The Impact of Target Volatility Strategies on Variable and Indexed Annuities

Transcription

1 Equity-Based Insurance Guarantees Conference November 18-19, 2013 Atlanta, GA The Impact of Target Volatility Strategies on Variable and Indexed Annuities Mark Hadley

2 The Impact of Target Volatility Strategies on Variable and Indexed Annuities Mark Hadley FSA, CFA, CAIA, FRM Equity Based Investment Guarantees 19 November 2013 ( Hours) 1

3 Agenda The structure of investment guarantees The structure of volatility Target volatility strategies Impact of target vol strategies on Variable and Indexed Annuities Some enhancements to target volatility Capital protection Fee Indexing 2

4 The Structure of Variable and Indexed Annuities Exposure, Potential Future Exposure, and Potential Future Greeks STRUCTURE OF INVESTMENT GUARANTEES 3

5 Indexed Annuity Crediting Mechanics Indexed Annuities are short-dated call-like equity derivatives priced with participation rates and caps Point-to-point and monthly cliquet designs are the most popular Asianing and basket features are common, as well There are non equity linked products eg Gold and Treasury There are non-equity linked products, eg Gold and Treasury rates, although they are less common 4

6 12/3/2013 Exposure Exposure is the future values of a trade, over its life and the full sample space. It is a big data quantity. Sample p Paths of a Point-to-Point 12% 10% Future Values 8% 6% 4% 2% 0% Trading Days

7 Potential Future Exposure (PFE) of a Point-to-Point PFEs are the percentiles of exposure, over time. A Pointto-Point has PFE bounds at 0 and its cap, here 10% 20% Potential Future Exposure of Point-to-Point 15% PFE 5/99 10% 5% 0% Trading Days 6

8 PFE: Point-to-Point vs Cliquet Though the cliquet and point-to-point start at the same PV, the upper bound on possible cliquet payoffs is noticeably higher 20% Potential Future Exposure of Cliquet and Point-to-Pointto 15% PF FE 5/95 10% Cliquet 5% P2P 0% Trading Days 7

9 Potential Future Exposure of a GLWB Rider Specification Withdrawal rate based on age at withdrawal: 4%/5%/6% vs ages <65/75> Annual ratchet; 5% simple annual rollup during deferral period up to 10Y Rider fee is 0.70%; M&E is 1.20%; fund fees are assumed zero 20/80 PFE Per-Leg GLWB PFE Claim Leg Fee Leg Actuarial Assumptions Varied ages and withdrawal cohorts, so a block-level l l perspective Mortality is A2000 with Scale G improvement Base lapse based on 7Y surrender charge period One-sided dynamic lapse function Months from Issue 8

10 Potential Future Greeks: Point-to-Point In general, vega decays as an option approaches expiry. Gamma can be quite explosive near maturity if the option is at-the-money 9

11 Potential Future Greeks: GLWB The claims leg has decaying vega, as expected. But the fee leg doesn t have any significant vega exposure, as it is delta-one Per-Leg GLWB Potential ti Future Vega PF Vega 20/ Claim Leg Fee Leg Months from Issue 10

12 An assortment of volatility s empirical behaviors A robust vol estimation model THE STRUCTURE OF VOLATILITY 11

13 Volatility is THE Measure of Risk Risk is a fundamental pillar of modern finance. People approach it in at least two philosophically distinct ways: Markowitz s (1952) focus on the volatility and correlations of a portfolio s constituents Black-Scholes (1973): quadratic variation of a continuous process 12

14 Measuring Volatility Vol is not directly observable, which makes measuring it difficult Sample standard d deviation, though h in wide use, is a noisy estimate of a process s volatility Range-based statistics ti ti have much better statistical ti ti efficiency (about 4x); also they are approximately Gaussian which makes them more useful in model estimation High-frequency tick data, which obviously has many more sample points, can also reduce standard errors for volatility measurement 13

15 Modeling Volatility Of course, this hasn t stopped academics from proposing dozens of models The literature is replete with vol models Stochastic Vol GARCH Dynamics Continuous Discrete Risk neutral fit Closed form solutions/approximat With rare exception, eg Heston Nandi ions are common (2000) Real world fit Maximum likelihood estimation is very difficult MLE is easy 14

16 Trading Volatility And it certainly hasn t stopped traders and bankers from making derivatives markets, bundling vol in many packages Options: Black-Scholes showed volatility is a main component of an option s value Variance swaps: Derman showed the variance of a continuous diffusion can be manufactured with the log-contract, which can be approximated by a range of options VIX derivatives: CBOE changed their VIX calculation methodology to one consistent with variance swaps, and since then have been able to launch many successful derivatives contracts 15

