**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 6, authored by Mr. Stolyarov. This is Section 65 of the Study Guide. See an index of all sections by following the link in this paragraph.**

**Formula 65.1:**

LDF from nth report to ultimate for excess layer from c to d = (f(c) – f(d))/(ec,n – ed,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

ec,n = ratio of nth-report losses in excess of c to ultimate ground-up losses

ed,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))/(ec,n – ed,n)

In this case, the desired LDF is (f(25000) – f(50000))/(e25000,39 months – e50000,39 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 ec,n and ed,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, e25000,39 months = (15/17)/2.55

e50000,39 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))/(ec,n – ed,n)

In this case, the desired LDF is (f(25000) – f(100000))/(e25000,39 months – e100000,39 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 ec,n and ed,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, e25000,39 months = (15/17)/2.55

e100000,39 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))/(ec,n – ed,n)

In this case, the desired LDF is (f(100000) – f(200000))/(e100000,39 months – e200000,39 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 ec,n and ed,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, e100000,39 months = (6/17)/3.35

e200000,39 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))/(ec,n – ed,n)

In this case, the desired LDF is (f(100000) – f(500000))/(e100000,39 months – e500000,39 months)

We calculate f(100000) = $1,200,000/$3,400,000 = 6/17.

We calculate f(500000) = $350,000/$3,400,000 = 7/68.

Thus, e100000,39 months = (6/17)/3.35

e500000,39 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))/(ec,n – ed,n)

In this case, the desired LDF is (f(200000) – f(500000))/(e200000,39 months – e500000,39 months)

We calculate f(200000) = $500,000/$3,400,000 = 5/34.

We calculate f(500000) = $350,000/$3,400,000 = 7/68.

Thus, e200000,39 months = (5/34)/4.13

e500000,39 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****.**

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