Calculation of Loss Development Factors for Excess Loss Layers: Practice Questions and Solutions

Formula 65.1:
LDF from nth report to ultimate for excess layer from c to d = (f(c) - f(d))/(e_c,n - e_d,n)

c = Lower bound of excess layer
d = Upper bound of excess layer
f(c) = ratio of ultimate losses in excess of c to ultimate ground-up losses
f(d) = ratio of ultimate losses in excess of d to ultimate ground-up losses
e_c,n = ratio of nth-report losses in excess of c to ultimate ground-up losses
e_d,n = ratio of nth-report losses in excess of d to ultimate ground-up losses

Source:
Pinto, E.; and Gogol, D.F., "An Analysis of Excess Loss Development," PCAS LXXIV, 1987, pp. 227-255.

Original Problems and Solutions from The Actuary's Free Study Guide

The following information applies to all problems in this section.

Ultimate ground-up losses: $3,400,000
Ultimate losses in excess of $25,000: $3,000,000
Ultimate losses in excess of $50,000: $2,100,000
Ultimate losses in excess of $100,000: $1,200,000
Ultimate losses in excess of $200,000: $500,000
Ultimate losses in excess of $500,000: $350,000

Excess loss development factor from 39 months to ultimate for losses in excess of $25,000: 2.55
Excess loss development factor from 39 months to ultimate for losses in excess of $50,000: 2.89
Excess loss development factor from 39 months to ultimate for losses in excess of $100,000: 3.35
Excess loss development factor from 39 months to ultimate for losses in excess of $200,000: 4.13
Excess loss development factor from 39 months to ultimate for losses in excess of $500,000: 5.56

Problem S6-65-1. What is the 39-month-to-ultimate excess loss development factor for the layer $25,000 in excess of $25,000?

Solution S6-65-1. We use Formula 65.1:
LDF from nth report to ultimate for excess layer from c to d = (f(c) - f(d))/(e_c,n - e_d,n)

In this case, the desired LDF is (f(25000) - f(50000))/(e_{25000,39 months} - e_{50000,39 months})
We calculate f(25000) = $3,000,000/$3,400,000 = 15/17.
We calculate f(50000) = $2,100,000/$3,400,000 = 21/34

We recall that the e_c,n and e_d,n are ratios of nth-report excess losses to ultimate ground-up losses. To use the excess loss development factors (nth report excess loss)/(ultimate excess loss), we need to multiply f(c) and f(d) by (nth report excess loss)/(ultimate excess losses), or, equivalently, divide by the excess loss development factors.

Thus, e_{25000,39 months} = (15/17)/2.55
e_{50000,39 months} = (21/34)/2.89

Thus, the desired LDF is (15/17 - 21/34)/((15/17)/2.55 - (21/34)/2.89) = 2.000769231

Problem S6-65-2. What is the 39-month-to-ultimate excess loss development factor for the layer $75,000 in excess of $25,000?

Solution S6-65-2. We use Formula 65.1:
LDF from nth report to ultimate for excess layer from c to d = (f(c) - f(d))/(e_c,n - e_d,n)

In this case, the desired LDF is (f(25000) - f(100000))/(e_{25000,39 months} - e_{100000,39 months})
We calculate f(25000) = $3,000,000/$3,400,000 = 15/17.
We calculate f(100000) = $1,200,000/$3,400,000 = 6/17.

We recall that the e_c,n and e_d,n are ratios of nth-report excess losses to ultimate ground-up losses. To use the excess loss development factors (nth report excess loss)/(ultimate excess loss), we need to multiply f(c) and f(d) by (nth report excess loss)/(ultimate excess losses), or, equivalently, divide by the excess loss development factors.

Thus, e_{25000,39 months} = (15/17)/2.55
e_{100000,39 months} = (6/17)/3.35

Thus, the desired LDF is (15/17 - 6/17)/((15/17)/2.55 - (6/17)/3.35) = 2.199785408

Problem S6-65-3. What is the 39-month-to-ultimate excess loss development factor for the layer $100,000 in excess of $100,000?

Solution S6-65-3. We use Formula 65.1:
LDF from nth report to ultimate for excess layer from c to d = (f(c) - f(d))/(e_c,n - e_d,n)

In this case, the desired LDF is (f(100000) - f(200000))/(e_{100000,39 months} - e_{200000,39 months})
We calculate f(100000) = $1,200,000/$3,400,000 = 6/17.
We calculate f(200000) = $500,000/$3,400,000 = 5/34.

We recall that the e_c,n and e_d,n are ratios of nth-report excess losses to ultimate ground-up losses. To use the excess loss development factors (nth report excess loss)/(ultimate excess loss), we need to multiply f(c) and f(d) by (nth report excess loss)/(ultimate excess losses), or, equivalently, divide by the excess loss development factors.

Thus, e_{100000,39 months} = (6/17)/3.35
e_{200000,39 months} = (5/34)/4.13

Thus, the desired LDF is (6/17 - 5/34)/((6/17)/3.35 - (5/34)/4.13) = 2.951798232

Problem S6-65-4. What is the 39-month-to-ultimate excess loss development factor for the layer $400,000 in excess of $100,000?

Solution S6-65-4. We use Formula 65.1:
LDF from nth report to ultimate for excess layer from c to d = (f(c) - f(d))/(e_c,n - e_d,n)

In this case, the desired LDF is (f(100000) - f(500000))/(e_{100000,39 months} - e_{500000,39 months})
We calculate f(100000) = $1,200,000/$3,400,000 = 6/17.
We calculate f(500000) = $350,000/$3,400,000 = 7/68.

We recall that the e_c,n and e_d,n are ratios of nth-report excess losses to ultimate ground-up losses. To use the excess loss development factors (nth report excess loss)/(ultimate excess loss), we need to multiply f(c) and f(d) by (nth report excess loss)/(ultimate excess losses), or, equivalently, divide by the excess loss development factors.

Thus, e_{100000,39 months} = (6/17)/3.35
e_{500000,39 months} = (7/68)/5.56

Thus, the desired LDF is (6/17 - 7/68)/((6/17)/3.35 - (7/68)/5.56) = 2.878825348

Problem S6-65-5. What is the 39-month-to-ultimate excess loss development factor for the layer $300,000 in excess of $200,000?

Solution S6-65-5. We use Formula 65.1:
LDF from nth report to ultimate for excess layer from c to d = (f(c) - f(d))/(e_c,n - e_d,n)

In this case, the desired LDF is (f(200000) - f(500000))/(e_{200000,39 months} - e_{500000,39 months})
We calculate f(200000) = $500,000/$3,400,000 = 5/34.
We calculate f(500000) = $350,000/$3,400,000 = 7/68.

We recall that the e_c,n and e_d,n are ratios of nth-report excess losses to ultimate ground-up losses. To use the excess loss development factors (nth report excess loss)/(ultimate excess loss), we need to multiply f(c) and f(d) by (nth report excess loss)/(ultimate excess losses), or, equivalently, divide by the excess loss development factors.

Thus, e_{200000,39 months} = (5/34)/4.13
e_{500000,39 months} = (7/68)/5.56

Thus, the desired LDF is (5/34 - 7/68)/((5/34)/4.13 - (7/68)/5.56) = 2.581056575

See other sections of The Actuary's Free Study Guide for Exam 6.