Chapter 10: Measurement of Risk and Property Insurance Premiums (PDF)

**[Chapter (10)]** **[Measurement of Risk ]** **[and Property Insurance Premiums]** **Average Frequency (Chance) of Loss** **Average Severity (Size) of Loss** ------------...

**[Chapter (10)]** **[Measurement of Risk ]** **[and Property Insurance Premiums]** **Average Frequency (Chance) of Loss** **Average Severity (Size) of Loss** ---------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- It is the mean (average) **probability** of loss occurrence متوسط احتمال الخسارة. It tells us "How often the event insured against can be expected to occur?". It is the mean (average) size of loss متوسط حجم الخسارة when the risk occurs. It tells us "How much loss we can face?". [\$\\mathbf{Average\\ Frequency\\ =}\\frac{\\mathbf{\\text{Number\\ of\\ Losses\\ }}}{\\mathbf{\\text{Number\\ of\\ Exposure\\ Units\\ }}}\$]{.math.inline} [\$\\mathbf{\\text{Average\\ Severity}} = \\frac{\\mathbf{\\text{The}}\\ \\mathbf{\\text{Amount}}\\ \\mathbf{\\text{of}}\\ \\mathbf{\\text{All}}\\ \\mathbf{\\text{Losses}}\\ }{\\mathbf{\\text{Number}}\\ \\mathbf{\\text{of}}\\ \\mathbf{\\text{Losses}}\\ }\$]{.math.inline} **2- Calculating Pure and Gross Premiums in Property Insurance** +-----------------------------------+-----------------------------------+ | **Pure** **Premiums** القسط | **Gross Premiums** القسط الاجمالي | | الصافي | | +===================================+===================================+ | - Pure premium is the net | - Gross premium (Office | | premium (disregarding the | premium) is the premium | | expenses) which each insured | considering the loading. | | must pay. | | | | - **[Loading]** is | | - pure premium represents the | the amount needed to pay all | | cost of loss only without any | expenses (including | | loading. | commissions, managerial | | | expenses, and allowances for | | | contingencies...etc.) and | | | profit for the insurance | | | company. | +-----------------------------------+-----------------------------------+ | \ | \ | | [**Pure** **Premium** **=** **Ave | [**Gross** **Premium** **=** **Pu | | rage** **Frequency** **x** **Aver | re** **Premium** **+** **(Gross** | | age** **Severity**]{.math |  **Premium** **x** **Loading** ** | |.display}\ | Ratio)**]{.math | | |.display}\ | | [\$\\mathbf{Pure\\ Premium | | | =}\\frac{\\mathbf{\\text{Number\\ | \ | | of\\ | [**Pure** **Premium** **=** **Gro | | Losses}}}{\\mathbf{\\text{Number\ | ss** **Premium** **(** **1** **−* | | \ | * **Loading** **Ratio)**]{.math | | of\\ Exposure\\ |.display}\ | | Units}}}\\mathbf{\\times \\ | | | }\\frac{\\mathbf{\\text{Amount\\ | \ | | of\\ All\\ Losses\\ | [\$\$\\mathbf{Gross\\ \\ Premium | | }}}{\\mathbf{\\text{Number\\ of\\ | = \\ | | Losses\\ }}}\$]{.math.inline} | }\\frac{\\mathbf{\\text{Pure\\ | | | Premium\\ }}}{\\mathbf{1 - | | \ | Loading\\ Ratio\\ \\ | | [\$\$\\mathbf{Pure\\ Premium = \\ | }}\$\$]{.math.display}\ | | }\\frac{\\mathbf{\\text{Amount\\ | | | of\\ All\\ Losses\\ | | | }}}{\\mathbf{\\text{Number\\ of\\ | | | Exposure\\ Units\\ }}}\$\$]{.math | | |.display}\ | | +-----------------------------------+-----------------------------------+ **3- Equality Principle** It