for Life for Ages in Short Life 1 Dr. Juan 2 1 Actuarial Science Department ITAM 2 School of Bussiness ITAM October 4, 2012 1 / 32
Table of contents for Life 1 2 3 4 5 Conclusions 2 / 32
distributions for Life There are two approaches that are commonly employed in the estimations of survival models: The parametric approach: This approach uses an analytic formula, such as the de Moivre, Gompertz or Makeham models, to calculate a survival distribution. The unknown parameters of the formula are estimated. The life table approach: A life table is used to compute the survival probabilities for integral ages, and an assumption for the distribution of future lifetime between integers is adopted. 3 / 32
Commonly used assumptions for Life Commonly used assumptions include a uniform distribution of deaths throughout the year, a constant force of mortality, and the Balducci assumption. None of these assumptions in the previous literature account for the seasonal effects observed in the mortality data, such as the excess mortality that occurs during the winter. 4 / 32
Empirical monthly distribution of deaths for Life We study the empirical monthly distribution of deaths in Mexico. Figure: Mexican mortality data: Observed (empirical) monthly death rates by age of death 5 / 32
Future lifetime between integers for Life of interest: T(x) =, the future-lifetime of an individual of age x, K(x) =, the curtate-future-lifetime of an individual of age x (K(x) = T(x) ) S(x) = T(x) K(x), the fractional portion of the individual s age at death. We will model S(x) K(x) = k as a circular random variable. 6 / 32
random variable for Life Origin of a circular random variable: Directional data (Θ [0, 2π)), Cyclical data (hours in a day, days in a week, months in a year...) (S [0, 1)), A circular observation is a point on the unit circle. 7 / 32
Particularities for Life -At what time do you fell asleep? -Sometimes at 23:00 and sometimes 01:00 -So, on average, you fell asleep at 23 + 1 = 12 : 00! 2 Time must be wrapped around the circle. 8 / 32
Particularities for Life -At what time do you fell asleep? -Sometimes at 23:00 and sometimes 01:00 -So, on average, you fell asleep at 23 + 1 = 12 : 00! 2 Time must be wrapped around the circle. 8 / 32
Particularities for Life -At what time do you fell asleep? -Sometimes at 23:00 and sometimes 01:00 -So, on average, you fell asleep at 23 + 1 = 12 : 00! 2 Time must be wrapped around the circle. 8 / 32
Properties of a circular random variable for Life The density function of a circular random variable must satisfy: f Θ (θ) 0, 2π 0 f Θ (θ)dθ = 1, f Θ (θ) = f Θ (θ + 2πk) for any integer k, Implementing the following variable transformation for S = θ 2π [0, 1) we get f S (s) = f Θ (θ) dθ ds = 2πf Θ(2πs). 9 / 32
Properties of a circular random variable for Life The density function of a circular random variable must satisfy: f Θ (θ) 0, 2π 0 f Θ (θ)dθ = 1, f Θ (θ) = f Θ (θ + 2πk) for any integer k, Implementing the following variable transformation for S = θ 2π [0, 1) we get f S (s) = f Θ (θ) dθ ds = 2πf Θ(2πs). 9 / 32
