Understand capital calculations
Concept of economic measures of value and capital and their uses in corporate decision-making process
Develop a capital model for a representative financial firm
Understand how to allocate capital across an organization
Reasons a business hold capital
Manage its cashflow (working capital)
Facilitate growth/new ventures (development capital)
Cover unexpected losses arising from exposure to risk (risk capital)
Will look at the concept of risk capital and consider a number of methods of assessing and allocating this capital
Company will typically hold (as a minimum) the larger of:
Company’s own assessment of the risk capital required (i.e. to cover unexpected losses from exposure to risk), sometimes referred to as economic capital
Risk capital determined by 3rd parties as being required to be held by the company (regulatory or required capital)
Capital required by regulators (e.g. based on the Basel or SII formula)
Capital required by rating agencies to maintain a given credit rating
We will consider both the development of capital models and issues relating to the allocation of capital across separate parts or segments of an organization
Module focuses on the concept of economic capital, but the principles (modeling techniques and capital allocation approaches) can apply similarly to regulatory required capital
Recall Module 7’s S&P paper “Insurance Criteria: Evaluating the Enterprise Risk Management Practices of Insurance Companies” discussed capital models, their potential shortcomings
Main concern is with risk capital (or economic capital, or simply just capital)
Common definitions of capital in use:
Capital should provide sufficient surplus to cover adverse outcomes with a given level of risk tolerance over a specified time horizon
3 main interpretations of adverse outcomes result in the following definition of capital:
Surplus needed to cover all potential outgoings, reductions in assets and/or increases in a company’s liabilities at a given level of risk tolerance over a specified time horizon`
Surplus needed to maintain a given level of solvency at a given level of risk tolerance over a specified time horizon
The XS of the value of the assets over the value of the liabilities at a given level of risk tolerance at a specified time horizon
This definition focuses on the values of the assets and liabilities at the specific time horizon
Rather than the funding position (cashflow or surplus) of the company throughout the time period concerned
Metrics used to set an appropriate level of risk tolerance
A certain percentile of the loss distribution
Extreme loss values
Possibility of some key indicator (e.g. credit rating) falling outside an acceptable level
There may be other ways to assess capital requirements within an org, such as
Regulatory standard formula (or other prescribed calculations)
Rating agency factor-based models
Where relevant these are also an important part of the capital management process
CMs can be used for a number of purpose within an org
Best-practice models are also able to allocate the capital across the company
Used by capital providers and regulators to gain a consistent assessment of capital requirement across different firms
Typically very simple in application (e.g. factor-table approaches determine capital as a multiple of business volume)
Generic model are becoming more complex:
Increase sophistication of risk management practices adopted internally at companies
Previous models failed to deal properly with all risks
Increasing pressure on companies to optimize their capital resources
Internal CM is use to simulate a company specific view of the capital needed
Aim is to cover all risks faced by a company in a consistent way allowing for the interaction between the various risks
Internal CMs can also be used to calculate regulatory capital (instead of using generic CM), but will require detailed scrutiny by the regulator
Internal CM typically comprises of an asset and liability model, which may be dynamic in nature
Key outputs of an internal CM:
Desirable features
Asset model allows for correlations between different asset classes over time
Liability model considers reinsurance and correlations between classes of risk
Asset and liability model should be integrated and allow for correlation between asset and liabilities
Model is dynamic
Dynamic asset-liability model has the following benefits:
(i.e. Asset and liability cashflows are linked by equivalent economic variables)
Improved understanding of the dynamics of the current strategy
Consideration of the impacts of implementing different strategies
(i.e. mix of business)
Examination of the impact of using different source of capital
Useful for due diligence for corporate transaction
Assesses the risk-adjusted performance of different BUs
Determines an optimal asset mix
Helps understand the impact of extreme events
Useful in producing Financial Condition Reports
Regulators require long term projections of the capital needs of the business and these are often incorporated into a Financial Condition Rerport (FCR)
