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Joint Extremes in Temperature and Mortality: A Bivariate POT Approach

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This paper contributes to insurance risk management by modeling extreme climate risk and extreme mortality risk in an integrated manner via extreme value theory (EVT). We conduct an empirical study using monthly temperature and death data in the U.S., and find that the joint extremes in cold weather and old-age death counts exhibit the strongest level of dependence. Based on the estimated bivariate generalized Pareto distribution, we quantify the extremal dependence between death counts and temperature indexes. Methodologically, we employ the cutting edge multivariate peaks over threshold (POT) approach, which is readily applicable to a wide range of topics in extreme risk management.
Joint Extremes in Temperature and Mortality:
A Bivariate POT Approach
Han Li[a]and Qihe Tang[b],[c]
[a]Department of Actuarial Studies and Business Analytics, Macquarie University
[b]School of Risk and Actuarial Studies, UNSW Sydney
[c]Department of Statistics and Actuarial Science, University of Iowa
September 10, 2020
This paper contributes to insurance risk management by modeling extreme climate
risk and extreme mortality risk in an integrated manner via extreme value theory
(EVT). We conduct an empirical study using monthly temperature and death data in
the U.S., and find that the joint extremes in cold weather and old-age death counts
exhibit the strongest level of dependence. Based on the estimated bivariate generalized
Pareto distribution, we quantify the extremal dependence between death counts and
temperature indexes. Methodologically, we employ the cutting edge multivariate peaks
over threshold (POT) approach, which is readily applicable to a wide range of topics
in extreme risk management.
Keywords: Extreme value theory; Bivariate POT; Climate change; Actuaries Cli-
mate Index; Mortality risk.
1 Introduction
Extreme weather events have become one of the top risks that pose a serious threat to the
global economy and health. Climate extremes have been extensively studied in the recent
decades (see, e.g. Easterling et al., 2000; Reichstein et al., 2013). In the insurance context,
effective management and mitigation of climate risks are increasingly important for sol-
vency considerations. In February 2020, the European Insurance and Occupational Pensions
Authority (EIOPA) emphasized the importance of adding climate risk to solvency capital
requirements calculation, and urged insurers to integrate climate change scenarios in their
catastrophe modeling. There have been many earlier works studying the impact of climate
change on the growth and stability of insurance industry, especially in the area of general
insurance (for a detailed discussion, see Mills, 2005).
Weather-related hazards can lead to severe consequences for human lives (Forzieri et al.,
2017). Taking heat waves as an example, the 2003 European heat wave led to more than
70,000 excess deaths from June to August. The 2009 Southeastern Australia heat wave
caused 374 deaths in just 16 days. According to the World Health Organization (WHO),
during the period 2030–2050, climate change is expected to cause approximately 250,000
additional deaths annually due to intense short-term temperature fluctuations and climate
sensitive diseases (WHO, 2014). However, we notice that little scholarly attention has been
paid to the adverse impact of climate change on life insurance. Moreover, there have been
very few studies on extreme mortality modeling, and these existing studies have so far been
developed quite separately from studies on climate extremes (see, e.g. Watts et al., 2006;
Gbari et al., 2017). In this paper, we aim to fill in the gap by jointly modeling the extremes
in temperature and mortality.
A key challenge in modeling extreme risks is the scarcity of extremal observations, which
makes it difficult to empirically estimate the probabilities of rare events. Extreme value
theory (EVT) tackles this problem by providing limiting results beyond observed values.
There has been a vast literature on EVT, with a growing usage in insurance and finance.
The two main approaches in EVT are often referred to as “Block Maxima” and “Peaks Over
Threshold”(POT). The former characterizes the distribution of the sample maximum, and
the latter characterizes the distribution of values over a high threshold. In this paper, as we
are interested in the joint distribution of temperature and mortality extremes over certain
thresholds, the bivariate POT method is more relevant.
We conduct an empirical study based on monthly region-specific climate measures and death
counts in the U.S. during the period 1968–2017. To collect data for temperature extremes,
we consider the T10 (frequency of temperatures below the 10th percentile) and T90 (fre-
quency of temperatures above the 90th percentile) components from the Actuaries Climate
Index (ACI). Six continental U.S. regions are included in the analysis, namely Central West
Pacific (CWP), Southwest Pacific (SWP), Southern Plains (SPL), Midwest (MID), South-
east Atlantic (SEA), and Central East Atlantic (CEA).1For the corresponding death counts,
we first collect state-level age-specific data from the National Center for Health Statistics
1For more details, see Actuaries Climate Index Executive Summary 2018, available at: http://
(NCHS) and the Centers for Disease Control and Prevention (CDC) WONDER database.
We then aggregate the state-level death data into the six aforementioned continental regions
to be consistent with the climate index, and apply a bivariate POT approach to each region.
Our results show that positive extremal dependence exists between monthly T10 and death
counts at older ages. While in the case of T90, there is no evidence of extremal dependence
across all age groups and regions.
The paper contributes to the existing literature in three ways. First, we are among the first to
investigate the impact of extreme climate on population mortality from the angle of actuarial
science. Most of the previous studies on this topic are in the area of biology, epidemiology,
and environmental health (see e.g. D´ıaz et al., 2005; McKechnie and Wolf, 2009; Armstrong
et al., 2011). In contrast to those studies which normally focus on certain causes of death,
in this paper we look at the joint extremes in temperature and aggregated mortality experi-
ence for different age groups. Second, we employ a bivariate POT approach to climate and
mortality data which are both non-stationary time series. Since the POT method is built on
independent and identically distributed variables, in this study we first remove trend, sea-
sonality, and heteroscedasticity from the data via time series models. Bivariate generalized
Pareto distributions are then fitted to the residuals, and thus we define “extreme” observa-
tions as unexpected large values in climate index and death counts. By doing so, we focus
on the short-term dynamics rather than the long-term relationship between climate change
and mortality experience. Third, our results provide important implications for extreme
risk management and mitigation. From an insurance point of view, catastrophic weather
events and mortality experience can simultaneously trigger large losses across different lines
of insurance business. Furthermore, in the longevity capital market various mortality-linked
securities can be triggered by extreme mortality events. Therefore, the contribution of this
project is valuable to the broader financial services industry as well as the insurance industry.
