The Interaction of Implied Equity Volatility, Stochastic Interest, - PDF document

Equity-Based Insurance Guarantees Conference Nov. 5-6, 2018 Chicago, IL The Interaction of Implied Equity Volatility, Stochastic Interest, and Volatility Control Funds for Modeling Variable Products Mark Evans SOA Antitrust Compliance Guidelines SOA Presentation Disclaimer Sponsored by

The Interaction of Implied Equity Volatility, Stochastic Interest, and Volatility Control Funds for Modeling Variable Products EBIG Conference (Chicago) 6 Nov 2018 (830 – 1000 Hours) Mark Evans, FSA, MAAA Applied Stochastic

Topics: • Impact of stochastic interest rates on variable annuities Stochastic interest rate provision in implied volatility • • Adjusting implied volatility for the impact of stochastic interest rates • Impact of adjusted implied volatility combined with use of stochastic interest rates upon variable annuity modeling • Accounting for stochastic interest rate impact with volatility control funds 2

Black Scholes • Black Scholes bases the equity return on the following formula. exp[(r- σ^2/2)t + ϵ σ √t] • • If we reflect stochastic interest rates, the formula becomes • exp[(r (ϴ) - σ^2/2 )t + ϵ σ √ t ] • where r(ϴ) is a stochastic process. 3

Interest Impact • r impacts the amount of the put option payoff associated with a GMxB. Unlike a vanilla put option, there are secondary compounding impacts. • • Lower interest environments result in not only a lower expected return, but also policyholders are more likely to persist. • Account value is lower relative to guarantee. • Future payment stream has a higher present value. In extreme cases, policyholders are more likely to utilize (except for • GMDB). 4

Interaction with Implied Volatility • Thus using stochastic interest rates results in higher GMxB costs. If modeling with implied volatility, however, one may want to adjust for • the provision the market makes for interest rate uncertainty in the implied volatility. • If using stochastic interest and implied volatility, then we are accounting for implied volatility twice. 5

Interaction with Implied Volatility • Since a vanilla put option price is not impacted by policyholder behavior, the impact of stochastic interest rates is less than for a variable annuity. • So reducing the volatility for the stochastic interest rate provision and using stochastic interest rates will result in a higher GMxB cost than using implied volatility and deterministic interest rates. 6

Example: • Determine the size of the stochastic interest rate adjustment in implied volatility for a 10 year put. • 1) Calculate the put price based on the swap curve and other market assumptions. • 2) Use a stochastic interest rate model to generate a set of 120 month interest rate paths. • 3) Calculate the put prices for the set of paths from step 2. • 4) Determine the average put price from step 3. • 5) Lower the volatility and redo step 3. Iterate until result from step 4 matches the result from step 1. Iteration should be quick as this is a well behaved function. 7

Details: • Stochastic Interest Rate Model Simple recombining BDT • • Monthly time steps • Calibrated to swap curve (about 3% with slight upward slope) • 30% lognormal interest rate volatility 8

Details: • Calculate 10 Year Put Along Set of Paths from Step 2 2% Continuous Dividend • • 23% Implied Volatility • 110% Strike • 10 year spot rate from 121 ending present value factors from recombining BDT tree Calculate weighted average from Pascal triangle probability • distribution 9

Results: • Adjusted volatility ends up being about 22.5% for a .5% adjustment. If modeling a variable annuity without volatility control funds, we use • 22.5% equity volatility and stochastic interest rates. 10

But what happens with a volatility control fund where the equity (or risky asset) percent is chosen to produce a target volatility?

Volatility Control Fund • Let’s assume that the volatility target is 10%. Then if using stochastic interest rates we would project equity returns • using 22.5% volatility. • Now the fund manager will observe volatility and use that observation to set the equity percent. • The stochastic interest rate has to introduce some volatility into this observation, but it turns out not to be .5%. 12

Black Scholes Again • When using stochastic interest rates, a lot of relationships that hold with fish bowl Black Scholes assumptions no longer work. • exp[(r (ϴ) - σ^2/2 )t + ϵ σ √ t ] • r(ϴ) is a function of prior periods interest rate changes so the expected return along a given path now depends on the prior actual returns. The fish bowl is broken. 13

Example: • Let’s work through an example. There are different ways of doing this, but let’s assume the volatility • control procedure bases the target on the last month’s observed volatility. • We can go into our interest rate model and look at the impact upon return volatility due to the interest rate volatility in month 120. 14

Month 120 • The interest rate at the start of month 120 is a function of what happened in the previous 119 months. • But, the interest rate volatility in the previous 119 months does NOT impact the volatility the fund manager is going to see during the one month observation period. 15

10 Years Versus One Month • So when we observe interest rate volatility standing at time zero and look out over the next 10 years, it creates a wide range of possible outcomes. • But for each of those possible interest rate paths, at any future point along that separate path, the fund manager will see a much smaller impact from interest rate volatility because he only sees the last month. 16

+/- 1 Standard Deviation MM Growth 0.6 0.5 0.4 Series1 Series2 0.3 Series3 Series4 0.2 0.1 0 1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 17

Interest Volatility in Month 120 18

Impact on Control Fund • If we look at the volatility due to the impact of stochastic interest rates along each path, we can do a direct measure of the result. • Under a lognormal interest rate model, this varies dramatically by interest rate level. • It is small except for very high interest rate paths which have a low probability and are less likely to produce GMxB claims anyway. 19

Method Equity Additional Risky Volatility Volatility Allocation Ignore Stochastic Interest 22.5% 0% 44.4% Pure Monthly Volatility 22.5% .0003% 44.4% Zero Mean Volatility 22.5% .052% 44.3% Implied Volatility 22.5% .5% 43.5% 20

Summary: • Variable annuities are particularly sensitive to interest rate volatility. Implied volatility used for pricing vanilla options includes a provision for • interest rate volatility. • The provision can be estimated from market data. • The provision can be removed from equity volatility when modeling variable annuities using stochastic interest rates. But this provision is likely much smaller when modeling volatility control • funds. • Numbers shown here are for illustrative purposes only. They are not intended to reflect current market conditions. 21