Solvency Ratios Under Different Valuation Bases

Why Valuation Bases Matter

A solvency coverage ratio is a comparison of available financial resources with a capital requirement. When the valuation basis changes, the ratio can change because assets are valued differently, liabilities are valued differently, the capital requirement changes, or more than one of these changes at once. A ratio comparison is useful only when its numerator and denominator are identified and measured consistently.

This tutorial treats valuation bases as alternative, internally consistent ways of measuring the same hypothetical insurer at the same reporting date. It does not assume a particular jurisdiction’s regulatory rules. Any stated capital amounts are treated as the required capital under the specified basis. The analysis is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $F$ denote eligible own funds available to meet the capital requirement, and let $K$ denote that requirement, with $K>0$. The solvency coverage ratio is $C=F/K$. It may be reported as a multiple, such as $1.50$, or as a percentage, such as $150\%$. If assets and liabilities are measured on a common basis and all net assets are eligible, then $F=A-L$, where $A$ is the value of assets and $L$ is the value of liabilities. If eligibility limits or other adjustments apply, $F$ need not equal $A-L$.

A valuation basis specifies how relevant balance-sheet items are measured. For this comparison, the basis must identify both the asset and liability values used, and the capital requirement must be the one calculated under that same basis. If the analysis holds the requirement constant, that is an explicit simplifying assumption, not a general property of solvency ratios.

Separate the Numerator and Denominator

The numerator is the amount of eligible own funds; the denominator is the required capital. Keeping them separate prevents a common interpretive error: a higher ratio does not necessarily mean that the insurer has more own funds. It may instead reflect a lower requirement. Conversely, a lower ratio can occur even if own funds rise, if required capital rises by a larger proportion.

When the eligible own funds are the net balance-sheet amount, changes in asset and liability values affect the numerator through their difference:

$$ F_2-F_1=(A_2-A_1)-(L_2-L_1). $$

Thus, an increase in assets raises net own funds if liabilities are unchanged, while an increase in liabilities reduces them if assets are unchanged. If both move, the net effect—not either movement alone—determines the change in own funds. Eligibility adjustments, if present, must also be reflected before the ratio is calculated.

Worked Example: Liability Revaluation with a Fixed Requirement

A hypothetical insurer has assets of $120$ million dollars under both valuation bases. Under Basis A, liabilities are $100$ million dollars and required capital is $10$ million dollars. Under Basis B, liabilities are $105$ million dollars and required capital remains $10$ million dollars. All net assets are eligible own funds; there are no eligibility adjustments. These are reported values at one date, with no projection, interest assumption, mortality assumption, or probability distribution needed.

Method. Calculate own funds as assets minus liabilities under each basis, then divide by the requirement under that basis.

Under Basis A, own funds are $120-100=20$ million dollars. The coverage ratio is $20/10=2.00$, or $200\%$.

Under Basis B, own funds are $120-105=15$ million dollars. The coverage ratio is $15/10=1.50$, or $150\%$.

The ratio falls by $200\%-150\%=50$ percentage points. The source is the $5$ million dollar increase in liabilities, which reduces own funds by $5$ million dollars while the requirement is unchanged. As a check, the own-funds change is $(120-120)-(105-100)=-5$ million dollars, matching the direct calculation $15-20=-5$ million dollars.

Common mistake avoided: Comparing the liability values alone does not give the ratio movement. The asset value and required capital must also be checked. Here, the requirement is explicitly fixed, so the entire ratio change comes through own funds.

Deriving a Ratio-Change Reconciliation

Suppose there are two valuation bases, indexed 1 and 2, with positive required capital $K_1$ and $K_2$. Their coverage ratios are $C_1=F_1/K_1$ and $C_2=F_2/K_2$. One exact way to explain the change is to change the numerator first while keeping the original denominator, then change the denominator while holding the new numerator fixed.

C_2-C_1 =\left(\frac{F_2}{K_1}-\frac{F_1}{K_1}\right) +\left(\frac{F_2}{K_2}-\frac{F_2}{K_1}\right) =\frac{F_2-F_1}{K_1}+F_2\left(\frac{1}{K_2}-\frac{1}{K_1}\right).

The first term is the numerator effect at the original requirement. The second is the denominator effect after the numerator has moved. This is an algebraic identity: the intermediate terms $F_2/K_1$ cancel, leaving $F_2/K_2-F_1/K_1$. The chosen order makes the reconciliation exact, but the individual contributions depend on the order of attribution. If the numerator is changed after the denominator instead, the split can differ even though the total change cannot.

This decomposition is useful for diagnosis, not a claim that the underlying effects are causally independent. For example, a change in liability valuation might also affect the calculation of required capital. If so, the reported changes in $F$ and $K$ should be measured under the stated bases, and the full ratio difference reconciled rather than attributing every movement to only one side.

Worked Example: Asset, Liability, and Requirement Changes Together

A hypothetical insurer reports the following values in millions of dollars. Under Basis A, assets are $500$, liabilities are $455$, and required capital is $30$. Under Basis B, assets are $510$, liabilities are $466$, and required capital is $22$. Assume all net assets are eligible, both bases refer to the same date and business, and the listed requirements are applicable to their respective bases. No interest, mortality, or probability assumptions are needed because the balance-sheet and capital amounts are provided.

Method. First calculate own funds and each ratio. Then use the numerator-first identity to allocate the exact ratio change.

Under Basis A, $F_1=500-455=45$ million dollars. Thus $C_1=45/30=1.50=150\%$.

