SISA Research Paper 
Title: Calculating the discounted YPLL  annotated. The basic equation for calculating the YPLL after removal of a particular cause of death k for a particular age group i equals (1): 1) YPLL_{k.i}=((number of death at a given age)*(weight for that age))=D_{k.i}w_{i} Thus, the not discounted wi for any year of life lost equals 1. If the end of productive life ends at the age of 65, then if a person dies at his 60^{th} birthday: w60+w61+w62+w63+w64=1+1+1+1+1=5 years of productive life lost D60=2, if 2 persons die at their 60^{th} birthday 2w60+2w61+2w62+2w63+2w64=2*1+2*1+2*1+2*1+2*1=10 years of productive life lost It is customary to summarize the individual age YPLL(i) , to represent loss of life in a range of ages between the start of the index i and the endpoint x: 2) YPLL_{k.i..x}= å _{(i..x) }YPLL_{k.i} = å _{(i..x) }D_{k.i}w_{i} Thus, if two persons die at their 60^{th} birthday, and three persons die at their 62^{nd} birthday, thus: D60=2 and D62=3: 2w60.60+2w60.61+2w60.62+2w60.63+2w60.64+3w62.62+3w62.63+3w62.64=2*1+2*1+2*1+2*1+2*1+3*1+3*1+3*1=10+9=19
years of productive life lost Often the age range of productivity, 15 to 65 (or 70) years of age is used For the following discussion it is considered that: 3) weight for that age W_{i}=Sum(weight for each year of life remaining) =å w_{j} whereby j is in the range i..x We
can rewrite 2w60+2w61+2w62+2w63+2w64 into
2*(w60+w61+w62+w63+w64)=2*(1+1+1+1+1)=10. and we can rewrite
2w60.60+2w60.61+2w60.62+2w60.63+2w60.64+3w62.62+3w62.63+3w62.64 into 2*(w60.60+w60.61+w60.62+w60.63+w60.64)+3*
(w62.62+w62.63+w62.64)= 2*(1+1+1+1+1)+ 3*(1+1+1)= 19 years of
productive life lost. For example, in the case of calculating the YPLL for the productive age groups 15 to 64, w(j) equals zero for years of potential life below the age of 15, zero for years of potential life above the age 64, and w(j) is one for the age groups 15 up to and including age 64. W(i) in these age groups equals å w_{j} whereby j=i..64; i>14. In the case of discounting w(j) will be a proportion which will be smaller for larger differences between i and j: representing that more time has elapsed between observed death and a potential year of life lost and the decreasing value of future life. In the case of mortality adjusting w(j) will be a proportion representing the probability to die from causes of death which "compete" with the cause of death k in a certain age category. Discounting the YPLL. Considering that the weight for the first year of potential life saved equals 1, the value of the weight for the second year at a discount rate of r equals approximately 1/(1+r) and the value of the jth year saved equals approximately (5): For discounting we need to lower the individual weights, the 1ns in 2*(1+1+1+1+1)+ 3*(1+1+1) 4) w_{j}=(1+r)^{(j1)} Please read the note on this formula If the discount rate is 1.5% (=0.015) the multiplication factor is 1/(1+r)=1/(1+0.015)=0.985221674 Following economists’ practice. the first year is not discounted, the second year is discounted at 1*0.985221674=0.985221674. the third year is discounted at 1*0.985221674*0.985221674=0.985221674 ^{2}= 0.970662; the fourth year at 1*0.985221674*0.985221674*0.985221674=0.985221674 ^{3}=0.956317; and the fifth year at 1*0.985221674*0.985221674*0.985221674*0.985221674=0.985221674 ^{4}=0.942184 and so forth Than, for a person who dies at his 60^{th} birthday the discounted (at 1.5%) number of years lost equals one not discounted year, followed by four increasingly discounted years:
W_{60}=w60+w61+w62+w63+w64=1+0.985221674+0.970662+0.956317+0.942184 =4.854385 years of life lost
And for two deaths D=2 it is twice that number: 2*4.854385=9.70877 and j is in the range 1..xi1 (whereby i is the age at which the deaths at interest took place and j is the jth year of potential life saved for a death at age i) For example, the weight w_{10} for the 10th year saved at a discount rate of 1.5% (r=0.015) equals 0.875. Because 0.875=1*0.985221674*0.985221674*0.985221674*0.985221674*0.985221674*0.985221674*0.985221674*0.985221674*0.985221674=0.985221674 ^{9}. (And it is ^{9 }and not ^{10} because formulae 4) above says ^{(j1)} and that is because we don’t discount the first year lost). Considering that death does not take place at a single moment in time at the beginning of the year, in each age period there is a fraction a_{i} of