Review of the literature

At the end of the last century and the beginning of this century, the World Climate Research Program proposed several indices that could provide useful information on climate change; see Peterson, Folland, and Plummer (2001). In 2012, the Climate Index Working Group (CIWG) of the CAS Climate Change Committee summarised scientific knowledge on climate change, and proposed in a report to develop a composite index, termed the Actuaries Climate Change Index (ACCI), to include information from several climate variables, (Solterra Solutions, 2012). The development of the Actuaries Climate Index™ (ACI), which studies climate change in the United States and Canada, was proposed by this working group in 2014 and was launched in November 2016.

The Actuaries Climate Index™ was jointly developed by the Canadian Institute of Actuaries (CIA), the Society of Actuaries (SOA), the Casualty Actuarial Society (CAS), and the American Academy of Actuaries (AAA), from climate data in North America; see ACI (2018). It is intended to provide actuaries, public policymakers, and the general public with a neutral, factual and helpful climate change index monitoring tool, to learn about climate change and its associated risks. Just as the Consumer Price Index (CPI) tracks changes in the cost of a standard basket of goods and services over time, the ACI measures climate risks through a basket of extreme climate events and changes in sea level. The focus is on extreme weather events rather than averages; extremes have a greater impact on policyholders and their insured goods, as well as on society and the economy. The index consists of six components, each forming a monthly time series starting in 1961, from records of the National Oceanic and Atmospheric Administration (Menne, Durre, & Houston, 2012), GHCNDEX1 (Donat, Alexander, & Caesar, 2013), and the Permanent Service for Mean Sea Level (PSMSL, see Permanent Service for Mean Sea Level, 2023).

Curry (2015) investigates extending the use of the ACI formula to the UK and Europe. He reviews the ACI definition and methodology and finds that it applicable to this region without change, although it would need appropriate regional data.

In 2018, the Institute of Actuaries of Australia developed the Australian Actuaries Climate Index (AACI), using the ACI methodology, to monitor climate change in Australia; Aaci (2018). In addition,Nevruz, Atici, and Yi̇ldi̇rak (2022) propose to apply the ACI in Turkey, adapting it to the conditions in Ankara, developing an index suitable for this region. They suggest that choosing the best grid dataset for their ACI is more important than changing how the index is calculated. The ACI is designed to present climate trends to actuaries, policymakers, and the general public.

Pan, Porth, and Li (2022) investigate how effective the ACI is in predicting crop yields for (re)insurance ratemaking. They find that the ACI has significant predictive power for crop yields and yield losses, and argue that a high-resolution index could benefit the insurance industry. Most recently, in an oral presentation at the 80-th anniversary event of the Spanish Institute of Actuaries (IAE), a first draft defines a Spanish version of the ACI, over a shorter, more recent reference period than the ACI (1975-1995 in general and 1993- 2000 for sea levels). Their temperature variables also differ, using average highs and average lows, and the composite index formula. This index is not yet published, but some details can be found at IAE (2023).

Extension to Iberian climate data

Here the Iberian Peninsula is defined as the land area ranging from 36° to 43.7° north latitude and -9.5° to 3.3° east longitude, excluding France. Balearic and Canary Islands are not included in the data set. We calculate the Iberian Actuarial Climate Index (IACI) for the Iberian Peninsula using data from the ERA5-Land reanalysis dataset; see Copernicus Climate Change Service (C3S) Climate Data Store (CDS) (2022). Unlike the ACI for the United States and Canada which is calculated for 12 sub-regions, here we compute the IACI for a total of 6,526 cells forming the Iberian Peninsula grid; see Figure 1. The graph shows the approximate land inclusions of the grid cells along the borders and seacoast, and how the islands and other non-continental territories of Spain and Portugal are excluded from the IACI calculations.

We compare the Iberian and North American cases and show how the IACI tracks well the climate change of the Iberian Peninsula through the years. You can see also the high-resolution IACI, showing the evolution of each grid by seasons through the years.

