Comparing Reinsurance Alternatives on Total Cost of Risk

Why Compare Total Cost Rather Than Premium?

A reinsurance alternative with a lower premium does not necessarily have a lower economic cost. It may leave more expected losses with the insurer, provide less protection against severe outcomes, or expose the insurer to a larger counterparty shortfall. Conversely, a higher-premium design may reduce expected retained claims and the cost of holding capital enough to be preferable overall.

A useful comparison starts by putting all alternatives on the same basis: the same loss distribution, time horizon, currency, capital method, cost-of-capital rate, and treatment of counterparty risk. The framework below is a one-year expected-cost comparison. The amounts in the examples are hypothetical and stated in thousands of dollars.

Definition: Let $L$ be the insurer’s gross loss during the period, $C$ the contractual reinsurance recovery, and $Z=L-C$ the contractual retained loss. Let $A$ be the part of the promised recovery that is not collected because of counterparty default. If $P$ is reinsurance premium, $Q$ is a risk load, $K$ is required economic capital, and $r$ is the annual cost-of-capital rate, define the comparison measure $$ \operatorname{TCOR}=P+E[Z]+E[A]+Q+rK. $$ Here $E[Z]+E[A]$ is expected cash loss retained after counterparty shortfall. The risk load $Q$ is a separate allowance for pricing or model uncertainty; $rK$ is the annual charge for holding capital. Include additional costs, such as administration, only if they are measured consistently for every alternative.

This measure is a decision framework, not a universal accounting identity. In particular, $K$ must be defined consistently. In the examples, it is a specified retained-loss capital measure, and it does not include an additional capital charge for counterparty default. If a comparison explicitly models credit capital, it must include that effect consistently and avoid counting the same exposure twice.

Expected Retention and Capital on a Consistent Basis

For each loss outcome, determine the contractual recovery from the treaty terms, then subtract that recovery from gross loss to find $Z$. Expected values follow by weighting each outcome by its probability. If the loss outcomes are $l_j$ with probabilities $p_j$, then

E[Z]=\sum_j p_j(l_j-c_j),\qquad E[C]=\sum_j p_jc_j,

where $c_j$ is the contractual recovery in outcome $j$. By linearity of expectation, $E[Z]=E[L]-E[C]$. This identity is a useful check, but it does not account for default: expected cash loss is $E[Z]+E[A]$, not merely $E[Z]$.

For the discrete examples, define $\operatorname{VaR}_p(Z)=\inf\{z:F_Z(z)\ge p\}$, and use the chosen economic-capital measure

K=\operatorname{VaR}_p(Z)-E[Z].

This is one possible capital convention, not the only one. The confidence level and capital definition should come from the decision framework being applied. For a discrete loss distribution, the quantile is the smallest loss value whose cumulative probability is at least $p$; it is not obtained by interpolating between loss outcomes.

Why the total-cost formula works: The insurer pays the premium $P$, bears contractual retained losses $Z$, and bears any promised recovery that is not collected, $A$. Therefore expected direct cash outgo is $P+E[Z]+E[A]$. Adding the separately specified uncertainty allowance $Q$ and annual capital charge $rK$ gives the stated comparison measure. The components represent different costs; they should not be silently substituted for one another.

Worked Comparison With a Quota Share and Excess-of-Loss Treaty

The first example illustrates why a premium ranking is not a total-cost ranking. Both alternatives use the same hypothetical gross-loss distribution and one-year basis.

Worked Example: Premium Ranking Versus Total Cost

A hypothetical insurer’s annual gross loss $L$, in thousands of dollars, is 0 with probability 0.50, 40 with probability 0.30, and 100 with probability 0.20. Compare two alternatives at the 90% capital confidence level. Alternative A is a 40% quota share with premium 17. Alternative B is an excess-of-loss treaty with retention 30, limit 70, and premium 24. For both, conditional expected unrecovered recovery is 2% of contractual recovery in each loss outcome. The assessed risk loads are 3 for A and 1 for B. Use a 10% annual cost-of-capital rate and the capital definition above. All amounts are in thousands.

Method. Calculate contractual recovery and retention outcome by outcome, then expected retention, capital, and counterparty shortfall. Finally add every cost component. This avoids treating the cheapest premium as the cheapest program.

