21.3 Reinsurance as Capital

Capital is held to pay for claims when the premium is not sufficient to do so

  • Highly variable results will require more capital

    \(\Downarrow\)

    Reinsurance reduce volatility of results \(\Rightarrow\) reduce capital required

    \(\Downarrow\)

    Reinsurance can be thought of as borrowed capital

We value the reinsurance by the impact it has in reducing Capital (or Cost of Capital)

\(\therefore\) We compare the reduction in cost of capital to the cost of the reinsurance

Measure the amount of capital reduction that the insurer receives from the reinsurance program

  • Compare the cost of capital reduction from reinsurance program

  • i.e. Compare the marginal cost of reinsurance to the marginal capital to judge the value of reinsurance

Definition 21.8 Cost of capital reduction: Metrics 1

Net Benefit = |\(\Delta\)Cost of Capital| - |Net Cost of Reinsurance|

  • Net benefit > 0 then accept

  • Net cost of reinsurance = \(\text{Reinsurance Premium} - \mathrm{E}[Recovery]\) on a PV basis

  • Alt formula: Net benefit = Net Cost of Reinsurance - \(\Delta\)Cost of Capital

  • Review other criteria when the dollar benefit is the same

Definition 21.9 Cost of captial reduction: Metrics 2

\(ROE = \dfrac{\text{Net Cost of Reinsurance}}{\Delta \text{Capital}}\)

  • Compare return on equity

  • We want \(ROE\) below the cost of capital here and that lower is better since we are reducing capital

    • Think about it if you’re going from the plan interested to gross, if the \(ROE\) is lower you don’t want to change so the current plan is better

    • Tink of this as investing negative capital, and \(\therefore\) we want a low ROE

21.3.1 Change in Capital

2 different methods to determine the change in capital from the reinsurance structure

  1. Theoretical models:

    Use risk measures we discussed earlier (e.g. VaR, TVaR)

  2. Practical models:

    Use captial requirements from rating/regulatory agencies

21.3.1.1 Theoretical Models

A risk metric is selected as a proxy to represent the economic capital required

  • e.g. \(VaR\), \(TVaR\), multiples of \(VaR\), \(XTVaR\), etc

Procedure:

  1. Calculate Expected Earnings

\[\text{Expected Earnings} = \text{Net Premiums} - \text{Expense} - \text{Expected Losses}\]

  1. Calculate Economic Captial

\[\text{Economic Capital} = \text{Net Premium} - \text{Expenses} - \text{Risk Measures}\]

  1. Measure the \(\Delta\) in Economic Capital and Expected Earnings to determine the comparison metrics above (21.8 and 21.9)

    • We measure the \(\Delta\) against a base (e.g. bare or current)

    • Here \(\text{Net Cost of Reinsurance} = \Delta \text{Expected Income}\)

    • Remember that lower \(ROE\) is better

21.3.1.2 Practical Models

Use capital requirements from rating agencies (e.g. BCAR, S&P, CAR, RBC, ICAR, etc)

  • Typically set target as a multiple of a given regulatory metric

    (e.g. 175% BCAR or 4 times RBC authorized control level)

Reinsurance reduces the capital required due to:

  1. Lower net premium and net reserves

  2. With a slight offset from increase counterparty risk

Pros/Cons

  • Pros: Easier to implement (no need to model loss distn and correlation)

  • Cons: Not as accurate because the measurement of risk is based on proxies and not the actual risk itself

    • e.g. stop loss only have small impact on premium but can have large impact on a risk (like the tail) that the method might not pick up

    • Alternatively, model a year out and predict the probability of falling below certain thresholds

      \(\Rightarrow\) Capital could be set so that the probability is lower

    • But then we’ll have to model the I/S and reinsurance structure and everything

21.3.2 Theorectical Model Example: Marginal ROE

Compare the marginal cost of reinsurance to the marginal capital to judge the value of reinsurance

  1. Pick a risk metric as proxy for capital required

  2. Calculate marginal capital relative the current program

  3. Calculate \(\Delta\) NPV Net Benefit = \(\Delta\) NPV Ceded Loss - \(\Delta\) NPV Ceded Premium

  4. Calculate \(ROE = \dfrac{\Delta \text{NPV Net Benefit}}{\text{Marginal Capital}}\)

  5. Calculate after-tax \(ROE\)

  • Compare the \(ROE\) (We still want it lower?)

    • Bare in mind that a high ROE is meaningless if the amount of capital you can invest is small

21.3.2.1 XTVaR

Might be better to look at a multiple of XTVaR at a lower percentile

  • XTVaR is the unfunded part of TVaR

\[XTVaR_{\alpha} = \mathrm{E}[Y \mid F(Y) > \alpha] - \mathrm{E}[Y]\]

  • VaR is a single point and can be volatile at high percentiles

21.3.3 Accumulation Risk

So far we implicitly assumed that the capital is held for only 1 year

  • Capital calculated so far was static without regard to how long it takes to run it down as claims are paid

  • Capital needs to be held until all claims are paid

21.3.3.1 “Life-time” Amount of Capital

\[\begin{align} \text{"Lifetime" amount of capital} \:\: = \:\: & \text{Capital for new business} \\ &+ \text{Capital to support all unpaid claims from prior years} \\ \end{align}\]

\(\therefore\) Set the capital requirements based on premium and reserves

  • Caveat: Current reserves may be based on business that is different from this year’s book (e.g. size, mix of business, etc)

  • Use “as-if” reserves for capital calculation

Definition 21.10 (As-if Reserves)

  • Calculate the reserves as if the company had been writing the same book the last several years

  • Adjust for inflation

  • Especialy important for long tailed lines where capital requirement for the accumulated outstanding reserve can be much large than the requirement for new business

Focus here is to determine the impacts on capital so we can assign a cost to each reinsurance program

  • When comparing reinsurance program, the reserves would be calculated under each reinsurance program

  • Capital calculated for both the new premium, and the loss reserves is a surrogate for the PV of the capital required for the current book over its lifetime

2 Advantages for the “as-if” reserves

  1. Measure the impact of accumulated risk caused by correlated factors

    • Likely means “correlated risk factors” that can influence all those accident years (e.g. inflation, change in law)

  2. Alternative reinsurance programs can be applied to the premium and as‐if reserves, providing a more valid measure of the accumulated risk and capital used over the full life of the AY

When aggregating AYs, we get temporal risk (risk based on time) that affect all AYs at the same time (e.g. severity trend and auto correlation of severity trend)

Including all AYs create a bigger spread of possible outcomes

21.3.4 Capital Consumed

Concept: How much capital is needed to pay deficiencies

Plot/compare PDF and CDF of Capital Consumed:

\[\begin{equation} \text{Capital Consumed} = - PV[\text{Premium} + \text{Reserves} - \text{Losses} - \text{Expenses}] \tag{21.1} \end{equation}\]

Apply Reinsurance Example

When modeling XoL contracts we need to model severity trend

  • Severity trend impacts all open claims

  • Payment pattern is important as claims that are open longer will have more severity trend applied

We can base the capital consumed on different risk measures (e.g. VaR and TVaR)

\[ROE = \dfrac{\text{Expected Profit}}{\text{Capital Consumed}}\]

  • Do we want increase or decrease ROE? Probably increase

Comment from notes that author is not sure about