17 Volatility Smirks SPX Option Vol Nov 14, % V400% Implied Vol 400% 200% 0% Jul14 Feb15 Aug15 Maturity Jan14 16

18 And Volatility Smiles VIX Option Vol Nov 14, % Implied Vo ol 400% 200% 0% Strike Apr14 Feb14 Maturity Jan14 17

19 Volatility Parameters The chart below illustrates how volatility dynamics manifest themselves in option prices 40% Volatility Skews 35% Imp plied Volatility 30% 25% VolofVol=0 VolofVol>0 Rho<0 20% Moneyness 18

20 A Day in the Life of Volatility Volatility has a 9-5 And it takes off weekends and holidays 19

21 Volatility has Volatility The log-normal distribution fits vol-of-vol best 0.8 Stein&Stein (1991) Normal LLF: llf: Heston (1993) CIR LLF: llf: Frequency 0.4 Frequency Daily VIX Movement Daily VIX Movement 0.8 GARCH Log-Normal LLF: llf: /2 LLF: llf: Frequency 0.4 Frequency Daily VIX Movement Daily VIX Movement 20

22 And the Volatility Volatility has that has Volatility, too The VIX of VIX index is computed from VIX option skews. 150 VVIX Index May05 Oct06 Feb08 Jul09 Nov10 Apr12 Aug13 Dec14 21

23 Volatility Remembers A process s persistence is measured with the autocorrelation function (ACF). In the context of volatility, this is referred to as clustering, and it has remarkably long memory statistically significant for more than a year! Sample: end-of-day fixings for S&P500 from 31OCT

24 Volatility Remembers A LOT! Mandelbrot (1963) observed this behavior for cotton prices and figured a power law fit it well but powers decay too slowly and exaggerate the memory Stochastic vol models have geometric decay, for example Heston s ACF is: 23

25 And Volatility Forgets It is clear that volatility has two components: one that is temporary but highly explosive (blue) and another that is less volatile, but persistent for years (black) 24

26 Volatility is no stranger to hard times The leverage function measures the impact of current returns with future volatility -50% Leverage Function of SPX Returns -10% Sample Le everage Function -20% -30% -40% Leverage(lag) Bartlett Bounds Heston Law -50% Lag (Trading Days) 25

27 Volatility is Predictable! The following multi factor E GARCH calibrated to daily range data provides excellent forecasts of near term volatility. Brandt & Jones (2006) Transient component mean reverts around a persistent component Persistent component mean reverts around a long-run average Estimated with historic daily SPX fixings from October 1981, using maximum likelihood 26

28 Motivation Mechanics Backtests TARGET VOLATILITY STRATEGIES 27

29 Why Target Volatility? It is much easier to hedge derivatives on a return sequence with constant volatility. Note we have said nothing about the derivative (yet): it could be anything First-Order Risk Second-Order Risk Profit and Loss Hedge Assets Profit and Loss Hedge Assets Delta * Fund Easily hedged with index futures Gamma * Vol Volatility is targeted so gamma losses are more stable Rho * Rates EQVega * EQVol Easily hedged with IRS or /Treasury futures Vega hedges, often economically disadvantageous, are unnecessary because the fund has constant vol IRVega * IRVol EQVolGamma * VolofVol Efficiently hedged with swaptions No contribution, as there is constant fund vol December 3,

30 Volatility Targeting If an asset has predictable volatility, then you can construct a selffinancing portfolio with locally constant volatility The portfolio consists of a risky-asset like the S&P 500 and a low-risk asset, here it is a constant maturity zero coupon bond The allocation changes dynamically based on future expected volatility and the volatility target The strategy includes some rebalancing frictions 29

31 Spectrum of Target Vol strategies Volatility Target Most insurance companies target 10-12% (lower the better!) The optimum target is the asset s average volatility Volatility Estimators Rolling standard deviation Exponentially weighted moving average GARCH Multi-factor E-GARCH VIX Published Target Volatility Indices S&P: two-scale EWMA Russell indices: sample standard deviation 30

32 Backtests The strategy is able to achieve a near-constant return volatility. It works well on both SPTR (left) and cotton (right) Fund Vol vs Benchmark Fund Benchmark Fund Vol vs Benchmark Fund Benchmark Month Rolling Vol Month Rolling Vol Mar93 Dec95 Sep98 May01 Feb04 Nov06 Aug09 May12 Feb15 0 Mar93 Dec95 Sep98 May01 Feb04 Nov06 Aug09 May12 Feb

33 Backtests Giese (2012) shows that risk controlled strategies have superior Sharpe ratios Target Vol SPTR Target Vol Cotton Return 8.52% 8.70% 2.03% 0.99% StDev 15.08% 19.36% 15.12% 25.66% Sharpe Kurtosis

34 The Volatility Skew of Target Volatility Target volatility strategies flatten the volatility skew of the return distribution. This is equivalent to a closer-to- Gaussian return distribution a la Breeden-Litzenberg (1978) 25% Vol Skew (1Y) S&P500 and Target Vol S&P500 20% 15% 10% 5% S&P500 Target Vol 0% 80% 100% 120% 140% Moneyness 33 33