A formal assessment of the financial viability of the insurer
FCR includes output from internal CM and other hard-to-quantify factors
(e.g. reputational risk and effectiveness of the ERM framework)
Model is a subjective tool and the results should be used with care
Internal CM can help to provide a sound basis for the development of a risk management strategy targeting the agreed risk objectives
(e.g. reducing earnings volatility)
Determining company or product risk profile:
Can be just one part of a wider RM process
Capital budgeting:
Ensure capital are allocated appropriately to establish the optimal business mix and establishing capital outflow/inflow policies
Capital needed in M&A:
One way to assess the level of risk is potential M&A and divestment transaction
Insurance pricing:
Assess the true cost of issuing a product
Comparing the rate of return on the capital held to back the product against the pricing hurdle rate used in setting premiums
Risk tolerances/constraints:
Monitor requirements and to impose constraints on the level of risk permissible for certain lines of business or considering reinsurance programs
Setting investment strategy:
Measure the effectiveness of any AL-matching strategies
Calculate risk-adjusted rate of return on capital (RAROC):
See Section 3
Performance measurement:
Used to determine the appropriate level of capital to used in Embedded Value calculations
(i.e. the calculations required for the formal reporting by insurance companies)
Incentive compensation:
Incentivise management
(although not that common yet)
Alternative to rating agency/regulatory requirements:
The required levels of capital set by external bodies may not be an acceptable monitoring measure
Disaster planning
Likely to have similar {underlying assumptions and parameters} but likely to have different {risk measures and calibrations} associated with them
Key differences
Views as to volatility of various classes of business
Allowances for diversification between risk type
Objectives of the model
Inclusion of different risk types or different treatment of the same risks
Different views regarding the availability of certain types of assets as capital
Within regulatory and economic capital models there may also be different scenarios run to allow for some of the accounting requirements of specific countries that are not appropriate when considering either regulatory or economic capital requirements
6 stages required in operating a successful capital model:
Identify purpose:
Clear purpose is required for the model
Influence matters such as:
Whether or not it is assumed that the business remains open to new business (going concern) or simply run off
Whether contingent management actions will be modeled
Level of resourcing required and the accuracy of the results
Identify and rank risks:
The dominant risks will vary by insurer, including: CAT, underwriting, reserving, pricing, liquidity, market, credit, and operational risks
Choose the simulation approach for each risk:
Deterministic
(e.g. stress scenarios)
Stochastic
(e.g. parametric, empirical)
Depending on cost/time considerations and benefits gained
Defined the risk metrics:
Include VaR or Tail VaR, the time horizon and the confidence interval
Select the modeling criteria:
Should use multiple criteria such as:
Exit value as measured by absolute ruin
Attaining a certain investment rating
Some ongoing business criteria as measured by superivsory intervention
Decide on the method of implementation:
Type of model should be appropriate to the nature, scale and complexity of the insurer’s business
Could be a set of univariate models together with a method by which to combine them (e.g. copula) or a single fully integrated model
IAA Note (Key Feature 7):
“As part of the ORSA and insurer should analyse its ability to continue in business, and the risk management and financial resources required to do so over a longer time horizon than typically used to determine regulatory capital requirements”
Such continuity analysis should:
Address a combination of quantitative and qualitative elements
In the medium and longer term business strategy of the insurer
Include projections of the insurer’s future financial position and modeling of the insurer’s ability to meet future regulatory capital requirements
Factors to consider when using a capital model as part of such a continuity analysis
What time period should be used
Longer time periods means the models results need more interpretation and the model’s limitation need to be clearly articulated
Different models may need to be developed for the longer term
Should the financial position of the insurer be assessed at a particular point in time, or once specific liabilities have run off
What capital reduction/injection policies can be assumed
What management actions need to be modeled
In general (e.g. premium setting, asset allocation, discretionary benefits, dividend policy and risk mitigation)
In times of crisis
Reliability of the insurer’s long term forecasts?
