ANTMACK215 × WEALTHWISEGUYS
WealthWiseGuys Practical Mathematics

Chapter 20

Insurance, Risk, and Financial Protection

Understand what is transferred and what you still pay

Insurance math connects everyday premiums with potentially large losses. Learn to read simplified coverage terms, split health costs, and compare expected losses with financial capacity. All people, policies, prices, and probabilities are hypothetical; these models are not coverage advice, and money is rounded to cents.

20.1

Premiums, Deductibles, and Coverage Limits

Learning objectives

After this section you will be able to:

  1. Compute annual premiums.
  2. Apply a deductible before an insurer-payment cap.
  3. Separate premiums from claim costs.

Real-life situation

Protecting a bicycle

A hypothetical Philadelphia rider compares a covered theft loss with a policy summary. All claims in this section are eligible; the limit caps insurer payment after the deductible.

ItemAmount or quantity
Monthly premium$15.00
Per-claim deductible$200.00
Insurer-payment limit$1,500.00
Covered loss$1,800.00
Figure 20.1 Simplified bicycle policy (hypothetical)

The concept

Insurance transfers specified financial risks in exchange for a premium. A premium pays for coverage during a period; it is not a deposit automatically returned when no claim occurs. Policy terms determine which events and property qualify.

A deductible is the part of an eligible loss the policyholder must absorb before payment begins. For these exercises there is one deductible per claim, and the insurer-payment limit applies after subtracting it. Other contracts can define limits differently.

Compute the covered loss above the deductible, never less than zero, then apply the insurer-payment cap. The policyholder bears the remainder of the loss, which can exceed the deductible when a limit applies. Excluded losses would receive no payment.

Annual cash cost in a claim year includes premiums plus retained loss. Keep the premium separate from the claim settlement to avoid subtracting it twice. These simplified policies omit depreciation, sublimits, and claim disputes; round money to cents.

Premiums, Deductibles, and Coverage Limits: solution map

  1. 1

    Identify the question

    Use the figure to find payment and retained loss.

  2. 2

    Calculation 1

    Loss above deductible = $1,800 − $200 = $1,600.00.

  3. 3

    Calculation 2

    Payment = minimum of $1,600 and $1,500 = $1,500.00.

  4. 4

    Calculation 3

    Retained loss = $1,800 − $1,500 = $300.00.

  5. 5

    Interpret and check

    The insurer pays $1,500.00; the rider retains $300.00.

Figure 20.1.V Premiums, Deductibles, and Coverage Limits: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • insurance premium

    The amount paid, often monthly, to keep an insurance policy active.

  • deductible

    The amount you pay yourself before an insurance policy starts paying.

  • risk

    The chance that an investment or decision will turn out worse than expected.

Key terms

Premium
Price paid to maintain coverage.
Deductible
Eligible loss retained before insurer payment begins.
Coverage limit
Maximum insurer payment under the stated model.
Exclusion
An event or loss the contract does not cover.
Retained loss
Loss not reimbursed by insurance.

Formulas & rules

  • Annual premium

    Annual premium = monthly premium × 12

  • Claim payment

    Payment = minimum of limit and maximum of 0 and (covered loss − deductible)

  • Retained loss

    Retained loss = covered loss − payment

  • Annual cash cost

    Cash cost = annual premiums + retained losses

Worked examples

Example 1A loss above the limit

Use the figure to find payment and retained loss.

Solution

  1. 1Loss above deductible = $1,800 − $200 = $1,600.00.
  2. 2Payment = minimum of $1,600 and $1,500 = $1,500.00.
  3. 3Retained loss = $1,800 − $1,500 = $300.00.

Answer: The insurer pays $1,500.00; the rider retains $300.00.

Example 2Add the cost of coverage

Using the same policy and one claim from Example 1, find annual cash cost.

Solution

  1. 1Annual premiums = 12 × $15 = $180.00.
  2. 2Retained claim loss = $300.00.
  3. 3Annual cash cost = $180 + $300 = $480.00.

Answer: Annual cash cost is $480.00.

