**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 4 / Exam C****, authored by Mr. Stolyarov. This is Section 25 of the Study Guide. See an index of all sections by following the link in this paragraph.**

An **ordinary deductible**, d, modifies the amount X that an insurance company needs to pay for a loss by transforming X into X – d. Deductibles can be considered *per payment* (the corresponding payment variable is YP) or *per loss* (the corresponding payment variable is YL).

The per-payment payment variable is as follows:

YP = undefined when X ≤ d. (This is so because the insurer does not need to make a payment.)

YP = X- d when X > d.

This corresponds to an excess loss random variable.

The per-loss payment variable is as follows:

YL = 0 when X ≤ d.

YL = X – d when X > d.

This corresponds to a left-censored and shifted random variable.

Here are the properties of these two random variables:

fY^P(y) = fX(y+d)/SX(d), y > 0

SY^P(y) = SX(y+d)/SX(d)

FY^P(y) = (FX(y+d) – FX(d))/(1 – FX(d))

hY^P(y) = fX(y+d)/SX(y+d) = hX(y+d)

fY^L(y) = fX(y+d), y > 0. (Note: The probability fY^L(0) is a discrete probability.)

SY^L(y) = SX(y+d), y ≥ 0

FY^L(y) = FX(y+d), y ≥ 0

**Source:** *Loss Models: From Data to Decisions,* (Third Edition), 2008, by Klugman, S.A., Panjer, H.H. and Willmot, G.E., Chapter 8, pp. 179-180.

**Original Problems and Solutions from The Actuary’s Free Study Guide**

**Problem S4C25-1.** Losses from falling cows follow an exponential distribution with mean θ = 340. Falling Cow Insurance Company requires policyholders to have an ordinary deductible of 200. Find fY^P(y), the pdf of the per-payment payment variable under this policy.

**Solution S4C25-1.** We use the formula fY^P(y) = fX(y+d)/SX(d), y > 0.

For an exponential distribution, fX(x) = e-x/θ/θ, and SX(x) = e-x/θ. Here, d = 200.

Thus, fY^P(y) = e-(y+200)/340/340/(e-200/340) = **fY^P(y) = e-y/340/340,** **y > 0**.

Why did we get as our answer the pdf of the given exponential distribution, except as applied to YP instead of X? This is so because the exponential distribution has the memoryless property, and so any random variable that is undefined when X ≤ d and is equal to X- d when X > d will have the same pdf, irrespective of the value of d.

**Problem S4C25-2.** Losses from falling cows follow an exponential distribution with mean θ = 340. Falling Cow Insurance Company requires policyholders to have an ordinary deductible of 200. Find fY^L(y), the pdf of the per-loss payment variable under this policy. Do not forget to consider what happens when y = 0!

**Solution S4C25-2.** We use the formula fY^L(y) = fX(y+d), y > 0.

For an exponential distribution, fX(x) = e-x/θ/θ. Here, d = 200.

Thus, fY^L(y) = e-(y+200)/340/340, y > 0.

We also want to find fY^L(0), which is the probability that the loss will be less than or equal to 200, which is the same as FX(200) = 1 – e-200/340 = 0.444693627.

Our answer is as follows:

**fY^****L****(y) =** **0.444693627 when y = 0;**

**fY^L(y) =** **e-(y+200)/340/340** **when y > 0**.

**Problem S4C25-3.** Losses from house-smashing snakes (HSSs) follow a Pareto distribution with α = 3 and θ = 1000. HSS Mutual Insurance Company requires policyholders to have an ordinary deductible of 500. Find SY^P(y), the survival function of the per-payment payment variable under this policy.

**Relevant properties for Pareto distributions:** S(x) **=** θα/(x + θ)α.

**Solution S4C25-3.** We use the formula SY^P(y) = SX(y+d)/SX(d), for d = 500.

Thus, SX(y+d) = 10003/(y + 500 + 1000)3 = 10003/(y + 1500)3 and

SX(d) = 10003/(500 + 1000)3 = 8/27.

Hence, SY^P(y) = 27*10003/(8(y + 1500)3) = **SY^P(y) =** **3.375*109/(y + 1500)3****=** ** 30003/(2y + 3000)3**.

**Problem S4C25-4.** Losses from house-smashing snakes (HSSs) follow a Pareto distribution with α = 3 and θ = 1000. HSS Mutual Insurance Company requires policyholders to have an ordinary deductible of 500. Find FY^L(y), the cumulative distribution function of the per-loss payment variable under this policy.

**Relevant properties for Pareto distributions:** S(x) **=** θα/(x + θ)α.

**Solution S4C25-4.** We use the formula FY^L(y) = FX(y+d), y ≥ 0. Here, d = 500 and

FX(x) = 1 – SX(x) = 1 – θα/(x + θ)α = 1 – 10003/(x + 1000)3.

Thus, FX(y+d) = 1 – 10003/(y + 1500)3, so

**FY^L(y) =** **1 – 10003/(y + 1500)3, y ≥ 0**.

**Problem S4C25-5.** Global warming/cooling/no-temperature-change is a serious environmental problem. Annual dollar losses from this peril follow a binomial distribution with m = 4 and q = 0.23. The Untied Nations has decided to implement an insurance policy against this peril, with an ordinary deductible of 1. Find FY^P(y), the cdf of the per-payment payment variable under this policy.

**Solution S4C25-5.** We use the formula FY^P(y) = (FX(y+d) – FX(d))/(1 – FX(d)).

FX(d) = FX(1) = fX(0) + fX(1), since the binomial distribution is discrete.

fX(0) = (1-q)m = (1-0.23)4 = 0.35153041.

fX(1) = C(m,1)*q*(1-q)m-1 =m*q*(1-q)m-1 = 4*0.23*0.773 = 0.42001036.

Thus, FX(1) = fX(0) + fX(1) = 0.35153041 + 0.42001036 = 0.77154077.

We now need to find FX(y+d). We note that the only possible losses occur in whole dollar amounts, and the value of FY^P(y) will depend on the amount of loss that has occurred.

We already know that FX(0+d) = FX(0+1) = FX(1) = 0.77154077.

fX(2) = C(4,2)*0.232*0.772 = 0.18818646.

Thus, FX(1+d) = FX(2) = 0.77154077 + 0.18818646 = 0.95972723.

fX(3) = C(4,3)*0.233*0.77= 0.03747436.

Thus, FX(2+d) = FX(3) = 0.95972723 + 0.03747436 = 0.99720159.

Since no loss can be greater than 4, FX(4) = FX(3+d) = 1.

Now we can apply our formula:

FY^P(0) = (FX(0+d) – FX(d))/(1 – FX(d)) = FY^P(0) = 0.

FY^P(1) = (FX(1+d) – FX(d))/(1 – FX(d)) = (0.95972723 – 0.77154077)/(1 – 0.77154077) = FY^P(1) = 0.823720101.

FY^P(2) = (FX(2+d) – FX(d))/(1 – FX(d)) = (0.99720159 – 0.77154077)/(1 – 0.77154077) = FY^P(2) = 0.9877509436.

FY^P(3) = (FX(3+d) – FX(d))/(1 – FX(d)) = (1 – 0.77154077)/(1 – 0.77154077) = FY^P(3) = 1.

Thus, we have the following answer:

**FY^P(0) = 0;**

**FY^P(1) =** **0.823720101;**

**FY^P(2) =** **0.9877509436;**

**FY^P(3) = 1**.

**See other sections of** **The Actuary’s Free Study Guide for Exam 4 / Exam C****.**