Capital management is more than a company simply holding more capital to cover possible losses
Principle of capitalism:
Market will allocate capital to activities by their propensity to provide a return on that capital, allowing for potential risks
IAA Note (Key Feature 6):
“As part of its ORSA and insurer should determine the overall financial resources it needs to manage its business given its own risk tolerance and business plans and to demonstrate that supervisory requirements are met. The insurer’s risk management actions should be based on consideration of its economic capital, supervisory capital requirements and financial resources”
Effective capital management can help transform risk into increased shareholder value
Pricing competitively:
Ensure an adequate return on capital
Reserving:
Improve estimates of reserves needed for outstanding claims
Performance management:
Enabling business outcomes to be measured and processes to be adapted accordingly
Risk Management:
Establishing their overall level of risk tolerance, identifying and assessing risks present and keeping all risks under control
Capital requirements is a function of 2 quantities:
Desired solvency standard
Its risk
Higher solvency standard or more risk means more capital needs to be held
\(K_t\): Capital required at time \(t\) is the difference between value of assets (\(A_t\)) and liabilities (\(L_t\))
\(K_0 = A_0 - L_0\) where \(K_0 \geq 0\) is known with certainty
\(K_t = A_t - L_t\) is an unknown r.v.
To avoid ruin, \(K_t\) must be greater than 0 for all \(t\geq 0\)
Strategy to avoid ruin may be to set \(K_0\) at a high enough level so as to ensure:
\(\Pr(K_t \geq kL_t) \geq 1 - \alpha\)
or equivalently \(\Pr(A_t - (1+k)L_t \geq 0) \geq 1 - \alpha\)
Where \(\alpha\) is the risk tolerance level and \(k\geq0\) is a comfort ratio
We would expect \(K_t\) varies as follows:
More volatile assets (relative to the liabilities) increase the required capital
Higher expected return on assets (relative to the liabilities) reduces the required capital
Higher positive correlation between the asset and liability reduces the required capital
Limitation of the approach
Difficulty in obtaining consistent valuations for both asset and liabilities
(Whether they be all market-to-market or market-to model)
Need to select an appropriate risk measure
(Ruin is not the only one)
Difficulty in formulating the necessary assumptions
(e.g. how assets and capital will be invested)
Large number of parameters needed
(Increases exponentially with the number of variables)
Difficulty in deriving robust estimates of the various parameters needed
(e.g. Correlations and variance etc)
Consideration of \(K_t\) only at discrete points in time
(Possible that ruin occurs between these time points)
Basic bottom-up approach generalized from the preceding example:
Generate stand-alone distributions of changes in the enterprise’s value due to each source of risk
Combine the distribution
(allowing for any diversification effects)
Calculate the total capital for the combined distribution at the desired standard for the selected risk metric(s)
(e.g. SCR)
Attribute capital to each activity based on the amount of risk generated by each activity
All models should allow for the correlation between risk factors
Allowance could be made implicitly or explicitly
Adjustment could be a micro (i.e. determining covariance matrices between all risk) or at a macro level (i.e. considering the correlation between certain risk types only)
Following sections will discuss different methods to assess capital requirements
Ruin:
When the market value of a company’s liabilities > MV of asset
Advantages
Disadvantages
Economic cost of ruin:
Amount key stakeholder (e.g policyholder of an insurance company or depositors with a bank) can be expected to lose in the event of ruin
Can be expressed as an absolute amount or as a proportion
(e.g. ratio of policyholder benefits)
Advantage
Disadvantages
Other methods to assess capital requirements
Full economic scenarios
A chosen set of economic scenarios is detailed
(e.g. interest rate, inflation, etc)
Each BU estimates their profitability under each scenario
Results are then aggregated across all units and assessed under a given risk measure (e.g. VaR)
Stress test
Factor tables
Tables of capital requirement per unit risk exposure by risk type are published by supervisors
Capital requirement is then calculated as the number of units of each activity undertaken by the business multiplied by the relevant factor from the table
Stochastic models
Either univariate or multivariate
Scenarios are generated at random and capital calculated on the basis of the results of these scenarios
Statistical methods
Credit risk methods
