Solvency Projection and Capital Run-Off

Why Project Solvency Coverage?

A solvency coverage ratio is a snapshot: it compares eligible own funds with required capital at one valuation date. For a long-term contract, however, both quantities can change as benefits run off, premiums and expenses emerge, and capital requirements evolve. A projection makes those changes visible and helps distinguish a temporary rise in coverage from a sustained decline.

The projection method in this tutorial is deliberately general. It does not prescribe a particular regulatory capital formula or threshold. Instead, it takes the eligible own funds and required capital at each date as inputs, explains how to project own funds in a stated financial model, and shows how to interpret the resulting coverage. The examples use hypothetical values and are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: At projection date $t$, let $E_t$ denote eligible own funds and $K_t$ denote required capital, measured on the same valuation basis and in the same currency units. When $K_t>0$, the solvency coverage ratio is $R_t=E_t/K_t$. It is often reported as $100R_t\%$. The excess of own funds over required capital is $E_t-K_t$. If $K_t=0$, the ratio is undefined; report the amounts rather than treating the ratio as a meaningful infinite value.

A projection consists of values at successive dates, such as $t=0,1,2,\ldots,n$, where time is measured in years. At every date, use values consistent in timing, valuation basis, and scope. For example, comparing own funds after a distribution with capital measured before that distribution would not be a like-for-like coverage calculation. Required capital may be given directly as a projected schedule, or calculated under a separately specified capital model; it should not be inferred from the coverage ratio itself.

Rolling Eligible Own Funds Forward

A simple annual model is useful when the question supplies investment return, operating results, and distributions. Let $E_t$ be own funds at the start of year $t$. Assume an effective annual investment return $i$ is credited to those funds, $G_{t+1}$ is the net operating contribution during the year excluding that investment return, and $D_{t+1}$ is a distribution paid at year-end. Under this timing convention,

E_{t+1}=E_t(1+i)+G_{t+1}-D_{t+1}.

This is a model assumption, not a universal accounting identity. It assumes the stated return is earned on the opening funds for the full year, and that the stated operating contribution and distribution are reflected at year-end. If cash flows occur at different times, or if the return applies to a changing asset balance, the projection must use those timings instead. The net operating contribution should also be defined clearly so that items such as claims or expenses are not counted twice.

Set the projection basis.
Specify dates, units, valuation basis, return assumption, and the timing and meaning of each cash flow.

Project eligible own funds.
Apply the stated roll-forward consistently, including any distributions that reduce own funds.

Obtain required capital at each date.
Use the given capital schedule or the stated capital calculation; do not assume it falls at the same rate as own funds.

Calculate and interpret coverage.
Divide own funds by positive required capital at each date, compare with any stated threshold, and identify changes over time.

How Coverage Rises or Falls

Coverage depends on the relative movement of own funds and required capital, not on either amount in isolation. Both amounts can fall while the ratio rises, if required capital falls proportionally faster. Conversely, own funds can increase while coverage falls if required capital increases even faster.

The following derivative gives a precise local test when own funds $E(t)$ and capital $K(t)$ are differentiable and $K(t)>0$:

R(t)=\frac{E(t)}{K(t)} \qquad\Longrightarrow\qquad R'(t)=\frac{E'(t)K(t)-E(t)K'(t)}{K(t)^2}.

The denominator is positive, so the sign of $R'(t)$ is the sign of $E'(t)K(t)-E(t)K'(t)$. If both amounts are positive, divide this expression by $E(t)K(t)$. Coverage rises exactly when the proportional growth rate of own funds exceeds the proportional growth rate of required capital:

R'(t)>0 \quad\Longleftrightarrow\quad \frac{E'(t)}{E(t)}>\frac{K'(t)}{K(t)}.

This result follows directly from the quotient rule and explains why comparing only the dollar changes can be misleading: a change of 2 is proportionally large for a balance of 10 but small for a balance of 100. For projections at discrete dates, calculate the ratios at each date. To compare two dates with positive capital, $R_{t+1}>R_t$ precisely when $E_{t+1}K_t>E_tK_{t+1}$; comparing these cross-products avoids prematurely rounding the ratios.

Worked Example: Finding the Highest Coverage Date

A hypothetical insurer projects eligible own funds and required capital, both in thousands of dollars, at three annual dates. The values are $E_0=24$, $E_1=25$, $E_2=21$, and $K_0=22$, $K_1=20$, $K_2=18$. Calculate the coverage at each date, identify the highest coverage, and find the first date at or above a 120% threshold.

