Investment Stress in Asset Shares and Solvency

Why Investment Stress Must Be Traced Through the Whole Balance Sheet

An adverse investment return can reduce the assets supporting a policy, lower its asset share, and weaken the insurer’s solvency position. It does not necessarily reduce every liability by the same amount. A guaranteed benefit may remain fixed even as the fund backing it falls; a discretionary terminal bonus may be reduced if the product terms permit it. A sound stress calculation keeps these effects separate.

This tutorial uses annual, end-of-year projections for hypothetical products. The examples isolate investment experience: premiums, expenses, mortality, lapses, and other assumptions are held unchanged unless stated otherwise. This makes the effect of the investment stress visible without combining it with other stresses.

Definition: An asset share is a per-policy accumulation of premiums and other allocated amounts, less allocated expenses and benefits, with investment returns applied according to the specified timing. It is an experience-based measure of funds attributed to a policy; it is not automatically equal to the policy reserve or to the insurer’s total assets. Own funds are, in a simplified balance-sheet calculation, the value of assets less the value of liabilities. A solvency surplus is own funds less the stated capital requirement.

Projecting an Asset Share Under a Stressed Return

Let $A_{t-1}$ be the asset share at the start of policy year $t$. Suppose the premium $P_t$ is received at the start of the year, the allocated expense $E_t$ is paid then, and any benefit $C_t$ is paid at the end of the year. If the effective investment return for the year is $j_t$, the annual recursion is

A_t=(A_{t-1}+P_t-E_t)(1+j_t)-C_t.

The timing is part of the model. In this formula, the premium net of the start-of-year expense earns the full year’s return, while the end-of-year benefit does not. A different cash-flow timing requires a different accumulation.

Derivation: Immediately after the start-of-year premium and expense, the accumulated amount is $A_{t-1}+P_t-E_t$. A one-year effective return $j_t$ multiplies that amount by $1+j_t$. Subtracting the end-of-year benefit $C_t$ gives the closing asset share. If there is no benefit during the year, set $C_t=0$; if a benefit is paid earlier, it must be accumulated or deducted at its actual payment time.

Worked Example: Projecting a Two-Year Asset Share

A hypothetical endowment policy has a zero opening asset share. At the start of each of two years, it receives a premium of $1,000 and incurs an allocated expense of $100. There are no claims or other benefits during the projection. The investment return is 5% in year 1 and 5% in year 2 under the base scenario. Under the stress scenario, the return is −10% in year 1 and 3% in year 2. Calculate both closing asset shares and the stress-related reduction.

Method. Apply the annual recursion separately in each year, carrying the unrounded result forward. This avoids treating a two-year return as though it were a single-year rate.

Base scenario, year 1. The amount invested after the premium and expense is $0+1{,}000-100=900$. Thus $A_1=900(1.05)=945$.

Base scenario, year 2. The amount invested is $945+1{,}000-100=1{,}845$. Thus $A_2=1{,}845(1.05)=1{,}937.25$.

Stress scenario, year 1. The same $900 is invested, but the return is −10%. Thus $A_1^{\mathrm{stress}}=900(0.90)=810$.

Stress scenario, year 2. The amount invested is $810+1{,}000-100=1{,}710$. Thus $A_2^{\mathrm{stress}}=1{,}710(1.03)=1{,}761.30$.

The closing asset-share reduction is $1{,}937.25-1{,}761.30=\$175.95$. As a check, each year’s invested amount is the opening asset share plus $900, and the final stress asset share is lower than the base value. The calculation does not imply that a policy benefit falls by exactly $175.95; that depends on the benefit design and guarantees.

From Asset Share to Benefits and Reserves

An asset share is a projection measure. A reserve is a liability valuation on a specified valuation basis, and the insurer’s balance-sheet assets are measured separately. Consequently, an investment shock can reduce the asset share while leaving a guaranteed benefit—and therefore its reserve—unchanged. If a discretionary benefit depends on surplus or fund performance, its projected amount may change, but only in accordance with the product terms and the valuation assumptions.