Historical data from rating agencies is used, or
Merton model approach
Operational risk methods
Option pricing theory/ Black Scholes
Effective where the risk event can be assumed to occur only at one certain point in time
Loss situation is considered to be an option and appropriate models used to calculate its value
Recall (Module 5) that the Basel capital requirements are set out in 3 pillars
One of the most important risk for a bank
Was the main target under Basel I
Basel II adopts a similar approach to Basel I when estimating the credit risk in a bank’s balance sheet
Bank calculates the risk-weighted value of its assets (categorized by credit quality)
OECD Gov bonds are excluded as being deemed free from credit risk
Other risky assets are given varying risk-weightings
2 methods of categorizing and risk-weighting assets
Standard approach
Factor table approach: where almost every credit rating relates to a risk-weighting category and risk weighting
Internal Rating Based (IRB) approach
Basel II allows banks to categorized and risk weight their assets based upon credit ratings determined by using their own IRB mode
A thorough credit assessment is required, and the methodology needs to be approved by the regulator
Calculate captial requirement
After determining the risk-weighting for each asset, the bank can calculate the total value of its asset portfolio allowing for these risk-weighting
\(\hookrightarrow\) Minimum capital required to be held is based on the risk-weighted value of assets
Benefits of diversification within the portfolio is not allowed
Many off-b/s credit related assets (e.g. securitized loans) must be treated as on b/s to avoid regulatory arbitrage
Banks may invest their capital in risky securities and Basel II requires banks to hold additional capital if such additional risks are taken on
Usually measured by modeling the assets and calculating a 10 day 99% VaR
This lead to a regulatory capital requirement under Pillar I, which is a multiple of this VaR loss
Banks can also use a standardized approach rather than use an internal VaR model
New and controversial aspect to the Basel II Pillar I calculation
Difficult to measure, but capital has to be set aside to cover risk of technological failure, mismanagement, fraud, litigation, and other operational risk
The calculation leads to a regulatory capital requirement where the capital held is calculated as a function of the gross income over the previous three years
Basel II definition of Op-risk:
Risk of loss resulting from inadequate or failed internal processes, people, and systems or from external events
Includes legal risk but exclude strategic and reputational risk
Op-risk has no potential upside unlike many of the other risk
Recall (Module 24), the alternative approaches for calculating regulatory op-risk capital under Basel II:
Basic Indicator Approach
Standardized Approach
Advanced Measurement Approach
Under Basel II a bank’s capital is in tiers
Tier 1: Bank’s equity and disclosed reserves
Tier 2: Other reserves and various debt instruments
Tier 3: Certain types of shorter-dated capital (e.g. unsecured, subordinated debt with a minimum maturity of 2 years)
Certain b/s assets such as goodwill are not allowed as sources of capital
Minimum capital under Basel II:
% of total risk-weighted assets (RWAs)
Starting 1/1/2015 Basel III strengthened the requirements to :
Tier 1 & 2 capital \(\geq\) 8% of RWAs
Tier 1 capital \(\geq\) 6% of RWAs (with common equity at least 4.5% of RWAs)
Conservative buffer (that a bank can draw upon during times of financial stress) of 2.5% RWAs (fully phased in 1/1/2019)
Additional deductions from common equity, which includes investments in financial institutions and deferred tax assets (fully phased in 1/1/2018)
Aim of Basel III
Does not deal with the concentration risk arising when banks pursue similar business strategies
Under SII, Pillar 1 is calculated using the twin peak approach
First peak
Market consistent valuation, the Solvency Capital Requirement (SCR)
Second peak
Basic valuation based on the SI Minimum Capital Requirement (MCR)
All assets are taken at fair value
Capital consists of the XS of assets over liabilities and is classed as tier 1 to 3 depending on how readily the capital can be called upon
SCR Must be achievable with 99.5% confidence over 1 year horizon and maybe based on standard formula or approved internal model
Standard formula
Based on specific deterministic basis but with some stochastic elements (e.g. valuation of guarantees)
Market risk:
Limited admissibility of some assets plus a number of stress test
Credit risk:
Limiting exposure to individual counterparties
Underwriting risk:
