ h w M t I t + V h t 1 + r h t - C ( t ) (6)

Indexes h and p identify housing and pension funds, T represents their sum, and r refers to the rate of return. Social security tax rates did not change during the period ( τ = 6.5 % ). The system includes a social fee paid by the federal government to each fund, which we set at a 5.5% rate ( τ s ) applied to the inflation-indexed minimum salary wm in July 1997 (adjusted in February of each year). The social fee was made progressive after a 2007 reform, but we do not model that detail. The function C ( t ) measures the use of the housing fund; it is the result of the flow of credit received and paid, and we do not attempt to model it in detail because public data on its components do not exist.

SIEFORE share prices allow the direct calculation of returns after commissions. Also, CONSAR (2019a) provides a history of AFORE commissions. Between 1997 and 2003, the regulation changed from a relatively free pricing framework to one of stringent control. AFOREs were free to adopt any combination of percent charges on deposits, assets, or real returns. Most had a two-part commission, charging a percentage on assets and a percentage on the flow of new savings. Only one had a commission on real returns. By 2003, commissions on real returns disappeared, and, since 2008, only commissions on asset values have been allowed. A proposed reform may allow more flexibility in the future.

To calculate the value of the funds after commissions (equation 7), let c V ( t ) be the percent commission charged on the value of the fund, c w ( t ) the commission charged on the value of contributions, and c r ( t ) the commissions charged on real returns. I π ( t ) is an index function that takes the value of unity if real returns are above inflation and otherwise equals 0:

V T t + 1 = 1 - c w t τ w t + τ s w M t I t + V p t 1 - c V t 1 + r p t - c r t r t - π ( t ) V p t + V h t 1 + r h t - C ( t ) (7)

Calculations were made for a worker earning the average taxable salary, with depos-its accruing interest beginning the first day of the month after the obligation to contribute is met. These calculations are not exact because deposits can be made up to the 17th of the month and are bimonthly. However, we ignored this variation because the effect is small. When commissions were charged on real returns, we assumed that accruals were monthly and only relevant when returns were positive.

We used data on the five AFORE that have operated continuously. Our calculations represent the AFORE’s actual take. The CONSAR Index of Net Returns refers to moving averages of returns after commissions and is not useful for obtaining the calculations pre-sented below. CONSAR information is confusing because it is based on the market prices of the shares of SIEFOREs and not on cash flows entering the system and the part taken by providers. Neither the CONSAR nor the AFORE provides a money value on commissions. The information that the regulatory body offers to consumers is doubly misleading. First, the buyer of a financial service is interested in the return on the full cash amount invested. Second, the consumer is interested in how much money the service costs. Should banks inform them about commissions as a percentage of total deposits or withdrawals during the last three years? Such averages hide the costs of services and confuse consumers.

Figures 1.a and 1.b show the ratio of commissions to monthly contributions as a per-cent of monthly contributions and in money. For the average worker, there is a convergence to a commission on contributions of 20%, or 200 pesos per month (at January 2019 prices).

Figure 1 Source: Own elaboration

In Figures 2.a and 2.b, we see that regulatory changes forced a substantial conver-gence since 2003. We present data under two frameworks of time to underscore the initial differences and eventual merging of the commissions. In the first months of operations, the AFOREs took up to 20% of assets. By 2002 (5 years after the foundation of the system), most were between 3 and 4% of assets, and by 2012 they had all converged at around 1% (corresponding to approximately one-thousandth of assets per month). Currently, the regu-lator bases policy on a “commissions on assets,” a non-stationary series that will be declin-ing for the foreseeable future. This decline will occur because, on average, the de-mographics of the labor force determine average asset growth for several decades (Mexico is a young country and fertility rates are above replacement, so the labor force will grow significantly in the long run). Thus, the regulatory goal of reducing commissions on assets is soft, because improvements are guaranteed even if costs do not fall.

