## Rolling Tail Hedges: The Dynamic Tradeoff between Cost and Potency

- In our hypothetical illustration, rebalancing tail risk hedges more frequently than once a year offers some benefit under all volatility curves (ignoring transaction costs) – but the benefits are greatest when the volatility curve is flat or steep.
- Some of the benefits of rebalancing can quickly disappear if the transactions costs are large.
- Two key points for investors to consider: Hedging has to be a systematic, repeated, asset allocation decision to obtain best long-term benefits, and the hedge program has to be active and consider pricing levels so efficient rebalancing can be implemented.

When we implement tail hedges, we have traditionally used one year as the hedge horizon. To calculate the premium for these hedges, we start by pricing the explicit one-year hedge and then explore ways of reducing the premium by using indirect hedges. Some of these indirect hedges might consist of cheaper “option-like” positions from other markets that take advantage of potential tail correlations. In addition, we actively manage the hedges in an effort to reduce the realized cost of hedging over this investment horizon.

Many tail risk clients have expressed a desire to move toward more frequent rebalancing of hedges. One popular frequency is quarterly. There are numerous reasons for this: First, implementing more frequent hedges “averages” out the premium spent and the attachment point when the hedge kicks in (for example, a 15% attachment point will hedge the portfolio against losses greater than 15%). Second, it avoids a “cliff” effect, i.e., the preset hedge strike becomes more distant as it nears expiration. Finally, more frequent rebalancing might allow a recalibration of the portfolio exposures within the one-year period and capture other rebalancing activity.

Of course, we cannot make a blanket statement about the efficacy or superiority of any one hedge algorithm. Even without conducting a lot of work analyzing these algorithms, one can anticipate that trade-offs will arise simply because of the variations in volatility term structures (i.e., the variation of implied volatility during the time to maturity). Since option prices are roughly proportionate to implied volatility as well as the square root of time to expiration, we can expect a more frequent hedge to do better if the volatility curve is steep and upward-sloping (shorter-term implied volatility lower than longer-term implied volatility), or when the volatility term structure is flat. However, if the volatility curve is inverted, as usually happens after a market shock, hedging more frequently costs more than it does when the curve is steep or flat. This observation also supports the argument for maintaining some flexibility when adjusting the frequency of rebalancing in the aftermath of market events. Transaction costs also should play an important role in any investment decision and some of the benefits of rebalancing can quickly disappear if the transactions costs are large.

**The trade-offs associated with more frequent hedging **As a hypothetical example of the key considerations involved in tail hedging, let’s look at a one-year hedge on the S&P 500 Index with a strike that is 25% out of the money. This is a common “benchmark” direct hedge because when hedging a 60/40 portfolio of stocks and bonds with a 15% attachment point, the effective strike for the direct S&P 500 put option is calculated as 15%/0.6, which is 25%. Let’s also assume that the one-year volatility for this 25% out-of-the-money strike is 30%. According to the Black Scholes method of pricing this option, the one-time “bullet” cost would be 250 basis points (bps), or 2.5%.

Now let’s take this one-year hedge and consider the trade-offs we might expect to make if we increased the frequency of the hedge (cost, sensitivities, etc.). For theoretical clarity, we will ignore the transaction costs of unwinding existing hedges and replacing them with new hedges, although these cost considerations are critical in running an actual hedge program. We also keep the hedge horizon fixed at the initial one-year, i.e., we will not replace the expiring shorter-dated hedges. This assumption allows us to focus on the sensitivities of the original portfolio of hedges.

Suppose first that the volatility of the S&P 500 options market is flat at 30% for the whole year. Then we can think of replacing the annual option with a string of daily options expiring in one day, two days, three days etc. all the way to 365 days, each of which would have 1/365th of the notional value of the option associated with it. Clearly no one would want to have 365 daily options for practical reasons, but as with all problems, taking the limit sheds important light on the nature of this problem: Since the time value of options with shorter expirations is lower than those of longer-dated ones, the total cost of the daily options is only 94 bps (see Figure 1 and Figure 2) – which is obtained by pricing each option and dividing by 365. Also, the shorter-dated options will expire with the passage of time and will need replacement, while the longer-dated options remain alive. If we consider the more sensible approach of quarterly rebalancing (where 250/4 of the premium is used each quarter), the total cost is only 127 bps, or roughly half of the “bullet” one-year hedge of 250 bps mentioned earlier. Figure 1 also shows that the pricing depends on the shape of the volatility curve.

For this exercise, we created three idealized curves: The “steep” curve starts at 15% volatility and goes to 30% volatility linearly, which is usually associated with a calm equity market (note all volatilities here refer to the out-of-the-money put options). A flat volatility curve stays at 30% all the way, and an inverted curve starts at 50% volatility for the shortest option (a one-day option) and ends up at the 30% volatility for the one-year option with linear interpolation, i.e., a straight line between the two coordinate points.