The rest of the paper is organised as follows. In Section 2 the Actuaries Climate Index™ (ACI) and its components are reviewed, discussing the possible application of the ACI to Iberian data, which is presented in Section 3. The comparison of the IACI with the North American ACI is presented in Section 4. Then, Section 5 splits and compares the IACI for Portugal to that of Spain. Finally, Section 6 gives a seasonal analysis of the IACI in each cell during the last 12 years. Some conclusive remarks and future developments wrap-up the paper.

Temperature components, T90, T10

The two temperature components are defined as the frequency of temperatures above the 90-th percentile and below the 10-th percentile, relative to the data from the reference period of 1961 to 1990. These frequency variables are detailed in Table 2.

For warm temperatures, because warmer days and warmer nights tend to occur together, the ACI measures the average of the percentage of days when daytime and nighttime high temperatures are greater than the corresponding 90-th percentile, centred on a 5-day window, over 1961-1990:

The cool temperature measure is calculated using a process similar to that used to calculate the frequency below the 10-th percentile:

where j = Jan, Feb, . . . , Dec, and k = 1961, 1962, . . . , 2022.

As the climate warms, the occurrence of cooler extreme temperatures decreases, and the temperature distribution shifts to the right. Hence, the sign of T10 is reversed in the calculation of the ACI to correctly reflect the risk (increasing the ACI when T10 decreases). Fewer extreme cold events indicate that the climate is changing, affecting plant and animal ecology and weather patterns. Increased melting of permafrost, increased infectious diseases transmission, and populations of pests and insects that were previously less likely to survive cooler temperatures are all thought to be associated with a reduction in the frequency of extreme cold events.

The standardised anomalies for warm temperatures are calculated as the difference between the monthly frequency of warm temperatures > 90-percentile and the average monthly frequency of warm temperature during the reference period, divided by the standard deviation during the reference period. The warm temperature T90 and the cool temperature T10 use the same algorithm:

where j = Jan, Feb, . . . , Dec, and k = 1961, 1962, . . . , 2022, ${\mu }_{ref}T90\left(j\right)$ is the average monthly frequencies of warm temperature for all months during the reference period, ${\sigma }_{ref}T90\left(j\right)$ is the standard deviation of warm temperature during the reference period.

The standardised anomalies calculation method for cold temperatures is consistent with that for warm temperatures:

where j = Jan, Feb, . . . , Dec, and k = 1961, 1962, . . . , 2022.

Precipitation component, P

The precipitation component focuses on extreme rainfall rather than average precipitation, using the maximum rainfall in any 5 consecutive days in the month, notated as $Rx5day\left(j,k\right)$ . The percentage anomaly of maximum 5-day rainfall in a month, relative to the reference period value for a given month, is given by:

where j = Jan, Feb, . . . , Dec, k = 1961, 1962, . . . , 2022, ${\mu }_{ref}Rx5day\left(j\right)$ is average in all months during the reference period, ${\sigma }_{ref}Rx5day\left(j\right)$ is the standard deviation during the reference period.

Drought component, D

Droughts are measured by the maximum number of consecutive dry days in each year, that is when the precipitation is less than 1 millimetre (denoted CDD(k)). Monthly values are obtained by linear interpolation of annual values:

The standardised anomalies calculations of CDD are similar to those for $Rx5day$ :

Wind speed component, W

Daily wind speed measurements are converted to wind power, WP, using the relationship WP (i, j) = 1/2ρw3, where w is the daily mean wind speed and ρ is the air density (taken to be constant at 1.23kg/m3). Wind power is used because the destructive potential of a wind has been shown to be related to wind power rather than wind speed. The wind component measures the monthly frequency of daily mean wind power above the 90-th percentile, denoted as WP 90.