Figure 2 Source: Own elaboration

The service that the system produces is the value of funds. Considering only the funds managed by the SIEFORE, Figure 3.a shows the value funds that would have ac-crued in the absence of commissions (Equation 6), and Figure 3.b shows the results after commissions (Equation 7). Figure 4.a shows the results with savings including the housing fund, when no mortgage is obtained (Equation 7 using C(t) = 0) and of the retirement fund invested at the largest AFORE; there is more money in the SIEFORE than in the IN-FONAVIT fund because contribution rates are higher for the first. Figure 4.b shows the ratio of values of pension and housing funds adjusted by the percentage of contributions; when the ratio is above 1, it indicates that the cumulated returns in the SIEFORE fund were higher. Conversely, when it was below 1, the housing fund paid higher cumulated returns. During the first ten years, SIEFOREs outperformed the housing fund, but since 2012 the housing fund has taken the lead. As shown in Figure 4, housing fund values are not fully funded; at the end of 2018, total liabilities were $1.132 trillion, but financial assets were only $0.166 trillion. The rest of the money-most of it earmarked for pensions-is to be paid from the INFONAVIT mortgage portfolio. This agency has reported large losses: according to the 2018 financial statement (INFONAVIT, 2018), its book value was $1.28 trillion, but $0.231 trillion was already written-off as unrecoverable, and $0.106 trillion was marked as non-performing. Thus, according to its financial statement, INFONAVIT was short at least $0.174 trillion to pay its “housing sub-account” obligations. In summary, the returns on this component of savings are subject to an idiosyncratic risk because the deci-sion on rates of return is discretionary (see p. 260 of INFONAVIT, 2019b). For example, INFONAVIT decided to set the rate at 8.32% for 2017 but only at 6.38% for 2018, even as 1-year government bonds were moving in the opposite direction (with average rates of 7.12% in 2017 and 8.09% in 2018).

The behavior of V ( t ) by AFORE depends upon the investment strategies used. Figure 5 shows that the trends and variation of share prices of SIEFOREs were similar for all fund managers except one.

Figure 3 Source: Own elaboration

Figure 4 Source: Own elaboration

Figure 5 Source: Own elaboration

The value of pension funds can be measured from different perspectives. One meas-urement used is the value of savings under the historical commission and investment strate-gies. Another is the historical profiles of the contributions. Alternative measurements are counterfactual since they assume strategies that were feasible but did not take place.

We explored the results of three investment strategies: (a) a historical SIEFORE; (b) Mexican government debt (a risk-free peso strategy); and (c) the SP500 index after chang-ing the inflow to foreign currency monthly (a dollar equity strategy). We have not even considered banking options because the rates were too low. Between 1997 and 2019, aver-age rates in banking deposits maturing at 91 days were 4.3%, compared with 8.4% for CETES of the same maturity (Banco de México, 2019).

The two low-cost counterfactual strategies are to invest in foreign assets and to in-vest in safe national assets. Figure 6 shows the results of investing passively in a large inter-nationally traded index (the SP500) and 28-day CETES. Investing in the SP500 industrial index would have resulted in a lower funds value throughout most of the period, but it would also have yielded significant gains in recent years following the substantial deprecia-tion of the peso and a boom in the American stock market. For investments abroad, contri-butions are exchanged at the interbank dollar rate, and the same rate is used to express the dollar value in pesos. Figure 7 shows the difference between historical values and those arising from the dollar equity strategy. The CETES strategy would have been the loser overall, as is evident in Figure 6. The difference in values yielded by different approaches is not small, ranging in the order of several thousand pesos up or down. The passive strategies under consideration are highly liquid, which means that the relative gains of SIEFOREs are smaller or their losses larger.

Regulation is based upon VaR statistics that cannot be deemed relevant for individ-uals because they measure maximum expected losses over a time scale that is usually much shorter than the retirement horizon. There are two main long-term risks for investments in pension funds in Mexico: the interest rate risk in government bonds and the exchange rate risk. The first risk is small. Given that AFOREs were required to hold most of their assets in such debt (54.7%) in the third quarter of 2019 (CONSAR, 2019b), managers cannot do much wrong. On other investments, the exchange rate risk has not been managed adequate-ly, as demonstrated by the wide variations in Figure 7. Of more concern is that the regula-tion allows “structured” and “securitized” investments authorized by the regulator and fi-nanced almost exclusively by the SIEFOREs (98.6% for structured and 68.0% for securit-ized investments; CONSAR, 2019b, Table 3.5). The result is a moral hazard that cannot be resolved by regulating portfolios but only by taking the regulator out of the business of au-thorizing investments.

Figure 6 Source: Own elaboration

The calculations for the passive strategies assume the same costs and commissions as the existing investment strategies and regulations. Likely, passive strategies involve much lower administrative costs. However, AFOREs provide two services: account management and fund management. Account management refers mainly to the realm of customer ser-vice, such as providing account statements, updating information, and providing support to process social security claims. Customer service tends to have fixed costs per worker, and commissions cover those costs. IRAs are not conventional brokerage accounts because AFOREs must manage many of them with small and infrequent deposits. Current regula-tions assume that AFOREs are large enough to diversify the profiles of their clients and impose cross-subsidization because tariffs take a percent of assets regardless of the size of the individual fund.

Figure 7 Source: Own elaboration

Forecasts of the short-term interest rates are required to relate returns on pension funds with the spread on interest rates at longer terms and purge entity-specific returns from market-wide returns. Table 1 shows the average real returns of SIEFOREs during three financial moments: the high-interest, 1997-2002 period; the low inflation-stable and declining rates period (2002-2008); and the post-panic near-zero real rates period (2009-2019). Table 1 also shows that the post-2008 volatility was significant, with monthly real returns often above 3% or below -3% during a period when annual real rates in government bills were hovering near zero.

