1. Concept of Survivor Curve and Probable Life Curve

2. From Chapter 9, Page 207 of Winfrey’s Text Book
“The survivor curve is a curve which shows the number of units of property that survive in service at given ages. The area under the curve is a direct measure of the average service life of the property units. The probable life of the surviving units at any age can also be calculated from the remaining area by diving the remaining area by the amount surviving at that age.”

3. As per Winfrey’s Text Book
Probable Life = (Shaded area)/(% Survived at that age)

4. From Winfrey’s Reference 9-13*
“Service remaining at any age is equal to the area under the curve to the right of the ordinate erected at that age.”
*Statistical Analyses of Industrial Property Retirements,
by Robley Winfrey, 1967.

5. As per Winfrey’s Reference 9-13
Expectancy at 5 yrs = (Shaded area at 5 yrs)/(% Survived at 5 yrs)

6. From Winfrey’s Reference 9-13*
“The expectancy at any age is a function of the remaining service which is obtained by summing the areas for each age interval, starting at the age of the last survival (0% surviving) and working to the left to the age in question.”
*Statistical Analyses of Industrial Property Retirements,
by Robley Winfrey, 1967.

7. From Winfrey’s Reference 9-13*
“The expectancy of life at any given age is then obtained by dividing the remaining service at that age by the percent surviving at the same age.”
“The probable average life of the survivors at any given age is equal to the sum of the expectancy and the age for which the expectancy is computed.”
*Statistical Analyses of Industrial Property Retirements,
by Robley Winfrey, 1967.

8. Example Problem

9. Frequency Curve
Choose X-axis for Age in Years (N) and Y-axis for Number of Units Retired (f)
Draw the Frequency Distribution Curve

10. Frequency Distribution Curve
Number of Units Retired (f)
Age, N (Years)

11. Average Service Life (N*f) 1889 f 100
Average Service Life, M = = = 18.89 f
100

12. Survivor Curve
Survivor Curve is drawn with X-axis for Age in Years and Y-axis for Percent Surviving

13. Survivor Curve

14. Total Remaining Service
Total Remaining Service in %-Years is calculated at any age by considering a triangle under the Survivor Curve to the right of the ordinate at that age
The area of a triangle is given by:
Area = ½*(Base)*(Altitude)
Total Remaining Service
= ½*(Number of Years Remaining)*(% Surviving)

15. Survivor Curve and Average Life Line
Average Service Life=18.89

16. Calculation of Total Remaining Service (%-Years)
Total Remaining Service at any time, t
At=1/2*(Number of Years Remaining)*(% Surviving)
At t=0, A0=1/2*(40-0)*(100) = 2000
At t=2, A2=1/2*(40-2)*(98) = 1862
At t=5, A5=1/2*(40-5)*(92) = 1610
At t=10, A10=1/2*(40-10)*(77) = 1155
At t=15, A15=1/2*(40-15)*(52) = 650

17. Calculation of Total Remaining Service (%-Years)
Total Remaining Service at time t,
At=1/2*(Number of Years Remaining)*(% Surviving)
At t=20, A0=1/2*(40-20)*(32) = 320
At t=25, A2=1/2*(40-25)*(17) =  128
At t=30, A5=1/2*(40-30)*(7) = 35
At t=35, A10=1/2*(40-35)*(2) = 5
At t=40, A15=1/2*(40-40)*(0) = 0

18. Average Service Life
Average Service Life can be calculated by dividing the total area (%-Years) below the Survivor Curve (A0) by Total % (100%).
Average Service Life = A0/100 = (2000)/(100) = 20.0 Yrs
Average Service Life as calculated from the Frequency Distribution Curve differs a little (18.89 Yrs), since the area calculation gives approximate results.

19. Expectancy As Per Winfrey
Expectancy at the start of any age is calculated by dividing Total Remaining Service by % Surviving

20. Expectancy at the Start of the Year
Expectancy at time t,
Et= (Total Remaining Service, At)/(% Surviving)
At t=0, E0= (2000)/(100)=20.0
At t=2, E2= (1862)/(98)=19.0
At t=5, E5= (1610)/(92)=17.5
At t=10, E10= (1155)/(77)=15.0
At t=15, E15= (650)/(52)=12.5
At t=20, E20= (320)/(32)=10.0
At t=25, E25= (128)/(17)=7.5

21. Expectancy at the Start of the Year
Expectancy at time t,
Et= (Total Remaining Service, At)/(% Surviving)
At t=30, E30= (35)/(7)=5.0
At t=35, E35= (5)/(2)=2.5
At t=40, E40= (0)/(0)=Undefined. However, considering 40 years as the end of service life after which no life is remaining, we consider E40= 0.