Practice exercises 20.1

  1. Premium is $22 monthly. Find its annual cost.
  2. An eligible $150 loss has a $200 deductible. Find payment.
  3. An eligible $900 loss has a $250 deductible and $2,000 payment limit. Find payment and retained loss.
  4. An eligible $3,000 loss has a $500 deductible and $2,000 payment cap. Find payment and retained loss.
  5. Annual premiums are $240 and retained loss is $125. Find annual cash cost.

Recap quiz 20.1

1. An eligible $700 loss with a $200 deductible and adequate limit yields payment of:
2. A $10 monthly premium costs annually:
3. If a loss is entirely excluded, the simplified insurer payment is:

Section 20.1 recap

  • Read the deductible and the limit before calculating payment.
  • Total cash cost includes both premiums and unreimbursed losses.
Answer key with explanations: Section 20.1

Practice exercises

  1. 12 × $22 = $264.00.
  2. Maximum of 0 and ($150 − $200) is $0.00; the policyholder retains $150.00.
  3. Above deductible = $900 − $250 = $650; payment = $650.00; retained loss = $250.00.
  4. Above deductible = $2,500; payment capped at $2,000.00; retained loss = $3,000 − $2,000 = $1,000.00.
  5. $240 + $125 = $365.00; do not subtract the premiums from the insurer payment.

Recap quiz

  1. $500: $700 − $200 = $500.00.
  2. $120: 12 × $10 = $120.00.
  3. $0: Excluded losses are outside coverage.
20.2

Health Cost Sharing and Out-of-Pocket Maximums

Learning objectives

After this section you will be able to:

  1. Apply a deductible and coinsurance sequentially.
  2. Use a remaining out-of-pocket maximum.
  3. Distinguish billed and allowed amounts.

Real-life situation

Reading a medical benefit summary

A hypothetical student has one covered in-network bill. The plan uses an allowed amount, then an annual deductible and 20% patient coinsurance; all patient cost sharing here counts toward the annual maximum.

ItemAmount or quantity
Allowed amount$5,000.00
Deductible still owed$1,000.00
Patient coinsurance20% after deductible
Remaining out-of-pocket maximum$1,500.00
Figure 20.2 Simplified health-plan claim (hypothetical)

The concept

The billed charge is not necessarily the amount used to compute a covered claim. These exercises start with the allowed amount, and assume no balance billing. The plan pays the remainder of that allowed amount after patient cost sharing.

Apply the remaining deductible first. Coinsurance then assigns the stated percentage of the remaining allowed amount to the patient. A 20% patient coinsurance rate means the plan generally pays 80% of that post-deductible amount, not 80% of the original claim.

A remaining out-of-pocket maximum limits the patient cost sharing that counts toward it. Compute ordinary deductible plus coinsurance and then cap that amount at the remaining maximum. Premiums and excluded services do not count toward the maximum in this simplified model.

Real plans can have copays, separate networks, family deductibles, and services with different rules. Do not treat this illustration as a universal benefits formula. Round money to cents and keep the remaining deductible and remaining maximum distinct.

Health Cost Sharing and Out-of-Pocket Maximums: solution map

  1. 1

    Identify the question

    Use the figure to find patient and plan payments.

  2. 2

    Calculation 1

    Apply the $1,000 remaining deductible to the $5,000 allowed amount.

  3. 3

    Calculation 2

    Post-deductible amount = $5,000 − $1,000 = $4,000.00.

  4. 4

    Calculation 3

    Coinsurance = $4,000 × 0.20 = $800; ordinary share = $1,800.00.

  5. 5

    Calculation 4

    Patient share is capped at $1,500; plan pays $5,000 − $1,500 = $3,500.00.

  6. 6

    Interpret and check

    Patient pays $1,500.00 and plan pays $3,500.00.

Figure 20.2.V Health Cost Sharing and Out-of-Pocket Maximums: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • liability

    Money or obligations a person or business owes to others.

  • collateral

    Property a borrower promises to a lender if the loan is not repaid, such as a car.

  • asset

    Something valuable that a person or business owns, like cash, equipment, or a building.