Spreads between short-term and longer-term government bonds have been small and unpredictable: (a) spreads were not higher for longer government debt during the inflation-ary period; (b) spreads became smaller when inflation lowered; (c) during the post-panic period, the yearly gains for investing in one-year Cetes (above 28-day Cetes) were 0.4-0.5%, down from 0.8% during the inflationary period; (d) real one-month losses for invest-ing in long-term Cetes could be higher during the inflationary period; and (e) during the more recent period, one-month losses related to Cetes investment were as high for short-term bonds as they were for long-term bonds (Table 2).

Table 1

Table 2

Forecasting r t g ( 1 ) was required to estimate equation 5. We tested two specifications. One assumed perfect foresight and used actual future rates; the other used the forecast from a Box-Jenkins model. For the ARIMA model, the AC and PAC of the 28-day CETES rate series, r t g ( 1 ) , suggested using a differenced model with only a one-lag autoregressive term. This, in fact, produces white-noise residuals (Ljung-Box Q-test, 47.63; Prob > chi2(40) = 0.1899). Thus, the short-term real interest rate is highly predictable when using available information, with only a one-month lag. These projections were used to build the variable in equation 4, which is carried over into equation 5. Figure 8 shows the calculated variables. Equation 2 gives these quantities the interpretation of risk premia (see Hall, Anderson, and Granger, 1992; Clements and Galvão, 2003). Figure 8 conveys that the risk premium in Mexican bonds is minimal (the dotted lines are near zero most of the time).

The considerable risk for pension funds lies in long-term trends that depend upon inflation and global real rates and the exchange rate, and these should be the focus of risk regulation.

Figure 8 Source: Own elaboration.

We tested equation 5, which relates spreads in interest rates with short-term rates, using fixed effects models and a SUR model. The estimated equations include or exclude the variable of government debt.

For our first estimation, we used a SUR model that includes the nominal monthly return of each SIEFORE as the dependent variable and the constructed quantities R ^ and L ^ as regressors (Table 3). This model used data only for the five large corporations that have operated since 1997. The coefficients for R ^ and    L ^ , which measure expected returns and risks, have the expected sign, but they were statistically significant for only one of the five corporations. This result means that regulations to encourage longer-term investments are not relevant for savers. There are, however, significant differences-between 0.30 and 0.63 for R ^ and -0.21 to -0.57 for w i d e h a t    L -across corporations. The risk premium predicted with the ARIMA model and the perfect-foresight model yielded different coefficients for the risk variable, but they were not statistically significant. The coefficient for the public debt variable has a negative impact and is statistically significant, at the 90% level, in two cases and at the 95% level in three cases.

A second estimation restricted the coefficients to be the same across firms and allowed for fixed effects (Table 4). These estimates include information on all firms. The coefficient for the 364-day Cetes forecasted rate was in the range estimated by the SUR model for R ^ and was statistically significant. The value of the coefficient for R ^ changed when public debt was included in the regression, which suggests that omitting public debt generates bias. The coefficient for L ^ changed signs depending upon the inclusion of the public debt variable, but it was not statistically significant. Thus, public debt and risk are related, and public debt reduces returns. We included a dummy, taking the value of unity after September 2008; the coefficient for this post-2008 crisis variable is significant. The regulatory change capping commissions at a percentage of assets took place in 2008, so the post-2008 dummy may have confounded the impacts of the post-crisis environment and regulatory reform. However, the effect is negative, in harmony with the fall in rates shown in Table 2, and contrary to the regulatory goal of improving returns.

To gauge the size of the estimated effects, we calculated the rates of return predicted at 90, 100, and 110% of the value of public debt. Table 5 shows our calculations for the SUR model, and Table 6 shows our results for the pooled model. We consider rates of return around 0.007 (0.7% per month) or yearly rates of 8.7%. In the combined model results, the difference in returns between the 90 and the 110% public debt scenarios was 0.0004873 (0.0070-0.0065) monthly, or 0.0063 (0.63%) per year. Since the mandatory fund assets were $3,425.5 billion (by May of 2019), the yearly impact between the 90 and 110% scenarios is approximately $21.6 billion ($1.13 billion per year at an exchange rate of 19). Between the 90 and 100% debt scenarios, the differences are 0.0026 monthly, or 0.34% yearly, for an impact of $11.68 billion ($615 million).

Finally, the SUR and the fixed effect models showed systematic differences in the rate of return paid by different corporations, suggesting the lack of healthy competition in the system. The competition authority ruled in 2015 against a cartel to set limits to the transfer of accounts between providers. The agreement included at least three of the AFORE whose data were used in this research (COFECE, 2015).²

Table 3

Table 4

Table 5

Table 6