The rest of the paper is organized as follows. Section 2 provides a detailed description of
the data collection and modeling procedure. Section 3 reviews some fundamental theories
in EVT with a focus on the multivariate setting. Section 4 presents results of our empirical
study based on U.S. data. Section 5 discusses the implication of these results for insurance
industry and concludes the paper.
2 Data Description and Modeling
Our empirical study concerns U.S. monthly temperature and death data during 1968–2017.
We have adopted a thorough data collecting and modeling procedure before applying the
bivariate POT approach. As the primary objective of our research is to analyze how short-
term climate shocks affect mortality experience, we apply the bivariate POT approach to the
“noise” (residuals) in the climate and mortality data. In this section we obtain the residuals
from ACI and death counts via time series models.
2.1 Actuaries Climate Index
In response to the increasing risks imposed by climate change onto the insurance sector,
the American Academy of Actuaries, the Casualty Actuarial Society (CAS), the Canadian
Institute of Actuaries (CIA), and the Society of Actuaries (SOA) jointly developed a series
of climate indexes to measure and monitor climate trends, namely the Actuaries Climate
Index (ACI). This recently compiled dataset provides an objective measure of climate risks,
taking into account the frequency of extreme weather and the extent of sea level change. It
is a useful monitoring tool for climate trends and is available since 1961 for the U.S. and
Canada and 12 subregions thereof. Based on the ACI, it is found that climate extremes are
already much more frequent than historical levels in the past 20 years (Actuaries Climate
Index Executive Summary, 2018).
By definition, the ACI averages zero over the 30-year reference period 1961–1990. It consists
of six components as follows:
T90: Frequency of temperatures above the 90th percentile;
T10: Frequency of temperatures below the 10th percentile;
P: Maximum rainfall per month in five consecutive days;
D: Annual maximum consecutive dry days;
W: Frequency of wind speed above the 90th percentile;
S: Sea level changes.
In this paper, we focus on the monthly T90 and T10 (unsmoothed and unstandardized val-
ues), as we want to quantify the impact of unexpected short-term temperature fluctuations
on mortality experience.2As illustrated in Figure 1, the six continental U.S. regions included
in this study are defined along state borders. A list of states included in each continental
region is given in Appendix 3 of the Actuaries Climate Index Executive Summary (2018),
which is used to aggregate the state-level death data. We acknowledge that there might be
within-state variations in weather extremes, and the analysis will be more complete and ac-
curate with data based on finer geographical units (e.g. county-level). However, it should be
noted that our proposed approach can be easily adopted to such datasets once they become
readily available to use.
Figure 2 plots the monthly T90 (left panel) and T10 (right panel) indexes for six continental
U.S. regions over the period 1968–2017. It can be seen that for all six regions there has been
an upward trend in the value of T90 and a downward trend in the value of T10, with the
downward trend being less apparent in some cases. These trends are consistent with our
observations that average temperatures have been going up over most of the U.S. regions in
recent decades. We can also see some degree of seasonality in both T90 and T10 indexes.
2We have also applied the same EVT approach to the other four ACI components as well as the ACI
itself, and found no tail dependence based on these indexes and mortality data. The additional results are
available upon request.
Figure 1: Six continental U.S. regions (Source: Actuaries Climate Index Executive Summary
(2018), Page 4, Figure 2)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
above the 90th percentile (%)
CEA (a)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
below the 10th percentile (%)
CEA (b)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
above the 90th percentile (%)
CWP (a)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
below the 10th percentile (%)
CWP (b)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
above the 90th percentile (%)
MID (a)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
below the 10th percentile (%)
MID (b)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
above the 90th percentile (%)
SEA (a)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
below the 10th percentile (%)
SEA (b)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
above the 90th percentile (%)
SPL (a)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
below the 10th percentile (%)
SPL (b)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
above the 90th percentile (%)
SWP (a)
1968 1978 1988 1998 2008 2018
Frequency of temperatures
below the 10th percentile (%)
SWP (b)
Figure 2: Frequency of extreme temperatures in the six U.S. regions: (a) T90; (b) T10
Before analyzing the short-term dynamics between extreme temperature and mortality, we
need to make sure that any trend and seasonality are removed from the data and we end
up with stationary time series. In this paper, we adopt the seasonal ARIMA model which
incorporates both non-seasonal and seasonal factors in a multiplicative model. The model
can be expressed as follows:
ARIMA(p, d, q)×(P, D, Q)S,(1)
p,d, and qdenote the order of the AR model, the order of differencing, and the order
of the MA model in the non-seasonal part, respectively,
P,D, and Qdenote the order of the AR model, the order of differencing, and the
order of the MA model in the seasonal part, respectively, and
Sis the time span of repeating the seasonal pattern. Since we are modeling monthly
T90 and T10 time series, Sis set to 12.
The Akaike Information Criterion (AIC) is used to select the optimal seasonal ARIMA model
for T90 and T10 in each of the six continental regions (Akaike, 1974). We then perform the
Ljung–Box test on residuals from the selected models to check if the residuals look like
white noise. The model selection results are shown in Table 1, with all models passing the
Ljung–Box test at 5% level of significance.3
Table 1: Selected seasonal ARIMA models for T90 and T10 based on AIC
T90 T10
Region Non-seasonal Seasonal Region Non-seasonal Seasonal
p d q P D Q p d q P D Q
CEA 0 1 3 0 0 0 CEA 0 1 5 1 0 2
CWP 1 0 1 0 0 0 CWP 0 1 3 0 0 0
MID 3 1 1 2 1 2 MID 1 1 1 0 0 0
SEA 0 1 4 0 0 0 SEA 2 1 2 0 0 2
SPL 2 1 1 0 0 1 SPL 0 1 2 0 0 2
SWP 2 1 2 1 1 1 SWP 0 1 2 2 1 2
2.2 Mortality Data
The U.S. regional-level death data for the time period 1968–2017 are obtained from two
main sources listed as follows:
3Rpackage forecast (Hyndman and Khandakar, 2008) is used to select optimal models and conduct
residual tests.