Properties of a circular random variable for Life The density function of a circular random variable must satisfy: f Θ (θ) 0, 2π 0 f Θ (θ)dθ = 1, f Θ (θ) = f Θ (θ + 2πk) for any integer k, Implementing the following variable transformation for S = θ 2π [0, 1) we get f S (s) = f Θ (θ) dθ ds = 2πf Θ(2πs). 9 / 32
Examples Turtles Data for Life 90 180 + 0 270 Figure: Turtles directions in their way to the sea 10 / 32
Examples for Life nntsplotint (x) 0.7 0.9 1.1 1.3 Day of the week M T W T F S S S Month nntsplotint (x) 0.5 1.0 1.5 J F M A M J J A S O N D S Figure: Accidents Mexico-Cuernavaca Highway 11 / 32
Examples for Life nntsplotint (x) 0.4 0.8 1.2 Days of the Week M T W T F S S S Time of the day nntsplotint (x) 0.5 1.0 1.5 3 6 9 12 15 18 21 24 S Figure: Car theft, Amis data 12 / 32
for Life The seasonal mortality assumption will be stated in terms of the density function f b S(x) K(x)=k (s) where b represents the individual s birth month. If S(x) and K(x) are independent then, the same seasonality pattern is expected throughout the different ages and f b S(x) K(x)=k (s) = f b S(x) (s) 13 / 32
Commonly used assumptions for Life f b S(x) K(x)=k (s) = f S(x) K(x)=k(s), b Uniform distribution of deaths: f S(x) K(x)=k (s) = 1. Constant force of mortality: f S(x) K(x)=k (s) = 1 q x+k µe µs where µ = ln(1 q x+k ). Balducci assumption: f S(x) K(x)=k (s) = 1 q x+k (1 (1 s)q x+k ) 2. S(x) and K(x) are independent for the uniform assumption, but they aren t for the other two. 14 / 32
Monthly rates of mortality for Life Density Density 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.90 0.95 1.00 1.05 1.10 1.15 1.20 Uniform 0.0 0.2 0.4 0.6 0.8 1.0 Year begins in January Uniform 0.0 0.2 0.4 0.6 0.8 1.0 Year begins in July Density Density 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.90 0.95 1.00 1.05 1.10 1.15 1.20 Constant force 0.0 0.2 0.4 0.6 0.8 1.0 Year begins in January Constant force 0.0 0.2 0.4 0.6 0.8 1.0 Year begins in July Density Density 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.90 0.95 1.00 1.05 1.10 1.15 1.20 Balducci 0.0 0.2 0.4 0.6 0.8 1.0 Year begins in January Balducci 0.0 0.2 0.4 0.6 0.8 1.0 Year begins in July Figure: Mexican mortality data: density function, varying the month in which the policy year begins, calculated using the uniform, constant force and Balducci mortality assumptions. The observed (empirical) rates of mortality are also included. Typical assumptions do not reflect the variability in death rates during the policy year seasonal mortality assumption. 15 / 32
NNTS distribution (Durán, 2004 and 2007) for Life Fejér (1915) expressed a univariate nonnegative trigonometric sum (a Fourier series) for a variable θ as M 2 c k e ikθ. (1) k=0 The parameters (a k, b k ) for k = 1,..., M of T(θ), the trigonometric sum of order M, as stated below: T(θ) = a 0 + M (a k cos(kθ) + b k sin(kθ)) k=1 are expressed in terms of the complex parameters in equation (1), c k, for k = 0,..., M. In particular, a k ib k = 2 n k ν=0 c ν+kc ν. The additional constraint of n k=0 c k 2 = 1 2π = a 0 is imposed to ensure that the trigonometric sum integrates to 1. 16 / 32
for Life The probability density function for a circular (angular) random variable is then defined as follows (Durán (2004)): f(θ; c, M) = 1 2π + 1 π M (a k cos(kθ) + b k sin(kθ)). k=1 Note that the trigonometric sum can also be expressed as a double sum: M M c k c m e i(k m)θ. k=0 m=0 This form is particularly useful in actuarial calculations involving forces of interest. 17 / 32