states that: **Insured Payments = Insurer Payments** **(No. of Exposure Units × Pure Premium) = (No. of Losses × Average Size of Loss)** **Example (1):** Assume that based on experience, an insurer is able to predict that 10 of 1,000 automobiles. On the average, destroy or damage by collision each year incurred losses of \$200,000. Each automobile is valued at \$30,000. Calculate the following: 1. The average frequency of loss. 2. Average severity of loss. 3. The pure premium. 4. The gross premium assuming that loading ratio is 25%. 5. Indicate the equality principle. **Solution:** \(1) Calculating the average frequency of loss \ [\$\$Average\\ Frequency = \\frac{\\text{Number\\ of\\ Losses\\ }}{\\text{Number\\ of\\ Exposure\\ Units\\ }} = \\frac{10}{1,000} = 0.01 = 1\\%\$\$]{.math.display}\ \(2) Calculating the average severity of loss \ [\$\$Average\\ Severity\\ of\\ Loss = \\frac{\\text{The\\ Amount\\ of\\ All\\ Losses\\ }}{\\text{Number\\ of\\ Losses\\ }} = \\frac{\\\$ 200,000}{10} = \\\$ 20,000\$\$]{.math.display}\ NB The value of each car is \$30,000 while the average size of loss is only \$20,000 \(3) Calculating the pure premium \ [*Pure* *Premium* = *Average* *Frequency* × *Average* *Severity*]{.math.display}\ \ [*Pure* *Premium* = 0.01 *x* 20, 000 = \$200]{.math.display}\ OR \ [\$\$Pure\\ Premium = \\ \\frac{\\text{Amount\\ of\\ All\\ Losses\\ }}{\\text{Number\\ of\\ Exposure\\ Units\\ }} = \\ \\frac{\\\$ 200,000}{1,000} = \\\$ 200\$\$]{.math.display}\ NB: For the insured, risk is reduced from \$30,000 (the value of the car) to \$200 (the value of pure premium) \(4) Calculating the gross premium if the loading ratio = 25% \ [\$\$Gross\\ \\ Premium = \\ \\frac{\\text{Pure\\ Premium\\ }}{1 - Loading\\ Ratio\\ \\ } = \\ \\frac{200}{1 - 0.25} = \\\$ 266.7\$\$]{.math.display}\ \(5) The Equality Principle Insured Payments = Insurer Payments (No. of Exposure Units × Pure Premium) = (No. of Losses × Avg. Size of Loss) (1,000 × \$200) = (10 × \$20,000) \$200,000 = \$200,000 **[Important Notes:]** 1\) Risk is reduced from \$30,000 (the value of each car on average) to \$200 (the pure premium). 2\) Pure premium = cost of losses only 3\) Equality principle is applicable on pure premium (not the gross premium) - Total pure premium = pure premium per unit × no. of exposure units - Total Gross premium = Gross premium per unit × no. of exposure units - Expected Profit = Total Gross premium per unit - Total pure premium - Insured Payments = pure premium per unit × no. of exposure units - Insurer Payments = no. of losses × Average Size of losses (losses severity) **[Example (2): ]** Out of 100,000 houses insurance against fire, 5,000 houses were damaged. The total amounts of all houses destroyed (damaged) by the fire incurred losses of \$10,000,000. Calculate the following: 1. The average frequency of loss. 2. The average Severity of loss. 3. The pure premium. **[Solution:]** - Number of exposure units = 100,000 - Number of losses = 5000 - Total amount of all losses = 10,000,000 1. [\$Average\\ Frequency = \\frac{\\text{Number\\ of\\ Losses\\ }}{\\text{Number\\ of\\ Exposure\\ Units\\ }} = \\frac{5000}{100,000} = 0.05 = 5\\%\$]{.math.inline} 