The National Center for Health Statistics (NCHS):
The NCHS records information of all individual deaths in the U.S. since 1959, includ-
ing gender, date of birth, month of death, and geographical identifier. It provides
micro-level death data based on information abstracted from death certificates filed in
vital statistics offices in the U.S. We collect state-level monthly death data from the
NCHS “Multiple Cause of Death Data, 1959–2017” vital statistics database for the
period 1968–2004.4From 2005, the geographic identifier is omitted due to a restriction
imposed on release of sub-national mortality data.
Centers for Disease Control and Prevention (CDC) WONDER online database:
The CDC WONDER online database provides a rich query system for public health
data and vital statistics in the U.S. We collect summarized state-level monthly death
data for 2005–2017 via the online query “Multiple Cause of Death, 1999–2017 Request”.
Due to privacy constraints, as of 23rd May 2011, all sub-national data representing
fewer than ten deaths are suppressed 5.
As there is no publicly available data on monthly age-specific population exposure, particu-
larly at the state level, in this study we choose to directly model death counts rather than
mortality rates. After removing any underlying trends and seasonality in the age-specific
death counts, we obtain the residuals which can be considered as “excess deaths”. We in-
terpret these residuals as the number of deaths above or below what would otherwise be
expected in normal conditions. For illustrative purposes, in this paper, we choose to focus
on death counts of five broad age groups, namely 0, 1–24, 25–54, 55–84, and 85+. This is
also to minimize the issue caused by data suppression.6
It is widely recognized that mortality rates have been declining across all age groups over
the past decades. However, the same conclusion cannot be simply applied to death counts
as the number of deaths is determined by both population exposure and mortality rates.
Therefore, with rapid population growth, death counts can be increasing over time even
with decreasing mortality rates. Figure 3 shows the size and age distributions of the U.S.
population in 1969 and 2017.7It is shown that there has been population growth across all
age groups, in particular for ages 25–54 and 55–84. Proportionally, the increase in age group
85+ is the highest (almost a 5-fold increase in less than 50 years), reflecting the aging of the
population in U.S. society. On the other hand, the number of newborns in the U.S. is at a
comparable level in 1969 and 2017. For each of the six continental U.S. regions defined in
the previous section, detailed population size and age structure are reported in Tables 6 and
7 of the Appendix, for 1969 and 2017 respectively. Similar conclusions can be made about
changes in the population structure for these continental regions.
4Note that records on individual deaths between 1959 and 1967 are also available, while due to data
quality concerns, we decide to only use NCHS data after 1968. For more details, see http://www.nber.
5See more details at
6In the CDC WONDER database, all sub-national data representing fewer than ten deaths are suppressed
due to privacy constraints.
7The U.S. population data is downloaded from the Survey of Epidemiology and End Results (SEER)
U.S. State and County Population Data at
90 60 30 0 0 50 100 150
1969 Age 2017
millions millions
Figure 3: U.S. population pyramid
In Figures 4–8, we plot the monthly death counts in the six U.S. continental regions for
five broad age groups. We can see that for infant age group and ages 1–24, Figures 4 and
5 show an overall downward trend in the number of deaths, although the rate of decline
tends to vary across the six regions. Moreover, the trends in death counts for these two
age groups look fairly similar, with ages 1–24 showing a much higher level of volatility. For
the age groups 25–54 and 55–84, we observe some mixed results: In CEA and MID, there
has been a slight decreasing trend in monthly death counts; while in CWP, SEA, SPL, and
SWP, the number of deaths has gradually increased in the last few decades. Finally, for the
age group 85+, all six regions have experienced a steady growth in the number of deaths,
which directly reflects the increase in life expectancy and population exposure of older age
groups. Due to the differences in population size, overall CWP has the smallest death
counts for all age groups, and SEA has the largest death counts for almost all age groups.
As the purpose of this paper is not to discuss the geographical heterogeneity in regional-
level mortality experience, we do not further comment on the trends observed in Figures 4–8.
It is worth noting that seasonality is a prominent feature exhibited in many death-count
time series. In particular, seasonal patterns are most apparent for the age groups 1–24,
55–84, and 85+. On the other hand, infant age group and ages 25–54 seem to show very
weak seasonality compared to the other age groups. Taking a closer look at the data, we find
that in the U.S. peaks in death counts generally happen in winter (November, December,
and January) for the age groups 55–84 and 85+, while peaks are more likely to happen in
summer (May, June, and July) for the age group 1–24. Our observations are all consistent
with those in Rau et al. (2017).
1968 1978 1988 1998 2008 2018
No of deaths
Age 0
Figure 4: Monthly death counts for the age 0 in six U.S. regions
1968 1978 1988 1998 2008 2018
No of deaths
Age 1−24
Figure 5: Monthly death counts for the age group 1–24 in six U.S. regions
1968 1978 1988 1998 2008 2018
No of deaths
Age 25−54
Figure 6: Monthly death counts for the age group 25–54 in six U.S. regions
1968 1978 1988 1998 2008 2018
No of deaths
Age 55−84
Figure 7: Monthly death counts for the age group 55–84 in six U.S. regions
1968 1978 1988 1998 2008 2018
No of deaths
Age 85+
Figure 8: Monthly death counts for the age group 85+ in six U.S. regions
To obtain a more complete picture of death data across age groups, and a better understand-
ing of the underlying patterns in the data-count time series, we further segment death data
by cause-of-death. The 10 leading causes of death for the five age groups during 1999–2017
are summarized and presented in Table 2.8In the CDC WONDER database, the codifi-
cation of cause-of-death is based on the 10th revision of the International Classification of
Diseases (ICD10), which came into effect since 1999.9We have also provided the detailed
ICD10 codifications for these leading causes of death in Table 8 of the Appendix.