Key terms

Allowed amount
The claim amount used for the covered-service calculation.
Coinsurance
A percentage share of covered cost after a deductible.
Remaining deductible
The annual deductible not yet satisfied.
Out-of-pocket maximum
An annual cap on specified patient cost sharing.
Copay
A stated fixed patient charge for a service, not used in this model.

Formulas & rules

  • Deductible applied

    Applied deductible = minimum of allowed amount and remaining deductible

  • Ordinary patient share

    Ordinary share = applied deductible + (allowed amount − applied deductible) × patient coinsurance

  • Capped share

    Patient share = minimum of ordinary share and remaining out-of-pocket maximum

  • Plan payment

    Plan payment = allowed amount − patient share

Worked examples

Example 1A capped claim

Use the figure to find patient and plan payments.

Solution

  1. 1Apply the $1,000 remaining deductible to the $5,000 allowed amount.
  2. 2Post-deductible amount = $5,000 − $1,000 = $4,000.00.
  3. 3Coinsurance = $4,000 × 0.20 = $800; ordinary share = $1,800.00.
  4. 4Patient share is capped at $1,500; plan pays $5,000 − $1,500 = $3,500.00.

Answer: Patient pays $1,500.00 and plan pays $3,500.00.

Example 2Deductible already met

A $600 allowed claim has no remaining deductible, 20% patient coinsurance, and $1,000 remaining maximum. Find the split.

Solution

  1. 1Applied deductible = $0.00.
  2. 2Ordinary patient share = $600 × 0.20 = $120.00.
  3. 3The $1,000 remaining maximum does not bind; plan payment = $600 − $120 = $480.00.

Answer: Patient pays $120.00; plan pays $480.00.

Practice exercises 20.2

  1. A $400 allowed claim is below the $700 remaining deductible. Find the split with an ample maximum.
  2. With the deductible met, a $1,000 allowed claim has 10% patient coinsurance and ample maximum. Find patient cost.
  3. Ordinary patient cost is $600 but the remaining maximum is $250. Find capped patient cost.
  4. A $2,000 allowed claim gives $300 patient cost. Find plan payment.
  5. A person pays $1,200 annual premiums and $800 covered cost sharing. Find combined spending; do premiums satisfy the maximum here?

Recap quiz 20.2

1. After a $500 deductible on $1,500 allowed cost, 20% coinsurance is applied to:
2. Ordinary patient share is $900; remaining maximum is $400. Patient pays:
3. Which does not count toward the maximum in this model?

Section 20.2 recap

  • Deductible comes before coinsurance in this model.
  • Cap eligible patient cost sharing, not premiums, at the remaining maximum.
Answer key with explanations: Section 20.2

Practice exercises

  1. Applied deductible = $400; post-deductible amount = $0; patient pays $400.00 and plan pays $0.00.
  2. $1,000 × 0.10 = $100.00 patient cost; the plan pays $900.00.
  3. Minimum of $600 and $250 = $250.00; the maximum reduces patient cost by $350.00.
  4. $2,000 − $300 = $1,700.00 plan payment.
  5. Combined spending = $1,200 + $800 = $2,000.00. Only the $800 cost sharing counts in this model, not premiums.

Recap quiz

  1. $1,000: $1,500 − $500 = $1,000.00 remains.
  2. $400: The share is capped at $400.00.
  3. Monthly premium: Premiums buy coverage and are excluded from the cost-sharing maximum.
20.3

Expected Loss and Financial Protection

Learning objectives

After this section you will be able to:

  1. Compute expected annual loss.
  2. Compare a premium saving with extra retained risk.
  3. Build a reserve for an uninsured exposure.

Real-life situation

Protecting a mobile work kit

A hypothetical student estimates a 5% annual chance of one $2,000 covered equipment loss and otherwise no loss. A policy costs $140 annually, has a $300 deductible, and has enough coverage for this loss.

ItemAmount or quantity
Loss probability5%
Loss if event occurs$2,000.00
Annual premium$140.00
Deductible$300.00
Figure 20.3 One-year risk model (hypothetical)

The concept

Expected loss is a probability-weighted average, not a prediction of the next year. In a two-outcome model, multiply the event probability by its loss and include the no-loss outcome with zero cost. A household may experience zero loss or the entire loss, rather than the average.