Figures 1 and 2 show the inverted volatility curve requires almost twice as much premium for daily rebalancing as the flat volatility curve and almost three times as much as the steep volatility curve. For quarterly rebalancing, the inverted volatility curve creates a 60% higher cost than the flat volatility curve and the steep volatility curve creates a discount of 25%. In terms of higher frequency rebalancing, a flat volatility curve makes the total cost of quarterly hedging 127 bps compared with 95 bps for the steep volatility curve and 202 bps for the inverted volatility curve. In all three cases, the more frequent hedging is cheaper than locking in the one-year hedge.

But there is no such thing as a free lunch: As Figure 3 shows, the more frequent hedging has a lower delta than the longer-term hedge. Since the delta represents the sensitivity of the hedge to the movements in the underlying security, this analysis shows that the sensitivity of a hedge portfolio goes down as costs are lowered.

Intuitively, it is clear to see why this happens: First, more frequent hedging means hedges expire more frequently, so the daily hedge will have 1/365th of the notional amount expire each day. Second, the shorter hedges have lower delta. Since the cumulative probability of a large market move is much lower in a shorter time period, the hedge is theoretically less likely to move into an “in-the-money” position. Of course, one might argue that if event risk is higher in the short term, using cheaper short-term hedges might be the better approach. This short-term approach is a reflection of an “active” view on the level of the volatility term structure – and an approach one might want to take around the time of a major policy decision, election or any similar event with potential unpriced jump risk.

Figure 4 illustrates the ratio of the potency, or deltas, of the various frequencies of hedges relative to the benchmark one-year hedge. Obviously, the quarterly delta ratio of 0.86 for the inverted volatility curve is higher than the quarterly delta ratio of 0.47 when the volatility curve is steep and upward-sloping.

**Bringing it all together**

To bring all these discussions about trade-offs together, Figure 5 shows the cost for the same unit of delta (the delta-adjusted cost) under the three volatility curves and various hedging frequencies. In other words, if we want the same delta regardless of the frequency of hedging, what would the total equivalent cost be? The conclusion is that rebalancing more frequently than once a year offers some benefit under all volatility curves (ignoring transactions costs) – but the benefits are greatest when the volatility curve is flat or steep.

Also, under all scenarios, new extension hedges have to be added to replace hedges when they expire. Since the normal volatility term structure is expected to appear more than 75% of the time (inversions are unusual and are typically associated with large downside market events), we generally expect to improve hedging efficiencies by moving to a reasonable (quarterly) hedging program.

This also highlights two other major points that we think any tail hedging client should consider: One, hedging has to be a systematic, repeated, asset allocation decision to obtain best long-term benefits, and two, the hedge program has to be active and consider pricing levels so efficient rebalancing can be implemented.

**Past performance is not a guarantee or a reliable indicator of future results.** **Tail risk hedging** may involve entering into financial derivatives that are expected to increase in value during the occurrence of tail events. Investing in a tail event instrument could lose all or a portion of its value even in a period of severe market stress. A tail event is unpredictable; therefore, investments in instruments tied to the occurrence of a tail event are speculative. **Derivatives** may involve certain costs and risks such as liquidity, interest rate, market, credit, management and the risk that a position could not be closed when most advantageous. Investing in derivatives could lose more than the amount invested.** **

Attachment point is the targeted maximum loss at the Investment Hedge Horizon of the combined Investment Portfolio and the Investment Hedge Portfolio. Not a guaranteed number, the Loss Threshold should be considered an investment objective.

No representation is being made that any account, product, or strategy will or is likely to achieve profits, losses, or results similar to those shown. Hypothetical or simulated performance results have several inherent limitations. Unlike an actual performance record, simulated results do not represent actual performance and are generally prepared with the benefit of hindsight. There are frequently sharp differences between simulated performance results and the actual results subsequently achieved by any particular account, product, or strategy. In addition, since trades have not actually been executed, simulated results cannot account for the impact of certain market risks such as lack of liquidity. There are numerous other factors related to the markets in general or the implementation of any specific investment strategy, which cannot be fully accounted for in the preparation of simulated results and all of which can adversely affect actual results. The S&P 500 Index is an unmanaged market index generally considered representative of the stock market as a whole. The index focuses on the Large-Cap segment of the U.S. equities market. It is not possible to invest directly in an unmanaged index.

This material contains the opinions of the author but not necessarily those of PIMCO and such opinions are subject to change without notice. This material has been distributed for informational purposes only and should not be considered as investment advice or a recommendation of any particular security, strategy or investment product. Information contained herein has been obtained from sources believed to be reliable, but not guaranteed. No part of this material may be reproduced in any form, or referred to in any other publication, without express written permission. PIMCO and YOUR GLOBAL INVESTMENT AUTHORITY are trademarks or registered trademarks of Allianz Asset Management of America L.P. and Pacific Investment Management Company LLC, respectively, in the United States and throughout the world. ©2012, PIMCO.

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