2. [\$Average\\ Severity\\ = \\frac{\\text{The\\ Amount\\ of\\ All\\ Losses\\ }}{\\text{Number\\ of\\ Losses\\ }} = \\frac{\\\$ 10,000,000}{5,000} = \\\$ 2,000\$]{.math.inline} 3. Pure premium = Average frequency × Average severity *=0.05×2000= \$100* **Example (3):** Suppose that among 100,000 insured houses, 2000 houses will be burn in the next year. The average size of loss is \$5000 per house. Pure premium is \$100. Indicate the equality principle. **Solution:** Insured Payments = Insurer Payments (No. of Exposure Units x Pure Premium) = (No. of Losses x Average Size of Loss) (100,000 x \$100) = (2,000 x \$5,000) \$10,000,000 = 10,000,000 **Example (4):** Rewrite and complete the following table with respect to an insurer that sells 2 types of insurance (Indicate your calculations). **Types of Insurance** **Fire** **Auto** ---------------------------------- ---------- ---------- **No. of exposure units** 10,000 50,000 **Number of losses** 20 \(8) ? **The probability of losses** \(1) ? 0.02 **Total losses (Paid claims)** \(2) ? \(9) ? **Average loss** \$45,000 \$30,000 **Pure or net premium per unit** \(3) ? \$600 **Pure premiums** \(4) ? \(10) ? **Loading ratio** 15% \(11) ? **Gross premium per unit** \(5) ? \$1,000 **Gross premiums** \(6) ? \(12) ? **Expected profit** \(7) ? \(13) ? **Solution:** \(1) The probability of losses (Fire) \ [\$\$Average\\ Frequency\\ (Prob.of\\ loss) = \\frac{\\text{Number\\ of\\ Losses\\ }}{\\text{Number\\ of\\ Exposure\\ Units\\ }} = \\frac{20}{10,000} = 0.002\$\$]{.math.display}\ \(2) Total Losses (Fire) \ [\$\$Severity\\ of\\ Loss = \\frac{\\text{The\\ Amount\\ of\\ All\\ Losses\\ }}{\\text{Number\\ of\\ Losses\\ }}\$\$]{.math.display}\ \ [\$\$45,000 = \\frac{\\text{The\\ Amount\\ of\\ All\\ Losses\\ }}{20\\ }\$\$]{.math.display}\ \ [∴ *The* *Amount* *of* *all* *losses*= 45, 000 *x* 20 = \$900, 000 ]{.math.display}\ \(3) Pure Premium per unit (Fire) [*Pure* *Premium* = *Average* *Frequency* *x* *Average* *Severity* = 0.002 *x* 45, 000 = \$90]{.math.inline} OR \ [\$\$Pure\\ Premium = \\ \\frac{\\text{Amount\\ of\\ All\\ Losses\\ }}{\\text{Number\\ of\\ Exposure\\ Units\\ }} = \\ \\frac{\\\$ 900,000}{10,000} = \\\$ 90\$\$]{.math.display}\ \(4) Pure Premiums (Fire) \ [*Total* *Pure* *Premiums*= *No*.*of* *Exposure* *Units* *x* *Pure* *Premium* *per* *unit*]{.math.display}\ \ [*Total* *Pure* *Premiums*= 10, 000 *x* \$90 = \$900, 000..............(*Equality* *principle*)]{.math.display}\ \(5) Gross Premium per unit (Fire) \ [\$\$Gross\\ \\ Premium = \\ \\frac{\\text{Pure\\ Premium\\ }}{1 - Loading\\ Ratio\\ \\ } = \\ \\frac{90}{1 - 0.15} = \\\$ 105.8\$\$]{.math.display}\ \(6) Gross Premiums (Fire) \ [*Total* *Gross* *Premiums*= *No*.*of* *Exposure* *Units* *x* *Gross*  *Premium* *per* *unit*]{.math.display}\ \ [*Total* *Gross* *Premiums*= 10, 000 *x* 105.8 = 1, 058, 000]{.math.display}\ \(7) Expected Profit (Fire) \ [*Expected* *Profits*= *Gross* *Premiums* − *Pure* *Premiums*]{.math.display}\ \ [*Expected* *Profits*= 1, 058, 000 − 900, 000 = 158, 000]{.math.display}\ \(8) Number of Losses (Auto) \ [\$\$Average\\ Frequency\\ (Prob.of\\ loss) = \\frac{\\text{Number\\ of\\ Losses\\ }}{\\text{Number\\ of\\ Exposure\\ Units\\ }}\$\$]{.math.display}\ \ [\$\$0.02 = \\frac{\\text{Number\\ of\\ Losses\\ }}{50,000\\ }\\ \\ \\ \\ \\ \\therefore Number\\ of\\ Losses = 50,000\\ x\\ 0.02 = 1,000\\ cars\$\$]{.math.display}\ \(9) Total Losses (Auto) \ [\$\$Severity\\ of\\ Loss = \\frac{\\text{The\\ Amount\\ of\\ All\\ Losses\\ }}{\\text{Number\\ of\\ Losses\\ }}\$\$]{.math.display}\ \ [\$\$30,000 = \\frac{\\text{The\\ Amount\\ of\\ All\\ Losses\\ }}{1,000\\ }\$\$]{.math.display}\ \ [∴ *The* *Amount* *of* *all* *losses*= 30, 000 *x* 1, 000 = \$30, 000, 000 ]{.math.display}\ \(10) Pure Premiums (Auto) \ [*Total* *Pure* *Premiums*= *No*.*of* *Exposure* *Units* *x* *Pure* *Premium* *per* *unit*]{.math.display}\ \ [*Total* *Pure* *Premiums*= 50, 000 *x* \$600 = \$30, 000, 000..............(*Equality* *principle*)]{.math.display}\ \(11) Loading Ratio (Auto) \ [\$\$Gross\\ \\ Premium = \\ \\frac{\\text{Pure\\ Premium\\ }}{1 - Loading\\ Ratio\\ \\ }\$\$]{.math.display}\ \ [\$\$1,000 = \\ \\frac{600\\ }{1 - Loading\\ Ratio\\ \\ }\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\$\$]{.math.display}\ \ [1, 000 (1−*Loading* *Ratio*)= 600            ]{.math.display}\ \ [\$\$\\left( 1 - Loading\\ Ratio \\right) = \\ \\frac{600}{1000} = 0.60\\ \$\$]{.math.display}\ \ [*Loading* *Ratio* = 1 − 0.60 = 0.40 = 40% ]{.math.display}\ \(12) Gross Premiums (Auto) \ [*Total* *Gross* *Premiums*= *No*.*of* *Exposure* *Units* *x* *Gross*  *Premium* *per* *unit*]{.math.display}\ \ [*Total* *Gross* *Premiums*= 50, 000 *x* 1, 000 = 50, 000, 000]{.math.display}\ \(13) Expected Profit (Auto) \ [*Expected* *Profits*= *Gross* *Premiums* − *Pure* *Premiums*]{.math.display}\ \ [*Expected* *Profits*= 50, 000, 000 − 30, 000, 000 = 20, 000, 000]{.math.display}\ **Exercise (1):** Insurance company is able to predict that 200 of 10,000 houses were damaged by fire. Calculate the average frequency of fire (loss). **Solution:** [\$Average\\ Frequency = \\frac{\\text{Number\\ of\\ Losses\\ }}{\\text{Number\\ of\\ Exposure\\ Units\\ }} = \\frac{200}{10,000} = 0.02 = 2\\%\$]{.math.inline} This means that the chance of house fire is 2%. **Exercise (2):** Insurance company pays on average each year about \$4,000,000 for house owners who their houses burn. The number of burn houses is 200 houses. Calculate the severity of loss. **Solution:** [\$Severity\\ of\\ Loss = \\frac{\\text{The\\ Amount\\ of\\ All\\ Losses\\ }}{\\text{Number\\ of\\ Losses\\ }} = \\frac{4,000,000}{200} = 20,000\\\$\$]{.math.inline} This means that the average loss in case of fire is 20,000 per house **Exercise (3):** An insurance company pays on average each year about \$12,000,000 for car owners who their cars damaged. The number of insured cars is 60,000 cars. The number of damaged cars is 600 cars. The average loss per care is 20,000. Calculate the pure premium of automobile collision insurance. **Solution:** \ [\$\$Pure\\ Premium = \\ \\frac{\\text{Amount\\ of\\ All\\ Losses\\ }}{\\text{Number\\ of\\ Exposure\\ Units\\ }} = \\ \\frac{12,000,000}{60,000} = \\\$ 200\\ \$\$]{.math.display}\ This means that each insured will pay \$200 to buy the insurance policy against collision. Exercise (4): Calculate the gross premium for