Tutorials › CAS Exam 9 - Risk Management for Actuaries › Economic Balance Sheets and Capital

Home  /  Tutorials  /  CAS Exam 9 / Economic Capital Foundations

CAS Exam 9 · Economic Capital Foundations · Tutorial 307 of 995

Economic Balance Sheets and Capital

Calculate economic surplus from consistently valued assets and liabilities, and see how cash-flow timing and valuation assumptions affect the result.

Intermediate 10 min read

What You'll Learn

  • Define economic asset value, economic liability value, and economic surplus at a valuation date
  • Discount specified future cash flows to a common date using the stated effective annual rates
  • Combine market-valued assets with assets valued from projected cash flows without double counting
  • Reconcile changes in surplus when a liability cash-flow assumption changes
  • Identify common errors in signs, timing, and use of rounded values

Why an Economic Balance Sheet Matters

A capital assessment needs a clear view of the resources available to meet obligations. An economic balance sheet provides that view by measuring assets and liabilities at a common valuation date on a specified economic basis. The difference between the two values is economic surplus. It is a useful starting measure of financial strength, but it is not, by itself, a measure of how much capital is required for a particular risk or confidence level.

The essential discipline is consistency: value both sides as of the same date, use cash flows appropriate to each item, and apply the valuation basis stated for the calculation. Economic values may be observed market values, or they may be estimated from projected cash flows when a direct market value is not available. The method depends on the information and assumptions supplied; there is no single discount rate that applies to every asset and liability in every setting.

Definition: At a specified valuation date, an economic asset value is the value assigned to the entity's resources on the stated economic basis. An economic liability value is the value assigned on that same date to its obligations on the stated basis. Economic surplus is economic asset value minus economic liability value:
$$ \text{Economic surplus}=A_{\text{econ}}-L_{\text{econ}}. $$
Here \(A_{\text{econ}}\) and \(L_{\text{econ}}\) denote the total economic values of assets and liabilities. A positive surplus means the measured assets exceed the measured liabilities; a negative surplus means the measured liabilities exceed the measured assets.

A balance sheet is a netting calculation, not simply a list of cash amounts. An asset cash flow expected in the future is not generally interchangeable with cash held today: its timing and valuation basis matter. Similarly, a liability due in the future is not automatically equal to its undiscounted amount at the valuation date. Each item must first be valued, then the totals can be compared.

Present Value of Specified Cash Flows

For a deterministic cash flow \(C_t\) due \(t\) years from the valuation date, with a stated effective annual discount rate \(i\) applying over those years, its present value is \(C_t/(1+i)^t\). If the cash flows are due at several dates and each has an applicable stated rate, add their present values. This calculation assumes the cash flows and rates are known for the purpose of the example; it does not, on its own, prescribe how uncertain cash flows should be valued.

$$ PV_0=\sum_{t=1}^{n}\frac{C_t}{(1+i_t)^t}. $$

The subscript 0 identifies the valuation date. If a single rate applies to all dates, \(i_t=i\) for each \(t\). The effective annual rate is compounded once per year; a cash flow two years away is divided by \((1+i)^2\), not by \(1+2i\).

The reason for discounting follows directly from the accumulation relationship. At effective annual rate \(i\), one unit invested today grows to \(1+i\) after one year and to \((1+i)^t\) after \(t\) years. Therefore an amount \(C_t\) due at time \(t\) corresponds to the amount \(PV_0\) today that accumulates to it:

$$ PV_0(1+i)^t=C_t \quad\Longrightarrow\quad PV_0=\frac{C_t}{(1+i)^t}. $$

For several cash flows, present values add because the total value of the separate obligations or receipts is the sum of their individual values under this valuation approach. This justification applies to the specified deterministic cash flows and rates. If cash flows are uncertain, valuation requires an appropriate treatment of that uncertainty and the applicable valuation basis; one should not use this deterministic formula as though the uncertainty had disappeared.

Worked Example: Calculating Surplus From Two-Sided Cash Flows

At the valuation date, a hypothetical insurer has an asset consisting of $10 million in cash and an asset paying $100 million with certainty in two years. Its liabilities are $80 million due with certainty in one year and $20 million due with certainty in two years. For this calculation, use a 5% effective annual rate for the two-year asset and the two-year liability, and a 4% effective annual rate for the one-year liability. All amounts are in millions of dollars. Calculate economic surplus.

