Worked Contingency Provision Comparison

Why Compare Contingency Methods?

A contingency provision can be set with different objectives. One method relates the provision to the standard deviation of losses; another chooses it to meet a specified probability of underwriting deficit; a third prices the cost of holding capital against adverse outcomes. These methods can use the same underwriting data and still produce different provisions because they answer different questions.

To make a comparison meaningful, hold the underwriting results and baseline fixed, state the risk measure and all parameters, and distinguish risk capital from the cost charged for that capital. The numerical methods in this tutorial are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $L$ be aggregate loss, with $E[L]=\mu$ and $\operatorname{SD}(L)=\sigma$. Let $M$ be the expected underwriting surplus before any contingency provision, and let $C$ be an added provision that increases underwriting surplus dollar for dollar. The resulting underwriting surplus is $U=M+C-(L-\mu)$. A deficit occurs when $U<0$. This formulation separates the expected result from the random deviation of actual loss around its mean.

The examples use a hypothetical one-year property portfolio with aggregate loss $L\sim N(800{,}000,100{,}000^2)$. Thus expected loss is $800,000 and the standard deviation of aggregate loss is $100,000. Before a contingency provision, expected premium less expenses and expected loss leaves an expected underwriting surplus $M=\$20{,}000$. There is no mortality assumption: the portfolio covers property losses. No discounting or investment income is included in the examples unless specified; where a cost-of-capital rate is used, it is an annual rate applied for one year.

Standard-Deviation Provision

The standard-deviation method sets the provision as a selected multiple of the standard deviation of aggregate loss. The multiple reflects a pricing or risk-management judgment; it does not, by itself, specify a particular probability of deficit.

Formula: Under the standard-deviation method, $C_{\mathrm{SD}}=k\sigma$, where $k\geq 0$ is a selected factor. This is an added provision, not the total expected underwriting surplus. The formula is appropriate when the selected factor and the loss standard deviation are defined on a consistent basis. It does not guarantee a target deficit probability unless the distribution and baseline surplus are also considered.

Worked Example: Standard-Deviation Provision and Its Implication

Method: Apply a standard-deviation factor to the hypothetical portfolio above, then calculate the deficit probability implied by the result. Loss is $N(800{,}000,100{,}000^2)$; $M=\$20{,}000$, and the selected factor is $k=0.50$. The contract is one year, with no discounting, investment income, or mortality assumption.

Step 1: Calculate the provision. The loss standard deviation is $\sigma=\$100{,}000$, so

C_{\mathrm{SD}}=k\sigma=0.50(\$100{,}000)=\$50{,}000.

Step 2: Find the total expected cushion. After adding the provision, expected underwriting surplus is $M+C_{\mathrm{SD}}=\$20{,}000+\$50{,}000=\$70{,}000$.

Step 3: Convert that cushion to a probability. A deficit occurs when the loss deviation exceeds $70,000. Standardizing gives $70{,}000/100{,}000=0.70$. For a standard normal variable $Z$, $P(U<0)=P(Z>0.70)\approx 0.2420$. The implied probability of deficit is about 24.20%.

Check and common mistake avoided: The provision is $50,000, while the total cushion is $70,000. Using $0.50\sigma$ as the total cushion would ignore the existing $20,000 expected surplus and incorrectly give a deficit probability of $P(Z>0.50)$. This method does not promise a specific deficit probability; it reports the consequence of the selected factor under the assumed model.

Target-Deficit-Probability Provision

A target-probability method starts with a chosen maximum probability of deficit and solves for the provision needed to attain it. Under a normal loss assumption, the required cushion is a normal quantile times the loss standard deviation. The provision is the amount of that cushion not already supplied by $M$.

Formula: If $L\sim N(\mu,\sigma^2)$, the target deficit probability is $\alpha$, and $z_{1-\alpha}$ satisfies $P(Z\leq z_{1-\alpha})=1-\alpha$ for $Z\sim N(0,1)$, then the required total cushion is $z_{1-\alpha}\sigma$. The required added provision is $C_{\mathrm{target}}=z_{1-\alpha}\sigma-M$. If a provision is constrained to be nonnegative, use $C_{\mathrm{target}}=\max(0,z_{1-\alpha}\sigma-M)$. If this maximum binds, the achieved deficit probability may be below the target; a negative provision would be needed to attain the target exactly.

