# **_4.3.1   Unadjusted Annuity Rates_**

We repeated the rolling estimation exercise using the (unadjusted) annuity interest rate received by the pensioner. Results are presented in <mark>figure 4.B.</mark> They are essentially similar to those found using the adjusted interest rate or TVA. It even looks as if these unadjusted interest rates began their adjustment earlier.

# **_4.3.2   Relevant Market Interest Rate_**

LICOs might consider a different interest rate as the relevant one for determining the annuity cost considering that the payment schedule of the PRC20 is flat and biannual, whereas expected annuity payments are monthly and decreasing.

To consider this possibility, we performed the following exercise. First, we estimated the end-of-month term structure of interest rates between 1993 and 2003 using bonds issued by the central bank of Chile, based on the Nelson and Siegel (1987) parametric representation. With the estimated term-structure parameters, we obtained a full set of monthly zero-coupon prices. Then, we used these prices to determine the monthly present value of the annuity and the corresponding series of internal rates of return.<sup>12</sup> <mark>Figure 5.A</mark> shows the evolution of the estimated annuity cost and <mark>Figure 5.B</mark> its internal rate of return (IRR). The regression R<sup>2</sup> of running the IRR against the PRC20 rate is 0.99, and the slope coefficient is insignificantly different from one with a p-value of 15 percent. Thus, for the purposes of this study, no term-structure-related errors seem to affect the results.

# **5. INTERPRETATION**

Summarizing, so far we have documented the following facts: first, at market interest rates the annuity costs increased by 30 percent, whereas the total price charged to pensioners increased only by 12 percent. This happened because spreads between the adjusted annuities interest rates and the reference rates (PRC20) were negative at the beginning of the sample period and became positive toward its end (figure 1.B).<sup>13</sup> Second, reported commissions have dropped (table 1 and figure 2). Third, the annuity interest-rate elasticity to market interest rates was significantly smaller than one and turned statistically indistinguishable from one. Thus, we should see what changes in the industry might explain these noteworthy adjustments in the annuity parameters.

12 Calculation details are available from the author upon request.

13 Given the reasoning presented in the previous sections, it is hard to understand why annuity rates end above the risk-free rates, especially if they are underestimated in outdated mortality tables. It is as if competition may have taken annuity rates “too far.” The evidence analyzed in James, Song, and Vittas (2001) is consistent with the idea that annuity rates in Chile are “too high” toward the end of the sample period. There are several possible explanations for this: (1) risk is much higher than what we assume it to be; (2)) irrationality (or agency problems) influence LICO behavior; (3) there are generalized (market-timing) bets on interest-rate increases (which itself may be rational or irrational, depending on the efficiency of the local financial market); (4) some LICOs may also be assuming conscious losses in order to acquire dominant market shares. This needs further study. However, even if, from the LICO’s perspective, annuities are sold with negative net present value, this need not imply bankruptcy or be reflected as negative margins in the financial statements.

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