Under Basis B, $F_2=510-466=44$ million dollars. Thus $C_2=44/22=2.00=200\%$. Own funds fell by $44-45=-1$ million dollars. The asset increase was $510-500=10$ million dollars, while the liability increase was $466-455=11$ million dollars, giving the same net change $10-11=-1$.

The total coverage-ratio increase is $200\%-150\%=50$ percentage points. The numerator contribution, using the original requirement, is $(44-45)/30=-1/30=-0.033333\ldots$, or $-3.33$ percentage points. The denominator contribution, using the new own funds, is $44(1/22-1/30)$. Since $1/22-1/30=8/660=2/165$, this term is $88/165=0.533333\ldots$, or $53.33$ percentage points. Together, $-3.33+53.33=50.00$ percentage points.

The higher ratio is therefore not evidence of increased own funds: own funds decreased. The lower capital requirement more than offsets that decrease in the ratio calculation. As a check, the directly calculated ratios are $150\%$ and $200\%$, and their difference agrees with the two-part reconciliation.

Common mistake avoided: It would be incorrect to attribute the whole increase to stronger own funds or to describe a $50$-percentage-point increase as a $50\%$ increase in own funds. The numerator fell, while the denominator fell enough to produce the higher coverage ratio.

Reading the Comparison Correctly

A disciplined comparison starts with the same reporting date and a clearly specified basis for each side of the ratio. It should show the asset value, liability value, resulting eligible own funds, and capital requirement. If own funds are adjusted for eligibility, that adjustment should be shown rather than silently assuming that all net assets count.

It is also important to distinguish a relative percentage change from a percentage-point change. Moving from $150\%$ to $200\%$ is an increase of $50$ percentage points. Relative to the initial ratio, it is an increase of $50/150=33.33\%$. These describe different comparisons and should not be substituted for one another.

The denominator must be positive for the ordinary coverage ratio and its comparison to be meaningful. A zero requirement makes division undefined; a negative requirement is not interpreted as a conventional solvency coverage ratio. If the requirement is positive but a ratio is near zero, the percentage form may be less informative than showing the underlying amounts as well.

When comparing two organizations, products, or dates, a ratio alone can conceal differences in valuation choices or capital calculation. The ratio does not standardize the inputs automatically. A responsible interpretation identifies whether a difference comes from asset measurement, liability measurement, eligibility adjustments, required capital, or a combination. Later analysis may examine further comparisons, but the arithmetic here is the essential first reconciliation.

Key takeaway: For each valuation basis, calculate eligible own funds and required capital on a consistent basis, then form $F/K$. Explain the change by separating numerator and denominator movements; do not infer the source from the ratio alone.

Common Exam Trap: Mixing a Liability Value with the Wrong Basis

Suppose assets are $200$ million dollars under both bases. Basis A has liabilities of $170$ million dollars and required capital of $15$ million dollars. Basis B has liabilities of $176$ million dollars and required capital of $12$ million dollars. Assume all net assets are eligible. A common incorrect calculation uses Basis A’s liabilities with Basis B’s requirement, producing $(200-170)/12=250\%$, and labels this the Basis B ratio.

That calculation mixes bases. Under Basis B, own funds are $200-176=24$ million dollars, so its ratio is $24/12=200\%$. Under Basis A, own funds are $200-170=30$ million dollars, so its ratio is $30/15=200\%$. The coverage ratios are equal, even though both own funds and requirements differ. The check is to write the numerator and denominator for each basis before dividing.

Check Your Understanding

For each question, use consistent values for each basis and distinguish percentage points from relative percentage changes where relevant.

A hypothetical insurer has assets of $90$ million dollars, liabilities of $80$ million dollars, and required capital of $5$ million dollars. Assume all net assets are eligible. Calculate own funds and the coverage ratio. (Answer: own funds $=\$10$ million; coverage ratio $=200\%$.)
Under Basis A, assets are $160$ million dollars, liabilities are $140$ million dollars, and required capital is $8$ million dollars. Under Basis B, assets are $158$ million dollars, liabilities are $142$ million dollars, and the requirement remains $8$ million dollars. Assume all net assets are eligible. Calculate both ratios and the change in percentage points. (Answer: Basis A $=250\%$; Basis B $=200\%$; change $=-50$ percentage points.)
Under Basis A, assets are $136$ million dollars, liabilities are $100$ million dollars, and required capital is $24$ million dollars. Under Basis B, assets are $133$ million dollars, liabilities are $100$ million dollars, and required capital is $22$ million dollars. Assume all net assets are eligible. Calculate both ratios and explain why they are equal. (Answer: both ratios $=150\%$; own funds and required capital each fall by $3$ million dollars.)
Under Basis A, assets are $300$ million dollars, liabilities are $270$ million dollars, and required capital is $15$ million dollars. Under Basis B, assets are $303$ million dollars, liabilities are $273$ million dollars, and required capital is $12$ million dollars. Assume all net assets are eligible. Calculate the ratios and identify the source of their difference. (Answer: Basis A $=200\%$; Basis B $=250\%$; own funds remain $\$30$ million, and the lower requirement causes the $50$-percentage-point increase.)
In a numerator-first reconciliation, own funds change from $48$ to $45$ million dollars and required capital changes from $32$ to $30$ million dollars. Calculate the starting and ending ratios and the two contributions to the change. (Answer: both ratios $=150\%$; numerator contribution $=-3/32=-9.375$ percentage points; denominator contribution $=+9.375$ percentage points; net change $=0$ percentage points.)

What You'll Learn