Worked Example: Solving for a 5% Deficit Probability

Method: Use the normal quantile for the target probability and subtract the existing expected surplus. Use the same one-year hypothetical portfolio: $L\sim N(800{,}000,100{,}000^2)$, $M=\$20{,}000$, and no discounting, investment income, or mortality assumption. The specified target deficit probability is $\alpha=0.05$.

Step 1: Select the quantile. For a standard normal distribution, $z_{0.95}\approx1.6449$. This is the loss-deviation multiple with 5% of the distribution above it.

Step 2: Calculate the required total cushion.

z_{0.95}\sigma=1.6449(\$100{,}000)=\$164{,}490.

Step 3: Subtract the existing expected surplus.

C_{\mathrm{target}}=\$164{,}490-\$20{,}000=\$144{,}490.

Step 4: Check the probability. With the provision, the total cushion is $164,490, or $1.6449\sigma$. Thus $P(U<0)=P(Z>1.6449)\approx0.0500$, as required.

Check and common mistake avoided: The provision is $144,490, not $164,490, because $20,000 of expected surplus already contributes to the cushion. Adding the full quantile amount as a new provision would produce a larger total cushion than required and a deficit probability below 5%.

Cost-of-Capital Provision

A cost-of-capital approach distinguishes the capital needed to support risk from the charge for providing that capital. A stated tail measure can define required risk capital $K$; a cost-of-capital rate then determines the charge for holding it over the relevant period. The charge is not the capital itself and does not, on its own, specify a target probability of deficit.

Formula: In the one-year, undiscounted setup used here, define risk capital as $K=z_{1-\alpha}\sigma$, the selected loss quantile in excess of expected loss. If $c$ is the annual cost-of-capital rate, the one-year charge is $C_{\mathrm{CoC}}=cK$. This simplified formula assumes the stated capital is held for a full year and that the charge is not adjusted for taxes, investment income, or changes in capital during the year. Other timing, tax, or capital conventions require corresponding adjustments.

Worked Example: Cost-of-Capital Charge on the Same Risk

Method: Calculate 95th-percentile risk capital, then charge for holding that capital for one year. Use $L\sim N(800{,}000,100{,}000^2)$, so $\sigma=\$100{,}000$, and use the same $M=\$20{,}000$. Set the annual cost-of-capital rate to $c=6\%$. There is no discounting, investment income, tax adjustment, or mortality assumption.

Step 1: Calculate the capital amount. For a 95th-percentile measure, $z_{0.95}=1.6449$, so

K=z_{0.95}\sigma=1.6449(\$100{,}000)=\$164{,}490.

Step 2: Calculate the one-year charge.

C_{\mathrm{CoC}}=cK=0.06(\$164{,}490)=\$9{,}869.40.

Step 3: Interpret the amount. The $9,869.40 is the charge for providing $164,490 of risk capital for one year under the stated assumptions. It is not a replacement for the capital amount. Adding the charge to the existing expected surplus gives a total cushion of $29,869.40.

Check and common mistake avoided: Multiplying by 6% gives the cost charge, not the capital requirement. Treating $9,869.40 as the capital amount confuses the price of capital with the capital supporting the risk.

Derivation and Comparison

The target-probability formula follows directly from the definition of deficit. Write the actual loss as its mean plus its deviation, $L=\mu+(L-\mu)$. Then $U=M+C-(L-\mu)$, so $U<0$ exactly when $L-\mu>M+C$. Under the normal model, $Z=(L-\mu)/\sigma\sim N(0,1)$, and therefore

P(U<0)=P\left(Z>\frac{M+C}{\sigma}\right) =1-\Phi\left(\frac{M+C}{\sigma}\right),

where $\Phi$ is the standard normal cumulative distribution function. Setting this probability equal to $\alpha$ gives $(M+C)/\sigma=z_{1-\alpha}$, hence $C=z_{1-\alpha}\sigma-M$. This derivation also shows why the expected surplus must be included: the probability depends on the total cushion $M+C$.