Fitting the distribution for Life To estimate the seasonal mortality effects from the monthly aggregated incidence data from INEGI, The likelihood function where L(c, M N) = 12 r=1 (F(2πu r ; c, M) F(2πu r 1 ; c, M)) Nr u r is the fraction of the year (u 0 = 0 and u 12 = 1), N k be the total number of death during the k-th month (k = 1,...,12), c = (c 0,...,c M ) is the vector of c parameters, M is the integer order of the trigonometric sum N = (N 1,..., N 12 ) is the vector of observations, and F(θ; c, M) is the accumulated NNTS circular distribution function. 18 / 32
for Life The fitting of the distribution was accomplished using CircNNTSR, an R software package. : S(x) and K(x) are independent. NNTS density NNTS density NNTS density Jan Apr Jul Oct May Aug Nov Feb Sep Dec Mar Jun Figure: Mexican mortality data: the fitted NNTS circular density function for the mortality occurrences in the population of individuals that are 20 or more years of age. 19 / 32
Short Term for Life Short term life insurance is usually issued at the time of a change in employment. Changing jobs usually means a gap in group insurance coverage that typically lasts 1-6 months. Short term life insurance is the same as longer life insurance policies except that coverage is issued entirely online and the application process is completed on the same day. The easy underwriting process allows coverage to be issued immediately and coverage can be cancelled whenever a replacement plan is provided by an employer. Another reasons for short term insurance: travel, dangerous expedition, short term debt,... 20 / 32
Monthly Term Life for Life Consider a 1-year term life insurance policy that pays a monetary unit at the moment of death to an insured who is x years old at the moment of issue and has a birth month of b. Let δ be the annual force of interest. Ā 1 x: 1 b = E [ e δt(x)] = 1 0 e δt f b T(x) (t)dt where ft(x) b is the probability density function of the future-lifetime of a person with an age of x and birth s month b. 21 / 32
for Life = = 11 Ā 1 1 b = x: 11 h=0 11 h=0 h=0 h+1 12 h 12 e δt f b T(x) (t)dt e δ h 12 P(K(x) = 0) 1 12 e δ h 12 P(K(x) = 0) 1 12 0 0 e δs fs(x) K(x)=0 b ( h + s K(x) = 0)ds 12 e δs f b S(x) ( h 12 + s)ds. 22 / 32
for Life On the other hand, Therefore, Ā 1 x+ h 12 : 1 11 Ā 1 x: 1 b = e δ h 12 h px b Ā1. 12 x+ h 12 : 1 12 b 12 b = q x h=0 h px b 12 1 12 0 e δs f b S(x) ( ) h 12 + s ds. 23 / 32
NNTS Distribution for Life f b S(x) (s) = 2πf Θ(2π(s + b)) = 1 + 2π M M k=1 m=1,m k c k c m e i(k m)2π(s+b) and the integral that we wish to examine is equal to and 1 12 0 + 2π e δs f b S(x) M M k=1 m=1,m k ( ) h 12 + s ds = 1 δ (1 e δ 12 ) c k c m e i(k m)2π( h 12 +b) [ 1 e (i(k m)2π δ)( 1 δ i(k m)2π h p b x = 1 qx h M 12 12 + 2π M c k c me i(k m)2πb 1 h ei(k m)2π 12 i(k m)2π k=1 m=1,m k 12 ) ] 24 / 32
for Life This monthly premium result can be compared with the corresponding monthly premiums obtained from the other assumptions for fractional ages: Uniform Constant force Ā 1 x+ h 12 : 1 12 b = Ā 1 x+ h 12 : 1 12 b = µ ( ) q x 1 ( 1 e δ 12 ) h 12 qx δ ( ) 1 e (δ+µ) 1 12 δ + µ 25 / 32
