Description of the Data
For segregated fund and variable-annuity contracts, the relevant data for a diversified equity fund or subaccount are the total returns on a suitable stock index. For the U.S. variable annuity contracts, the S&P 500 total return (that is with dividends reinvested) is often an appropriate basis. For equity-indexed annuities, the usual index is the S&P 500 price index (a price index is one without dividend reinvestment). A common index for Canadian segregated funds is the TSE 300 total return index (the broad-based index
of the Toronto Stock Exchange); and the S&P 500 index, in Canadian dollars, is also used. We will analyze the total return data for the TSE 300 and S&P 500 indices. The methodology is easily adapted to the price-only indices, with similar conclusions. For the TSE 300 index, we have annual data from 1924, from the Report on Canadian Economic Statistics (Panjer and Sharp 1999), although the TSE 300 index was inaugurated in 1956. Observations before 1956 are estimated from various data sources. The annual TSE 300 total returns on stocks are shown in Figure 2.1. We also show the approximate volatility, using a rolling five-year calculation. The volatility is the standard deviation of the log-returns, given as an annual rate. For the S&P 500 index, earlier data are available. The S&P 500 total return index data set, with rolling 12-month volatility estimates, is shown in Figure 2.2. Monthly data for Canada have been available since the beginning of the
TSE 300 index in 1956. These data are plotted in Figure 2.3.We again show
the estimated volatility, calculated using a rolling 12-month calculation. In
Figure 2.4, the S&P 500 data are shown for the same period as for the TSE data in Figure 2.3. Estimates for the annualized mean and volatility of the log-return process are given in Table 2.1. The entries for the two long series use annual data for the TSE index, and monthly data for the S&P index.
Monthly total returns and annual volatility, S&P 500 1956–2000.
the shorter series, corresponding to the data in Figures 2.3 and 2.4, we use monthly data for all estimates. The values in parentheses are approximate 95 percent confidence intervals for the estimators. The correlation coefficient between the 1956 to 1999 log returns for the S&P 500 and the TSE 300 is 0.77. A glance at Figures 2.3 and 2.4 and Table 2.1 shows that the two series are very similar indeed, with both indices experiencing periods of high volatility in the mid-1970s, around October 1987, and in the late 1990s. The main difference is an extra period of uncertainty in the Canadian index in the early 1980s.
Selecting the Appropriate Data Series for Calibration
There is some evidence, for example in French et al. (1987) and in Pagan
and Schwert (1990), of a shift in the stock return distribution at the end of
the great depression, in the middle 1930s. Returns may also be distorted by
the various fiscal constraints imposed during the 1939–1945 war. Thus, it is attractive to consider only the data from 1956 onward. On the other hand, for very long term contracts, we may be forecasting distributions of stock returns further forward than we have considered in estimating the model. For segregated fund contracts, with a GMAB, it is common to require stock prices to be projected for 40 years ahead. To use a model fitted using only 40 years of historic data seems a little incautious. However, because of the mitigating influence of mortality, lapsation, and discounting, the cash flows beyond, say, 20 years ahead may not have a very substantial influence on the overall results.
TSE 300 1956–1999 0.082 0.013 – 0.024 S&P 500 1956–1999 0.027 – 0.057 0.032 Current Market Statistics 22 risk-neutral Investors, including actuaries, generally have fairly short memories. We may believe, for example, that another great depression is impossible, and that the estimation should, therefore, not allow the data from the prewar
period to persuade us to use very high-volatility assumptions; on the other
hand, another great depression is what Japan seems to have experienced in
the last decade. How many people would have also said a few years ago that such a thing was impossible? It is also worth noting that the recent implied market volatility levels regularly substantially exceed 20 percent. Nevertheless, the analysis in the main part of this paper will use the post-
1956 data sets. But in interpreting the results, we need to remember the implicit assumption that there are no substantial structural changes in the factors influencing equity returns in the projection period.
In Hardy (1999) some results are given for models fitted using a longer 1926 to 1998 data set; these results demonstrate that the higher-volatility assumption has a very substantial effect on the liability.