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Concept of Survivor Curve and Probable Life Curve

Concept of Survivor Curve and Probable Life Curve. From Chapter 9, Page 207 of Winfrey’s Text Book.

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Concept of Survivor Curve and Probable Life Curve

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  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 Average Service Life, M = = = 18.89 100 f

  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 Survivor Curve 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) = 127.5  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+0=20.0 At t=2, P2= 19.0+2=21.0 At t=5, P5= 17.5+5=22.5 At t=10, P10= 15.0+10=25.0 At t=15, P15= 12.5+15=27.5 At t=20, P20= 10.0+20=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= 7.5+25=32.5 At t=30, P30= 5.0+30=35.0 At t=35, P35= 2.5+35=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 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 • Trend Line • 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 = 0.0413t2 – 4.4574t + 107.74

  37. t=40  ydt  Area at age 0 yrs = t=0 t=40 =  [0.0413t2 – 4.4574t + 107.74]dt t=0 t=40 = [0.0413t3/3 – 4.4574t2/2 + 107.74t] t=0 = [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] - [0.0413(0)3/3 – 4.4574(0)2/2 + 107.74(0)] = 1624.75 %-Years

  38. t=40  ydt  Area at age 2 yrs = t=2 t=40 =  [0.0413t2 – 4.4574t + 107.74]dt t=2 t=40 = [0.0413t3/3 – 4.4574t2/2 + 107.74t] t=2 = [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] - [0.0413(2)3/3 – 4.4574(2)2/2 + 107.74(2)] = 1624.75 – 197.7605= 1426.989 %-Years

  39. t=40  ydt  Area at age 5 yrs = t=5 t=40 =  [0.0413t2 – 4.4574t + 107.74]dt t=5 t=40 = [0.0413t3/3 – 4.4574t2/2 + 107.74t] t=5 = [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] - [0.0413(5)3/3 – 4.4574(5)2/2 + 107.74(5)] = 1624.75- 484.70 = 1140.05 %-Years

  40. t=40  ydt  Area at age 10 yrs = t=10 t=40 =  [0.0413t2 – 4.4574t + 107.74]dt t=10 t=40 = [0.0413t3/3 – 4.4574t2/2 + 107.74t] t=10 = [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] - [0.0413(10)3/3 – 4.4574(10)2/2 + 107.74(10)] = 1624.75- 868.297 = 756.453 %-Years

  41. t=40  ydt  Area at age 15 yrs = t=15 t=40 =  [0.0413t2 – 4.4574t + 107.74]dt t=15 t=40 = [0.0413t3/3 – 4.4574t2/2 + 107.74t] t=15 = [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] - [0.0413(15)3/3 – 4.4574(15)2/2 + 107.74(15)] = 1624.75- 1161.05 = 463.645%-Years

  42. t=40  ydt  Area at age 20 yrs = t=20 t=40 =  [0.0413t2 – 4.4574t + 107.74]dt t=20 t=40 = [0.0413t3/3 – 4.4574t2/2 + 107.74t] t=20 = [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] - [0.0413(20)3/3 – 4.4574(20)2/2 + 107.74(20)] = 1624.75- 1373.453 = 251.297%-Years

  43. t=40  ydt  Area at age 25 yrs = t=25 t=40 =  [0.0413t2 – 4.4574t + 107.74]dt t=25 t=40 = [0.0413t3/3 – 4.4574t2/2 + 107.74t] t=25 = [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] - [0.0413(25)3/3 – 4.4574(25)2/2 + 107.74(25)] = 1624.75- 1515.667 = 109.083%-Years

  44. t=40  ydt  Area at age 30 yrs = t=30 t=40 =  [0.0413t2 – 4.4574t + 107.74]dt t=30 t=40 = [0.0413t3/3 – 4.4574t2/2 + 107.74t] t=30 = [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] - [0.0413(30)3/3 – 4.4574(30)2/2 + 107.74(30)] = 1624.75- 1598.07 = 26.68 %-Years

  45. t=40  ydt  Area at age 35 yrs = t=35 t=40 =  [0.0413t2 – 4.4574t + 107.74]dt t=35 t=40 = [0.0413t3/3 – 4.4574t2/2 + 107.74t] t=35 = [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] - [0.0413(35)3/3 – 4.4574(35)2/2 + 107.74(35)] = 1624.75- 1630.988= - 6.238%-Years

  46. t=40  ydt  Area at age 40 yrs = t=40 t=40 =  [0.0413t2 – 4.4574t + 107.74]dt t=40 t=40 = [0.0413t3/3 – 4.4574t2/2 + 107.74t] t=40 = [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] - [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)] = 1624.75- 1624.75 = 0 %-Years

  47. Summary of Calculations

  48. Survivor Curve and Probable Life Curve

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