Portfolio Changes as Capital Alternatives

Why Portfolio Changes Are Capital Alternatives

An insurer can respond to an unfavorable risk profile by changing what it writes, how it invests, or how much exposure it retains. These actions differ from external risk transfer: a portfolio change modifies the underlying business or assets, while a transfer contract shifts some loss to another party in exchange for a price. The effects are therefore not captured by looking only at a reduction in tail loss. A decision may also change premium income, investment earnings, expenses, and the expected level of losses.

A sound comparison states the time horizon, currency, loss definition, and capital measure first. The examples below use hypothetical one-year amounts in dollars, with no discounting. For underwriting, expected earnings mean premium less expected claims and stated expenses. For investment, expected earnings mean expected annual investment profit or loss. These are simplified measures, not a substitute for a full business plan.

Definition: A portfolio change is an action that alters the insurer’s retained underwriting exposures, investment holdings, or other risk-bearing activities. Its effect is assessed by comparing the distributions and expected earnings before and after the action on a consistent basis. In the examples, economic capital is defined as $\operatorname{TVaR}_{\alpha}(X)-E[X]$ for the specified loss variable $X$. This is an analytical measure used for comparison; it is not asserted to equal a regulatory or rating-agency capital requirement.

Measure Tail Loss on a Consistent Basis

For a loss variable $X$, the expected loss $E[X]$ describes its average, while a tail measure describes losses in a specified adverse part of its distribution. For confidence level $\alpha$, define $\operatorname{VaR}_{\alpha}(X)$ as the $\alpha$-quantile of loss. Define $\operatorname{TVaR}_{\alpha}(X)$ as the average loss in the worst $1-\alpha$ fraction of outcomes. In a continuous distribution this is the conditional mean above the quantile, subject to the usual quantile convention. For a discrete distribution, the tail may cut through a probability mass, so only the fraction needed to fill the tail is included.

\operatorname{TVaR}_{\alpha}(X) = \frac{1}{1-\alpha}\int_{\alpha}^{1}\operatorname{VaR}_{p}(X)\,dp

The integral definition explains the discrete calculation: the tail contains exactly $1-\alpha$ probability, including a fractional part of an outcome’s probability if necessary. Multiplying each included loss by its included probability gives the tail loss-weighted amount; dividing by $1-\alpha$ gives the tail average. This avoids counting too much probability just because an outcome has a large atom.

Comparison rule: State whether losses are gross or net of premium, reinsurance, or other offsets, and apply the same convention before and after the action. Report expected earnings separately from tail loss. If using economic capital, state its formula and apply it to the same loss variable and confidence level in both cases.

Underwriting Actions: Lower Tail Loss Can Change Earnings Too

Underwriting changes can include tightening eligibility, changing limits or deductibles, or declining a segment of business. Their effect depends on how both frequency and severity change, and on the premium and expenses lost or retained. The following example isolates a change in severity while holding the probability of each claim outcome and premium constant except as explicitly stated.

Worked Example: Tightening Terms on a High-Severity Segment

A hypothetical insurer reviews one year of a small segment. Before a proposed underwriting change, annual claims $X$ are $0$ with probability $0.90$, $\$100$ with probability $0.08$, and $\$700$ with probability $0.02$. Annual premium is $\$35$, and expenses are $\$5$. The change reduces the largest claim outcome to $\$400$, with the probabilities unchanged, and premium falls to $\$30$; expenses remain $\$5$. Calculate expected underwriting earnings, 95% TVaR of claims, and economic capital defined as $\operatorname{TVaR}_{0.95}(X)-E[X]$, before and after.

Method. Calculate the expected claim amount from all outcomes. For 95% TVaR, include the worst 5% of probability. The $\$700$ outcome occupies 2% of that tail, so only 3% probability from the $\$100$ outcome is needed.

Before the change. Expected claims are $0(0.90)+100(0.08)+700(0.02)=8+14=\$22$. Expected underwriting earnings are $35-22-5=\$8$. The loss-weighted amount in the worst 5% is $700(0.02)+100(0.03)=14+3=\$17$. Therefore, $\operatorname{TVaR}_{0.95}(X)=17/0.05=\$340$, and economic capital is $340-22=\$318$.

After the change. Expected claims are $0(0.90)+100(0.08)+400(0.02)=8+8=\$16$. Expected underwriting earnings are $30-16-5=\$9$. The worst 5% has loss-weighted amount $400(0.02)+100(0.03)=8+3=\$11$, so $\operatorname{TVaR}_{0.95}(X)=11/0.05=\$220$. Economic capital is $220-16=\$204$.

