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

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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

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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.

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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).

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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.

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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.

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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.

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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

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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

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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.

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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
• bservation, but it turns out not to be .5%.

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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.

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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.

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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.

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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

• utcomes.
• 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.

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+/- 1 Standard Deviation MM Growth

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0.1 0.2 0.3 0.4 0.5 0.6 1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 Series1 Series2 Series3 Series4

Interest Volatility in Month 120

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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.

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Method Equity Volatility Additional Volatility Risky 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%

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.

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