for Life Balducci assumption Ā 1 b x+ 12 h : 1 = 12 1 (1 h)q x (1 q x)(12 11q x)q x ( q x(12 11q x) 12q x(1 q x)e δ 12 ( 1 qx ( ( ) ( )) +δe δ qx )(11q ) 2x δ(12 11qx) δ(1 qx) 23qx + 12) Ei Ei 12q x q x where Ei is the exponential integral defined by Ei(z) = 1 e tz dt for all real values of z. t 26 / 32
Results for Life The NNTS circular density fitted to the data for the population that is 20 or more years of age. Table shows the monthly premiums, for different values of policy month s initial date (h = 0,..., 11) and month s birth (b = 0,...,11). The annual force of interest equals 4%, an insured sum of 100,000 monetary units and q x = 0.01. 27 / 32
for Life Monthly premiums ÆÆÌË ¼ ½ º ¾ º ¼º º º º¼ ¼º ¾ ¾º¾½ º ¾º º º ¾ º ¼º º¾ º ¾º ¼º ¾º½ º ¼ ¾º º º º ¼ ½ ¾ ½¼ ½½ ¾ ¼º ½ º½ º ½ ¾º ¼ ¼º ¾º¼ º ¾º ½ º¾ º º ¼ º º½¾ º ¾º ¼º ¾ ¾º¼½ º ¾º º½ º¾ º ½ º ½º½ º ¾º ¼º ½º º¾ ¾º º½¼ º½ º º¾ ½º¼ º ¾º ½ ¼º ½ ½º º¾ ¾º ¼ º¼ º½¼ º º¾½ ¼º º º ¼ ¼º ½º º½ ¾º º º¼ º¾ º½ ¼º ¾ º º º ¼ ½º º½ ¾º º ½ º º¾¾ º¼ ¼º º ½ º¾ º ½º¾¾ ½¼ º¼ ¾º ¾ º º ¼ º½ º¼½ ¼º º º¾ º¾ ½º½ ¾º ½½ ¾º¾ º º ¾ º¼ º ¼º ¾ º º½ º¾ ½º½½ ¾º ¼ º º ¼ º º¼¼ º ¼º º ½ º½½ º½ ½º¼ ¾º º ¾ º¼¾ ÓÒ Ø ÓÖ º º º º º º º º º º º º ÍÒ ÓÖÑ º º ½ º ¼º º º¼ º½½ ¼º ¾º¾ º ¾º º ¾ Ð Ù º½ º¾ º º ¼ º º º ½ º º º ¾ º º º º¼ º½¼ º½ º¾ º ½ º º º ¾ º ¼ º º 28 / 32
S(x) and K(x) not independent for Life The same analysis can be accomplished if we do not assume independence among S(x) and K(x). We illustrate the fitted NNTS distribution for different age groups and the corresponding monthly premiums for an insured individual in each age group who was born in January and purchases the monthly policy at different months of the year. 29 / 32
for Life NNTS density ages 20 24 Jan Mar May Jul Sep Nov NNTS density ages 40 44 Jan Mar May Jul Sep Nov Tabla20[1, ] Tabla40[1, ] 0.81 0.84 24.5 26.0 Monthly premiums ages 20 24 2 4 6 8 10 12 Index Monthly premiums ages 40 44 2 4 6 8 10 12 Index NNTS density ages 60 64 Monthly premiums ages 60 64 Jan Mar May Jul Sep Nov 120 135 2 4 6 8 10 12 Figure: Mexican mortality data: the fitted NNTS circular density function for the mortality occurrence of the population segments between 20-24, 40-44 and 60-64 years of age and the corresponding monthly premiums. Tabla60[1, ] Index 30 / 32
Conclusions for Life Main conclusions: Many actuarial variables need to be modeled by circular random variables. The NNTS distribution is a useful model for circular random variables. 31 / 32
Bibliography for Life Fejér, L. (1915) Über trigonometrische Polynome. J. Reine Angew. Math. 146, pp. 53-82. Durán, J.J. (2004) Based on Nonnegative Trigonometric Sums. Biometrics, 60, pp. 499-503. Durán, J.J. (2007) Models for -Linear and - Data Constructed from Based on Nonnegative Trigonometric Sums. Biometrics, 63, pp.579-585. Durán, J.J. and M.M. (2010) Maximum Likelihood Estimation of Nonnegative Trigonometric Sum Models Using a Newton-like Algorithm on Manifolds. Electronic Journal of Statistics, 4, pp. 1402-1410. Durán, J.J. and M.M. (2012) CircNNTSR; An R Package for the Statistical Analysis of Data Based on Nonnegative Trigonometric Sums 32 / 32