# A Shortcut to Calculating Return on Required Equity and It s Link to Cost of Capital

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2 Statutory Reserve (S) = N x GAAP Reserve (G) Where N > 1 In theory every reserve dollar saved will be spent on benefits or paid out in annuity payments, thus over the lifetime of the block total payments will equal the reserve, G, we just defined. Total Benefits (B) = GAAP Reserve (G) = Premiums (P) x TLR There are several studied numbers needed to complete the calculation 1. The equivalent Ci factors used to compute the block s Risk Based Capital (RBC) calculation. These factors are applied to premium or statutory reserve balances. For example to cover the C1 risk a company might need to hold 6% of premium and 2% of statutory reserves. 2. A measure of the covariance among the Ci factors. To continue the example the summation of the C1-C4 factors might be 30% of premium and 10% of reserves, but the factors could have a 20% covariance between each other, a covariance factor of 80%. 3. The target RBC percentage (RBC%), this is a enterprise level business decision to hold a given premium over the minimum RBC capital needed for solvency. For instance a company may decide to maintain a 400% RBC ratio to target a specific credit rating on its debt. 4. Valuation capital, which is the percentage difference between Statutory and GAAP reserves that will have to be held as capital. For example if the corporate tax rate is 35% and the specific products only obligation to hold extra capital above its RBC level is to fund the deferred tax asset then the total amount of valuation capital is 35%x(statutory GAAP reserves). Depending on the accounting treatment this capital is held against the reserve differences, DAC amortization, or deferred tax asset. Sometimes all three. The idea of the process is to write the numerator and denominator of RORE, post-tax AOI and required equity respectively, in terms of their percentage of GAAP reserve, then when taking the ratio all GAAP reserves will cancel out, leaving an RORE defined purely in terms of the pricing, RBC, and valuation ratios. The working equation for RORE, on a present value basis, is Post Tax Adjusted Operating Income (AOI) RORE= Required Equity Note that the numerator is dependent on the denominator, part of AOI is the investment return on required equity. Due to this dependency the equations below first expand required equity, writing everything in terms of the GAAP reserve. Required Equity = RBC Capital+Valuation Capital

3 Which can be expanded to Equity = ( i=1 to 4 Cp i P + i=1 to 4 Cr i STAT Reserve)(Target RBC)(Covariance)+ X%(STAT Reserve GAAP Reserve) After substituting the relationships from the first paragraphs of this section every term can be related to the GAAP reserve, which is then factored out 1 Equity = GAAP Reserve Cp i + Cr TLR i N (Target RBC)(Covariance) + X% N- 1 Let the first term in the second factor generically be called the RBC capital ratio. This simplifies the result to GAAP Reserve [RBC Capital Ratio +X%(N- 1)] Proceeding further, the adjusted operating income numerator can be written in terms of the GAAP Reserve as well. AOI=Premium-Bene its-expenses+net Investment Income AOI=GAAP Reserve 1 Expense Margin 1 + I TLR TLR 1 + I 2 (Equity) Where the expense margin is the SG&A expenses as a percent of premium, I1 is the rate of return on GAAP reserves and I2 is the rate on required equity. When calculating RORE as the ratio of the post tax AOI to required capital the GAAP reserves now cancel out, leaving an estimate based completely on the pricing ratios, capital targets, and tax rate. RORE = 1 TLR Margin 1 Expense +I TLR 1 +I 2 (RBC Capital Ratio +X%(N 1)) RBC Capital Ratio +X%(N 1) 1-Tax Rate (1) Equation 1 above has a number of important features that describe the business RORE is defined solely in terms of the business ratios targeted by the company without reference to volume. The number of factors used in the calculation is minimal, requiring the TLR, expense margin, expected returns on GAAP reserves and required equity, and RBC target, with the X% factor, N factor, C factors and covariance the result of straightforward studies. There is a feedback component in that increases in required equity also increase AOI in the numerator, dampening the impact of equity increases. AOI is a GAAP based reporting metric of income statement earnings while required equity is a balance sheet quantity calculated using statutory principles and adjusted to a GAAP basis. Hence the AOI is a cumulative sum over a quarter, year, etc while the