22. Probable Life
Probable Life at the start of any age is the Sum of Expectancy and Number of Years of Life at that age

23. Probable Life at the Start of the Year
Probable Life at time t,
Pt= Expectancy (Et) + Number of Years of Life (N)
At t=0, P0= =20.0
At t=2, P2= =21.0
At t=5, P5= =22.5
At t=10, P10= =25.0
At t=15, P15= =27.5
At t=20, P20= =30.0

24. Probable Life at the Start of the Year
Probable Life at time t,
Pt= Expectancy (Et) + Number of Years of Life (N)
At t=25, P25= =32.5
At t=30, P30= =35.0
At t=35, P35= =37.5
At t=40, P40= 0+40=40.0

25. Probable Life Curve
Probable Life Curve can be drawn by taking X-axis for Age in Years and Y-axis for Probable Life at any time

26. Probable Life Curve

27. Summary of Calculations

28. Survivor and Probable Life Curve
Survivor Curve
Average Service Life=18.89

29. Area Calculation by Integration
Enter the Data in MS Excel Spreadsheet

30. Area Calculation by Integration
Draw a Scatter Plot for % Surviving against Age
using the following Excel option

31. Area Calculation by Integration
Scatter Plot in order to draw the Survivor Curve

32. Area Calculation by Integration
We find the best fit curve by Regression using Excel
Right click on any dot and select Add Trend Line,
which gives the following window
Select Polynomial of Order 2

33. We get the Trend Line as it fits best with the data

34. Right click on Trend Line and select Format
Click on Options and check the boxes as shown

35. We find the 2nd Degree Polynomial Equation
of the Survivor Curve as shown below

36. Area Calculation by Integration
Area at any age can be calculated by integrating
the equation of the Survivor Curve between
two age limits of consideration
t=40
Area at any age t1 =  ydt
t=t1
where y = t2 – t

37. t=40
 ydt
 Area at age 0 yrs =
t=0
t=40
=  [0.0413t2 – t ]dt
t=0
t=40
= [0.0413t3/3 – t2/ t]
t=0
= [0.0413(40)3/3 – (40)2/ (40)] - [0.0413(0)3/3 – (0)2/ (0)]
= %-Years

38. t=40
 ydt
 Area at age 2 yrs =
t=2
t=40
=  [0.0413t2 – t ]dt
t=2
t=40
= [0.0413t3/3 – t2/ t]
t=2
= [0.0413(40)3/3 – (40)2/ (40)] - [0.0413(2)3/3 – (2)2/ (2)]
= – = %-Years

39. t=40
 ydt
 Area at age 5 yrs =
t=5
t=40
=  [0.0413t2 – t ]dt
t=5
t=40
= [0.0413t3/3 – t2/ t]
t=5
= [0.0413(40)3/3 – (40)2/ (40)] - [0.0413(5)3/3 – (5)2/ (5)]
= = %-Years

40. t=40
 ydt
 Area at age 10 yrs =
t=10
t=40
=  [0.0413t2 – t ]dt
t=10
t=40
= [0.0413t3/3 – t2/ t]
t=10
= [0.0413(40)3/3 – (40)2/ (40)] - [0.0413(10)3/3 – (10)2/ (10)]
= = %-Years

41. t=40
 ydt
 Area at age 15 yrs =
t=15
t=40
=  [0.0413t2 – t ]dt
t=15
t=40
= [0.0413t3/3 – t2/ t]
t=15
= [0.0413(40)3/3 – (40)2/ (40)] - [0.0413(15)3/3 – (15)2/ (15)]
= = %-Years

42. t=40
 Area at age 20 yrs =
 ydt
t=20
t=40
=  [0.0413t2 – t ]dt
t=20
t=40
= [0.0413t3/3 – t2/ t]
t=20
= [0.0413(40)3/3 – (40)2/ (40)] - [0.0413(20)3/3 – (20)2/ (20)]
= = %-Years

43. t=40
 ydt
 Area at age 25 yrs =
t=25
t=40
=  [0.0413t2 – t ]dt
t=25
t=40
= [0.0413t3/3 – t2/ t]
t=25
= [0.0413(40)3/3 – (40)2/ (40)] - [0.0413(25)3/3 – (25)2/ (25)]
= = %-Years

44. t=40
 ydt
 Area at age 30 yrs =
t=30
t=40
=  [0.0413t2 – t ]dt
t=30
t=40
= [0.0413t3/3 – t2/ t]
t=30
= [0.0413(40)3/3 – (40)2/ (40)] - [0.0413(30)3/3 – (30)2/ (30)]
= = %-Years

45. t=40
 ydt
 Area at age 35 yrs =
t=35
t=40
=  [0.0413t2 – t ]dt
t=35
t=40
= [0.0413t3/3 – t2/ t]
t=35
= [0.0413(40)3/3 – (40)2/ (40)] - [0.0413(35)3/3 – (35)2/ (35)]
= = %-Years

46. t=40
 ydt
 Area at age 40 yrs =
t=40
t=40
=  [0.0413t2 – t ]dt
t=40
t=40
= [0.0413t3/3 – t2/ t]
t=40
= [0.0413(40)3/3 – (40)2/ (40)] - [0.0413(40)3/3 – (40)2/ (40)]
= = 0 %-Years

47. Summary of Calculations

48. Survivor Curve and Probable Life Curve