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Soa Equity Based Insurance Guarantees Conference 2008
 

Soa Equity Based Insurance Guarantees Conference 2008

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    Soa Equity Based Insurance Guarantees Conference 2008 Soa Equity Based Insurance Guarantees Conference 2008 Document Transcript

    • Equity–Based Insurance Guarantees Conference March 31-April 1, 2008 Quantifying Risks Associated With Guarantees CF Yam Simina Cana and David Gott
    • Quantifying Risks Associated withEquity-Based GuaranteesC.F. YamContents • Introduction – Equity-Based Guarantees • Risk Management Considerations • Basics of Dynamic Hedging • Quantification of Risks • Practical Limitations • Concluding Remarks 2
    • Equity-Based Guarantees • Equity-Based Guarantees, namely: - Fixed Annuities Variable Annuities • Variable Annuities = Unit-linked Investment Product with given Fund Choices (Unit Trusts / Mutual Funds rather than Market Indices); and Investment Allocation / Fund Switching made by Customer; (may be on the advice of Distributor / Financial Planner); (can be on an unrestricted manner / frequency); Plus Investment Return / Benefit Guarantees offered at a given fee for the specific guarantee; which is normally fixed (less dependent on recent investment condition); (and may not be changeable, regardless of changes in investment conditions in the future). 3Risk Management • Other than Strategic, Operations, Skill-set, Reputation, Litigation, Counterparty Risks, the Key Risk for managing Variable Annuities is Financial (Balance Sheet) Risk • If not Outsourcing the VA management, typical Financial Risk Management will be: - Asset Liability Security Investment ← → Risk Management Static Hedging ← → Pricing Trading Dynamic Hedging ← Desks / Asset ALM Financial → Valuation Models Static & Dynamic Hedging ← Management ↔ → Financial Reporting ArmCapital Management (Naked) ← → Capital Planning 4
    • Risk Management• Financial Modeling to quantify Guarantee Risk: - Black Box Judgemental ← Historical Data Quantification Results ← Models ← Assumptions ← Empirical Experiences ← Allowance for Unknown ← Unknown of Unknown • Nested Stochastic Projections (Number of Scenario Runs) • Partial Differential Functions to provide Analytical Results • Heuristic Approach to save Computation Time • Inter-dependence of Parameters / Path-dependent Scenarios• Risk on Risk (i.e. Risk Management on Financial Modelling to make sense for Quantification Results). 5Risk Management• Risk Management typically involves Identification, Assessment, Response, Control, Monitoring• Practically, Quantification of Risks by Financial Models require - Understanding of Basics - Systematic Building of Blocks and Comparing Relativity - Using Boundary Conditions to help checking reasonableness of Outputs / Signals - Identifying Exotic Payoff Movements for small changes in assumptions / underlier values - Tradeoff of Risk Alternatives to assess Model Relativity / Consistency - Reasoned Comparisons of Outputs across different Financial Market disciplines - Due Diligence / Disciplined Processes / Actions 6