Require additional solvency margins, generally calculated by reference to business volumes (e.g. premiums) or risks (e.g. claims incurred, sums assured)
Internal Model must satisfy certain standards including:
Use test:
Company must actually use the model in its decision making and risk management systems
Statistical quality standards:
Ensure assumptions are realistic and reliable
Calibration standard:
Ensure the output can be used to properly calculate the SCR
P&L attribution
Validation standards
Documentation standards
MCR is €3 plus a margin based on premium or reserve
Must be achievable with 80-90% confidence over 1 year horizon
Failure to maintain the MCR will result in withdrawal of the company’s authorization
Recall the qualitative requirements of Pillar 2 from Module 5
Recall (Module 26) the objective of risk management is to optimzie the balance between risk and return (not simply minimizing risk)
Understanding return on capital is one way of ensuring that the company is putting its limited capital resources to the best use
\(RAROC = \dfrac{\text{Risk-adjusted return}}{\text{Capital}}\)
Can be calculated for the enterprise as a whole
Can be used to compare different and diverse business activities
(i.e. to identify activities that are creating or destroying s/h value by comparing RAROC vs cost of capital)
Can be based on actual or expected return or capital
There is no single definition of either return or capital so RAROC is not well defined
EIC captures the quantity of return generated by a unit of activity:
\(EIC = (RAROC - hurdle rate) \times Capital\)
Hurdle rate: standard against which business activities must be measured
If a proposed activity does not offer a RAROC above the hurdle rate then that is one basis upon which it might be rejected
Considerations when determining an appropriate hurdle rate
Reflect the cost of capital (e.g. WACC)
Allow not only for the risks inherent within a proposal, but also the degree to which those risk diversify existing risks
EIC is a monetary amount and can be used to encourage marginal growth opportunities
SHV and SVA assess the intrinsic economic value of a business as a going concern (i.e. over an extended period)
SHV captures the PV of all future cashflows (i.e. as a perpetuity):
\(\begin{align} SHV &= \text{Discounted value of all future cashflows} &= Capital \times \left( \dfrac{RAROC - g}{hurdle - g} \right) \end{align}\)
SVA measures the extent that SHV exceeds the capital invested
\(\begin{align} SVA &= \text{Discounted value of economic value added} &= Capital \times \left( \dfrac{RAROC - g}{hurdle - g} -1 \right) \end{align}\)
After the overall required capital is calculated, next is to translate this in a fair way into capital requirements at the level of individual business units, produce and other segments
Important for business planning, performance measurement and pricing
There is no single way of achieving this allocation of capital, with a combination of methodologies generally resulting in a better overall approach
Notional allocation of risk capital
We generally consider a notional allocation of risk capital (output from model), not a physical distribution of money (in the sense of working capital)
Exception is if we are looking at allocation across regulated entities (e.g. different geographies) then capital will be physically allocated (but then each entity will have its own regulatory capital requirement and allocation issues)
The notionally allocated capital amounts are important, as the values are then applied to management processes, which in turn will have a significant impact on business decision-making
e.g. total allocated capital may be used as a basis for pricing, risk control limits, and performance measurement etc
How alloation of capital affects decision making
Capital allocation process link risk to performance measurement
e.g. BU’s success should be measured relative to the risk it takes in its operations, which should in turn reflect the amount of capital the company is willing to allocate to the BU
Amount of capital that is allocated to each BU:
Determines the BU’s performance
(e.g. under RAROC)
Could affect, directly, or indirectly, the remuneration of the unit’s managers and consequently, their level of motivation and behavior
Dictates the amount of business the BU can write
(As each product written consumes capital and the total amount of capital is limited)
Determines in part the price at which business can be written
(e.g. a minimum price might be determined by a stipulated minimum RAROC)
Allocation needs to allow for concentration / diversification of risk between the BUs
Not all business areas have the same degree of risk
Some allocation of capital may result in the less risky areas subsidizing the more risky area
