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Hedging Variable Annuities: A Dealer's Perspective

Hedging Variable Annuities in the Capital Markets: A Dealer's Perspective

by Edward A. Mirsepahi

A focus on the capital markets perspective and some of the products dealers offer to hedge and help mitigate capital costs and tail exposure risks.

In March 2004, The Actuary newsletter presented a lead article, "Managing the Risks from Variable Annuities: The Next Phase", by Phil Bieluch and Herbert Mueller. The authors provided a useful and accurate depiction of the equity exposure that insurers face due to embedded optionality sold in variable annuity products. Specifically, the four main risks arise from guaranteed minimum death benefits (GMDB), guaranteed minimum accumulation benefits (GMAB), guaranteed minimum withdrawal benefits (GMWB), and guaranteed minimum income benefits (GMIB). While the characteristics of these guarantees aid marketing and drive sales growth, they effectively leave the issuer with a short embedded-equity put exposure.

More than ever, insurers are using derivatives for hedging annuity liability risks. The equity indexed annuity (EIA) market has used equity derivatives for hedging since the mid-1990s. In addition to the EIA market, hedging variable annuities (VAs) with capital market products has gained in popularity as insurers have grown more comfortable using derivatives. Interest in equity derivative products as tools for hedging has also increased with the diminished availability of reinsurance alternatives and increased scrutiny from rating agencies regarding unhedged insurance risks.

Market View
Some insurers have remained unhedged or partially hedged with the belief that extreme downside risks would be remote, temporary and thus manageable in the overall scheme of their product business. The negative implication of leaving this unprotected embedded optionality on the books is that fat tail risk may arise. The sensitivity to tail risk and the long-dated nature of the VA liability also make it difficult to hedge. However, with recent scrutiny growing toward unhedged risks, more insurers are likely to consider and adopt static or dynamic hedging approaches, or a combination. In this article, discussion will focus on the capital markets perspective and some of the products dealers offer to hedge and help to mitigate capital costs and tail exposure risks.

For a background on the basic features of each of the four key aforementioned guaranteed benefits, the reader may refer to the March 2004 article. However, the fundamental conclusion is that the VA guarantees equate to what capital markets refer to as sold, or "written" equity index look-back puts. For example, assume a product guarantees a policyholder that, after ten years, the policy will pay its accumulated value (based on crediting rates and S&P 500 total return performance), but at no time pay less than the principal amount. Consider the following interpretation:

Product payoff is: Max {0, 100,000 – AV10}
where AV10 = tenth year account value.
Simple algebra may show that:
= Max {0, 100,000 – 100,000 x
(SP10TR / SP0)}
= Max {0, [100,000 x SP0 – 100,000 x
SP10TR]/ SP0}
= (100,000 / SP0) x Max {0, SP0 –
SP10TR}

The last line matches the formula for calculating a payout for a 10-year at-the-money (ATM) put on S&P 500 total return (with put notional = $100,000). Thus, one can conclude that the GMAB payoff equates to an ATM put. Equating insurance and capital markets mathematics and terminology acts as a useful first step in identifying a possible hedging solution in the capital markets. Often, promises made in policies reflect similar payout profiles to options traded in the capital markets.

The problem is not solved so readily, however, since the ATM put in some cases may be exercised against the issuing insurer at unexpected times. Thus, the maturity is uncertain. To elaborate a bit further, a typical GMDB may have one, two or all of the following features: 1) principal–protects beneficiaries upon the insured's death, against a possible drop in value of underlying and also guarantees a pre-specified minimum equal to total value of premium paid; 2) rising floor–the initial investment grows at a minimally guaranteed rate (e.g. 5 percent); 3) ratchet (look-back option)–upon the insured's death, guarantees ensure the highest anniversary account value. For the issuer, the risks are primarily those of mortality, surrender and market movements (both embedded optionality and reinvestment rate risk). More importantly, in addition to the level of the underlying market (S&P 500, in this example), the embedded option value will vary based on other underlying forces, especially volatility and interest rates. Due to the complex nature of the option-like guarantees, actuarial approaches sometimes under-weigh the value of these liabilities.