For A, contractual recovery is $C=0.40L$, so the retained outcomes are 0, 24, and 60. Thus

$$ E[Z_A]=(0.50)(0)+(0.30)(24)+(0.20)(60)=0+7.2+12=19.2. $$

The 90% quantile is 60: cumulative probability is 0.50 at retention 0, 0.80 at 24, and 1.00 at 60. Therefore $K_A=60-19.2=40.8$. Expected contractual recovery is $E[C_A]=(0.40)E[L]$, where $E[L]=(0.50)(0)+(0.30)(40)+(0.20)(100)=32$. Hence $E[C_A]=12.8$, and $E[A_A]=(0.02)(12.8)=0.256$.

For B, contractual recovery is $C=\min((L-30)^+,70)$, where $x^+=\max(x,0)$. The recovery outcomes are 0, 10, and 70; retained outcomes are 0, 30, and 30. Thus

$$ E[Z_B]=(0.50)(0)+(0.30)(30)+(0.20)(30)=9+6=15. $$

The 90% quantile is 30, so $K_B=30-15=15$. Expected recovery is $E[C_B]=(0.30)(10)+(0.20)(70)=3+14=17$. The expected counterparty shortfall is $E[A_B]=(0.02)(17)=0.34$.

The complete totals are

\begin{aligned} \operatorname{TCOR}_A&=17+19.2+0.256+3+(0.10)(40.8)=43.536,\\ \operatorname{TCOR}_B&=24+15+0.34+1+(0.10)(15)=41.840. \end{aligned}

Alternative B has the higher premium but the lower total cost, by $43.536-41.840=1.696$ thousand dollars. As a check, $E[Z_A]+E[C_A]=19.2+12.8=32=E[L]$, and $E[Z_B]+E[C_B]=15+17=32$. The comparison adds expected counterparty shortfall separately. A common mistake is to compare premiums alone, which would select the quota share at 17 rather than the excess-of-loss treaty at 24 without considering the other costs.

Break-Even Tests and Sensitivity

A total-cost result depends on assumptions such as the capital charge and counterparty shortfall. A break-even calculation shows how much an assumption would have to change to reverse the ranking. It also makes the trade-off more useful than a single point estimate.

Worked Example: Maximum Premium at a Given Capital Charge

Use the loss distribution and treaty terms from the preceding example, but suppose the decision-maker wants to know the maximum premium for Alternative B at which it remains no more costly than A. Keep the 90% capital measure, the 2% expected recovery shortfall, the risk loads, and the 10% capital charge unchanged.

For A, the total remains 43.536. For B, the components other than premium are expected retention 15, expected shortfall 0.34, risk load 1, and capital charge $(0.10)(15)=1.5$. Therefore B’s cost excluding premium is $15+0.34+1+1.5=17.84$. Its break-even premium satisfies

P_B^*+17.84=43.536,\qquad P_B^*=25.696.

Thus B is no more costly than A when its premium is at most 25.696 thousand dollars, under these assumptions. Its actual premium of 24 is below that threshold. The check is $24+17.84=41.84$, the previously calculated total. The threshold is not an independent market price: it is conditional on the assumed losses, capital convention, credit cost, and risk loads.

The same comparison can solve for a break-even cost-of-capital rate. In the first example, the totals before capital charges are $17+19.2+0.256+3=39.456$ for A and $24+15+0.34+1=40.34$ for B. Equating costs gives

39.456+40.8r=40.34+15r \quad\Longrightarrow\quad r=\frac{0.884}{25.8}\approx 0.03426.

At a capital charge above about 3.426%, B has the lower total under the stated assumptions; below that rate, A has the lower total. At equality they tie. This sensitivity follows because A requires 25.8 thousand dollars more capital under the selected measure, even though its premium is lower.

Counterparty Costs Can Change the Comparison

A contractual recovery is not necessarily a collected recovery. The expected shortfall should reflect the probability and severity of nonpayment, preferably by loss outcome when credit quality is related to the same events that generate recoveries. Multiplying expected recovery by one average shortfall percentage is appropriate only when that percentage is an applicable conditional expected loss fraction across the relevant outcomes.

Worked Example: Outcome-Specific Counterparty Shortfall

A hypothetical insurer has annual gross loss, in thousands, of 10 with probability 0.70, 50 with probability 0.20, and 110 with probability 0.10. Compare a 20% quota share, premium 7, with an excess-of-loss treaty attaching at 40 and capped at a recovery limit of 60, premium 9. The quota share’s expected unrecovered fraction is 1% for each outcome. For the excess-of-loss treaty, the counterparty default probability conditional on the recovery being 10 is 5%, and conditional on the recovery being 60 is 10%; assume a default causes the full recovery for that outcome to be lost. Use a 95% capital level, an 8% capital charge, and risk loads of 1.5 for the quota share and 2 for the excess-of-loss treaty.

