Multi-Year Capital Measurement

Why the Measurement Horizon Matters

A one-year capital measure describes losses over one year; it does not automatically describe the losses an insurer may accumulate over several years. If a portfolio remains exposed for three years, losses can arise in each of those years. A multi-year measure must therefore define the total loss over the full period, not just extend a one-year number by an assumed multiplier.

The key modeling distinction is between an annual incremental loss and a cumulative horizon loss. An annual incremental loss records what is incurred during one particular year. The cumulative horizon loss adds those annual amounts over the period being measured. The loss basis must be consistent across years: for example, the same treatment of expenses, recoveries, and the portfolio being measured.

Definition: Let \(L_t\) be the incremental loss during year \(t\), for \(t=1,\ldots,n\). The cumulative loss over an \(n\)-year horizon is \(L^{(n)}=\sum_{t=1}^{n}L_t\). For a selected confidence level \(\alpha\), define horizon VaR as \(\operatorname{VaR}_{\alpha}(L^{(n)})=\inf\{x:F_{L^{(n)}}(x)\geq\alpha\}\). Under the unexpected-loss convention used here, horizon economic capital is \(\operatorname{VaR}_{\alpha}(L^{(n)})-E[L^{(n)}]\).

This is a VaR-based definition of unexpected-loss capital. It separates the selected tail loss from the expected loss over the same horizon. Other capital frameworks may define a different basis; the loss measure, horizon, confidence level, and treatment of expected loss must be specified before interpreting a capital result.

For all worked examples below, amounts are hypothetical and stated in thousands of dollars. Losses are measured on a common valuation basis with no discounting, corresponding to an effective annual interest rate \(i=0\). Mortality is not modeled. Each example states its loss distribution and any assumption about how annual outcomes are related.

Building the Cumulative Loss Distribution

To measure capital over several years, first determine the possible values and probabilities of the sum of annual losses. If annual losses are independent and discrete, the probability of a particular combination of outcomes is the product of the annual probabilities. Add the losses in that combination, then combine probabilities whenever different combinations produce the same total.

The expected cumulative loss is simpler to calculate. By linearity of expectation,

E[L^{(n)}] =E\left[\sum_{t=1}^{n}L_t\right] =\sum_{t=1}^{n}E[L_t].

This equality does not require independence. However, the distribution of the sum—and therefore its VaR—depends on how annual losses occur together. Independence is one possible assumption, not a default fact about a multi-year risk.

Why expected losses add: For two annual losses, the expectation of their sum is the sum of their expectations because expectation is linear: \(E[L_1+L_2]=E[L_1]+E[L_2]\). By contrast, VaR is a quantile of a distribution, not an expectation. The quantile of a sum is generally not the sum of the separate quantiles. To find horizon VaR, form the distribution of the cumulative loss and take its quantile.

Worked Example: Two Years of Identically Distributed Losses

A hypothetical portfolio has an incremental loss in each of two years. In each year, loss is \(0\) with probability \(0.75\) and \(40\) with probability \(0.25\), in thousands of dollars. The annual losses are independent and identically distributed. Calculate the two-year 95% VaR and unexpected-loss capital, and compare them with the one-year measures.

Method and assumptions. Construct the distribution of the sum of the two annual losses. The distribution in each year is a two-point distribution with the probabilities and amounts stated above. There is no discounting (\(i=0\)), and mortality is not modeled.

The possible two-year outcomes are:

Neither year has a loss: total \(0+0=0\), with probability \(0.75\times0.75=0.5625\).
Exactly one year has a loss: total \(0+40=40\), with probability \(0.75\times0.25+0.25\times0.75=0.3750\).
Both years have a loss: total \(40+40=80\), with probability \(0.25\times0.25=0.0625\).

The probabilities sum to \(0.5625+0.3750+0.0625=1.0000\). The cumulative distribution is \(0.5625\) at \(0\), \(0.9375\) at \(40\), and \(1.0000\) at \(80\). Since \(0.9375<0.95\leq1.0000\), the two-year 95% VaR is \(80\) thousand dollars.

The expected loss in one year is \(0.75(0)+0.25(40)=10\) thousand dollars, so the two-year expected loss is \(10+10=20\) thousand dollars. Therefore two-year unexpected-loss capital is \(80-20=60\) thousand dollars.