Insurance often costs more than expected claims because administration, reserves, and other expenses also require funding. Comparing average costs alone misses the value of avoiding a large unaffordable loss. Risk capacity concerns what a person can afford; risk tolerance concerns comfort with uncertainty.

A higher deductible usually shifts more risk to the policyholder. Compare the annual premium saving with the possible extra retained cost and the expected extra cost under explicit probabilities. The comparison is a model, not a recommendation, and estimates may be wrong.

A reserve can cover deductibles and excluded losses without assuming an insurer will pay everything. Subtract any money already set aside from the target, then divide the remaining amount needed by the months available, ignoring interest; if the reserve is already fully funded, no new saving is required. Insurance contracts, emergency savings, and loss prevention address different parts of financial protection; none guarantees that every loss is affordable. Round money to cents.

Expected Loss and Financial Protection: solution map

  1. 1

    Identify the question

    Compare expected annual costs with and without the figure's policy.

  2. 2

    Calculation 1

    Uninsured expected loss = 0.05 × $2,000 + 0.95 × $0 = $100.00.

  3. 3

    Calculation 2

    If insured, retained loss in the event is $300.00.

  4. 4

    Calculation 3

    Expected retained loss = 0.05 × $300 = $15.00.

  5. 5

    Calculation 4

    Expected insured cost = $140 + $15 = $155.00.

  6. 6

    Interpret and check

    Expected insured cost is $155.00 versus $100.00 uninsured, but insurance reduces the modeled event loss retained from $2,000.00 to $300.00.

Figure 20.3.V Expected Loss and Financial Protection: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • default

    Failing to repay a loan as agreed, which can harm credit and lead to collection.

  • deposit

    Money placed into a bank account.

  • balance

    The amount of money in an account at a given moment.

Key terms

Expected loss
Probability-weighted average of possible losses.
Risk capacity
Ability to absorb a financial loss.
Risk tolerance
Comfort with uncertain financial outcomes.
Reserve
Money set aside for possible future costs.
Risk transfer
Shifting specified financial exposure through a contract.

Formulas & rules

  • Expected loss

    Expected loss = sum of (outcome probability × outcome loss)

  • Expected insured cost

    Expected cost = annual premium + expected retained loss

  • Reserve saving

    Monthly saving = (reserve target − starting reserve) ÷ months available

Worked examples

Example 1Average cost versus large-loss protection

Compare expected annual costs with and without the figure's policy.

Solution

  1. 1Uninsured expected loss = 0.05 × $2,000 + 0.95 × $0 = $100.00.
  2. 2If insured, retained loss in the event is $300.00.
  3. 3Expected retained loss = 0.05 × $300 = $15.00.
  4. 4Expected insured cost = $140 + $15 = $155.00.

Answer: Expected insured cost is $155.00 versus $100.00 uninsured, but insurance reduces the modeled event loss retained from $2,000.00 to $300.00.

Example 2Fund a deductible reserve

A household wants a $900 reserve in 12 months, starting with $300 and ignoring interest. Find the monthly saving.

Solution

  1. 1Amount still needed = $900 − $300 = $600.00.
  2. 2Monthly saving = $600 ÷ 12 = $50.00.
  3. 3Check: $300 + 12 × $50 = $900.00.

Answer: Save $50.00 monthly.

Practice exercises 20.3

  1. A 2% chance of a $5,000 loss otherwise means no loss. Find expected loss.
  2. A higher deductible saves $60 annually but adds $400 retained cost in a claim. With a 10% chance of one claim, find expected extra retained cost and expected net saving.
  3. Build a $600 reserve over 8 months from zero with no interest.
  4. An annual premium is $240 and expected retained loss is $35. Find expected insured cost.
  5. A model has expected loss $80. Does it mean the next loss will be $80?

Recap quiz 20.3

1. A 4% chance of $1,000 loss gives expected loss of:
2. Saving $40 monthly for 10 months with no interest builds:
3. A lower expected cost proves a choice is affordable in the worst case.