4 required equity is an average of the balance sheet quantity over the corresponding period of time. The first term in the numerator is the underwriting gain priced into the business, company or product. For example if a product is sold for a \$100 premium with a target loss ratio of 90% there is an expectation to retain \$10. Given a GAAP reserve of \$90 we can anticipate the same \$10 retained by evaluating the formula: 90(1/90%-1)=\$10. The second term evaluates to the full expense margin of the product or business. We can expand the example above to show that if the TLR was 90% and expense margin 5% the total retention would be 15%, and that on \$100 of premium the expense margin is evaluated to 90(5%/90%)=\$5 as predicted by the 5% margin assumption The third term in the numerator equates to the net investment income expected to be generated, with an adjustment due to the equity ratio. Based on the points above the formula can be written in a more business conscious way RORE = Post Tax Underwiting Margin+Expense Margin+Net Investment Income Equity In this form a profit margin is not explicitly stated. It is assumed to be included in the target loss ratio but it can be embedded in the expense assumptions or listed separately without loss of generality. The remaining sections will discuss the implications and usages of this relationship in reference to total company value.

5 3. RORE Optimization and Cost of Capital While RORE can be calculated at a case or policy level, product, or reporting group level, it is useful in determination of company value through calculation at the highest level. When calculated for an entire company the evident question is this: which of the dependant ratios, TLR, expenses, interest, etc, is most controllable at a senior executive level? Target loss ratio: although TLR is controllable at the highest company level in theory actual pricing policy is often set by high competitive pressure and internal expertise at all levels of the company and can be difficult to change in practice. Achieving a different TLR also has a drastic impact on AOI, leading to high earnings volatility and RORE fluctuations. Expense margin: company policy does dictate expense margin at all levels, but there is constant pressure from external stakeholders to drop expenses to their lowest level possible. Increases in expense margins are often seen as temporary investments while expense cuts are deemed permanent targets. Indeed, the equation shows that RORE is positively correlated to expense margin as it has a direct impact on AOI. Investment Returns: Both the return on GAAP reserves and required equity are difficult to change as insurers work under restricted investment conditions and regulatory pressure. In addition these returns are usually shared with the policyholders in some way. Increases in investment income are usually parlayed into better pricing and market share components. The X%, N, and RBC Covariance factors: These factors are largely prescribed or derived from prescribed calculations and are difficult to change.

6 Tax Rate: Corporate tax rates are largely out of a company s control, but it is worth noting that the effective tax rate can be managed in varying ways. The assumption is that RORE will be calculated at the corporate rate and is not a changeable quantity. The Target RBC ratio: Of all the relevant factors this is the most relevant to debate. Although under external pressure for increases from rating agencies and decreases from investors there is a business decision on what ratio should be targeted at a company and often business unit level. As the target RBC ratio increases the equity will increase, putting pressure on the RORE, but since the numerator and denominator of the RBC ratio are both impacted by the target RBC ratio there is feedback created that dampens the decrease in RORE as equity rises, as an example below with a common set of ratios demonstrates. There is a balancing act that must be performed when setting the RBC target. Investors don t want a company that hordes capital to achieve an AAA rating when they could be paying them dividends and have a comfortable AA rating, hence there is pressure to lower the target. On the other hand regulators and rating agencies want as high a target as possible for solvency and credit rating s sake, not to mention that if similar companies raise or lower their target there is pressure to maintain the same level as the competition. Movements in RORE are hyperbolic in relation to the RBC target because the equation shows that any target change will fully impact the denominator but only partially impact the denominator to the extent that investment income is earned on the required equity and added to adjusted operating income.

7 At which point is the tradeoff between RORE and target RBC best balanced? The shape of the RORE curve suggests two things 1. There is an ultimate level of RORE that can be achieved as the target is increased. This is evident when the domain of our sample calculation is expanded. 2. Since the RORE decrease is large when the target is increased above 100% and small when the target is large there must be a point of inflection, where the tangent line to the curve has a slope of -1. At this point of inflection any decrease in the RBC target will lead to a larger increase in RORE than at higher target levels, due to a larger drop in equity than in AOI and any increase in target levels will cause a lower decrease in the RORE than all points at previous target levels, indicating that this point is the best balance between the forces pushing the RBC target up and down. Since a company s cost of capital represents the hurdle rate that the company must overcome before it can generate value, and the interest rate that a company can borrow at is linked to its RBC ratio, there must be a connection between RORE and cost of capital. A table mapping corporate debt rating to RBC level is an intermediate step to forming the link between RORE and cost of capital. This assumption of debt cost as a function of the RBC target set is key. Since we assume the company sets RBC targets to optimize RORE the cost of capital is the sum of the debt cost plus equity cost of its financing activities. Weighted Average Cost Of Capital= D V f(rbc Target) (1-Corporate Tax Rate)+ E Equity Cost V