For the quota share, retained losses are 8, 40, and 88, with probabilities 0.70, 0.20, and 0.10. Expected retention is

$$ E[Z_Q]=(0.70)(8)+(0.20)(40)+(0.10)(88)=5.6+8+8.8=22.4. $$

The 95% quantile is 88, so $K_Q=88-22.4=65.6$. Expected contractual recovery is $0.20E[L]$, with $E[L]=(0.70)(10)+(0.20)(50)+(0.10)(110)=7+10+11=28$; hence $E[C_Q]=5.6$. The counterparty shortfall is $E[A_Q]=(0.01)(5.6)=0.056$. The total is

\operatorname{TCOR}_Q=7+22.4+0.056+1.5+(0.08)(65.6)=36.204.

For the excess-of-loss treaty, contractual recoveries are 0, 10, and 60, leaving retained losses of 10, 40, and 50. Expected retention is $(0.70)(10)+(0.20)(40)+(0.10)(50)=7+8+5=20$. The 95% quantile is 50, so $K_X=50-20=30$. The outcome-specific expected shortfall is

$$ E[A_X]=(0.20)(10)(0.05)+(0.10)(60)(0.10)=0.10+0.60=0.70. $$

Thus

\operatorname{TCOR}_X=9+20+0.70+2+(0.08)(30)=34.100.

The excess-of-loss treaty has the lower total by $36.204-34.100=2.104$ thousand dollars. As checks, expected gross loss is 28; quota-share expected retention plus recovery is $22.4+5.6=28$, and excess-of-loss expected retention plus recovery is $20+[(0.20)(10)+(0.10)(60)]=28$. The shortfall is then added to contractual retention, not subtracted from it. This distinction is important when the default probability differs across recovery outcomes.

Practical Comparison and Common Traps

A defensible comparison keeps each component visible, so the decision-maker can see why an alternative wins and which assumptions matter. Before selecting a design, verify that the quoted premium and all modeled costs refer to the same period and that the same loss basis is used for every option. A multi-year price should not be compared directly with a one-year expected loss unless the amounts have first been converted to a consistent basis.

Do not rank by premium alone. A lower premium can be offset by higher expected retention, capital charges, or credit costs.
Do not count ceded recovery as certain cash. Include expected shortfall when the stated comparison includes counterparty risk.
Do not mix definitions of capital. Use the same confidence level and capital convention across alternatives, and state whether credit capital is included.
Do not merge risk load and capital charge without explanation. A risk load is an explicit allowance in the pricing comparison; $rK$ is the cost assigned to holding capital under the chosen method.
Check the treaty outcome by outcome. Apply attachment and limit to each loss before taking expectations; a limit caps recovery, not gross loss.

These comparisons are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus. Their main discipline is consistency: define the cost basis, calculate each component under the same assumptions, and then compare totals and sensitivities rather than relying on one attractive headline figure.

Check Your Understanding

For each problem, use the stated assumptions and keep the cost components separate until the final total.

A hypothetical quota-share option has premium 3, expected retained loss 5, risk load 0.5, expected counterparty shortfall 0.1, and required capital 8. The annual capital charge is 5%. Find its total cost of risk. (Answer: 9.00 thousand dollars)
Gross loss is 0, 30, or 90 with probabilities 0.40, 0.40, and 0.20. An excess-of-loss treaty has retention 20 and recovery limit 40. Find expected contractual recovery and expected contractual retained loss. (Answer: Expected recovery 12 thousand dollars; expected retained loss 18 thousand dollars)
A retained loss is 0 with probability 0.70, 10 with probability 0.20, and 30 with probability 0.10. Using $K=\operatorname{VaR}_{0.95}(Z)-E[Z]$, find $K$. (Answer: 25 thousand dollars)
A treaty pays a recovery of 5 with probability 0.40 and 20 with probability 0.10; it pays zero otherwise. Conditional default probabilities are 2% for the recovery of 5 and 10% for the recovery of 20, with full loss of recovery on default. Find expected counterparty shortfall. (Answer: 0.24 thousand dollars)
Option A has premium 6, expected retained loss 12, risk load 1, capital charge 2, and counterparty cost 0.2. Option B has corresponding amounts 10, 8, 0.5, 1, and 0.5. Which option has lower total cost, and by how much? (Answer: Option B, by 1.2 thousand dollars)

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