Method. Value each cash flow at the valuation date using its stated effective annual rate, total the asset and liability values separately, and subtract liabilities from assets. The cash flows are treated as certain, and no additional adjustments are specified.

Assets. Cash is already at the valuation date, so its value is $10 million. The two-year receipt has value

$$ \frac{100}{(1.05)^2} =\frac{100}{1.1025} =90.7029478458\text{ million}. $$

Thus,

$$ A_{\text{econ}}=10+90.7029478458 =100.7029478458\text{ million}. $$

Liabilities. The one-year payment has value \(80/1.04=76.9230769231\) million. The two-year payment has value \(20/1.1025=18.1405895692\) million. Therefore,

$$ L_{\text{econ}} =76.9230769231+18.1405895692 =95.0636664923\text{ million}. $$

Surplus. Subtract the unrounded totals:

$$ 100.7029478458-95.0636664923 =5.6392813535\text{ million}. $$

The surplus is $5.6393 million to four decimal places in millions, or $5,639,281.35 to the nearest cent. As a check, the asset value exceeds the liability value, so the surplus must be positive. The method also avoids the common mistake of subtracting rounded intermediate totals: subtracting rounded figures can change the last reported digit.

Combining Market Values and Cash-Flow Values

An economic balance sheet can contain items valued in different ways, provided each value is appropriate to the stated basis and is measured at the same valuation date. For example, an asset may have an observed market value, while a separate asset without a supplied market value may be represented by a specified future receipt. Do not both include an asset's market value and add the cash flows that are already represented by that market value; that would count the same resource twice.

Worked Example: Mixing a Quoted Asset Value With Projected Cash Flows

A hypothetical company has $1,200 thousand in cash, a bond with a supplied market value of $4,650 thousand, and a separate asset that will pay $1,800 thousand with certainty in two years. Its liabilities are $5,000 thousand due in one year and $2,000 thousand due in two years. Use a 4% effective annual rate for the one-year liability and a 5% effective annual rate for the two-year cash flows. Amounts are in thousands of dollars. Find the economic surplus.

Method. Use the supplied market value for the bond, do not revalue it from unprovided cash flows, and discount the other stated cash flows using their specified rates.

Assets. Cash and the bond are already valued at the valuation date. The separate two-year receipt has value

$$ \frac{1{,}800}{(1.05)^2} =\frac{1{,}800}{1.1025} =1{,}632.653061\text{ thousand}. $$

So the total asset value is \(1{,}200+4{,}650+1{,}632.653061=7{,}482.653061\) thousand dollars.

Liabilities. The one-year liability is worth \(5{,}000/1.04=4{,}807.692308\) thousand dollars. The two-year liability is worth \(2{,}000/1.1025=1{,}814.058957\) thousand dollars. Thus,

$$ L_{\text{econ}} =4{,}807.692308+1{,}814.058957 =6{,}621.751265\text{ thousand}. $$

Subtracting gives \(7{,}482.653061-6{,}621.751265=860.901796\) thousand dollars, or $860,901.80 to the nearest cent. The result is positive, as expected because the total asset value exceeds the total liability value. A common mistake would be to discount the bond again even though its market value was already given; that would replace a supplied valuation with a different one without justification.

How Assumptions Affect the Balance Sheet

The balance sheet is conditional on its inputs. Changing an expected liability payment, its timing, or its applicable discount rate changes the liability value and therefore changes surplus, even if the asset side is unchanged. A useful review is to identify the affected item, recompute its value, and then trace the effect through the subtraction. A larger liability, all else equal, reduces surplus; a smaller liability increases it.

Worked Example: Updating a Liability Estimate

A hypothetical insurer has assets worth $55 million at the valuation date. Its liability cash flows are $30 million due in one year and $25 million due in three years. Treat these projected cash flows as fixed for this calculation; use a 4% effective annual rate for the one-year payment and a 5% effective annual rate for the three-year payment. A revised estimate increases the one-year payment by 10%, with all other inputs unchanged. Calculate surplus before and after the revision.