For one year, the 95% VaR is \(40\) thousand dollars because the cumulative probability below \(40\) is only \(0.75\). One-year capital is \(40-10=30\) thousand dollars. The two-year result is not obtained merely by restating the one-year result: it follows from the two-year cumulative distribution. Common mistake avoided: treating a one-year VaR as if it already covered losses arising in both years.

Check: the two-year expected loss, \(20\) thousand dollars, equals the sum of the two annual expected losses, and the three aggregate probabilities total \(1.0000\).

Annual VaRs Do Not Generally Add

A tempting shortcut is to calculate a VaR for each year and add the results. That shortcut generally does not produce the VaR of cumulative loss. A horizon VaR depends on which annual outcomes can occur together and on the probability of each combined outcome. The next example shows the difference even when annual losses are independent.

Worked Example: Different Annual Loss Distributions

A hypothetical two-year portfolio has independent annual losses. In year 1, loss is \(0\) with probability \(0.80\) and \(20\) with probability \(0.20\). In year 2, loss is \(0\) with probability \(0.90\) and \(50\) with probability \(0.10\). Amounts are in thousands of dollars. Find two-year 95% VaR and capital, then compare the answer with the sum of the annual 95% VaRs.

Method and assumptions. Enumerate the combinations of the two independent annual outcomes, add losses within each combination, and then calculate the quantile from the resulting cumulative distribution. There is no discounting (\(i=0\)), and mortality is not modeled.

The four combinations are:

\(0\) in both years: total \(0\), probability \(0.80\times0.90=0.72\).
\(20\) in year 1 and \(0\) in year 2: total \(20\), probability \(0.20\times0.90=0.18\).
\(0\) in year 1 and \(50\) in year 2: total \(50\), probability \(0.80\times0.10=0.08\).
\(20\) in year 1 and \(50\) in year 2: total \(70\), probability \(0.20\times0.10=0.02\).

The probabilities sum to \(0.72+0.18+0.08+0.02=1.00\). The cumulative probabilities at totals \(0,20,50,70\) are \(0.72,0.90,0.98,1.00\), respectively. The first total at which the cumulative probability reaches at least \(0.95\) is \(50\). Thus, two-year 95% VaR is \(50\) thousand dollars.

Expected loss in year 1 is \(0.20(20)=4\) thousand dollars, and in year 2 it is \(0.10(50)=5\) thousand dollars. Two-year expected loss is \(4+5=9\) thousand dollars, so two-year capital is \(50-9=41\) thousand dollars.

For comparison, year 1 has 95% VaR \(20\), while year 2 has 95% VaR \(50\). Adding these annual VaRs gives \(20+50=70\), not the two-year VaR of \(50\). Common mistake and correction: the wrong calculation is \(\operatorname{VaR}_{0.95}(L_1)+\operatorname{VaR}_{0.95}(L_2)=20+50=70\). The correction is to form the aggregate distribution; its 95% quantile is \(50\). The annual VaRs identify separate annual thresholds, not the threshold for the sum.

Check: the expected cumulative loss is \(9\) thousand dollars both from the scenario distribution and from adding the annual expectations.

Reading a Multi-Year Result Carefully

A multi-year capital number is meaningful only in relation to its horizon and assumptions. A two-year VaR answers a different question from a one-year VaR, even when the underlying portfolio is unchanged. The longer horizon allows losses from multiple years to accumulate; it may also involve different annual loss distributions or a changing portfolio. A model must specify which of these features it includes.

The following example emphasizes the sequence of annual losses. It calculates the aggregate distribution directly rather than applying a multiplier to a one-year capital result.

Worked Example: Three-Year Cumulative Loss

A hypothetical portfolio has the same annual loss distribution in each of three years: loss is \(0\) with probability \(0.80\) and \(25\) with probability \(0.20\), in thousands of dollars. Annual losses are independent. Calculate the three-year 95% VaR and unexpected-loss capital.

Method and assumptions. Let \(K\) be the number of years, out of three, in which the \(25\)-thousand-dollar loss occurs. Under the stated independence assumption, \(K\) has a binomial distribution with three trials and probability \(0.20\) per trial. Cumulative loss is \(25K\). There is no discounting (\(i=0\)), and mortality is not modeled.