The wind power thresholds $W{P}_u\left(i,j\right)$ are determined for each day i and month j in the reference period at each grid point separately. The thresholds value is determined as the mean plus 1.28 standard deviations of $WP\left(i,j,k\right)$ , for all 30 values of the same day and month in the 30-year reference period:

The count of days where mean winds exceed the threshold is then expressed as a percentage of the number of days in the month, providing an exceedance frequency measure, for every month of every year, throughout the entire period:

where i represents the day, j the month, k the year, and n(j) the number of days in month j.

Then the standardisation is as usual:

Sea level, S

Monthly mean sea level measurements can be obtained from tide gauges, which record changes in sea level relative to a vertical datum (to the sea floor). To avoid negative values, a revised local reference datum is defined to be approximately 7000 millimetres below mean sea level; see Permanent Service for Mean Sea Level (PSMSL) (2023). It is important to note that the height of the land relative to sea level is subject to change due to crustal movements, and these measurements encompass the combined effects of land and ocean positional changes. The standard anomalies in sea level are calculated using the following equation:

The composite ACI index

The standardised anomalies of these six components are averaged when they are combined into the Actuaries Climate Index, though the cool temperature is subtracted because the temperature probability distribution curve shifts to the right, as explained earlier:

The Iberian Actuarial Climate Index

The warm and cool temperature components of each grid cell in the Iberian Peninsula (denoted IP in graph captions) are used to replace (1) and (2), respectively. The standardised anomalies of the temperature components, T90std and T10std, are from Equations (3) and (4).

The 5-year moving averages of these standardised anomalies are plotted in Figure 2, which depicts the changes in the frequency of extreme warm and extreme cold temperature events. It shows that the local cooling before the mid-1970s led to a decrease in the frequency of extreme warm temperatures and an increase in the frequency of extreme cold temperatures. Subsequently, climate change increased the frequency of extreme warm temperature events and decreased the frequency of extreme cold temperature events. This resulted in an increase in T90 and a decrease in T10, causing their difference to grow. The black line represents the 5-year moving average of T90std –T10std, which helps to better visualise the main trend. The change in the frequency of extreme warm temperature events is opposite to that of extreme cold temperature events, indicating that changes in air temperature have completely different effects on these two frequencies.

Then, Figure 3 illustrates the standardised anomalies of the precipitation component, Pstd, see (5). Positive values of the standardised anomalies show an increase in heavy precipitation events, compared to the reference period.

Conversely, when the value is -0.1, it means that the maximum 5-day rainfall has decreased by 0.1 standard deviations, relative to the mean of the reference period. From the figure, we see that heavy precipitation events on the Iberian Peninsula have slightly decreased compared to the reference period.

Figure 4 plots the Dstd time series, see Formula (7), representing the maximum consecutive dry days in the Iberian Peninsula. From the figure, it is clear that the maximum consecutive dry days have increased, in comparison to the reference period. This indicates longer periods without rainfall in the region. Furthermore, considering the monthly maximum five-day precipitation, shown in Figure 3, one can gather additional insights. The figure demonstrates that heavy precipitation events, as measured by the P component, have decreased slightly in the Iberian Peninsula, compared to the reference period. This implies a reduction in the amount of precipitation during such intense rainfall events. Combining these observations, we can infer that the Iberian Peninsula is experiencing a trend toward drier conditions. The increased occurrence of consecutive dry days, along with the decrease in heavy precipitations, suggests a shift towards reduced overall rainfall and an overall increase in dryness in the region.

Next, Figure 5 shows the changes in wind power in the Iberian Peninsula, which seems to have a similar trend to that of wind speed. From this figure, we can assume that, in general, winds in the Iberian Peninsula reach higher speeds after than during the reference period. Interestingly, the figure also suggests some kind of periodic trend in wind speeds in the Iberian Peninsula.