From Table 2, we can see that perinatal death accounts for almost 50% of the total infant
deaths. For the age group 1–24, the top leading cause-of-death is “Accidents” which com-
prises 41.18% of the total deaths. “Malignant neoplasms” makes up the highest proportion
of total deaths for both age groups 25–54 and 55–84. “Diseases of heart” is responsible for
around one third of the total deaths for ages 85+. It is also the second leading cause-of-death
for the age group 55–84 (24.90%), and the third leading cause-of-death for the age group
25–54 (16.90%). Moreover, it is worth noting that deaths from “Respiratory diseases” and
“Cerebrovascular diseases” are major components in the total number of deaths for both ages
55–84 and 85+. Finally, “Influenza and pneumonia” is included in the 10 leading causes of
death for all age groups except for ages 25–54, although its proportion is considerably bigger
in older age groups (1.91% for ages 55–84, and 3.61% for ages 85+ ) than in younger age
groups (0.88% for infants, and 0.94% for ages 1–24).
Based on these cause-of-death information, we are able to better comprehend the existence
of seasonality in age-specific death counts. The higher number of deaths during winter for
8We obtain cause-of-death information via the CDC WONDER online query “Multiple Cause of Death,
1999–2017 Request”. During our observation window 1968–2017, the ICD coding system has changed from
ICD8 (1968–1978) to ICD9 (1979–1998), and then to ICD10 (1999 onwards). To ensure consistency in the
codification, we choose to only report the leading causes of death during 1999–2017.
9See more details at
Table 2: 10 leading causes of death during 1999–2017 (crude rate per 100,000 population, non age-adjusted)
Rank 0 1–24
Cause of death Crude rate % of total Cause of death Crude rate % of total
1 Perinatal conditions 327.8 49.94 Accidents 17.5 41.18
2 Congenital anomalies 133.0 20.26 Assault (homicide) 5.7 13.41
3 Accidents 28.1 4.28 Suicide 4.9 11.53
4 Diseases of heart 9.6 1.46 Malignant neoplasms 3.0 7.06
5 Assault (homicide) 7.8 1.19 Diseases of heart 1.4 3.29
6 Septicemia 5.9 0.90 Congenital anomalies 1.3 3.06
7 Influenza and pneumonia 5.8 0.88 Influenza and pneumonia 0.4 0.94
8 Nephritis 3.1 0.47 Respiratory diseases 0.4 0.94
9 Cerebrovascular diseases 3.0 0.46 Cerebrovascular diseases 0.3 0.71
10 Malignant neoplasms 1.7 0.26 Septicemia 0.3 0.71
Rank 25–54 55–84
Cause of death Crude rate % of total Cause of death Crude rate % of total
1 Malignant neoplasms 51.4 21.51 Malignant neoplasms 601.3 29.64
2 Accidents 40.6 16.99 Diseases of heart 505.1 24.90
3 Diseases of heart 40.4 16.90 Respiratory diseases 138.8 6.84
4 Suicide 15.9 6.65 Cerebrovascular diseases 109.0 5.37
5 Liver disease and cirrhosis 9.0 3.77 Diabetes mellitus 72.1 3.55
6 Assault (homicide) 7.7 3.22 Accidents 54.0 2.66
7 Cerebrovascular diseases 6.7 2.80 Alzheimer disease 41.6 2.05
8 Diabetes mellitus 6.6 2.76 Influenza and pneumonia 38.7 1.91
9 HIV disease 6.2 2.59 Nephritis 38.4 1.89
10 Respiratory diseases 4.1 1.72 Septicemia 31.5 1.55
Rank 85+
Cause of death Crude rate % of total
1 Diseases of heart 4632.3 32.50
2 Malignant neoplasms 1709.0 11.99
3 Cerebrovascular diseases 1136.8 7.98
4 Alzheimer disease 953.0 6.69
5 Respiratory diseases 678.9 4.76
6 Influenza and pneumonia 514.1 3.61
7 Accidents 323.1 2.27
8 Diabetes mellitus 296.4 2.08
9 Nephritis 296.2 2.08
10 Hypertensive disease 210.4 1.48
older age groups are most likely due to “Diseases of heart”, “Cerebrovascular diseases”,
“Respiratory diseases”, as well as “Influenza and pneumonia”. Cold temperatures cause
blood vessels to narrow, and thus can restrict blood flow and reduce oxygen to the heart.
Furthermore, virus and bacteria tend to survive more easily in cold weather and thus increase
the risk of influenza infections (Dushoff et al., 2005; Rau et al., 2017). On the other hand,
the high proportion of accidental deaths for ages 1–24 could potentially explain the peak in
death counts during summer months. Studies have found that motor vehicle accidents tend
to peak around July and August, and extreme hot weather can also lead to an increased
number of workplace accidents (Rau et al., 2017; Rameezdeen and Elmualim, 2017).
Similar to T10 and T90, we want to obtain the “noise” in death counts via time series
models. Besides trends and seasonal patterns described in Equation (1), we also consider
a GARCH component in the model as it is a common practice to handle heteroscedasticity
in mortality in the literature (see e.g. Lee and Miller, 2001; Gao and Hu, 2009; Giacometti
et al., 2012; Lin et al., 2015; Chen et al., 2017; Li and Tang, 2019). We fit a seasonal ARIMA
model first and include the GARCH component if the resulting residuals fail the Ljung–Box
test at 5% level of significance or the variance of the residuals is clearly non-constant. The
optimal model is selected based on AIC, and we present the model selection results in Table
3 in which mand ndenote the orders of the ARCH and GARCH models, respectively.10
The seasonal ARIMA model component is expressed in the form of Equation (1) with Sset
to 12. From Table 3, we can see that the GARCH component is mostly required for the age
groups 0, 1–24, and 85+. For 25–54 and 55–84, the only case where a GARCH component is
included is the age group 55–84 in CEA. All standardized residual series from the estimated
ARIMA-GARCH models passed the Ljung–Box test at 5% level of significance.
