Chapter 24

Question 1

How are insurance liabilities valued at market value?

Answer 1

◦ Insurance liabilities aren’t directly traded on an exchange, so their market value is estimated by finding a replicating portfolio of traded assets that matches the expected timing and amount of liability cashflows.
◦ These liability cashflows are then discounted using a risk-free yield curve to calculate their present value.
* Risk-Free Discount Rate:
◦ Should represent the time value of money with minimal credit/default risk bc you can’t exactly earn a risk-premium on those liabilities
◦ Government zero-coupon bonds could be used, but in many markets long-dated government bonds are illiquid, thinly issued, or only available at certain maturities.
* so often, swap rates are used… which are better bc the swap market is often deeper, more liquid and offers high trading values at many more maturities => smoother and more reliable term structure for discounting

Question 2

What is the illiquidity premium, and when can insurers include it in their discount rate for valuing liabilities?

Answer 2

• Definition:
  ◦ The illiquidity premium is the extra yield investors demand on assets (e.g., corporate bonds) to compensate for price volatility and difficulty in selling them before maturity.
  ◦ It is often confusingly called the “liquidity premium.”
  
  • Corporate Bond Yield Breakdown:
    = Government bond (risk-free) yield
  • Default risk premium
  • Illiquidity premium
  • When Insurers Can Use It:
    ◦ For long-term predictable liabilities (e.g., annuities) where matching assets can be held to maturity.
    ◦ Since the insurer is not exposed to interim price volatility, the illiquidity premium is effectively a “free lunch.”
    ◦ Regulators may allow the risk-free discount rate = government bond yield + illiquidity premium.
  • Restriction:
    ◦ The default risk premium is never included, since insurers remain exposed to defaults even when bonds are held to maturity.

Question 3

What are the differences between the active and passive approaches to setting supervisory reserves?

Answer 3

• Active Approach:
• Supervisory authority sets best estimate assumptions for each insurer.
• Assumptions (e.g. interest rates, mortality, morbidity) are updated regularly.
• Reflects recent company/market experience → more realistic, but more volatile reserves.
• Greater work for supervisors, insurers must revalue frequently.
  
  • Passive Approach:
• Supervisory authority prescribes fixed/prudent assumptions (e.g. conservative mortality, low interest rate).
• Assumptions remain stable for long periods, only changed occasionally.
• Reserves are more stable, easier to administer.
• But less responsive to actual emerging experience, may not reflect reality.
  
  • Trade-off: Active = realistic but volatile; Passive = stable but less accurate.

Question 4

What are the advantages and disadvantages of the active vs passive approaches to setting supervisory reserves? (6 each method)

Answer 4

Active Approach
✔ Advantages:
* More realistic, uses current assumptions based on actual experience.
* Encourages insurers to monitor experience closely.
* Better reflection of true economic value of liabilities.
✘ Disadvantages:
* More volatile reserves → financial results fluctuate and therefore higher P(making knee jerk management decisions)
* High burden on supervisors (must regularly update bases).
* May require frequent system/model updates for insurers.

Passive Approach
✔ Advantages:
* Simpler, easier to administer and monitor.
* Produces stable reserves, less volatile reporting and also PROFIT emergence more stable
* Lower supervisory costs (assumptions only updated occasionally).
✘ Disadvantages:
* Less responsive to actual company/market experience.
* May result in systematically excessive or inadequate reserves.
* Can create distortions between insurers (those closer/further from true experience).

Question 5

When is an active approach particularly useful?

Answer 5

• options and guarantees

Question 6

What are the key advantages and disadvantages of using a valuation approach that is a combination of active and passive elements?

Answer 6

Advantages:
Stability: If the approach leans passive, it may benefit from being more straightforward to implement, involving less subjectivity, and resulting in relatively stable profit emergence.
* Informativeness: If the approach includes active elements, it may be more informative in terms of understanding the impact of current market conditions on the company’s ability to meet its obligations, especially regarding financial guarantees and options

Disadvantages:
* Greater Mismatch and Volatility: A combination of active and passive elements can result in a greater mismatch between assets and liabilities. This, in turn, leads to greater volatility of results.
* Volatile Financial Position: This increased volatility manifests as greater profits/losses or changes in free surplus when market conditions change.
* Specific Example of Mismatch Risk: An approach that uses a net premium valuation for liabilities (a passive technique, as the assumptions may be “locked in” or updated infrequently) and the market value of assets (an active technique) could experience greater volatility of results than a fully active approach (like using market-consistent valuation for both)

Question 7

How do insurers perform a market-consistent valuation of liabilities under a risk-based capital regime?

Answer 7

Liabilities valued as Best Estimate Liability (BEL) + Risk Margin.
* Discounting uses a risk-free curve (e.g. swap rates, government bonds, adjusted for illiquidity in some cases).
* Process for BEL:
* Project expected future liability cashflows (claims, expenses, options/guarantees).
* Construct a replicating portfolio of traded assets that mimics timing/amount of cashflows.
* Discount cashflows using the risk-free curve (can use swaps for long terms due to deep, liquid markets).
* Solvency Capital Requirement (SCR):
* SCR = capital to withstand a 1-in-200 year loss event over one year (99.5% confidence).
* Calculated via:
– Standard Formula: prescribed shocks for risks (market, credit, insurance, operational).
– Internal Model: company-specific stochastic modelling (if approved).
* SCR ensures insurer can survive extreme but plausible stresses.
* Risk Margin:
* Purpose: Ensure liabilities can be transferred to another insurer; reflects cost of holding capital for non-hedgeable risks.
* Calculated using Cost of Capital (CoC) approach:
– Estimate future SCR for non-hedgeable risks (longevity, morbidity, etc.).
– Multiply by the prescribed CoC rate (e.g. 6% in Solvency II).
– Discount back at the risk-free rate.
* Proxy models often used: simplified methods to approximate SCR run-off (e.g. scaling from current SCR using duration or proportionality).