The probabilities for \(K=0,1,2,3\) are:

\(P(K=0)=0.80^3=0.5120\).
\(P(K=1)=3(0.20)(0.80^2)=3(0.20)(0.64)=0.3840\).
\(P(K=2)=3(0.20^2)(0.80)=3(0.04)(0.80)=0.0960\).
\(P(K=3)=0.20^3=0.0080\).

These probabilities sum to \(0.5120+0.3840+0.0960+0.0080=1.0000\). The possible cumulative losses are \(0,25,50,75\) thousand dollars. Their cumulative probabilities are \(0.5120,0.8960,0.9920,1.0000\). Since \(0.8960<0.95\leq0.9920\), the three-year 95% VaR is \(50\) thousand dollars.

Expected annual loss is \(0.20(25)=5\) thousand dollars. Expected three-year loss is \(3(5)=15\) thousand dollars, so three-year unexpected-loss capital is \(50-15=35\) thousand dollars.

Common mistake avoided: multiplying a one-year VaR or capital amount by three without deriving the three-year loss distribution. The stated assumptions allow the aggregate distribution to be calculated directly; a different annual relationship could produce a different result.

Check: the expected number of loss years is \(3(0.20)=0.60\), giving expected cumulative loss \(0.60(25)=15\) thousand dollars, as calculated above.

Practical Checks and Common Traps

Before reporting a multi-year capital measure, verify that the time basis and probability model match the question being answered. In particular:

Keep annual and cumulative losses distinct. Do not label an annual loss as a multi-year total or count a cumulative amount again in a later year.
Use the same loss basis throughout. Annual amounts should be measured consistently, and any discounting or change in exposures should be explicit.
Check the joint assumptions. If annual independence is used, state it. If it is not given, do not silently multiply annual probabilities.
Find the quantile from the cumulative distribution. For a discrete loss, use the smallest loss amount whose cumulative probability is at least the selected confidence level.
Separate VaR from capital. Under the unexpected-loss convention here, subtract expected cumulative loss from horizon VaR. Do not subtract only one year of expected loss from a multi-year VaR.

This approach is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus: specify the horizon, construct the loss measure over that horizon, and apply the selected capital definition to that distribution. The result should always be interpreted alongside its confidence level, loss basis, and assumptions about annual outcomes.

Key takeaway: Multi-year capital is based on the distribution of cumulative loss, \(L^{(n)}=\sum_{t=1}^{n}L_t\). Expected losses add by linearity, but horizon VaR must be determined from the aggregate distribution; it is generally not the sum of annual VaRs.

Check Your Understanding

For each problem, use the stated horizon and assumptions. Amounts are hypothetical. Where asked for capital, use horizon VaR minus expected cumulative loss.

A portfolio has annual losses of \(12\) thousand dollars in year 1 and \(18\) thousand dollars in year 2, with certainty. There is no discounting. What is the two-year cumulative loss? (Answer: \(30\) thousand dollars.)
For two independent years, annual loss is \(0\) with probability \(0.90\) and \(10\) thousand dollars with probability \(0.10\) in each year. Find the two-year 95% VaR and capital. (Answer: VaR \(=10\) thousand dollars; expected loss \(=2\) thousand dollars; capital \(=8\) thousand dollars.)
For three independent years, annual loss is \(0\) with probability \(0.90\) and \(20\) thousand dollars with probability \(0.10\). Find the three-year 99% VaR and capital. (Answer: VaR \(=40\) thousand dollars; expected loss \(=6\) thousand dollars; capital \(=34\) thousand dollars.)
In each of two years, a portfolio loses \(40\) thousand dollars if a shared event occurs and zero otherwise. The event has probability \(0.25\); it causes a loss in both years or neither year. Find the two-year 90% VaR and capital. (Answer: VaR \(=80\) thousand dollars; expected loss \(=20\) thousand dollars; capital \(=60\) thousand dollars.)
Two independent annual losses are specified as follows: year 1 is \(0\) with probability \(0.80\) or \(30\) thousand dollars with probability \(0.20\); year 2 is \(0\) with probability \(0.90\) or \(60\) thousand dollars with probability \(0.10\). Find the two-year 90% VaR and capital. (Answer: VaR \(=30\) thousand dollars; expected loss \(=12\) thousand dollars; capital \(=18\) thousand dollars.)

What You'll Learn