Table 3: Selected seasonal ARIMA-GARCH models for death counts based on AIC
0 1–24 25–54
m n m n m n
CEA (4,1,1)(2,0,0) 2 2 (1,1,1)(2,1,2) 1 1 (1,1,3)(0,1,1) 0 0
CWP (1,1,2)(2,0,0) 1 1 (0,1,1)(0,1,1) 0 0 (1,1,2)(2,0,0) 0 0
MID (2,1,2)(2,0,0) 1 1 (1,1,1)(1,1,2) 1 2 (0,1,2)(0,1,1) 0 0
SEA (0,1,2)(2,0,0) 1 1 (2,0,2)(0,1,1) 1 2 (2,1,2)(0,1,1) 0 0
SPL (1,1,2)(2,0,0) 1 2 (1,0,3)(0,1,1) 1 1 (1,1,3)(2,1,2) 0 0
SWP (1,1,2)(2,0,0) 1 1 (0,1,1)(0,1,2) 0 0 (0,1,3)(2,1,1) 0 0
55–84 85+
m n m n
CEA (2,1,1)(0,1,2) 1 3 (1,1,2)(0,1,2) 0 0
CWP (2,1,1)(1,1,1) 0 0 (0,1,3)(0,1,1) 1 3
MID (0,1,2)(2,1,1) 0 0 (1,1,2)(1,1,1) 1 2
SEA (2,1,1)(0,1,2) 0 0 (1,1,2)(1,1,1) 2 3
SPL (2,1,1)(1,1,1) 0 0 (1,1,2)(0,1,2) 1 2
SWP (2,1,3)(1,1,2) 0 0 (2,1,1)(1,1,2) 1 3
10Rpackage rugarch (Ghalanos, 2019) is used to estimate the GARCH parameters in the model.
3 Multivariate Extreme Value Theory
3.1 Fundamentals
Before going into the details of multivariate EVT, we first provide a brief introduction to
some basic theory in univariate EVT. For a quick introduction to EVT, see Embrechts et al.
(1999), and for a more comprehensive review of EVT in the context of insurance and finance,
see Embrechts et al. (1997) and McNeil et al. (2015).
Consider a random variable Xwith distribution Fon Rand denote by Mnthe maximum
of a sample of size nfrom F. If there exist norming constants an>0 and bnRsuch that
the distribution of a1
n(Mnbn) converges to a non-degenerate distribution Gon R, that is,
n→∞ Pr Mnbn
y=G(y), y R,(2)
then we say that Fbelongs to the max-domain of attraction of G, denoted by FMDA(G),
and call Ga generalized extreme value (GEV) distribution. The GEV distribution function
Gmust be of the same type as
G(y) = exp (1 + γyµ
where µR,σ > 0, and γRare the location, scale, and shape parameters, respectively,
and c+= max(c, 0). The GEV distribution function Gin this standard form is supported
on the region of ydefined by 1 + γyµ
σ>0. For γ > 0, γ < 0, and γ= 0, the standard
GEV distribution Gcorresponds to the Fr´echet distribution, the Weibull distribution, and
the Gumbel distribution, respectively. For the case of γ= 0, by continuity Gtakes the form
G(y) = exp exp yµ
Following the classical works of Balkema and de Haan (1974) and Pickands (1975), we know
that for a random variable Xdistributed by FMDA(G), the conditional distribution of the
normalized exceedance over a high threshold converges to a generalized Pareto distribution
(GPD). That is,
n→∞ Pr Xbn
X > bn=H(y), y > 0,(5)
where His of the same type as
H(y) = 1 1 + γyµ
with the location, scale, and shape parameters µR,σ > 0, and γR. The GPD Habove
is supported on the region of ydefined by y > 0 and 1 + γyµ
σ>0. By continuity, when
γ= 0 we have the exponential distribution.
3.2 Multivariate Generalized Extreme Value Distribution
Extensions of the results in Section 3.1 to the multivariate setting have been realized fairly
recently, especially for the multivariate POT approach (see Beirlant et al., 2004; Rootz´en
et al., 2018a,b, and references therein).
Consider a d-dimensional random vector Xwith distribution Fon Rdand denote by Mn
the component-wise maximum of a sample of size nfrom F. The multivariate EVT aims
to establish that for some norming constant vectors an>0and bnRd, the distribution
of a1
n(Mnbn) converges to a distribution Gon Rdwith non-degenerate marginal distri-
butions, where all operations between vectors are interpreted as component-wise. For this
case, we write FMDA(G) as before. The limit distribution G, called a multivariate GEV
distribution, has marginal distributions Gifor 1 ididentical to
n→∞ Pr M(i)
which therefore is of the same type as Hdefined by Equation (6).
In practice, it is common to first transform the marginal distributions to a particular distri-
bution before fitting a multivariate GEV distribution. Such distributions include the uniform
distribution (see e.g. Coles et al., 1999), the inverted standard exponential distribution (see
e.g. Falk and Reiss, 2005), and the unit Fr´echet distribution (see e.g. Smith, 1994; Smith
et al., 1997; Bortot et al., 2000; Rootz´en and Tajvidi, 2006), with the latter being the most
commonly adopted. In this paper, we choose the unit Fechet transformation
log Gi(y).(8)
According to Propositions 5.10 and 5.11 in Resnick (1987), the representation of a multi-
variate GEV distribution with unit Fechet margins can be written as
G(y) = exp {−V(z)},(9)
where V(·), referred to as the exponent measure, has a functional representation
V(z) = ZSd
with φbeing a finite spectral measure on Sd={qRd:kqk= 1}, and k·k representing an
arbitrary norm in Rd. We also impose a constraint such that, for 1 id,
but beyond this the spectral measure φis generally unknown. Clearly, Vdefined by (10) is a
homogeneous function of order 1. It is worth noting that, contrary to the univariate EVT,
there are infinitely many choices for Vsuch that Equation (10) holds. Classical choices of the
parametric family for Vinclude the logistic, negative logistic, bilogistic, negative bilogistic,
mixed, and asymmetric mixed models.