8 Where D/V is the percentage of the firm that is financed by debt and E/V is the percentage financed by equity. For an example of this, if we associate credit ratings with debt costs as follows RBC Band Credit Rating Debt Cost 500%-600% AAA 1.5% % AA 2.0% % BBB-A 3.0% % B-BB 3.5% % C 4.0% A company can now create a menu of costs of capital as determined by RORE and the underlying target RBC ratio. For example if the following company has an equity cost of 10% and a debt percentage of 60%, then the potential costs of capital per RORE and RBC target are Target RBC RORE Debt Cost COC 100% 45% 4.0% 5.6% 200% 25% 3.5% 5.4% 300% 18% 3.0% 5.2% 400% 14% 2.0% 4.8% 500% 12% 1.0% 4.4% Given an RBC target, as determined by the need to optimize RORE, we can then derive the company s cost of capital.

9 4. RORE Decomposition Because the RORE metric has a clearly defined numerator and denominator connected to the financial statements it is possible to decompose it several ways. Two Factor Decomposition By interposing premium as the factor we can decompose RORE into two components, gross margin and equity ratio RORE = Post Tax AOI Post Tax AOI = Equity Premium Premium Equity =Gross Margin Equity Ratio For a given product or company the equity ratio is related to duration and is often managed to maintain little volatility at this level. Therefore an RORE ratio is highly correlated to a product s gross margin which is a measure of the operating efficiency of the company or product. Three Factor Decomposition We can additionally interpose assets as a factor in the decomposition RORE = Post Tax AOI Post Tax AOI = Equity Premium Premium Assets Assets Equity =Gross Margin Asset Turnover Equity Multiplier Which is analogous to a classical Dupont analysis, with the Gross margin representing operational efficiency, the premium to assets ratio representing asset turnover and assets to equity as the leverage in the balance sheet. The goal is to, over time, track what is really changing in the company or product being analyzed beyond the overall changes in the ratio. It is possible to have higher gross margins but lower asset turnover to the extent that they cancel out and don t impact the overall RORE ratio for instance.

10 5. Practical RORE Calculation The shortcut method above is a view of the particular product, case, or company as of the time of calculation, assuming that current and future premiums will perfectly fund future liabilities. In reality there can be reserve deficiencies in the aggregate or path wise combined with the effects of rate increases and lapsation. A detailed calculation of RORE requires Projecting lifetime premium, retention and expense factors to arrive at lifetime cash flows Performing a full reserve calculation on a GAAP and Statutory basis Calculating the path-wise RBC and subsequent required equity at each point in time based on the premium and reserve projections Adding return on equity to cash flows in step 1 and removing corporate taxes Discount the projected AOI and the average required equity back to time 0 at a relevant hurdle rate, take the quotient to arrive at the RORE. Note that since required equity is a balance sheet concept it is necessary to calculate its value at the endpoints of the calculation s granularity. For instance calculate required equity at the beginning and end of each month, take the average of the two to form the monthly average required equity then discount to time zero using a monthly rate of interest. Also not that changes in GAAP reserves are a component of the AOI calculation in this version. The shortcut method considers the fully discounted GAAP reserve at time zero, at which all reserve changes over time sum to zero. RORE is actually a species of internal rate of return (IRR) calculation using purely GAAP based measurements for AOI and equity.

12 RAROC= RAOI AOI+Exp Shock Losses+RAC to R.E. Capital Adjustment = RAC Required Equity + Impact of Shocks 7. Conclusion In its most general form RORE is the GAAP earnings by source (underwriting, expense, and investment margins) on a post tax basis as a percentage of average required equity. There is a shortcut method for calculating this using pure pricing and valuation ratios and of decomposing it into other financial ratios. If the target RBC ratio is considered the independent variable in the RORE shortcut equation then there is a solution to the optimal balancing point for that ratio and the subsequent cost of capital associated with it. RORE is a versatile metric that is an integral part of a company-wide risk management framework. It has the advantage of scalability from the company level down to the policy level and can form the basis of more complex calculations needed to define long-term capital needs.

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