As insurers ponder the best method to hedging these complex liabilities, the trend has begun to favor using a combination of approaches and products to isolate key risks and hedge individual components of the overall exposure to limit tail risk. C-3 Phase II should strongly impact capital differences required for hedgers versus non-hedgers, so capital market activity will likely continue to increase. Also, as more regulators and rating agencies utilize stochastic models, insurers that do not manage the risk of these policies may be penalized compared to those that hedge.

With this in mind, the most popular approaches have been to delta hedge using futures and to purchase over-the-counter (OTC) options from institutional dealers. For example, an insurer with GMDB risk offering the greater of the highest anniversary value or a credited interest rate roll-up could use a lognormal model and stochastic scenarios with shocks to determine delta and the other "greeks" (i.e., gamma for delta sensitivity, vega for volatility, rho for interest rates) to calculate roughly how many SPX futures it needs to hedge its delta risk. The dynamic hedging approach typically entails the issuer shorting futures contracts equal to the size of their guarantee liability, then tracking the market daily and adjusting periodically to rebalance the hedge. Transaction costs prohibit true dynamic hedging, which assumes continuous markets and rebalancing in a risk-neutral framework.

Assuming the insurer has the trading expertise and infrastructure in place to track and account for its positions, dynamic hedging should work well for small market movements. However, the expression "it works well until it doesn't" has some truth to it, as illiquidity and price jumps may occur during large market swings. Thus, while dynamic hedging is useful for small market moves, rebalancing and continuous monitoring is necessary for prudent risk management. This requires both human and technological resources as well as trading experience and ongoing operations tracking for accounting and regulatory reporting.

For this and other reasons, static hedging using OTC options may be used to hedge the tail risk and curvature (gamma) of market movements more effectively. The short futures positions are typically complemented with layering in options to hedge the curvature arising from larger and longer-dated market moves. Remember, the GMDB contract life may be estimated at 30 - 40 years, with an average life of approximately 20 years. Unfortunately, at the time of this writing, OTC options beyond 10 years are generally unavailable and highly illiquid.

In consideration of the above, and because insurers may redesign their products to include annual rate and cap resets, shorter-dated OTC options have become popular. These options enjoy high liquidity in the major U.S. indices as well as favorable pricing due to implied volatility trading at the historical lows of its range. Figure 1 illustrates the general decline in SPX volatility, which supports the case for purchasing options and getting long volatility on a relative value basis.

Most often, straight put options are purchased with varying maturities and strikes to reflect the term structure of liability exposure based on actuarial analysis. To reduce the cost of purchasing puts, some insurers have defined their exposures more precisely and have chosen to purchase put spreads. By purchasing an ATM put and selling an out-of-the-money (OTM) put, the insurer spends less premium while maintaining its hedge for a specified market move. Another variation of this cost reduction theme is the purchase of a collar. In this case, the insurer purchases an ATM put and funds the purchase, in whole or part, by selling an OTM call. In other words, the call strike may be set to equal the premium owed on the put, resulting in a ?costless? collar. If the insurer is not content with limiting its upside to the OTM call strike of the costless collar, it may decide instead to sell a further OTM call to keep more upside yet reduce the cost of the purchased put (but not to zero), but the reality of capital market pricing may be that not much value is derived from a deep OTM call. Figure 2 illustrates this point and provides indicative pricing for puts, collars and put spread collars, as of Dec. 9, 2004.


One considerable advantage of purchasing OTC options is that the carrier is effectively passing on the dynamic hedging risk and daily hedging maintenance to the dealer. In essence, credit risk replaces market risk. For the most part, insurers are comfortable with this risk since several major derivatives dealers have strong credit ratings and an established history of trading derivatives. As the comfort level has risen, so has the quest for discovering new approaches to hedging the complex variable annuity dilemma. In addition to delta and gamma risk, insurers are now keenly analyzing and addressing vega (volatility) and rho (interest rate) risks. Due to their long maturities, interest rate risk becomes very relevant to analyzing and hedging VA liabilities. Rho risk will be revisited, but this article focuses more on equity exposure, so vega risk is discussed in more detail.