Check and interpretation. Expected claims fall by $\$6$, while TVaR falls by $\$120$ and the stated economic-capital measure falls by $\$114$. Expected earnings rise by $\$1$, because the $\$5$ premium reduction is more than offset by the $\$6$ reduction in expected claims. The method avoids treating premium as a reduction in gross claims TVaR; premium is included in the earnings calculation, not in the stated claims-loss variable.

This example also shows why the expected-loss change alone does not describe tail improvement. A small change in a low-probability, high-severity outcome can have a modest effect on the mean but a much larger effect on TVaR. Conversely, a change that removes a substantial amount of expected premium may improve tail risk while reducing expected earnings.

Investment Actions: Compare Expected Return and Downside

An investment change can reduce portfolio downside while also lowering expected investment earnings. For a simple one-year comparison, define investment loss $X$ as the negative of investment profit: a profit is a negative loss, and a loss is a positive loss. The loss convention must be maintained when calculating both the mean and the tail measure.

Worked Example: Moving to a Lower-Risk Asset Mix

A hypothetical insurer considers replacing a higher-risk asset mix with a more conservative mix. Under the current mix, annual investment profit is $\$40$ with probability $0.90$ and an investment loss of $\$100$ with probability $0.10$. Under the proposed mix, profit is $\$20$ with probability $0.95$ and loss is $\$60$ with probability $0.05$. Calculate expected annual investment profit, 95% TVaR of investment loss, and economic capital using $\operatorname{TVaR}_{0.95}(X)-E[X]$.

Current mix. Expected profit is $40(0.90)-100(0.10)=36-10=\$26$. In loss terms, $X=-40$ with probability $0.90$ and $X=100$ with probability $0.10$, so $E[X]=-26$. The worst 5% lies entirely within the 10% probability of a $\$100$ loss; thus TVaR is $\$100$. Economic capital is $100-(-26)=\$126$.

Proposed mix. Expected profit is $20(0.95)-60(0.05)=19-3=\$16$, so $E[X]=-\$16$. The worst 5% is exactly the $\$60$ loss outcome, giving TVaR of $\$60$. Economic capital is $60-(-16)=\$76$.

Check and interpretation. Expected investment profit falls by $\$10$, TVaR falls by $\$40$, and the stated economic-capital measure falls by $\$50$. The negative expected loss is appropriate here because the portfolio is expected to earn a profit. A common mistake is to replace $E[X]$ by the positive expected profit in the capital formula; with this definition, $X$ is loss, so $E[X]=-\$26$ for the current mix.

The calculation is a one-period illustration. Actual investment decisions may also affect asset-liability matching, liquidity, and changes in risk over time. Those considerations require additional modeling; the two-outcome distributions here do not imply that investment outcomes have only two possible values in practice.

Exposure Reduction: Scale Only When the Model Supports It

Reducing the number of policies, insured values, or units of exposure can reduce both expected claims and tail claims. It may also reduce premium and variable expenses, while fixed expenses may remain. A proportional calculation is justified only when the assumed scaling relationship is appropriate. For example, the following example explicitly assumes each claim amount scales by the same factor while event probabilities remain unchanged. It does not assume that every real portfolio can be reduced this way.

Worked Example: Reducing a Homogeneous Exposure Block

A hypothetical block has annual claims of $\$0$ with probability $0.90$, $\$40$ with probability $0.08$, and $\$300$ with probability $0.02$. Annual premium is $\$80$. Expenses consist of a fixed $\$3$ and variable expenses of $\$12$. The insurer removes 25% of the exposure. Assume every claim amount and variable expense scale in direct proportion to retained exposure, while the probabilities and fixed expense do not change. Calculate expected earnings, 95% claims TVaR, and economic capital before and after.

Before reduction. Expected claims are $40(0.08)+300(0.02)=3.20+6.00=\$9.20$. Expenses total $3+12=\$15$, so expected earnings are $80-9.20-15=\$55.80$. The worst 5% consists of all 2% at $\$300$ and 3% at $\$40$. Its loss-weighted amount is $300(0.02)+40(0.03)=6+1.20=\$7.20$, giving TVaR $7.20/0.05=\$144$. Economic capital is $144-9.20=\$134.80$.

After reduction. Retained exposure is 75%, so the claim outcomes become $\$0$, $\$30$, and $\$225$. Premium becomes $80(0.75)=\$60$, and variable expenses become $12(0.75)=\$9$; total expenses are $3+9=\$12$. Expected claims are $30(0.08)+225(0.02)=2.40+4.50=\$6.90$. Expected earnings are $60-6.90-12=\$41.10$. Tail loss-weighted amount is $225(0.02)+30(0.03)=4.50+0.90=\$5.40$, so TVaR is $5.40/0.05=\$108$. Economic capital is $108-6.90=\$101.10$.