For the shared hypothetical portfolio, the methods produce the following results. The implied deficit probabilities use the same normal loss assumption and baseline surplus.

Method	Added provision	Total cushion $M+C$	Interpretation
Standard deviation, $k=0.50$	$50,000	$70,000	Implied deficit probability about 24.20%
Target deficit probability, $\alpha=5\%$	$144,490	$164,490	Designed for a 5% deficit probability under the normal model
Cost of capital, $c=6\%$ on 95th-percentile capital	$9,869.40	$29,869.40	Charge for capital; not a 5% deficit guarantee

The amounts differ because the methods do not measure the same thing. A standard-deviation load is a selected risk-related premium amount. A target-probability provision fills the gap between the current expected surplus and a quantile-based cushion. A cost-of-capital provision charges for holding a selected amount of capital; it does not add the full capital amount to the premium. In particular, the cost-of-capital provision above is much smaller than the target-probability provision, but that does not make it a competing estimate of the same 5% cushion.

Common Traps and Practical Checks

Ignoring the baseline surplus. Target-probability calculations require the total cushion $M+C$. Subtract $M$ when solving for the added provision.
Calling a cost charge “capital.” State both $K$ and $cK$. They are different quantities with different interpretations.
Treating a standard-deviation factor as a confidence level. A selected $k$ implies a probability only after the distribution and existing surplus are specified.
Mixing risk measures. A quantile of losses, a standard deviation, and the cost of capital are not interchangeable without an explicit model and conversion.
Forgetting model limitations. The normal calculation assumes the aggregate loss deviation is adequately represented by a normal distribution. Skewed or heavy-tailed losses may make normal tail probabilities unsuitable. The cost-of-capital result also depends on its chosen capital measure, rate, duration, and adjustments.

Key takeaway: Before comparing contingency provisions, identify what each amount represents, what risk measure it uses, and whether it is an added cushion, required capital, or a charge for capital. Matching the underwriting data alone does not make the objectives equivalent.

Check Your Understanding

For each problem, treat all portfolios and values as hypothetical. Assume normal aggregate loss where stated and ignore mortality, taxes, discounting, and investment income unless the problem specifies a cost-of-capital rate.

A one-year portfolio has $L\sim N(500{,}000,60{,}000^2)$, expected surplus before provision of $10,000, and a standard-deviation factor of 0.40. Find the provision and the implied deficit probability. (Answer: provision $24,000; $P(Z>0.5667)\approx0.2854$.)
A portfolio has $\sigma=\$80{,}000$, existing expected surplus $M=\$15{,}000$, and a 2.5% target deficit probability. Using $z_{0.975}=1.9600$, find the added provision and verify the total cushion. (Answer: provision $141,800; total cushion $156,800.)
For a one-year portfolio with $\sigma=\$50{,}000$, use the 99th-percentile normal capital measure, $z_{0.99}=2.3263$, and a 7% annual cost-of-capital rate. Calculate risk capital and its one-year charge. (Answer: capital $116,315; charge $8,142.05.)
A portfolio has $\sigma=\$120{,}000$, $M=\$25{,}000$, and a standard-deviation factor of 0.75. Calculate the added provision and express the implied deficit probability using a standard normal tail probability. (Answer: provision $90,000; $P(Z>0.9583)$, approximately 0.1691.)
A portfolio has $\sigma=\$40{,}000$, $M=\$70{,}000$, and a 5% target deficit probability, with $z_{0.95}=1.6449$. If provisions cannot be negative, find the selected provision and the resulting deficit probability. (Answer: provision $0; $P(Z>1.75)\approx0.0401$, below the 5% target.)

Method	Added provision	Total cushion \(M+C\)	Interpretation
Standard deviation, \(k=0.50\)	$50,000	$70,000	Implied deficit probability about 24.20%
Target deficit probability, \(\alpha=5\%\)	$144,490	$164,490	Designed for a 5% deficit probability under the normal model
Cost of capital, \(c=6\%\) on 95th-percentile capital	$9,869.40	$29,869.40	Charge for capital; not a 5% deficit guarantee

What You'll Learn