Finally, Figure 6 shows the seasonal standardised anomalies of sea level in the Iberian Peninsula. The value △S = S(j, k) − µref (j) has been positive since 1991 and has progressively increased over time. These findings indicate that, in general, the sea level around the Iberian Peninsula is rising yearly. The warming of the earth’s climate has melted ice sheets and glaciers, and the thermal expansion caused by warming seawater has further exacerbated sea level rise. Rising sea levels imply a series of risks for coastal areas of the Iberian Peninsula, such as coastal erosion, flooding, and salinisation of freshwater resources.

Curry (2015) uses a factor fS as the fraction of the number of coastal to total grid points, 0 < fS ≤ 1, to adjust the ACI as follows:

Considering the specificity of the geographical location of the Iberian Peninsula, we argue that the sea level variable S is important for the entire region. Hence, for now, we set fS = 1, as in the ACI, and defer to future work the study of estimating fS as in Curry’s extension of the ACI to the UK. The Iberian Actuarial Climate Index (IACI) for the whole Iberian Peninsula, given in (13), is hence obtained by averaging the IACI values from each grid cell.

Figure 7 reports the correlation coefficients between the IACI and its 6 components. As anticipated, the correlations among the components are generally weak, except for the frequency of extreme warm and extreme cool temperatures, which exhibit a notable association. Moreover, there is a significant positive correlation observed between the IACI and sea level changes, as well as the frequency of extreme warm temperatures.

Finally, Figure 8 shows the 5-year moving average of the seasonal IACI and its 6 components. The sign of the IACI is determined mainly by the sea level and the frequency of extreme warm temperatures. A positive IACI value means an increase in the included climate extremes since 1991, relative to the 1960-1991 reference period. The value is expressed as a standardised anomaly, implying that a component index of 0.5 means that the component has increased on average by 0.5 standard deviations. The IACI clearly shows that climate extremes in the Iberian Peninsula have become more frequent, compared to 1960-61.

Comparison with the North American ACI

Figure 9 shows the seasonal Actuaries Climate Index™ of the United States and Canada (here denoted as USC) from 1961 to 2022. Compared to the Iberian IACI, both have similar trends in temperature components (T90std, T10std), although their values are different. After to the reference period, the frequency of extreme warm temperature events in the United States and Canada is higher than in the Iberian Peninsula.

Unlike the small decrease observed in the Iberian Peninsula, North America experienced a significant increase in maximum five-day precipitations (Pstd) after the reference period. Furthermore, its increasing trend shows some periodicity. The land areas of the United States and Canada are very large; hence, each region has different dry periods. Overall, the USC drought component (Dstd) does not show a clear trend. However, the USC average annual maximum consecutive dry days, since the reference period, is slightly smaller. The trends in the precipitation (Pstd) and drought (Dstd) components indicate that the USC has become wetter, compared to the IP.

Another difference with the Iberian Peninsula, wind speeds in the United States and Canada did not change significantly during the post-reference period, and these wind speed changes are different for each region in the USC. The Actuaries Climate Index™ website provides regional graphs of the ACI and of its 6 components for the USC and its 12 sub-regions ¹ .

Sea level changes show a similar increasing trend in both the Iberian Peninsula and the USC. However, the sea level change data in the USC indicates that not all regions, within these countries, experience a sea level rise. In fact, some areas have sea levels lower than the 1961-1990 reference period average-level. In addition, some inland regions are not affected by sea level changes. These regional differences cause their average, that is, mean sea level change in the USC, to change less than in the seas around the IP.

Although the performance of each component of the Actuaries Climate Index™ and the Iberian Actuarial Climate Index is not exactly the same, due to the differences in geography, their composite index shows a similar increasing trend. Moreover, the values are also similar, ranging from -0.5 to 1.5. Particularly in recent years, their composite index values have consistently exceeded 1. This indicates that climate change has led to a similar factor of the standard deviation increase in the frequency of extreme weather events, both in the Iberian Peninsula and USC, compared to the 1961-1990 reference period. In summary, the occurrence of extreme climate events is becoming increasingly common in these two regions.