As in this study we focus on assessing the upper tail dependence between temperature and
mortality, we adopt the symmetric logistic model for the function V, which is a natural
candidate and a commonly used dependence model in bivariate POT studies (see e.g. Tawn,
1990; Coles et al., 1999; Rootz´en and Tajvidi, 2006). The logistic model originates from the
Gumbel copula (also known as the Gumbel–Hougaard copula), which is an Archimedean
copula exhibiting dependence in the positive tail but no dependence in the negative tail. As
one of the oldest multivariate extreme-value models, the logistic model is the only model that
is both Archimedean and extreme-value (Tawn, 1988; Genest and Rivest, 1989). Moreover,
it has been independently discovered and widely used in the survival analysis literature (see
e.g. Hougaard, 1986). Under the symmetric logistic model,
V(z1, z2) = (zr
2)1/r, r 1,(12)
which can be retrieved from Equation (10) with a suitably chosen spectral measure φon
S2. The exponent measure V(z1, z2) determines the strength of dependence between the two
margins. In particular, independence is obtained when r= 1, and perfect dependence is
obtained as r→ ∞.
3.3 Bivariate POT Theory
The multivariate POT theorem then states that, for a random vector Xdistributed by
FMDA(G), assuming 0 < G(0)<1 without loss of generality, the conditional distribution
of a1
n(Xbn) given Xbnconverges to the multivariate GPD as
H(y) = 1
log G(0)log G(y)
which is defined for all yRdsuch that G(y)>0. In particular, H(y) = 0 for y<0and
H(y) = 1 log G(y)
log G(0)for y>0(Rootz´en and Tajvidi, 2006; Rootz´en et al., 2018a,b).
Since our focus is on bivariate extremes, we illustrate in the following the bivariate case
only. To describe the strength of extremal dependence, we introduce the Pickands depen-
dence function A(Pickands, 1981) and the dependence measure χ(Coles et al., 1999) to
measure the strength of extremal dependence.
For the bivariate case, the Pickands dependence function A: [0,1] [0,1] is defined as
A(ω) = ZS2
max (ωq1,(1 ω)q2)(q),0ω1,(14)
which links the function Vthrough the relation
A(ω) = V(z1, z2)
with ω=z2
z1+z2. By Equation (11), it is clear that A(0) = A(1) = 1. If two random variables
with unit Fechet margins are independent, then A(ω) = 1 for all 0 ω1, while if they
are perfectly dependent, then A(ω) = max(ω, 1ω) for all 0 ω1.
Coles et al. (1999) developed the index χto measure extremal dependence for bivariate
random variables. Assuming that random variables Z1and Z2have the same marginal
distribution F, the index χis defined as
χ= lim
u1Pr (F(Z2)> u|F(Z1)> u),(16)
provided that the limit exists. Thus, χdenotes the probability of one variable reaching an
extreme value given that the other variable has already reached it. If χ= 0, the two variables
are said to be asymptotically independent. While for full tail dependence, we have χ= 1.
Furthermore, it can be verified (Coles et al., 1999) that,
χ= 2 V(1,1) = 2 2A(0.5).(17)
4 Empirical Results
In this section, we apply the bivariate POT approach introduced in Section 3 to the U.S.
data described in Section 2. Separate models are estimated for each of the six continental
U.S. regions, and for all unique combinations of temperature indexes and age groups.11 We
also demonstrate our results via Pickands dependence function and simulated bivariate GPD
variables based on the estimated models.
The selection of suitable thresholds is a critical step in EVT analysis, and it presents a
trade-off between variance and bias. We need to make sure that there are enough data
points to reduce the variance of our estimation, but at the same time not too many data
points to induce bias. While a vast literature exists for threshold selection in univariate
EVT (see, e.g. Embrechts et al., 1997; Coles, 2001), to our knowledge there has not been
a widely recognized threshold selection method for bivariate POT theory, as the theoretical
development of multivariate EVT is far behind its univariate counterpart.
In a similar application of multivariate EVT in general insurance, Brodin and Rootz´en (2009)
applied both univariate and bivariate POT approaches to wind storm insurance loss data.
They selected the univariate thresholds based on inspection of various diagnostic plots such
as the parameter stability plots, QQ-plots, and quantile plots. The bivariate threshold is
then formed using these univariate thresholds from each marginal distribution. However,
due to the limited sample size in our study, adopting the approach by Brodin and Rootz´en
(2009) results in too few data points for the bivariate GPD estimation. Thus, we instead
use a benchmark of top 5% to represent the joint exceedance and thus to estimate the bi-
variate GPD, with the “top 5%” of observations selected by a common threshold for each
unit Fechet marginal distribution. The details are illustrated in Tables 4 and 5.
In Table 4, we present the results for joint extremes in T90 and death counts for the age
groups 0, 1–24, 25–54, 55–84, and 85+. The extremal dependence between hot temperature
and death count across all age groups and regions is fairly weak, as the value of χis always
close to zero. The strongest dependence between T90 and death counts has been found in
the age group 1–24, regions SEA and SPL. On the other hand, extreme hot temperature
does not seem to have a noticeable impact on older populations. This result is rather
11Rpackage POT (Ribatet, 2019) is used to estimate the bivariate GPD distributions.
surprising as many studies have found evidence on the negative impact of heatwave on
population mortality (see e.g. Lye and Kamal, 1977; Rooney et al., 1998; Le Tertre et al.,
2006; Xu et al., 2016). Heat waves contribute directly to deaths from cardiovascular disease,
respiratory disease and diabetes-related conditions, to which elderly people are particularly
vulnerable as demonstrated in Section 2.2. However, it should be noted that high frequencies
of hot temperature do not necessarily imply the occurrence of heat waves. A heatwave is
generally defined in terms of a consecutive period of excessive heat events12. Therefore, the
T90 index does not fully capture and measure heat waves. Furthermore, as our analysis
is conducted based on monthly death counts, it is unclear to what extent the immediate
increase due to extreme heat is offset by “harvesting effect” or “mortality displacement”.