Important to remember that the level of dependency might be different during times of stress
As required capital is typically calculated with reference to extreme events, tail dependency will have to be considered carefully
Often the case that total required capital as well as the related capital allocation is very sensitive to the choice of dependency structure (e.g. copula) used to aggregate the risks
The diversification benefit can be shared across the business
Key consideration when allocating capital across the company
Some approaches calculate the capital for each BU followed by an adjustment for the benefit of diversification
The adjustment is retained at the level of the enterprise and is not passed on to the individual units
Alternative approach is to calculate the capital required at the enterprise level and then allocates the diversification benefit across all units
Fair approach is important as the allocation impacts the pricing and subsequent measures of performance
Appropriate method for allocation depends on a number of factors such as the purpose of the allocation and the stakeholders to the process
However, a significant element of judgement to ensure that a suitable answer is obtained
This is an area of continuing development and no specific methodology is currently an industry standard
Care should always be taken to consider the purpose of the allocation and the practical implications of the chosen methodology before results are used
Method of allocation should have regard to the use to which the results will be put and should consider desirable properties of the results (e.g. stability over time)
There is no one method that is best suited in all cases
Should compare results from several methods
Use judgement when recommending or setting the final allocation
There can be conflict between desired properties of capital allocation exercises (e.g. the financial principle of marginal pricing contrasted with fairness between BUs)
Approaches to allocate total capital to BUs
Total capital could be retained fully by the company centrally in the main corporate business line
Allocate using some risk measure
Marginal approach:
Each LoB receives the change in capital as the result of adding it to the diversified portfolio
Allocate using some game theory approach
Allocate on some pro-rate basis
(e.g. weighted by reserves or PV of future expected revenue from the business area)
Calculate capital for each LoB individually on a stand-alone basis and any remaining capital retained in the main corporate business line
Approach of not allocating capital
Will lead to a lack of understanding of the impact of each LoB’s action on the capital requirements of the business
Can lead to potential over-investment in risky BUs
Allocate capital by reference to some measure of risk
Allocation based on Euler principle (Sweeting Sec 18.8)
Euler’s homogeneous function theorem states:
If \(f(u_1,...,u_n)\) is homogeneous, then
\(f(u_1,...,u_n) = u_1 \left. \dfrac{\partial f}{\partial u_1} \right|_{(u_1,...,u_n)} + \dots + u_n \left. \dfrac{\partial f}{\partial u_n} \right|_{(u_1,...,u_n)}\) \(\dots (1)\)
The vertical bar and subscript means that each partial derivative is evaluated at \((u_1,...,u_n)\)
Theorem can be generalized to higher orders of homogeneity but we are only interested in order 1
Suppose that the organization has \(n\) BUs and that each unit \(i\) has an associated random loss variable \(L_i\) \(\Rightarrow\) Total loss is \(L = L_1 + \dots + L_n\)
For a coherent risk measure \(F\) (which satisfy the axiom of positive homogeneity):
\(F(kL) = kF(L)\) for any \(k>0\)
Here, the risk measure \(F\) is a measure of capital required
Consider there are \(p_i\) units (instead of having one unit of each loss \(L_i\)) and let \(L(\mathbf{p}) = p_1 L_1 + \dots + p_n L_n\)
From (1) it can be seen that
\(F(L) = \left. \dfrac{\partial F(L)}{\partial p_1} \right |_{\mathbf{p}=1} + \dots + \left. \dfrac{\partial F(L)}{\partial p_n} \right |_{\mathbf{p}=1}\)
So if \(F(L)\) is the total capital then a possible subdivision is to have a capital requirement of \(C_i\) for BU \(i\), where \(C_i = \left. \dfrac{\partial F(L)}{\partial p_i} \right |_{\mathbf{p}=1}\)
Example
Note the 2 risk measure used below satisfy the positive homogeneity axiom but not always satisfy the subadditivity axiom
It can be shown that the total capital requirement equals the sum of the individual allocations: \(F(L) = C_1 + \dots + C_n\)
Standard deviation risk measure:
\(F(L) = c \sqrt{Var(L)}\)
\(C_i = c \dfrac{Cov(L_i,L)}{\sqrt{Var(L)}}\)
VaR risk measure:
\(\rho(L) = VaR_{\alpha}(L)\)
Assuming that \((L_1,...,L_n)\) has a joint density \(L\) then \(C_i = \mathrm{E}[L_i \mid L = VaR_{\alpha}(L)]\)
