FINANCE, INVESTMENT & RISK MANAGEMENT CONFERENCE. SWAPS and SWAPTIONS Interest Rate Risk Exposures JUNE 2008 HILTON DEANSGATE, MANCHESTER

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1 FINANCE, INVESTMENT & RISK MANAGEMENT CONFERENCE 5-7 JUNE 8 HILTON DEANSGATE, MANCHESTER SWAPS and SWAPTIONS Interest Rate Risk Eposures Viktor Mirkin vmirkin@deloitte.co.uk 7 JUNE 8 HILTON DEANSGATE, MANCHESTER Outline Cash flow structure, optionality and pay-off Overview of the OTC market in 8 Valuation methodologies models of the yield curve Black s formula Implied volatilities Data sources the swap curve Hedging interest sensitive liabilities with swap and swaptions Rho, rhoga, vega, volga, rhova Eotic swaps GAO hedges: impact of mortality Liability valuation basis impact: marked to market calibration of yield curve and volatility. Evidence from WP insurers

2 Relevance to Life Insurance Liabilities Insurers are structurally long in interest rates on marked to market liabilities Guaranteed Annuity Options Mortality linked receiver swaption with an asset linked nominal Annuities Mortality linked bond portfolios Liability matching strategies utilise opposite eposures Gilts Corporate Bonds Receiver Swaps Receiver s The relevance of swaps and swaption is therefore two fold As hedging instruments As reference instruments for marked to market calibration Receiver Swap Cash Flows.7 Fied Rate Reference Floating Rate Reference Date Settlement Date.6 Swap agreement arranged Active leg of the swap Reference Interest Rate.5.4. Fied>Floating Reds pay Purples (Fied-Floating) year interest swap arranged in 7: Purples receive fied pay floating, Reds receive floating pay fied. Floating>Fied Purples pay Reds (Floating-Fied) Receiver Cash Flows.7 Fied Rate Reference Floating Rate Reference Date Settlement Date Eercised Date Reference Interest Rate agreement arranged Option leg Fied>Floating Reds pay Purples (Fied-Floating) Swap leg Floating>Fied Purples pay Reds (Floating-Fied) 5 into 5 interest rate swaption purchased by Purples. Receive fied pay floating. Purples eercised in

3 Receiver/Payer Relationships Payer Swap + Receiver Swap = Receiver - Payer =Deferred Receiver Swap ATM: Receiver = Payer ATM Strike = Forward Swap Rate Value Caps Ma Min Receiver Swap Receiver # Payments Strike Nominal # Payments Strike Nominal Nominal Pay-off at Eercise Pay-off per nominal.4. pay-off Floorlet pay-off Interest rate at maturity relative to ATM

4 Receiver Swap and : Parallel IYC Shift Response Function 5% 4% In-the- Money Stress/Base Case % Big upside % % % At-the- Money value> before eercise/optionality Receiver (Long) 5 Year Receiver Swap (Long) -% -% -% -4% After swaption eercise eposures match Interest Rates (bp) Shift Maimum loss = swap nominal Out-of-the- Money Overview of the OTC Swap and Market Daily average turnover 8, 7, 6, USD m. Nominal Amounts 5, 4,, Swaps 7 s (and other IR options) 7 Swaps 4 s (and other IR options) 4,, USA Germany France Japan UK Switzerland BIS Triennial Survey 7 Overview of the OTC Swap and Market Switzerland UK Turnover Change 7/4 Japan France s (and other options) Swaps Germany USA -5.%.% 5.%.% 5.%.% 5.%.% 5.% 4.% 45.% BIS Triennial Survey 7 4

5 Overview of the OTC Swap and Market by Country International Interest Rate Derivatives OTC Market USA Germany France Japan UK Switzerland BIS Triennial Survey 7 Overview of the OTC Swap and Market by Country Local GBP Swaps Turnover Non Financial Companies,,88 Other Financial Institutions, 8, Dealers Other Financial Institutions Non Financial Companies Dealers, 4,99 BIS Triennial Survey 7 Valuation Methods Swap Closed Form Fully Fitted Initial Yield Curve Closed Form/Monte Carlo/Finite Differences Fully Fitted Initial Yield Curve Stochastic Yield Curve Model Arbitrage Free Market Consistent Calibration 5