Method. Divide own funds by required capital at each date, then compare the resulting ratios. Required capital is positive throughout, so the ratio is defined.

At time 0, $R_0=24/22=1.090909\ldots$, or $109.09\%$ to two decimal places. At time 1, $R_1=25/20=1.25$, or $125.00\%$. At time 2, $R_2=21/18=1.166666\ldots$, or $116.67\%$. The highest coverage is therefore $125.00\%$, at time 1. The first projected date at or above $120\%$ is also time 1.

As a check, the excess own funds at time 1 are $25-20=5$ thousand dollars, and $5/20=25\%$; thus own funds are 25% above required capital, consistent with 125% coverage.

Common mistake avoided. Writing $25/20\times100=133.33\%$ is an arithmetic error: $25/20=1.25$, so the correct percentage is $125.00\%$. Do not choose a highest date or threshold crossing based on a miscomputed quotient.

Projecting the Ratio from a Funds Roll-Forward

When a question provides the financial components rather than future own-funds amounts, apply the roll-forward first. Only then compare projected funds with the required-capital schedule. The following example keeps investment return, operating contribution, and distribution separate so their effects can be checked.

Worked Example: Projecting Coverage with Annual Distributions

A hypothetical long-term contract has opening eligible own funds of \$50 thousand. Assume an effective annual return of $4\%$. In year 1, the net operating contribution excluding investment return is \$5 thousand and a \$3 thousand distribution is made at year-end. In year 2, the corresponding contribution is \$2 thousand and the distribution is \$4 thousand. Required capital at times 0, 1, and 2 is \$45 thousand, \$40 thousand, and \$38 thousand. Project own funds and coverage, and determine the first date at or above $130\%$.

Method. Apply the annual roll-forward with the same timing in each year. All monetary amounts below are in thousands of dollars.

At time 0, $E_0=50$. During year 1, investment return is $50(0.04)=2$. Therefore,

$$ E_1=50(1.04)+5-3=52+5-3=54. $$

The year 1 coverage is $54/40=1.35=135.00\%$. During year 2, investment return on opening funds is $54(0.04)=2.16$. Therefore,

$$ E_2=54(1.04)+2-4=56.16+2-4=54.16. $$

The time 0 coverage is $50/45=1.111111\ldots=111.11\%$. The time 2 coverage is $54.16/38=1.425263\ldots=142.53\%$. The projected sequence is thus $111.11\%$, $135.00\%$, and $142.53\%$. The first date at or above $130\%$ is time 1.

The arithmetic also checks against the cash flows: the year 1 increase in own funds is $54-50=4$, equal to return $2$ plus contribution $5$ less distribution $3$; the year 2 increase is $54.16-54=0.16$, equal to $2.16+2-4$. A common error is to omit the distribution or subtract it twice by including it already in the net operating contribution. The stated definition of each cash-flow input determines the correct treatment.

Thresholds Between Projection Dates

A projection may be reported only at annual dates, while a question asks when coverage first reaches a threshold. A time within an interval cannot be inferred uniquely from endpoint values alone. One must state an interpolation assumption. If, for example, own funds and required capital are each assumed to change linearly between two dates, interpolate the amounts and solve $E(t)=rK(t)$, where $r$ is the target coverage as a decimal. Interpolating the ratio itself is a different assumption and generally gives a different crossing time.

Worked Example: Locating a Decline Through a Threshold

A hypothetical run-off projection, in thousands of dollars, gives $(E_0,K_0)=(40,30)$, $(E_1,K_1)=(34,24)$, and $(E_2,K_2)=(26,20)$. Assume own funds and required capital each change linearly between annual dates. Determine whether coverage rises or falls over each interval, and find when it first falls to $135\%$ during the second year.

Method. Calculate endpoint ratios, then use the linearly interpolated amounts to solve the threshold equation. At times 0, 1, and 2, coverage is $40/30=133.33\%$, $34/24=141.67\%$, and $26/20=130.00\%$, respectively, to two decimal places.