    • Back to BasicsAssume No Transaction Costs/Lapses/MortalitiesPV (Guarantee Fee Income ≅ Black-Scholes Option Premium (10-on Single Premium year Vanilla Put with Strike 100)10-Year GMAB (Point-to-Point))PV (Guarantee Fee Income ≅ ∑ Discrete Probability Function (K) * K ≥100on Single Premium Black-Scholes Option Premium (10-Year Vanilla Put with Strike K)10-Year GMAB (Ratchet)) 7Option Pricing• Under the conditions of Constant Volatility and No Transaction Costs, Black-Scholes when publishing the option pricing formula, asserted that the Price of an Option should be the Discounted Value of the cost of Dynamically Hedging the exposure to Expiration.• Dynamic Hedging refers to: - - Delta Hedging of a non-linear position with linear investment(s) of the underlying - The deltas of the non-linear position and linear position offset, yielding a zero delta for the hedged portfolio (π) - The non-linear position f can generally be expressed in the parabolic form of the underlying (S): - f ( S ) = cS + bS + a 2 ⎛∂ f ⎞ ⎛ ∂f ⎞ 2 where Gamma ⎜ ⎟ = 2 c, Delta ⎜ ⎟ = b, a = f ( S = 0 ) ⎝ ∂S ⎠ 2 ⎝ ∂S ⎠ 8
    • Dynamic Hedging δ line Long Putf f Short Put f S S S0 S π = f −δ * S Zero Delta Short δ f stock S to f nullify Delta S0 S S 9 Dynamic Hedging • As the underlying (S) changes value, the delta of the non-linear position changes, but not for the linear instrument. The deltas no longer offset. Thus, the linear hedge has to be adjusted (increase or decrease exposure) to restore the delta hedge. The continual adjusting of the linear position to maintain a zero delta is Dynamic Hedging. • Writing a VA is equivalent to a Short Put. Short Put creates Negative Gamma on the non-linear exposure ( curvature opens downward) of the underlying. • Dynamic Hedging for a Negative Gamma position will lead to Gamma Loss. 10
    • Gamma Loss At time 0, for the delta hedged portfolio, ∂f π = f(s ) − δ * S and δ = S = S0 0 0 0 ∂S At time 1, a) If S moves up to S + Δ S , π 1 = f ( S + Δ S ) − δ * ( S + Δ S ) 0 0 0 0 ∂f ∂f Given Negative Gamma, S =S0 + ΔS < S < s< s0 + Δs < δ ∂S ∂S 0 > π − π = f (S + ΔS ) − δ * (S + Δ S ) − f (S ) − δ * S [ ( ) ] 1 0 0 0 0 0 = f S + ΔS − f (S ) − δ * ΔS 0 0 ⎡ ∂f ⎤ = ⎢ S0 <S <S0 +ΔS − δ ⎥ * Δ S = Negative Value * Δ S = Loss; ⎣ ∂S ⎦ b) If S moves down to S − Δ S , π = f (S − ΔS ) − δ * (S − ΔS ) 1 0 1 0 0 ∂f ∂f Given Negative Gamma, S =S0 −ΔS > S >S >S0 −ΔS > δ ∂S ∂S 0 > π − π 1 0 [ = f (S − Δ S ) − f (S ) − δ * ΔS 0 0 ] ⎡ ∂f ⎤ = ⎢ S 0 > S > S 0 − Δ S − δ ⎥ * ( − Δ S ) = Positive Value * (- Δ S ) = Loss ⎣ ∂S ⎦ 11Gamma Loss• For a Negative Gamma Portfolio, it always loses value on delta re-hedge due to Buy High / Sell Low phenomenon. It never gains it.• For each re-hedge, the Loss will equal to ½ * Gamma * (Change of Underlying Value)2.• The sum of Gamma Loses, till Expiration, is the actual cost of the option (before Transaction Costs).• The cash balance on dynamic hedging a Short Put (negative Г) reduces as follows: - 1 N −1 2 s −s= ∑ (r − σ 2Δt ) Γ⎛ ⎞ s 2, where r = i + 1 i ,σ = Implied Volatility 2 i =1 i i ⎜ i, s ⎟ i ⎝ i⎠ i s i i ← Experienced Volatility < Implied Volatility ← Experienced Volatility = Implied Volatility 12 Option Sold Option Expired
    • Pricing Considerations• As can be seen, at a higher volatility, the underlying will fluctuate more, and the delta hedge needs to be adjusted more frequently. The cash balance (arisen from the premium / fee income received) will lose more rapidly when dynamically hedging the non-linear exposure.