Expected shortfall risk measure:
There are other methodologies that use risk measures calculations as the basis for capital allocation and the methods vary in complexity:
Simple proportional spread:
(Overall capital is allocated in proportion to the risk measure applied to each line in isolation)
Complicated numerical implementations requiring complex modeling:
(Use of co-measure or the consideration of capital as a shared asset)
Results from different approaches can be very different, even for the same risk measure
Capital is allocated to LoBs in accordance with the marginal additional capital required for writing that business (given that the other LoB are already in place)
Advantage
Corresponds to the financial principle of marginal pricing
(Whereby a business will want to write additional business that covers its marginal costs)
Can be argued that this method allocates the true capital to each BU
Disadvantage
Potentially unfair as it is dependence on the order of consideration
Complicated to calculate and does not ensure that the total capital is allocated
(i.e. sum of the marginal capital requirements \(\neq\) sum of the total)
Shapley method
Allocates capital with reference to an average of the marginal capital requirements
Assuming that the segment under consideration is added to the overall portfolio first, second, third and so on
Advantage
Disadvantage
Computationally intensive for all but the smallest of portfolio
Similar answer can be achieved using other methods that are easier to implement
Allocation according to some basis:
Advantage
Disadvantage
Treating each BU in isolation
Disadvantage
Recall (Module 14) on coherent risk measures
Use of coherent risk measures is particularly relevant when considering capital allocation
Conditions for a risk measure to be coherent
Translation invariance:
If we add an amount to the observed loss, then the capital requirement needed to mitigate the impact of loss increases by the same amount
Subadditivity:
Merger of risk situations does not increase the overall level of risk
Positive homogeneity:
If we double the size of the loss situations we double the risk
Monotonicity:
Greater expected loss requires a greater amount of capital to be held
Subadditivity is considered a desirable property for risk measure as it will exhibit diversification benefits as risk portfolios are combined
An org would typically wish to calculate capital required at the aggregate level, however it might be much more convenient to calculate capital required at the level of individual LoBs
If a coherent (so subadditive) risk measure is used, then sum of the capital required for each LoB will exceed the capital that would be calculated for the aggregate business
This means that the risk tolerance levels for each LoB, based on the chosen coherent risk measure, can be separated out
If these tolerances are adhered to, then a related risk tolerance at the level of the org using the same risk measure will be satisfied automatically
\(\therefore\) subadditivity makes decentralization of risk management systems possible
Note that VaR is not subadditive so care must be taken when aggregating results of models based on VaR
It is desirable to use a different risk measure from that used in assessing the overall capital requirement when performing a capital allocation exercise, e.g.
Capital requirement maybe based on a high target percentile in the tail
But the diversified capital may be allocated down to individual product lines or companies within the group by reference to a lower percentile to prevent over-allocation to LoB with extreme outcomes
(e.g. CAT-prone general insurance classes)
Important to consider the purpose of the exercise and the implications of the results before making a final selection
Org may hold more capital than its capital model suggests is needed
XS capital may also be allocated between segments (e.g. pro rata to the allocation of risk-based capital or to certain components of it)
In reality, capital assessment and allocation can change dramatically from year to year
Can be due to modification of capital models from year to year as part of a standard development cycle as new techniques emerge
Results can be highly sensitive to the underlying choice of risk allocation and the correlation model which are not always transparent to all stakeholders
There aspects can create practical issues in terms of communication and gaining buy-in from business units and stakeholders
Implementing capital allocation in practice requires strong communication skills and a good understanding of the conflicts that the implementation can create
Understanding how these conflict should be managed and minimized is itself a part of a strong ERM framework