6 Swap Valuation V swap i: s i S = * ( * [ = N g ZCB ( s )] + ZCB ( S ) ) i i V swap = g ATM = ( ZCB ( S )) /[ i: s i = S i ZCB ( s )] i Eotic Swap Valuation Eample Receiver Swap with Asset Linked Nominal Designed to achieve a holistic hedge of a GAO liability V swap qτ τ Ne = * + ZCB( τ ) = t L τ f ZCB(, t)*( k g ( t, t + ))*ep( σ + ρσ σ ) B N B Receiver Valuation in Closed Form Short Rate Model Bond Pricing Equation Closed Form Solution Available Models Closed Form Solution for a Analytic Calibration to Yield Curves And Volatilities Realistic Yield Curve Dynamics 6

7 Receiver Valuation: Monte Carlo Fi t P s : s+ i i= = P s : s ( s, s, t) = k ( s, s, t) t Float + ( s, s, t) = Ma( Fi( s, s, t) Float( s, s, t),) n n (, s, t) = j ( s, s, t) D j () s ( ) j= D Receiver Valuation: Black s Formula i: s i V swaption = = i ZCB ( s i )[ g * N ( d + σ S d ln( f / g ) + σ T / = σ T Log-Normal swap rate Forward risk-neutral Universal quoting convention T ) f * N ( d )] Simple Forward Risk-Neutral Model for a GAO ln SA = ln MA μ σ ~BivN, μ ρσ σ ρσ σ σ GA ln( ) + (ρσ ) [ ] σ + σ E MA ln( ) + (ρσ σ + σ ) GA E[ MA] E[ SA]* GA E [ SA* ma{, }] = E [ SA]* N( ) + * N( GA ) MA σ E[ MA] σ 7

8 Receiver Valuation: Implied Volatilities Inverting closed-form solution for log-normal swap rate V V Etended Market Model swaption swaption = V = V Black ' s Black Formula ' s Formula swaption swaption Infer corresponding implied volatility Important benchmark for cross product/quote comparison Important for etended model validation Models and Prices Market Model Swap Curve Swap Price Swap Price Possible Insurer s Model Gilt Curve Black s Volatility Price Price Black s Volatility Swap over Gilts Spread Spread over Gilts (bps) No counter party risk at inception for ATM swaps Ma eposure = current swap value Demand/supply type arguments for gilts A AA Swaps 5 8//5 8//5 8/5/5 8/7/5 8/9/5 8//5 8//6 8//6 8/5/6 8/7/6 8/9/6 8//6 8//7 8//7 8/5/7 8/7/7 8/9/7 8//7 8//8 8//8 Dates 8

9 Data Sources: Swap Curves Pit falls Interpolation Spot rates vs swap rates Market compounding convention Day counts Yield Curves EUR Yield Curve Fit December 7 6.% 5.% 4.% Spread to Gilts Rates.%.%.% Input points Initial Spot Rates Swap Rates.% Term (Years) Methodology: /asmithyield.pdf Data Sources: Implied Volatilities Pit falls Log-Normal vs Normal vols 9

10 Implied Volatilities: //7 GBR Option Term Most volatile Swap Term Implied Volatilities: //7 GBR Decrease in Option Term Decreases in Swap Term Option Term Swap Term Implied Volatilities: //7 GBR nd Generation ESGs GBR Implied Volatility Fit 5 Goodness of Fit optimised to liabilities term and duration Option Term 5.%-45.% 5.%-5.% 5.%-5.% 5.%-5.% 95.%-5.% 85.%-95.% 75.%-85.% 65.%-75.% Swap Term

11 Interest Rate Sensitive Eposures Long Position/ Asset Short Position/ Liability Receiver Swap Payer Swap Bonds/Annuities Receiver /GAO Payer Net value moves in the opposite direction to interest rates Losses when interest rates go up Net value moves in the same direction as interest rates Losses when interest rates go down Interest Rate Volatility Sensitive Eposures Receiver Swap Payer Swap Long Position/ Asset Short Position/ Liability Bonds/Annuities Receiver /GAO Payer Net value moves in the opposite direction to volatility Losses when volatility goes up Net value moves in the same direction to volatility Losses when volatility goes down Eposure Hedging Offsetting eposure construction Liabilities Annuities GAOs Interest Rate Sensitivity Interest Rate Volatility Sensitivity Assets Bonds (Long) Receiver swap (Long) Receiver swaption (Long) Losses when the driver goes up Losses when the driver goes down