• Thus, the initial key consideration in assessing the sufficiency of VA (i.e. GMB) charge is: - - to evaluate the Г of the guarantee with change in value of the underlying till expiration -to consider the potential volatility of the underlying (as the ultimate hedging cost depends on the experienced volatility)• In addition, for the non-linear exposure f=f(S,σ,r,t) - By Taylor Expansion ∂f ∂f ∂f ∂f 1 ∂2 f Δf = Δt + ΔS + Δσ + Δr + (ΔS ) 2 + ..... ∂t ∂S ∂σ ∂r 2 ∂S 2 - By Delta Hedging using linear instruments 2 ∂f ∂f ∂f ∂f 1 ∂ f 2 Δf − ΔS = 0 = Δt + Δσ + Δr + ( ΔS ) + ..... ∂S 2 ∂S ∂t ∂σ ∂r 2 Theta Vega Rho └ Gamma Management ┘ Decay 13Pricing Considerations• For VA, the non-linear exposure F ~ ~ σ F = F(f( S i , i , r, t), Si , Ti , Transaction Cost, Profit / Capital Charge) ~ for all Fund Choices Si where S i is the proxy investment vehicle which can be used to dynamically ~ hedge Si ~ ~ Ti is the period under which S i is selected by the Customer for F to be subject to.• The additional pricing considerations or quantifications, therefore, include: - Basis / Gap Risk ( S i → s i ≅ 10 % extra cost) ~ - Policyholder Behavior (Product Design (MVA) or Fixed Penalty to address?) -Transaction Costs for Hedging - Fixed VA Charge vs Variable Theta Decay (Profitability & Profit Variation) on top of cost considerations for Vega, Rho, Gamma management, Underlying Volatility implications and Frequency of Hedging. 14
    • Vega• May estimate the Fair Strike Value of Variance Swap, based on raw option prices. 2 ΔK ΔK i erT Call ⎡ K ⎤ − 1 ⎜ FT −1⎟ ⎛ ⎞ σ =2 ∑ 2 i erT put ⎡ K ⎤ + 2 ∑ ⎢ ⎥ ⎢ i⎥ ⎜ ⎟ ⎣ i⎦ T ⎦ T ⎜K K ⎣ ⎟ T K ≤F K 2 K >F K 2 ⎜ ⎝ 0 ⎟ ⎠ i T i i T i S where F = e rT , K = i T i S 0 K is the first strike below the forward F 0 T• Issue: - Not sufficient option prices available in the market for longer term T. - Work best for shorter term T - Hong Kong: Warrant Transactions (20%-40% daily stock market turnover of US$10- 20bn), but mainly 6-9 months for T. 15Volatility Surface• Volatility is not constant (per Black-Scholes) and changes with St, k, T• When Market falls, price movements are usually sharp. Volatility will shoot up. Issuer of in-the-money VA (or GMB) will feel a double-hit.• Implied Volatility Volabtility Smile Volatility Skew Strike Strike• Volatility skew in equities reflects investors’ fear of market crashes which would potentially bid up the prices of options at strike below current market levels.• Volatility increases with reduction in expiration time. 16
    • Volatility Surface• The hedge ratio will be off if the non-linear position is dynamically hedged without incorporating the effect of Volatility Surface into the delta calculation.• Standard Models to allow these effects include: - Jump-Diffusion Model adds random / Poisson jumps to the GBM that the underlying assumes - Regime-Switching Model probabilistically selects different volatility bases in the modelling process - Stochastic Volatility Model models the underlying’s value & its volatility as stochastic processes 17Delta-Gamma Hedging• To subscribe another non-linear instrument (g) in addition to the linear instrument to hedge against the exposure on non-linear instrument (f). ∂f ∂f 1 ∂2 f 2 Δf = Δt + ΔS + (ΔS ) ∂t ∂S 2 ∂S 2 ∂g ∂g 1 ∂2g Δg = Δt + ΔS + (Δ S )2 ∂t ∂S 2 ∂S 2• Hedged Portfolioπ = f + α 1S + α 2 g ⎛ ∂ f ∂ g ⎞ Δ π = ⎜ + α ⎟ Δ t ∂ t ∂ t 2 ⎝ ⎠ ⎛ ∂ f ∂ g ⎞ + ⎜ + α + α ⎟ Δ S ∂ S ∂ S 1 2 ⎝ ⎠ ⎛ ∂ 2 f ∂ 2 g ⎞ ⎟ (Δ S ) 1 2 + ⎜ ⎜ + α ⎟ ∂ S ∂ S 2 2 2 2 ⎝ ⎠• To Make Terms for Δ S & (Δ S )2 = 0 ∂2 f ∂2g ⎛ ∂f ∂g ⎞ +α = 0;⎜ + α1 + α ⎟ = 0 ∂S 2 ∂S 2 ⎝ ∂S ∂S ⎠ 2 2 ∂ g 2 ∂ f 2 ∂f ∂g ∂2g ∂2 f α = - ;α 1 = −( − * ) ∂S 2 ∂S 2 ∂S ∂S ∂S 2 ∂S 2 2• f=Negative Gamma → g = Positive Gamma ; with g = Positive Gamma, direct δ exposure to S reduces 18