For a simple solvency comparison, let $A$ denote the insurer’s assets and $R$ the value of its liabilities. Own funds are $OF=A-R$. If a capital requirement $K$ is given, the solvency surplus is $OF-K$. These definitions are deliberately distinct from the policy’s asset share: the asset share may inform product analysis, but it cannot be substituted for $A$, $R$, or $OF$ without an explicit model linking them.

Worked Example: Lower Discretionary Bonus and the Own-Funds Change

One year before maturity, a hypothetical participating portfolio has assets of $10,500. Its projected maturity benefit comprises a guaranteed amount and a discretionary terminal bonus. On the valuation basis, the base-scenario benefit is $10,500, while an adverse investment scenario reduces the projected discretionary bonus so that the benefit is $9,000. The valuation discount rate is an effective annual rate of 4%, and each benefit is payable in one year. Assets fall to $8,400 under the investment stress. Calculate the reserves and own funds in both scenarios, then calculate the change in own funds.

Method. Discount each one-year benefit at 4%, subtract the resulting reserve from the assets in that scenario, and calculate the change using unrounded values. The bonus is treated as reducible here because the scenario explicitly says it is discretionary; a guaranteed amount could not be reduced on that basis.

The discount factor is $v=1/1.04$. The base reserve is $10{,}500/1.04=\$10{,}096.153846\ldots$, so base own funds are $10{,}500-10{,}096.153846\ldots=\$403.846153\ldots$.

The stressed reserve is $9{,}000/1.04=\$8{,}653.846153\ldots$. Stressed own funds are $8{,}400-8{,}653.846153\ldots=-\$253.846153\ldots$.

The change in own funds is calculated directly from the unrounded figures:

\left(8{,}400-\frac{9{,}000}{1.04}\right) - \left(10{,}500-\frac{10{,}500}{1.04}\right) =-657.692307\ldots

Therefore, to the nearest cent, own funds decrease by $657.69. The reserve decreases by $10{,}500/1.04-9{,}000/1.04=\$1{,}442.307692\ldots$, or $1,442.31. The $2,100.00 fall in assets is partly offset by that $1,442.31 reserve decrease.

Check and rounding trap. The rounded own-funds values are $403.85 and −$253.85. Subtracting those displayed values gives −$657.70, a one-cent rounding artifact. The correct change to the nearest cent is −$657.69 because it is calculated from the unrounded amounts. Carry full precision through intermediate calculations, then round the requested results.

When a Guarantee Does Not Fall with the Fund

A guarantee creates an important asymmetry. If an account value is below a guaranteed maturity amount, an investment loss can reduce the supporting assets without reducing the amount owed to the policyholder. The insurer may then face a larger funding gap. This differs from the discretionary-bonus example: a benefit can be reduced only where the contract and applicable valuation basis allow that treatment.

Worked Example: Guaranteed Maturity Benefit and Stressed Solvency

A hypothetical equity-linked policy is one year from maturity. Its fund value and the insurer’s assets attributable to the contract are initially $10,000. The contract guarantees a maturity payment of $9,000, and there is no additional discretionary benefit. An adverse investment return of −20% reduces the fund and assets to $8,000. The reserve basis uses an effective annual discount rate of 4%, the benefit is payable in one year, and the insurer’s stated capital requirement is $500. Calculate the maturity payment, reserve, own funds, and solvency surplus after the stress. Compare the stressed position with the position before the stress, assuming the reserve basis and guarantee are unchanged.

Method. First apply the investment return to the fund. Then value the contractual guarantee independently of the fund, discount the resulting benefit, and compare assets with that reserve. This avoids the mistake of treating the guaranteed amount as if it were simply the stressed account value.