Method. Discount each liability cash flow at its stated rate, sum the present values, and subtract the total from the $55 million asset value. Then recalculate only the changed cash flow.

Before revision. The one-year payment has present value \(30/1.04=28.8461538462\) million. The three-year payment has present value

$$ \frac{25}{(1.05)^3} =\frac{25}{1.157625} =21.59593996\text{ million}. $$

Hence \(L_{\text{econ}}=28.8461538462+21.59593996=50.4420938062\) million, and the initial surplus is \(55-50.4420938062=4.5579061938\) million.

After revision. A 10% increase changes the one-year payment from $30 million to \(30(1.10)=33\) million. Its revised present value is \(33/1.04=31.7307692308\) million. The three-year value is unchanged, so the revised liability value is \(31.7307692308+21.59593996=53.3267091908\) million. Revised surplus is \(55-53.3267091908=1.6732908092\) million.

The increase in the liability's present value is \(3/1.04=2.8846153846\) million, and surplus falls by the same amount: \(4.5579061938-1.6732908092=2.8846153846\) million. This agreement is a useful check. The common mistake is to reduce surplus by the full $3 million increase in the future payment; the valuation-date impact is its present value, not its undiscounted future amount.

Interpretation and Common Traps

Economic surplus answers a specific question: on the stated valuation basis and at the stated date, how much do valued assets exceed valued obligations? It does not say that the surplus is immediately available as cash, nor does it establish how much capital is needed to withstand adverse outcomes. Those questions require additional definitions and risk analysis. This tutorial is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Before interpreting a result, check the balance-sheet construction:

  • Common valuation date: asset and liability values must refer to the same date.
  • Correct time units: a payment due in two years is discounted for two years, not one.
  • Specified valuation basis: use the rates and valuation information supplied, rather than silently substituting a different basis.
  • No double counting: do not add projected cash flows already represented by a supplied market value.
  • Correct sign: surplus is assets minus liabilities, not liabilities minus assets.
  • Rounding discipline: retain unrounded values through the subtraction and round only the final result.

A quick directional check catches many errors. Holding all else fixed, increasing an asset value raises surplus; increasing a liability value lowers it. A result moving in the opposite direction usually indicates a sign error, an incorrectly applied change, or a mismatch in the valuation date.

Key takeaway: First value each asset and liability on the specified economic basis at a common date; then calculate \(A_{\text{econ}}-L_{\text{econ}}\). Economic surplus is meaningful only when its inputs, cash-flow timing, and valuation assumptions are clear.

Check Your Understanding

For each problem, use only the valuation assumptions stated. Keep intermediate values unrounded and report the final surplus to four decimal places in millions unless another unit is specified.

  1. A hypothetical firm has $3.2 million in cash and an asset paying $5.5 million with certainty in two years. Liabilities are $4 million due in one year and $1 million due in two years. Use 3% for the one-year liability and 4% for both two-year cash flows. What is economic surplus? (Answer: $3.4770 million)
  2. A company has cash worth $0.6 million and a bond with a supplied market value of $12.4 million. Its only liability is $8.2 million due in one year; use a 4% effective annual rate to value that liability. Calculate surplus. (Answer: $5.1154 million)
  3. A hypothetical insurer has $2 million in cash and an asset paying $10 million with certainty in three years. Liabilities are $7 million due in two years and $3 million due in three years. Use 4% for the two-year liability and 5% for both three-year cash flows. Calculate surplus. (Answer: $1.5750 million)
  4. Assets have a valuation-date value of $11 million. The liability is a fixed payment of $10 million due in two years, discounted at 5% effective annually. The payment estimate rises by 6%, with the rate and timing unchanged. What is revised surplus? (Answer: $1.3855 million)
  5. A hypothetical entity has assets comprising a market-valued security worth $18 million, $2 million in cash, and a separate certain receipt of $4 million in one year valued at 4%. Its liabilities are $12 million due in one year valued at 3% and $8 million due in three years valued at 4%. Calculate economic surplus. (Answer: $5.0837 million)

Previous

Next

Back to Tutorials
Clinical Biostats