Question 8

Why is a risk margin required in insurance BEL valuation?

Answer 8

• Market-consistent principle: liabilities must be valued as if they could be transferred to another insurer.
  
  • A new insurer taking on the liabilities would need capital to cover non-hedgeable risks (e.g. mortality, longevity, morbidity).
  • The risk margin ensures:
• Liabilities are not understated relative to a true “exit value.”
• The transfer value includes not only expected future cashflows (Best Estimate Liabilities, BEL) but also the cost of holding capital.
• Protects policyholders by ensuring an acquiring insurer is compensated for risk-taking.
  
  • Without risk margin:
• Liabilities would equal only BEL, ignoring cost of capital.
• Would not reflect reality of market transaction

Question 9

How is the risk margin calculated, and what proxy models are used in practice?

Answer 9

• Risk margin = present value of the future cost of capital needed to support non-hedgeable risks.
• Formula:
  RM =sum( CoCt * Capital t *Vt) at risk-free rate at time t
  – CoC: Cost-of-Capital rate (6% p.a. in Solvency II).
  – SCR_t^{NH}: SCR for non-hedgeable risks in year t (e.g. mortality, longevity, morbidity, lapse).
  – r_t: risk-free discount rate.
• Requires projecting future SCRs, which is data- and resource-heavy.
  
  • Practical difficulties:
• Full projection of SCRs each year is rarely feasible.
• Need to approximate future capital requirements without running a full internal model over the lifetime of liabilities.
  
  • Proxy models (approximations):
• Proportional to BEL: assume SCR declines in proportion to Best Estimate Liabilities.
• % of capital-at-risk: link SCR_t to the amount of capital-at-risk (e.g. sum at risk for mortality).
• Duration-based scaling: approximate run-off of SCR by applying a simple duration/decay factor to initial year SCR.
• Factor-based formulas: supervisors sometimes prescribe simple methods (e.g. fixed % of BEL).
• Aim: reduce complexity while giving a reasonable, auditable approximation.
  
  • Key point from notes:
• Proxies are permitted because projecting SCR_t exactly each year is too burdensome.
• But supervisory approval may be needed, and companies must demonstrate that proxies are not materially misstating the risk margin.

Question 10

What are the steps in calculating the SCR under the VaR approach, including aggregation and correlations?

Answer 10

1. VaR Definition and Setting
   * Based on a risk measure such as Value-at-Risk.
   * Typically 99.5% confidence over a 1-year horizon.
   * Capital ensures assets > liabilities in 1 year with probability 0.995.
   
   • 2. Stress Testing
        
        • Apply shocks to each risk factor (mortality, expenses, lapse, market, etc.).
        • Shocks are calibrated to the 99.5% 1-year level.
        • Recalculate surplus after stress.
        • Hold capital at outset so post-shock assets ≥ liabilities.
   • 3. Aggregation and Correlations
        
        • Each stressed risk gives an isolated capital requirement.
        • Combine them to form the total SCR.
        • Diversification benefits recognised via correlations.
        • Techniques: correlation matrices or copulas.
   • 4. Correlation Matrix Formula
        
        • Aggregated SCR must include correlations using correlation matrix or copulas
   • 5. Non-Linearity and Non-Separability
        
        • Correlation matrices may understate total risk.
        • Non-linear effects: capital ≠ linear function of risks.
        • Non-separability: joint impact > sum of parts.
        • Adjustments may be required.

Question 11

What are the steps in calculating the SCR using the run-off (or ruin probability) method?

Answer 11

• 1. Principle
• Focuses on long-term solvency rather than a 1-year VaR.
• Requires that the insurer has enough assets so that the probability of ruin over the lifetime of liabilities is below a defined threshold.
  
  • 2. Projection
• Project cashflows of assets and liabilities to extinction (run-off).
• Model stochastic scenarios for key risks (economic + demographic).
• Assess whether assets cover liabilities under these scenarios.
  
  • 3. Solvency Condition
• SCR is the capital required today so that the probability of default (ruin) over the lifetime of obligations is below the regulatory minimum (e.g., ≤ 0.5%).

Question 12

How are Solvency Capital Requirements (SCR) calculated specifically in relation to economic risks?

Answer 12

1. General Framework (VaR)
   * SCR based on Value-at-Risk (VaR), usually 99.5% confidence over 1 year.
   * Supervisory balance sheet (market-consistent) stressed against shocks to economic risk factors.
   * Insurer must hold enough capital so assets ≥ liabilities at year-end under stressed conditions.
   
   • 2. Stochastic Modelling
        
        • Capital requirements typically quantified using stochastic models.
        • Distribution must reproduce extreme behaviour (fat tails).
        • Must capture tail size accurately.
        • Must also capture path dependency (e.g., investment return trajectory before a crash).
   • 3. Model Calibration
        
        • Real-world asset model used for capital projections.
        • Model must be arbitrage free.
        • Calibrated with reference to historic parameters.
        • Advanced techniques used to ensure good fit in the tail — avoid understating extreme outcomes.
   • 4. Aggregation
        
        • Economic risk capital combined with other risks.
        • Diversification reflected through correlation matrices or copulas.
        • Adjustments needed for non-linearity and non-separability — correlation matrices alone may underestimate total capital.