Am I Halfway? Life Lived = Expected Life

Abstract

We have reached halfway in life when our age equals our remaining life expectancy at that age. This relationship in stable population models between life lived and life left has captured the attention of mathematical demographers since Lotka. Our paper aims to contribute to the halfway-age debate by showing its time trends under mortality models and with current data from high longevity countries. We further contrast the halfway-age results by sex, and between period and cohort perspectives. We find that in 1850 life expectancy at birth was higher than halfway-age by around 10 years (HMD-mean halfway-age of 33.3 and 32.2 against HMD-mean life expectancy of 44.3 and 41.4 for women and men respectively). Nevertheless, declines in mortality at young ages radically changed life expectancy and it is found today at the same level as the double of halfway-age. While the period perspective puts halfway-age for females and males at 41.8 and 39.5 in 2010, for cohorts born in 2010 this might be as high as 10 years more. The stage of midlife has always been considered an important step in the life of human beings. However, there is no agreement on which is the age or age-range that represents the middle phase. Here we have further added the notion that halfway-age is not a static index but a moving age. Current and future progress in reducing mortality at older ages will require redefining our notion of midlife.

“Nel mezzo del cammin di nostra vita. Mi ritrovai per una

selva oscura. Ché la diritta via era smarrita. [In the middle

of the journey of our life, I came to myself in a dark wood,

for the straight way was lost.]”

(Dante 1308–1320, translated by Durling ( 1996 )).

Appendix: The Forecasting “Gap Method”

Here we present a simplistic forecasting procedure of the cohort halfway-age which consists in a linear extrapolation of the observed gap between cohort and period halfway-ages (cohort minus period) for the time in which the two data sets overlap (1850–1920). Similar relations between cohort and period life expectancies have been previously studied (Goldstein and Wachter 2006; Canudas-Romo and Schoen 2005). Given the linearity of the cohort minus period halfway-ages trend, a basic linear extrapolation was the best model fitting the data and which returned much higher halfway-ages than the standard forecasting of the Lee-Carter model. The model links the value of halfway-age for period t, x _p,t with the value of halfway-age for cohorts, x _c,t , as:

$$ {x}_{c,t}={x}_{p,t}+{\kappa}_t, $$

(3.A1)

where κ_t is the line fitted to the cohort minus period halfway-age gap (1850–1920), and used for calculating this gap between (1921–2010) as:

$$ {\kappa}_t=5.64+0.05\;\left(t-1920\right). $$

(3.A2)

Figure 3.7 presents the lags and gaps between period and cohort perspectives of halfway-age for females. As shown in Fig. 3.7, the female cohort halfway-age is higher than the period one for any of the available years. However, we can observe a linearly increasing trend in the cohort-period gap , which we extrapolate, as described in Eqs. (3.A1) and (3.A2), and use to assert the simplistic prediction on the halfway-age for cohorts born in 2010.

Fig. 3.7

Female period and cohort halfway-ages and lines depicting the gaps and lags between these measures, means of HMD countries 1816–2010 (Source: Authors’ calculations, based on HMD)

Figure 3.8 presents the mean female life expectancy at birth and halfway-age for cohorts and periods. Also included in this Figure are the time trends for the two forecasting results, namely from the Lee-Carter model and from the “gap -method.” As shown in Fig. 3.8, the female cohort halfway-age is higher than the period one for all available years, and the linear prediction of this gap is higher than those values obtained by using the Lee-Carter model. Under the gap-model, the mean cohort female halfway-age will be the result of adding to the period halfway-age in 2010 the predicted cohort-period gap (see Eqs. (3.A1) and (3.A2) above): by 2010, this gap amounts to more than 10 years and, as shown in Fig. 3.8, the predicted halfway-age for the 2010 cohort is 52.2 years (41.8 years for the period 2010). Alternatively, the Lee-Carter model estimates levels of mean halfway-age of 44.9 years in 2010, which corresponds to a period-cohort gap of 3 years, similar to the gaps observed at the beginning of the twentieth century.

Fig. 3.8

Female period life expectancy at birth and halfway-age , cohort halfway-age and predicted cohort halfway-age under two scenarios, means of HMD countries 1850–2010 (Source: Authors’ calculations, based on HMD . Notes: The two predicting scenarios are based on the Lee-Carter model and the gap method explained in the Appendix)