The stressed fund value is $10{,}000(1-0.20)=\$8{,}000$. Since the guarantee is $9,000 and there is no other benefit, the policyholder receives $\max(8{,}000,9{,}000)=\$9{,}000$ at maturity. The insurer must provide a $1,000 top-up relative to the fund value.

The reserve is $9{,}000/1.04=\$8{,}653.846153\ldots$. After the stress, own funds are $8{,}000-8{,}653.846153\ldots=-\$653.846153\ldots$. The solvency surplus is $-653.846153\ldots-500=-\$1{,}153.846153\ldots$.

Before the stress, assets are $10,000 and the same guarantee produces the same reserve of $8,653.846153… . Base own funds are therefore $1,346.153846… and the base solvency surplus is $846.153846… . Thus the asset loss is $2,000, the reserve is unchanged, and own funds and solvency surplus each decline by $2,000. To the nearest cent, stressed own funds are −$653.85 and stressed solvency surplus is −$1,153.85; the policyholder still receives $9,000 at maturity. This check confirms that the unchanged guarantee, rather than the reduced fund, determines the benefit.

Reading an Investment Stress Correctly

The examples illustrate why an investment stress should be followed through a sequence rather than summarized as “assets fall.” First project the fund or asset share using the specified return and cash-flow timing. Next determine the contractual benefit, distinguishing guaranteed amounts from benefits that may be discretionary. Then revalue liabilities on the stated basis, update assets, and calculate own funds and any stated capital margin.

Apply the stressed return.
Use the effective return for each stated period and the correct timing of premiums, expenses, and benefits.

Recalculate policy benefits.
Apply contract terms: a fund-linked benefit may move with the fund, while a guarantee can impose a minimum payment.

Revalue liabilities.
Use the specified reserve basis and distinguish fixed guarantees from discretionary amounts that are permitted to change.

Reconcile solvency.
Calculate assets less liabilities, then subtract the stated capital requirement if a solvency surplus is requested.

A frequent exam trap is to assume that a fall in asset share automatically reduces the reserve by the same amount. That conclusion is not valid without a link between the asset-share projection and the liability valuation. Another trap is to apply a lower projected bonus to a guaranteed benefit. The benefit terms—not the direction of the market shock—determine what can change.

Key takeaway: Keep four quantities distinct: the policy asset share, the insurer’s assets, the policy benefit, and the reserve. An adverse return can lower the first two while leaving a guarantee unchanged; solvency depends on the resulting difference between assets and liabilities.

Check Your Understanding

For each problem, use the stated timing and assumptions, and carry unrounded values through any intermediate calculations.

A policy has an opening asset share of $600. At the start of the year it receives a premium of $400 and incurs an expense of $40. There are no benefits, and the annual effective return is −5%. Find the closing asset share. (Answer: $912.00)
A fund of $2,000 has no cash flows over two years. Its annual effective returns are −10% in year 1 and +10% in year 2. Find the ending fund and explain why the result is not the original $2,000. (Answer: $1,980.00; the sequential accumulation factor is $0.90(1.10)=0.99$)
A policy has an opening asset share of $5,000. At the start of the year it receives $1,000, incurs an expense of $100, and earns −8% over the year. At maturity it is guaranteed $5,600. Find the closing asset share and the guarantee top-up required at maturity. (Answer: Asset share $5,428.00; top-up $172.00)
An insurer has assets of $7,500, a reserve of $7,200, and a capital requirement of $250. An investment stress reduces assets by 12%; the reserve and capital requirement are unchanged. Find stressed own funds and the stressed solvency surplus. (Answer: Own funds −$600.00; solvency surplus −$850.00)
Before an investment stress, assets are $12,000 and the reserve is $11,500. Under the stress, assets fall by $1,500 and the reserve falls by $400 because a discretionary benefit is reduced. Calculate the change in own funds and the stressed own funds. (Answer: Own funds decrease by $1,100.00; stressed own funds are −$600.00)

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