an insurance policy against theft if you know that the pure premium is \$400 and loading ratio is 25%. **Solution:** \ [\$\$Gross\\ \\ Premium = \\ \\frac{\\text{Pure\\ Premium\\ }}{1 - Loading\\ Ratio\\ \\ } = \\ \\frac{400}{1 - 0.25} = 533.3\$\$]{.math.display}\ **[Questions (10)]** **[First: MCQ (10)]** - **[Given the following table with respect to an insurer that sell fire insurance in year 2023: ]** +-----------------------------------+-----------------------------------+ | **Types of insurance** | **Fire** | +===================================+===================================+ | **Expected number of exposure | **10,000** | | units** | | +-----------------------------------+-----------------------------------+ | **Expected number of losses** | **30** | +-----------------------------------+-----------------------------------+ | **The expected probability of | 1. **?** | | loss** | | +-----------------------------------+-----------------------------------+ | **Expected total loss (expected | 2. **?** | | paid claims)** | | +-----------------------------------+-----------------------------------+ | **Expected average loss** | **\$33000** | +-----------------------------------+-----------------------------------+ | **Pure or net premium per unit** | 3. **?** | +-----------------------------------+-----------------------------------+ | **Pure premium** | 4. **?** | +-----------------------------------+-----------------------------------+ | **Loading ratio** | **25%** | +-----------------------------------+-----------------------------------+ | **Gross premium per unit** | 5. **?** | +-----------------------------------+-----------------------------------+ | **Gross premiums** | 6. **?** | +-----------------------------------+-----------------------------------+ | **\$total loading** | 7. **?** | +-----------------------------------+-----------------------------------+ | **Actual number of losses** | **36** | +-----------------------------------+-----------------------------------+ | **Objective risk** | 8. **?** | +-----------------------------------+-----------------------------------+ 1. The probability of loss is a\) 3% b) 333.33 c) [0.003] d) none of the above 2. The total losses (expected paid claims) equals a\) \$330,000 b) \$[990,000] c) \$99 d) none of the above 3. pure or net premium per unit is a\) \$ b) \$990000 c) \$33000 d) none of the above 4. pure premiums equal a\) \$[990,000] b) \$2970 c) \$366.67 d) none of the above 5. Gross premium per unit is a\) \$123.75 b) \$ c) \$44 d) none of the above 6. Gross premiums equal a\) \$123750 b) \$742500 c) \$[1320, 000] d) none of the above 7. \$ the expected profit in fire insurance a\) \$110,000 b) \$440,000 c) \$[330,000] d) none of the above 8. \$ Insurer Payments a\) \$110,000 b) \$[990,000] c) \$330,000 d) none of the above 9. \$ Insured Payments a\) \$110,000 b) \$[990,000] c) \$330,000 d) none of the above 10. Objective risk is a\) -- 20% b) [20%] c) 0 d) none of the above 1. Gross premium represents the cost of loss only without any loading. 2. Loading is the amount needed to pay all expenses. 3. The net or pure premium represents the cost of loss only without any loading.