Next to delta and gamma exposure, volatility (vega) risk is prominent. As Table 1 shows, the value of a call or put is extremely sensitive to implied volatility assumptions.

Volatility increases as markets sell off. For example, the value of the VIX increases when the market declines and decreases when the market rises. This is caused by the stock market?s bullish bias, which assumes a rising stock market is less risky than a declining stock market. As the market declines, higher perceived risk is often answered with more put option buying. The increased demand for puts usually drives prices and implied volatility rates higher.

Since the VA writer is short vega and will experience an increase in (short) vega if markets decline, it may help to own an asset that will pay when actual volatility rises. The variance swap was originally developed to meet this need for dealers looking to hedge their short vega exposure. The variance swap market has evolved to serve hedge funds and, more recently, institutional investors including insurers.

Intuitively, the variance swap1 makes sense since the insurer may get long or short volatility at a specified strike level for an underlying index. Volatility for index price levels is calculated as the annualized standard deviation of the daily log returns of the closing price levels. Variance is simply volatility squared.

Variance swaps are usually traded for an amount of vega (sensitivity to a 1 percent increase in volatility). To price a variance swap, the traders must compute the weighted fair values of the options required to hedge the swap. For any given variance swap quote, then, the strike price is set to reflect the aggregate cost of the hedge portfolio. So, when pricing a variance swap, the major factor is the cost of the strip of options.

Note that historical volatility is not a factor in the pricing. Also note that, because a whole strip of options is used, the volatility skew has a major impact on pricing. The strike of the variance swap is not necessarily the implied volatility of an at-the-money option. In fact, under normal circumstances, the skew will result in the variance swap strike being higher than the at-the-money implied strike. These pricing nuances are important for insurers to keep in mind for hedging and product development analysis.

So, why use a variance swap? Buyers and sellers of volatility have typically traded straddles and range accrual notes. Both of these products are very sensitive to volatility. However, the payoff profiles of both products are not ideal and the products are both still very dependent on the absolute level of the underlying. Variance swaps have no dependence on the absolute level of the underlying, and are a pure volatility play. The formulaic payout allows the investor to benefit regardless of any factor outside of the pure price movements that may influence the variance calculation. Thus, a variance swap enables an investor to express a view on realized equity market volatility.

As mentioned above, volatility has a direct impact on dynamic hedging effectiveness. For this very reason, it is difficult to measure the overall success of a dynamic hedging strategy partly due to the uncertainty of volatility. Variance swap products could help isolate and offset this volatility risk.

For purposes of hedging VA exposure using variance swaps, a few different approaches may be used. The most straightforward view is just to enter into a variance swap to get long volatility (illustrated in the above flowchart diagram). On trade date, no cash flows are exchanged, but the mark-to-market on the trade will begin the day after trade date, and will flow through the insurer's earnings statement just as any other trading asset. These marks should offset changes in the value of the liabilities. Insurers should conduct their own accounting and statutory regulatory analysis prior to entering any transaction.

Two popular alternative approaches have been for insurers to enter into a forward starting variance swap or to purchase an option on a variance swap. The forward starting variance swap allows the VA issuer to hedge against future increases in volatility while locking in at today's relatively low strike levels. For example, the hedger could enter into a two-year SPX variance (long) swap starting one year forward. An additional benefit of this approach could be an early unwind for a windfall gain, if volatility increases noticeably before the swap start date.

An option on a variance swap presents the advantage of limiting downside costs if volatility moves against the purchaser, yet providing upside as necessary to hedge the liability. The up-front premium does raise the break-even level for the trade to have a positive payout, but should not affect the trade?s advantage in a high volatility environment. The payout profile would resemble the "hockey stick" shape of a call option, but would possess the additional benefit of a "curved handle" due to the positive convexity of the long variance, as illustrated earlier.