35 Impact on Indexed Annuity Pricing Impact on Variable Annuity Risk IMPACT ON INVESTMENT GUARANTEES 34

36 Indexed Annuities: Participation Rates Lower volatility targets make higher participation rates affordable 35

37 Indexed Annuities: Participation Rates This advantage exists over the full business cycle 180% 160% Annual Point To Point Fair Participation Rate 140% 120% 100% 80% 60% 40% 20% 0% 12/3/ Jan 05 Apr 05 1 Jul 05 Oct 05 Jan 06 Apr 06 1 Jul 06 Oct 06 Jan 07 Apr 07 1 Jul 07 Oct 07 Jan 08 Apr 08 1 Jul 08 Oct 08 Jan 09 Apr 09 1 Jul 09 Oct 09 Jan 10 Apr 10 1 Jul 10 Oct 10 Jan 11 Apr 11 1 Jul 11 Oct 11 Jan 12 Apr 12 1 Jul 12 1 Oct Jan 13 Apr 13 S&P 500 RC5 S&P 500 RC10 S&P 500 RC15 S&P 500 TR

38 Indexed Annuities: Caps Similarly, lower volatility targets make higher caps affordable 37 37

39 Indexed Annuities: Caps History Since 2005 we ve seen a secular decline in yields and a range of volatility regimes. In all these market environments, capped point-to-points are cheaper to manufacture on lower volatility Risk Controlled indices. Annual Point To Point Fair Market Cap 18% 16% 14% 12% 10% 8% 6% Index Average Cap S&P500 TR 7.33% 4% S&P500 RC % S&P500 RC % 2% 0% 38 12/3/ Jan 05 1 Apr 05 1 Jul 05 1 Oct 05 1 Jan 06 1 Apr 06 1 Jul 06 1 Oct 06 1 Jan 07 1 Apr 07 1 Jul 07 1 Oct 07 1 Jan 08 1 Apr 08 1 Jul 08 1 Oct 08 1 Jan 09 1 Apr 09 1 Jul 09 1 Oct 09 1 Jan 10 1 Apr 10 1 Jul 10 1 Oct 10 1 Jan 11 1 Apr 11 1 Jul 11 1 Oct 11 1 Jan 12 1 Apr 12 1 Jul 12 1 Oct 12 1 Jan 13 1 Apr 13 S&P500 TR S&P500 RC15 S&P500 RC10

40 Indexed Annuities: Potential Gamma Gamma is the second-order difference of a 3% shock. Gamma of the risk-controlled product is an order of magnitude less. In fact, there are scenarios when it is negative

41 Indexed Annuities: Potential Vega Vega is measured as a 100bps parallel shock to the SPX volatility surface. It is negligible for the duration of its life, though it has wider bounds near maturity. This is because target volatility strategies perform best over longer windows

42 GLWB: Target Vol Vega Reduction Target volatility is very effective at substantially reducing vega exposure from the claims leg 0.5 GLWB Calims: Potential ti Future Vega 0.4 PF Vega 20/ Target Vol Static Months from Issue 41

43 GLWB: Target Vol Gamma Reduction Similarly, target volatility reduces the claim leg s gamma 0.01 GLWB Calims: Potential Future Gamma PF Vega 20/ Static Target Vol Months from Issue 42

44 Target Vol and Capital Protection Target Vol and Fee-Indexing ENHANCEMENTS TO TARGET VOL 43

45 Capital Protection Dynamics The Capital Protection strategy involves an additional component in the fund s returns: the delta of a 5-year constant maturity put (around -25%), Target volatility (again) Capital protection 44

46 Capital Protection Future Greeks When Capital Protection is added to target vol, the claims leg has is slightly negative vega 0.5 GLWB Claims: Potential Vega Static vs TargetVol+CapitalProtection Vega 20/80 PF Static Target Vol + CapitalProtection Months from Issue 45

47 Fee Indexing Dynamics Fee-indexing in VA product is very similar to the indexing mechanics seen in Adjustable Rate Mortgages 46

48 Fee Indexing Future Greeks The vega term structure of Fees less Claims is net POSITIVE for a fee-indexed design with target vol in the funds GLWB Fees - Claims: Potential ti Future Vega PF Vega 20/ Months from Issue 47

49 Conclusion Volatility has many interesting empirical behaviors, but the most interesting is its predictability Target volatility strategies are intuitive, easy to execute, and have a material and beneficial impact on both indexed and variable annuities Additional enhancements like fee indexing and capital protection can further tailor risk profiles 48

50 The Nested Stochastic Design Pattern Happy to share this Nested Stochastic framework, which is implemented fully in one instance of 64-bit Excel 49