12 Hedged Position Construction Net Asset Value is a function of interest rate level and interest rate volatility. NAV ( r, σ ) Δ NAV NAV ( r, σ ) NAV ( r, σ ) = Δ r + Δ σ r σ NAV ( r, σ ) NAV ( r, σ ) NAV ( r, σ ) + ( ( Δ r ) + ( Δ rδ σ ) + ( Δ σ ) ) r r σ σ Net Asset Value will be unresponsive to changes in interest rate and volatility if the asset portfolio hedges liabilities. Greeks based hedge. % Interest Rate Sensitivity 8% 5- Receiver (Long) Stress/Base Case 6% 4% % % -% Base Case 5 Year Bond (Long) 5 Year Receiver Swap (Long) Whole of Life Annuity, =6 (Liability) -4 GAO =45 (Liability) Parallel IYC Shift -4% -6% -8% -% Parralel IYC shift (bp) Volatility Sensitivity 5% % Stress/Base Case 5% % 95% 9% Interest Rate Vol Shift 5- Receiver (Long) 5 Year Bond (Long) 5 Year Receiver Swap (Long) Whole of Life Annuity, =6 (Liability) -4 GAO =45 (Liability) 85% 8% Vol shift (bp)

13 Liability Book Annuity GAO Stress/Base Case Volatility Sensitivity: Why Second Order Terms Matter Sensitivity to Vol Increases as Interest Rates Increase Interest Rate Shift Interest Rate Vol Shift % NAV Model: A Simple Longevity Book GAO % Annuities % Bonds % of Base Case 5% % Big Downside Base Case Flat Eposure Profile Assets Liabilities Net Assets 5% Parallel IYC Shift % Parallel IYC Shift (bp) NAV Model: Addition of a Downside Swap Hedge 5% % % Bonds + ATM Receiver Swap ( Cost) % of Base Case 5% % Upside Assets Liabilities Net Assets Assets Base Liabilities Base NAV Base 5% Steep Eposure % Ruin % Parallel IYC Shift (bp)

14 6% 5% NAV Model: A Rho Hedge Bonds + Receiver 4% % of Base Case % Optimises Gap between Assets, Liabilities change Assets Liabilities Net Assets Assets Base Liabilities Base NAV Base % Base Case % % Parallel IYC Shift (bp) 5% % NAV Model: Two Bond And Rho+Vega Hedge Solvency II Up stress Solvency II Down stress % of Base Case Gap Closed Further 5% % 5% Base Case Is this the best we can do? Assets Liabilities Net Assets Assets Base Liabilities Base NAV Base % 5% % Parallel IYC Shift (bp) 5% Minimal Solvency II Hedge Solvency II Up stress Solvency II Down stress % % of Base Case 5% % Base Case Assets Liabilities Net Assets Assets Base Liabilities Base NAV Base 5% % Parallel IYC Shift (bp) 4

15 Minimum Hedge Composition Bond Bond Minimum Hedge Composition Bond Bond 6 5 Minimal Solvency II Hedge Dominates the simple strategy everywhere Bond Bond 4 NAV m. Euro % in Bond Interest Rate Parallel Shift Minimal Solvency II Hedge 6 Bond Bond 5 4 NAV m. Euro Stable under Volatility Stress Interest Rate Volatility Stress 5 Minimal Solvency II Hedge: Drastic Longevity Shock Impact Insolvency NAV m. Euro Opposite longevity eposure missing tp Longevity Shock Interest Rate Parallel Shift 5

16 Marked to Market Valuation: Significance of Assumptions Annuities GAO Bonds Swaps s Yield Curve Interest Rate Volatility Conclusions Non-dealer financial companies are important players in the UK swap market Insurers interest rate sensitive eposures can be profoundly manipulated with OTC long term interest rate derivatives Hedging schemes must be carefully evaluated Yield curve is the single most significant valuation input 6