Section 20.3 recap

  • Expected value is an average, not a guarantee or a complete risk decision.
  • Compare retained exposure as well as premiums, and plan reserves.
Answer key with explanations: Section 20.3

Practice exercises

  1. 0.02 × $5,000 + 0.98 × $0 = $100.00.
  2. Expected extra cost = 0.10 × $400 = $40; expected net saving = $60 − $40 = $20.00, but a claim year adds $400 before the premium saving.
  3. $600 ÷ 8 = $75.00 per month.
  4. $240 + $35 = $275.00.
  5. No. Expected loss is the probability-weighted average; for example, 0.04 × $2,000 = $80 while individual outcomes can be $0 or $2,000.

Recap quiz

  1. $40: 0.04 × $1,000 = $40.00.
  2. $400: $40 × 10 = $400.00.
  3. No: An average does not measure whether the large-loss outcome is affordable.

Chapter 20 Organizer and Review

Chapter Organizer

Every section at a glance: key formulas, key terms, and the main idea.

  • 20.1 Premiums, Deductibles, and Coverage Limits

    Key formulas

    • Annual premium: Annual premium = monthly premium × 12
    • Claim payment: Payment = minimum of limit and maximum of 0 and (covered loss − deductible)
    • Retained loss: Retained loss = covered loss − payment
    • Annual cash cost: Cash cost = annual premiums + retained losses

    Key terms

    Premium, Deductible, Coverage limit, Exclusion, Retained loss

    Remember

    Read the deductible and the limit before calculating payment.

  • 20.2 Health Cost Sharing and Out-of-Pocket Maximums

    Key formulas

    • Deductible applied: Applied deductible = minimum of allowed amount and remaining deductible
    • Ordinary patient share: Ordinary share = applied deductible + (allowed amount − applied deductible) × patient coinsurance
    • Capped share: Patient share = minimum of ordinary share and remaining out-of-pocket maximum
    • Plan payment: Plan payment = allowed amount − patient share

    Key terms

    Allowed amount, Coinsurance, Remaining deductible, Out-of-pocket maximum, Copay

    Remember

    Deductible comes before coinsurance in this model.

  • 20.3 Expected Loss and Financial Protection

    Key formulas

    • Expected loss: Expected loss = sum of (outcome probability × outcome loss)
    • Expected insured cost: Expected cost = annual premium + expected retained loss
    • Reserve saving: Monthly saving = (reserve target − starting reserve) ÷ months available

    Key terms

    Expected loss, Risk capacity, Risk tolerance, Reserve, Risk transfer

    Remember

    Expected value is an average, not a guarantee or a complete risk decision.

Cumulative Review

Mixed practice from this chapter and the two chapters before it.

  1. Section 20.1

    Annual premiums are $240 and retained loss is $125. Find annual cash cost.

    Show answer

    $240 + $125 = $365.00; do not subtract the premiums from the insurer payment.

  2. Section 20.2

    A person pays $1,200 annual premiums and $800 covered cost sharing. Find combined spending; do premiums satisfy the maximum here?

    Show answer

    Combined spending = $1,200 + $800 = $2,000.00. Only the $800 cost sharing counts in this model, not premiums.

  3. Section 20.3

    A model has expected loss $80. Does it mean the next loss will be $80?

    Show answer

    No. Expected loss is the probability-weighted average; for example, 0.04 × $2,000 = $80 while individual outcomes can be $0 or $2,000.

  4. Review of Section 18.3

    A snack sells for $5 and costs $2 in variable costs. Find contribution per snack.

    Show answer

    Contribution = $5 − $2 = $3.00 per snack.

  5. Review of Section 19.3

    A hypothetical business earns $18,000 revenue and has $6,000 allowed expenses. Find taxable profit.

    Show answer

    Taxable profit = $18,000 − $6,000 = $12,000.00.

All prices, stores, people, and accounts in this chapter are hypothetical and for teaching only. Product references are generic categories; no brand endorses or is endorsed by this textbook.

Chapter 19Chapter 21