Harvesting effect refers to the phenomenon where a compensatory decrease in mortality rates
has been observed in the subsequent weeks after a heat event, which suggests that extreme
hot weather severely affects those who are so ill that they “would have died in the short
term anyway” (Huynen et al., 2001). In fact, Deschenes and Moretti (2009) argued that
in the U.S., the increase in mortality following extreme heat is entirely driven by mortality
displacement. Therefore, temporal displacement of mortality is another possible explanation
of the weak dependence between extreme values of T90 and death counts, in the sense that
the monthly frequency of death counts is likely to include mortality displacement period.
Table 4: Results from bivariate POT analysis on T90 and death counts
0 1–24 25–54
% above threshold χ% above threshold χ% above threshold χ
Marginal Joint Marginal Joint Marginal Joint
CEA 0.21 0.048 0.001 0.22 0.050 0.001 0.22 0.048 0.001
CWP 0.20 0.048 0.001 0.23 0.048 0.001 0.20 0.052 0.001
MID 0.23 0.050 0.001 0.23 0.050 0.010 0.24 0.050 0.001
SEA 0.21 0.048 0.001 0.21 0.052 0.053 0.22 0.050 0.014
SPL 0.21 0.052 0.001 0.19 0.050 0.064 0.23 0.040 0.001
SWP 0.23 0.050 0.001 0.22 0.050 0.011 0.21 0.043 0.018
55–84 85+
% above threshold χ% above threshold χ
Marginal Joint Marginal Joint
CEA 0.21 0.048 0.002 0.24 0.052 0.006
CWP 0.22 0.048 0.004 0.21 0.048 0.010
MID 0.23 0.052 0.001 0.25 0.045 0.001
SEA 0.22 0.052 0.001 0.25 0.052 0.001
SPL 0.23 0.050 0.001 0.22 0.048 0.001
SWP 0.23 0.050 0.002 0.24 0.053 0.014
In Table 5, we present the results for joint extremes in T10 and death counts for the age
groups 0, 1–24, 25–54, 55–84, and 85+. Based on the values of χ, it is shown that the
12In the CDC WONDER database, three definitions are available for heat events: 95th percentile of daily
maximum air temperature, 95th percentile of daily maximum heat index, and 95th percentile of net daily
heat stress.
extremal dependence between low temperature and death counts for age groups 0 and 1–24,
is close to zero across all regions. For the age group 25–54, the extremal dependence remains
to be very weak except for SEA. On the other hand, we can see that extreme cold weather
clearly has a stronger impact on the mortality experience of older people aged 55–84 and
85+. In particular, for all regions the dependence measure increases from the age group 55–
84 to the age group 85+. Overall, SEA has the highest level of extremal dependence between
T10 and death counts. Cold weather can cause substantial short-term increase in mortality,
from thrombotic and respiratory disease, but more importantly influenza (Donaldson and
Keatinge, 2002; Dushoff et al., 2005). Epidemics of influenza are likely to be associated with
extreme cold weather, as viruses that cause flus may spread more easily in low temperatures,
and exposure to cold air may have an adverse impact on the immune system. It is widely
recognized that elderly are more fragile to extreme climate events (see e.g. Tillett et al., 1983;
Astr¨om et al., 2011). Therefore, these results are consistent with findings and conclusions
from existing research in epidemiology, as well as the information on leading causes of death
presented in Section 2.2.
Table 5: Results from bivariate POT analysis on T10 and death counts
0 1–24 25–54
% above threshold χ% above threshold χ% above threshold χ
Marginal Joint Marginal Joint Marginal Joint
CEA 0.24 0.050 0.001 0.22 0.0483 0.001 0.22 0.048 0.001
CWP 0.21 0.050 0.012 0.23 0.0483 0.018 0.24 0.048 0.001
MID 0.25 0.048 0.001 0.27 0.0517 0.001 0.23 0.050 0.001
SEA 0.22 0.048 0.043 0.24 0.0483 0.001 0.19 0.050 0.093
SPL 0.21 0.050 0.017 0.24 0.0517 0.018 0.21 0.047 0.027
SWP 0.22 0.052 0.008 0.26 0.048 0.001 0.22 0.048 0.047
55–84 85+
% above threshold χ% above threshold χ
Marginal Joint Marginal Joint
CEA 0.21 0.053 0.064 0.18 0.052 0.107
CWP 0.19 0.053 0.075 0.20 0.052 0.086
MID 0.22 0.052 0.037 0.20 0.053 0.066
SEA 0.16 0.050 0.153 0.18 0.055 0.154
SPL 0.19 0.048 0.087 0.20 0.052 0.088
SWP 0.20 0.048 0.058 0.20 0.052 0.067
For illustration purposes, in Figure 9 we plot the Pickands dependence function for T10 and
deaths in the age group 85+ for each continental U.S. region. The horizontal line in the
upside down triangle corresponds to independence while the other two sides correspond to
perfect dependence. It can be seen that SEA has the strongest extremal dependence, which
is consistent with the results in Table 5. Overall, we find that there exists certain extremal
dependence between T10 and death counts at older ages, though not significant. Finally,
for each estimated bivariate GPD for T10 and deaths in the age group 85+, we generate
5,000 sample points and present them in Figure 10, which discovers a similar phenomenon
of extremal dependence between T10 and death counts at older ages.
0.0 0.2 0.4 0.6 0.8 1.0
0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.5 0.6 0.7 0.8 0.9 1.0
Figure 9: Pickands dependence function for T10 and deaths in the age group 85+
0 2 4 6 8 10
0 2 4 6 8
0 2 4 6 8
0 2 4 6 8 10
0 2 4 6
0 2 4 6 8
Figure 10: Simulated bivariate GPD variables for T10 and deaths in the age group 85+
5 Conclusions and Insurance Implications
For insurance and reinsurance companies, the management of extreme risks has a significant
impact across many aspects of their business, as the nature of insurance is to minimize ad-
verse effects of risks associated with losses in wealth, health, and life. It is well recognized
that risks do not exist in isolation and thus should not be treated as being independent. We
propose a rigorous bivariate POT approach to modeling the joint behavior of two extreme
risks, aiming to improve our understanding of the short-term dynamics between extreme
climate and mortality risks. In the empirical study, we quantify the extremal dependence
between the frequency of low and high temperature and death counts for different age groups
in the U.S. Our results show that the strongest extremal dependence lies between T10 and
deaths in the age group over 85. The ideas underlying the proposed multivariate EVT ap-
proach are applicable to a wide range of topics such as multi-peril insurance and insurance
portfolio management.