Check and interpretation. Expected earnings decrease by $\$14.70$, claims TVaR decreases by $\$36$, and economic capital decreases by $\$33.70$. The mean and tail calculations both scale by 75%, as assumed; earnings do not, because the fixed expense remains $\$3$. The calculation would not be valid if the removed exposures had materially different claim probabilities or severities from those retained.

A Short Derivation: Why the Discrete Tail Uses Partial Probability

Suppose the worst tail has probability $q=1-\alpha$, and losses are ordered from largest to smallest. If the largest loss $x_1$ has probability $p_1$, it contributes $x_1\min(p_1,q)$ to the tail loss-weighted amount. If $p_1<q$, the next-largest loss contributes only enough probability to bring the included total to $q$. Repeating this process gives included probabilities $r_j$ satisfying $0\le r_j\le p_j$ and $\sum_j r_j=q$. The average loss in the selected tail is therefore $\sum_j x_jr_j/q$. This is precisely the discrete form of the quantile-integral definition: the quantile is integrated over a probability interval of width $q$, not over all the probability attached to the boundary outcome.

For instance, if a 5% tail contains a 2% probability of the largest loss and a further 3% of the next-largest loss, the numerator is $0.02x_1+0.03x_2$, not $0.02x_1$ plus the entire probability mass of the second outcome. This distinction is essential when a loss distribution is discrete.

Common Mistakes and Decision Use

Using all of a boundary atom. In the underwriting example, the worst 5% includes 2% at $\$700$ and only 3% at $\$100$. Including all 8% at $\$100$ would calculate $(700(0.02)+100(0.08))/0.05=\$440$, which incorrectly averages 10% probability while dividing by 5%. The correct TVaR is $(700(0.02)+100(0.03))/0.05=\$340$.
Mixing earnings and loss measures. Premium belongs in underwriting earnings, but it is not subtracted from gross claims TVaR unless the loss variable has explicitly been defined net of premium.
Ignoring fixed costs. When exposure falls, premium and variable expenses may scale while fixed expenses do not. Scaling the entire earnings result by the exposure factor can therefore be wrong.
Using a proportional model without support. A retained book may have a different risk mix from the business removed. A common scaling factor for all claim amounts is an assumption, not a general property of exposure reduction.
Calling one capital measure universal. TVaR minus expected loss is the stipulated economic-capital measure in these calculations. A different internal, rating-agency, or regulatory framework may define capital differently.

These measures make portfolio actions legible alongside external risk transfer: show the change in expected earnings, expected loss, tail loss, and the explicitly defined capital measure. They do not, by themselves, determine which alternative is preferable. The decision also depends on the insurer’s objectives, constraints, and the costs and terms of any transfer arrangement.

Check Your Understanding

Use the stated loss convention and include exactly the required probability in each tail.

A hypothetical underwriting block has annual claims of $\$0$ with probability $0.92$, $\$80$ with probability $0.06$, and $\$500$ with probability $0.02$. Find expected claims and 95% TVaR. (Answer: expected claims are $\$14.80$; 95% TVaR is $\$248$.)
A one-year investment portfolio earns $\$30$ with probability $0.80$ and loses $\$40$ with probability $0.20$. Define $X$ as negative investment profit. Find expected investment profit and 90% TVaR of loss. (Answer: expected profit is $\$16$; 90% TVaR is $\$40$.)
A block has claims of $\$0$ with probability $0.90$, $\$100$ with probability $0.08$, and $\$600$ with probability $0.02$. Under a stipulated proportional exposure reduction, every claim amount is multiplied by $0.60$, with probabilities unchanged. Find expected claims and 95% TVaR before and after. (Answer: expected claims fall from $\$20$ to $\$12$; TVaR falls from $\$300$ to $\$180$.)
Before an underwriting action, premium is $\$50$, expenses are $\$6$, and expected claims are $\$31$. Afterward, premium is $\$46$, expenses are $\$6$, and expected claims are $\$25$. Find expected earnings before and after and the change in earnings. (Answer: earnings rise from $\$13$ to $\$15$, an increase of $\$2$.)
A portfolio has expected loss $\$18$ and 95% TVaR of $\$260$ before an action. Afterward, expected loss is $\$15$ and TVaR is $\$215$. Using economic capital $\operatorname{TVaR}_{0.95}(X)-E[X]$, calculate capital before and after and the reduction. (Answer: capital falls from $\$242$ to $\$200$, a reduction of $\$42$.)

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