Various other exotic options have been structured in the past, with differing levels of success as considered from a hedge potency and cost effectiveness standpoint. For example, knock-out puts provide downside protection unless a trigger market level has been reached (e.g., a one year 100 percent SPX put that knocks out, or ceases to exist, if the S&P 500 closes at or above 103 percent of its spot level on trade date). Another product is the "option on an option", in which the issuer purchases, for example, a call on a put. Pricing on these options can be expensive and often does not meet the insurer's hedging budget, but the structures are worth tracking to see if relative value opportunities arise. For example, several months ago, a VA issuer could pay a dealer 2 percent up-front to purchase a four-year call struck at 12 percent on a 10- year 90 percent strike SPX put. Compare this to a straight 10-year 90 percent strike put, offered at 8.75 percent. Obviously, the insurer pays little premium today but, in this example, must be willing to pay a total of 14 percent (2 percent plus 12 percent) for protection if a severe downside scenario arises. Hybrid products represent another variation, and may encompass puts with strikes that adjust according to reference interest rate levels, or puts that pay out on a deferred basis so that reinvestment rate risk may be effectively locked in for the VA hedger. These products naturally pose correlation risk to the selling dealer and are quite sensitive to pricing assumptions.

Whether or not an insurer chooses to hedge using the capital markets, the important decision is to hedge versus not hedge. Hedging not only aids in stabilizing surplus sensitivity but also should result in capital savings and more favorable reviews from rating agencies and market analysts. Product development with hedging costs in mind should lead to long-term success since the policy's pricing component will align with market forces. An ongoing evaluation will then allow the insurer to develop new relevant products while using the most potent hedging tools for each product feature. It is important not only to evaluate hedge effectiveness in "real world" terms, but also to design products that stand up to this test.

In an increasingly mark-to-market regime, insurers will benefit from products that act to offset the volatility in their liabilities. Since risk minimization is generally superior to delta hedging due to prevention of realizing worst-case scenarios, the use of options will likely grow and prove more effective than hedging with the underlying. As the "GMAB equals put" example on a previous page demonstrated, the issuer is short a bundle of options, so should probably try to hedge with the same. Volatility and interest rate products also cannot be ignored, as they will complete the risk reduction formula for a long-term hedging program.

As capital markets converge, combinations of equity, interest rate and credit derivatives may be utilized to synthetically replicate and mirror embedded risks for insurers. The challenges for such products are numerous, especially since more complex products are typically very sensitive to forward volatility and correlation pricing assumptions. These risks for the dealer typically translate to higher premiums as well as relative illiquidity (in size as well as bid-ask spreads), and thus will probably not ever become the norm for hedging programs.

The ideal OTC products should be easy to understand, related to the liability hedging need at hand, and pertinent to generating cash flows when needed, whether directly or on a marked-to-market basis. Other preferred attributes include liquidity and model-friendly structures that provide a robust framework for back-testing and scenario analysis. Puts, variance swaps and hybrid puts with strikes driven by interest rates represent a few of the current solutions to consider.2

Many applications of capital markets products are growing in popularity with the insurance industry. Away from VA hedging using equity derivatives, bank assurance and capital efficiency structured solutions have been used steadily recently and promise to endure and evolve with the industry's need to enhance surplus sensitivity profiles. These structures will be discussed in detail in a future article.

1 The reason that variance swaps are traded is that the hedge for a variance swap is relatively simple. It is possible to trade volatility swaps but the hedge for these are a lot more complicated. As a result, there is a lot less liquid market in volatility swaps, and margins on volatility swaps are a great deal higher than they are for variance swaps.

2 OTC derivatives contracts can only be offered to accredited investors, and may not be appropriate to all clients. They are offered by Bank of America N.A., an affiliate of Banc of America Securities LLC. Potential clients will be required to represent that they have consulted with their advisor prior to executing any transaction.

Edward A. Mirsepahi is head of insurance risk management, Equity Finanacial Products, Banc of America Securities, LLC. He can be reached at: edward.mirsepahi@bofasecurities.com.