A potential application of our results is to assist the pricing of insurance linked securities
(ILS) in the longevity capital market, where the pay-off is usually linked to extreme mor-
tality events. For example, mortality catastrophe (CAT) bonds have been used as a risk
mitigation tool by many large insurance companies to transfer extreme mortality risk to
the capital market. By construction, if the underlying mortality index exceeds a certain
threshold during the risk period of the bond, the investors’ principal will be reduced or even
completely written off to compensate insurance losses. Therefore, these bonds are designed
to protect insurance companies from extreme mortality experiences. Conventionally, future
mortality levels are often projected under stochastic mortality models such as the Lee-Carter
model (Lee and Carter, 1992). However, such modeling frameworks generally do not incor-
porate environmental factors into the forecasting procedure. As climate-associated extreme
mortality has become a growing proportion of total mortality risk, the extremal dependence
between climate indexes and death counts will provide important insights into the pricing
of mortality CAT bonds. Having quantified the relationship between extreme weather and
extreme mortality, scenario-based analysis and forecast can be made. In this way, if extreme
cold/hot weather becomes much more frequent, the expected number of excess deaths that
will occur in the population can be estimated.
Moreover, based on the age-region-specific analysis of joint extremes in temperature and
mortality, we observe different levels of exposure and resilience to extreme weather across
geographical regions and age groups in the U.S. These results motivate the development of
innovative ILS products in the longevity capital market, targeting extreme mortality experi-
ence in specific regions and age groups as opposed to the total population, which may better
reflect the life insurance company’s risk portfolio. By identifying regions and age groups that
are most vulnerable to extreme climate, adaptations can be made in the government policy
and planning, as well as in the finance of health insurance to reduce the adverse effects of
climate catastrophes on vulnerable populations.
Finally, this paper is amongst the first to explore the unique relationship between climate
and mortality, by conducting an innovative quantitative analysis on the mortality risk of
extreme temperatures. The proposed approach will facilitate and lay the foundation of
future research on related topics (e.g. the impact of climate change on health insurance).
This future research will have a positive impact on society, not only in promoting climate-
resilience in the insurance sector, but also in protecting vulnerable populations in society.
This research is sponsored by the Society of Actuaries through the 2019 Individual Grant
Competition, project titled “An EVT Approach to Modeling Joint Extremes in Climate and
Mortality”. We thank the editor and the two anonymous referees for their valuable com-
ments and suggestions.
Han Li
Department of Actuarial Studies and Business Analytics
Macquarie University
Sydney, NSW 2109, Australia
Qihe Tang
School of Risk and Actuarial Studies
UNSW Sydney
Sydney, NSW 2052, Australia
Department of Statistics and Actuarial Science
University of Iowa
Iowa City, IA 52242, United States
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Table 6: Population structure for six U.S. regions in 1969
0 1–24 25–54 55–84 85+
Population size Percentage Population size Percentage Population size Percentage Population size Percentage Population size Percentage
CEA 918,575 1.65% 23,489,592 42.19% 20,115,133 36.13% 10,777,750 19.36% 369,570 0.66%
CWP 105,976 1.72% 2,734,438 44.49% 2,138,684 34.79% 1,121,625 18.25% 46,048 0.75%
MID 899,953 1.75% 22,934,307 44.72% 17,787,360 34.68% 9,304,319 18.14% 359,958 0.70%
SEA 732,376 1.77% 18,731,580 45.27% 14,205,219 34.33% 7,453,987 18.01% 254,851 0.62%
SPL 349,418 1.79% 8,843,890 45.25% 6,657,380 34.06% 3,552,384 18.17% 142,924 0.73%
SWP 466,099 1.79% 11,688,590 44.78% 9,516,711 36.46% 4,271,765 16.37% 158,538 0.61%
Table 7: Population structure for six U.S. regions in 2017
0 1–24 25–54 55–84 85+
Population size Percentage Population size Percentage Population size Percentage Population size Percentage Population size Percentage
CEA 735,449 1.11% 19,223,188 29.13% 26,142,682 39.61% 18,351,347 27.81% 1,541,860 2.34%
CWP 160,007 1.21% 3,977,965 29.99% 5,322,494 40.12% 3,561,537 26.85% 243,459 1.84%
MID 731,890 1.19% 19,068,250 30.89% 23,636,997 38.30% 16,959,056 27.48% 1,324,900 2.15%
SEA 965,553 1.18% 24,654,999 30.10% 31,999,050 39.07% 22,689,650 27.70% 1,589,967 1.94%
SPL 562,893 1.40% 13,662,419 33.88% 16,003,922 39.69% 9,447,273 23.43% 647,019 1.60%
SWP 754,454 1.25% 19,087,716 31.63% 24,622,158 40.80% 14,809,109 24.54% 1,074,582 1.78%
Table 8: ICD10 codification for 10 leading causes of death
Cause of death ICD10 codification
Accidents V01-X59, Y85-Y86
Alzheimer disease G30
Assault (homicide) U01-U02, X85-Y09, Y87.1
Cerebrovascular diseases I60-I69
Congenital abnormalities Q00-Q99
Diabetes mellitus E10-E14
Diseases of heart I00-I09, I11, I13, I20-I51
HIV disease B20-B24
Hypertensive disease I10, I12, I15
Influenza and pneumonia J09-J18
Liver disease and cirrhosis K70, K73-K74
Malignant neoplasms C00-C97
Nephritis N00-N07, N17-N19, N25-N27
Perinatal conditions P00-P96
Respiratory diseases J40-J47
Septicemia A40-A41
Suicide U03, X60-X84, Y87.0
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