For a more precise direction check, write $s$ for time elapsed within an interval. In year 0 to 1, $E=40-6s$ and $K=30-6s$, with $0\leq s\leq1$. The sign of the ratio derivative is the sign of

$$ (-6)(30-6s)-(40-6s)(-6)=60>0. $$

Thus coverage rises throughout the first interval. In year 1 to 2, let $s$ again denote time elapsed since the start of that interval. Then $E=34-8s$ and $K=24-4s$. The corresponding sign expression is

$$ (-8)(24-4s)-(34-8s)(-4)=-56<0. $$

Coverage therefore falls throughout the second interval. To find when it reaches $135\%$, solve $(34-8s)/(24-4s)=1.35$. Since the denominator is positive on the interval,

34-8s=1.35(24-4s)=32.4-5.4s, \qquad 1.6=2.6s, \qquad s=\frac{8}{13}=0.61538\ldots.

The threshold is reached $8/13$ of a year after time 1, or at time $1.62$ years to two decimal places. It is the first such crossing: coverage at time 1 is above $135\%$, rises during the first interval, and then decreases continuously through $135\%$ during the second. The interpolation assumption is essential; endpoint data alone do not establish this fractional-year timing.

Interpreting a Run-Off Projection

Run-off does not imply that coverage must move in one direction. Benefits and associated capital requirements may reduce as exposure declines, while own funds may also change through investment return, operating experience, or distributions. A useful interpretation reports both the ratio and its components. For example, a rising ratio caused by a faster reduction in required capital is not the same as a rise caused by additional own funds.

A stated threshold is a comparison point, not an automatic conclusion about every solvency framework. Apply the threshold to the date and basis specified in the problem. If the question asks for the first projected date meeting it, use the discrete dates; only report an intermediate crossing if an interpolation rule is supplied or explicitly assumed. When coverage is close to a threshold, retain sufficient precision during calculations and round only the final reported result.

Check units and basis: eligible own funds and required capital must be stated on a comparable basis.
Check cash-flow timing: an end-year distribution does not reduce opening funds in the roll-forward shown here.
Check both quantities: a falling own-funds amount does not by itself imply falling coverage.
Check the denominator: a zero required-capital amount makes the ratio undefined, not a valid coverage percentage.
Check threshold wording: “at or above” includes equality; “above” does not.

Key takeaway: A defensible solvency run-off projection keeps own funds, required capital, timing, and valuation basis consistent. Calculate coverage at each date, explain its movement through the relative changes in both amounts, and use an explicit interpolation assumption for any within-period threshold crossing.

Check Your Understanding

For each hypothetical projection, retain full precision during the calculation and round reported coverage to two decimal places.

At two dates, eligible own funds and required capital are $(18,16)$ and $(20,15)$, in thousands of dollars. Calculate both coverage ratios and identify the first date at or above $125\%$. (Answer: $112.50\%$ and $133.33\%$; the second date is the first at or above $125\%$.)
Opening own funds are \$80 thousand. At an effective annual return of $3\%$, the first-year net operating contribution excluding return is \$4 thousand and the year-end distribution is \$2 thousand. In year 2, these amounts are \$3 thousand and \$5 thousand. Required capital at times 0, 1, and 2 is \$70 thousand, \$65 thousand, and \$60 thousand. Project own funds and coverage, and identify the first date at or above $130\%$. (Answer: own funds are \$80.00 thousand, \$84.40 thousand, and \$84.932 thousand; coverage is $114.29\%$, $129.85\%$, and $141.55\%$; time 2 is the first date at or above $130\%$.)
Own funds decline linearly from 48 to 42 thousand dollars over one year, while required capital declines linearly from 40 to 32 thousand dollars. Find the first time coverage reaches $125\%$. (Answer: $0.50$ years after the start.)
Own funds and required capital move from $(32,24)$ thousand dollars to $(28,20)$ thousand dollars over one year. Calculate endpoint coverage and state whether coverage rises or falls, despite both amounts declining. (Answer: coverage rises from $133.33\%$ to $140.00\%$, an increase of $6.67$ percentage points.)
A run-off projection gives own funds of 60, 55, and 48 thousand dollars and required capital of 50, 42, and 40 thousand dollars at times 0, 1, and 2. Calculate coverage at each date, find the highest coverage, and identify the dates at or above $125\%$. (Answer: $120.00\%$, $130.95\%$, and $120.00\%$; highest coverage is $130.95\%$ at time 1, and only time 1 is at or above $125\%$.)

What You'll Learn