life lived by the death in that period. In life table analysis this fraction is often set at 0.5, considering that death takes place more or less equally spaced throughout an age period. For i=0 this fraction should be set at a_{0}=0.1, considering that in the first year of life deaths occur disproportionally often in the beginning of the year (6). It is unlikely that people who die at age 60 all die at their 60^{th} birthday. The lifetable parameter a gives the mean in the age group studied at which people die and the i in a_{i }is the age group which is studied. Often a is proposed to be 0.5, which means that on average people die six months into the age year of death. Because we do not discount the first year we correct for that in the last year of the series. The calculation of the weight W_{i} for the discounted YPLL_{i} for age i up to x would be: 5) W(i)= å _{(j=i...x) }w_{j }=1+å _{(j=2...xi1) }(1+r)^{(j1)} + a_{i} * (1+r)^{(xi1) } In formulae five there is first a not discounted year of benefit, followed by a number of years of benefits of decreasing importance and, lastly, a partial discounted year of relatively low benefit. Thus, undiscounted the expected number of life years lost for someone who dies at age 60 equals (if a_{60}=0.5): W_{60}=w60+w61+w62+w63+ a_{i} * w64=1+1+1+1+0.5*1=4.5 years of life lost Discounted at 1.5% this is: W_{60}=w60+w61+w62+w63+ a_{i} * w64=1+0.985221674+0.970662+0.956317+0.5*0.942184 =4.383293 years of life lost
And if we combine two deaths at age 60 with three deaths at age 62 this becomes:
2w60.60+2w60.61+2w60.62+2w60.63+2 * a_{i} * w60.64+3w62.62+3w62.63+3 * a_{i} * w62.64=2*1+2*0.985221674+2*0.970662+2*0.956317+2*0.5*0.942184+3*1+3*0.985221674+3*0.970662 =2*4.383293+3*2.470553=16.17824 years of life lost In the case of abridged tables, if ages are grouped in, for example, 5 or 10 year age categories, the weight of an age category is the sum of years contained in the individual age categories in that group. For an abridged table category c with width n and considering the age range i..i+n 6) w_{c} =n _{(j=i...} _{i+}_{n}_{ ) }w_{j} Abridged tables are the case that age groups are summarized in a more limited number of age categories. The lost year weight of an age category is the sum of weights of each of the age years contained in the category, according to the formulae above. But in fact the situation is rather complicated and we will work through an example. The abridged category method is difficult to program into a spreadsheet, the SISA lifetable computer program however will do it fine.
If an age category runs from i to i+n then the start age of the series equals truncate((i+i+n)*a_{(i..} _{i+}_{n)}), whereby a_{(i..} _{i+}_{n)} is the lifetable weight which tells you were the mortality in an age category is centered. Mostly this will be 0.5. The truncated start value is the first whole number integer below the value found. So, if a person dies in the five years age category 60 up to and including 64, the start age for the series equals truncate((60+60+5)*0.5)=62. To consider the partial year lived in the remainder left after the truncate procedure we need to calculate a new a_{i }which is defined as a_{i..last }=remainder((i+i+n)*a_{(i..} _{i+}_{n)})= remainder ((60+60+5)*0.5)=0.5.
For a person who dies between 60 and 65 we can develop the series w62.62+w62.63+ a_{i..last} * w62.64 as above.
Table one shows the abridged table method using U.S. cancer data for the year 2006. The table cells show the number of life years lost by a single individual for each of the age groups were life is lost, discounted at 1.5% . The third lowest green row gives for each of the ages in which people died the total number of years lost for a single individual. Thus, someone who died between the age of 25 and 30, loses 28.95 life years up to the age of 65. In the U.S. in 2006 in total 1390 individuals aged 25 up to 30 die, giving a total number of life years lost for this age group of 28.95x1390=40239. Same for all the other age groups. Lastly the age group data is summarized, giving a total number of life years lost between the ages of 25 up to 65 of 1504017.
The spreadsheet used to make table one can be found here
Table
one. Number death and YPLL from all malignant cancers in the United
States, 2006.

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SISA Research Paper  