    • Delta-Gamma Hedging• g=Positive Gamma means Long Volatility Option• Option Illiquid Higher Transaction Costs• Delta-Gamma Hedging = Dynamic + Static Hedging• The continuous dynamic hedging (including use of static hedges) will incur an infinite amount of transaction costs, no matter how small it is.• In the presence of transaction costs, the absence of arbitrage argument is invalid, market is incomplete, which leads to many solutions.• There is no definitive solution on VA management. The success in offering VA will therefore depend on the availability of the right skill sets, integrated processes, risk management capabilities by the underwriter to generate viable and sustained solutions. 19Hedging Methods• Finally, the desirable hedging method will, inter alia, depend on Transaction Costs and the Risk Tolerance / Appetite of the VA underwriter.• Assume a sale of δ shares of the underlying incurs transaction costs λ/δ/s (λ≥0). Below are 6 common hedging methods. a) The Black-Scholes Hedging at Fixed Regular Intervals. The balance account adjusted on reinstating the target hedge ratio: - ⎡⎛ ∂f ∂f ⎞ ⎛ ∂f ∂f ⎞⎤ ⎢⎜ − ⎟ − λ ⎜ − ⎟⎥S ⎜ ⎟ ⎜ ∂S ∂S t ⎟ t+ h ⎢⎝ ∂ S t+ ⎣ h ∂S t ⎠ ⎝ t+ h ⎠⎥⎦ b) The Leland Hedging at Fixed Regular Intervals As per (a) using a modified volatility in the model. (σ 2 m = σ 2 [1 − λ * Constant * Γ ] ) c) The Delta Tolerance Strategy ∂f Δ − > h (a given constant) ; Re - hedge to Target Hedge Ratio ∂S d) The Asset Tolerance Strategy S (t + Δ t ) − S (t ) > h (a given constant) ; Re - hedge to Target Hedge Ratio S (t ) 20
    • Hedging Methods (con’t)e) Hedging to a Fixed Bandwidth around Delta ∂f Δ = ± h per the Delta Tolerance Strategy ; ∂S Re-hedge to the Hedge Ratio to the nearest boundary of the Hedging Bandwidthf) The Asymptotic Analysis of Whalley and Wilmott ∂f 1 Δ = ± h ( e − r (T − t ) S Γ 2 ) 3 ; ∂S Similar to (e), the Size of the Hedging Bandwidth depends on the price of the underlying and the option gamma• h depends on the risk aversion of the VA underwriter• Empirical studies suggest e) & f) are outperforming methods. e) outperforms f) if the risk tolerance of the VA underwriter is higher. 21Concluding Remarks• The above quantification considerations will lead to a relevant Variable Theta Decay to breakeven the related risks & costs.• Assuming the risks associated with Policyholder Behavior to be mitigated by relevant Product Design, it still leaves the issue of Fixed Guarantee Fee for Variable Theta Decay in the ultimate pricing.• There are variances in practice within the financial service industry: Insurer Investment BankUse of Capital Capital at Risk(Naked) → Hedging Hedging → Capital for Extremes of Models r=μ μ −σGirsanov’s Theorem r= θRisk Premium Shareholder / Customer to share Customer to bear• Reasonable Check: How the Customer values the net results across different financial instruments / products. 22
    • Concluding remarks (con’t)• The challenge when structuring VA in Hong Kong US Variable Annuity Hedging via Investment Bank Long Term Volatility ~ 15% p.a. Implied Volatility ~ 50% p.a. (Term slope not very deep) + Basis / Gap Risk + Profit Load• Implications: - - Product Design → To find means to stabilize the σ for hedging → To reduce anti-selection Policyholder Behavior (Distributor Advice) - Skill-set → Risk Management and Capital Planning → Operations and Integration with Trading Desks• VA is a very Valued Product for Customer & for Retirement Planning / Use• Actuary’s Social Responsibility: - To research, develop, implement and risk manage the Product in a professional manner 23