*A version of this paper will appear in the Winter 2014 edition of The Journal of Portfolio Management.*

In the aftermath of the 2008 crisis, the phrase “tail risk hedging” has entered the vernacular of financial market participants with a speed and intensity unmatched by any other market concept in recent memory.

However, this concept so far has largely been limited to the hedging of risks related to rapid declines in equity-related markets. Policymakers are well aware of these equity market risks, which from a macro perspective are associated with a deflationary fat tail, and have flooded the markets with unprecedented amounts of liquidity – via low interest rates, an increasing quantity of money and outright asset purchases. These extreme measures, by most metrics, have resulted in a historically unprecedented set of market conditions and raise the question of whether there are other risk factors “under our noses” whose tail risks now need to be anticipated and proactively managed. This paper reviews what we know about observed and expected inflation, and then discusses practical techniques for hedging inflation tail risks.

Indeed, both theoretically and empirically there are reasons to believe that in addition to the equity risk factor, the duration risk factor (or sensitivity to interest rates) is the second key variable that drives investment risks over long horizons. This is economically sensible, since growth largely drives the returns of equity markets, and inflation largely determines the returns to credit-risk-free fixed income instruments. Clearly equity market risk is more volatile and has historically been higher frequency than inflation risk, and hedging techniques have to be adapted appropriately to these inherent differences. But we believe that while the last five years might have made it possible for us to ignore the tail risks from the rise in interest rates (given the tailwind of central bank liquidity injections of various sorts, low realized and expected inflation and low interest rates), continuing with this assumption might no longer be a very safe approach.

First, since global central banks around the world are injecting an unprecedented amount of liquidity into the financial markets to boost asset prices and raise inflation expectations, expecting inflation to remain subdued for a very long period of time does not seem to be an evenly biased bet. Other than the direct influence of a ballooning money supply, the demand for food and energy globally, not the least from newly emergent nations, could result in consequential inflation spikes not unlike those witnessed in the early 1970s. The direct purchase of mortgages by the Federal Reserve has resulted in a lowering of the mortgage interest cost that plays an important role in the computation of an inflation index such as the U.S. Consumer Price Index (CPI).^{1} A rise in these mortgage rates could result in a substantial percentage rise in the CPI as the cost of servicing the loans flows indirectly into the inflation indexes. Indeed, in today’s low-inflation environment these events seem much too remote and unlikely, both in terms of probabilities and potential severities, but so did the possibility of a rapid credit market widening prior to the 2008 crisis.

Second, rising inflation can affect not only fixed income assets, which occurs directly since yields and prices move inversely to each other, but also the prospective returns of all risky assets. This occurs conceptually because most risky assets derive a significant part of their value from the discounting of future cash flows, and the discount factor used for the present value computation is driven mostly by the path of policy rates, inflation expectations and the inflation risk premium.

Finally, even without inflation actually rising, the rise of inflation uncertainty can result in rising inflation risk premiums (there is plenty of empirical evidence and theoretical basis to justify that high levels of inflation will be correlated to high inflation volatility, which is the primary driver of inflation risk premia) and thus result in adverse performance of inflation-sensitive assets. This variability, while currently low, can amplify the impact of rising inflation and inflation expectations as and when they occur.

While the impact of rising inflation on liabilities is well understood and managed, the impact on the pricing of asset portfolios is equally relevant for inflation tail risk management. For the purpose of this paper, we do not fixate on predicting which part of portfolios inflation is going to have a larger impact on. We believe that the tail risks on the total value of the portfolio assets and liabilities, and the techniques described here, can be practically useful for both aspects.

**Hedging “at-the-money” inflation versus inflation tails
**Before diving into the details of inflation tail hedging, we would like to clarify some issues that become important for implementation.

First, when one thinks of inflation hedging, it is important to distinguish between two kinds of inflation risks: small, expected deviations from current inflation and large, unanticipated inflation spikes. Since asset markets discount expected inflation fairly efficiently, an approach that uses asset allocation techniques efficiently is best suited for relatively small, expected deviations from current inflation. We discuss how to find and scale allocation to such assets below.

Second, for hedging purposes it is not sufficient to find assets that have exhibited anecdotal contemporaneous positive correlations with inflation in particular episodes, since many of these correlation relationships are horizon- and period-sensitive. For the hedge to be robust, it is critically important that the cumulative returns over a period of time on the hedging instruments are positively correlated to the factors that drive the cumulative increase in inflation. Since most assets exhibit much higher volatility than the inflation rate over short periods of time, the efficacy of short-term correlations (or betas) for hedging becomes questionable if one relies too heavily on simplistic anecdotal and episodic correlation analysis.

Third, we should be prepared for the same assets to exhibit different hedging behavior as a function of the time horizon. For instance, there is a large amount of academic and practitioner literature that discusses the weak hedging nature of equities against inflation over short horizons but a more reliable hedging relationship over a longer horizon. Rather than thinking of this change of inflation hedging potency as a detractor, we should think of this horizon-dependent variability as an essential ingredient in the construction of actively managed portfolios that are robust over a variety of hedging horizons.

To illustrate these issues specifically, assume that the current inflation rate (as measured by the CPI) is 2.5%. We believe that dynamic asset allocation should be the first line of defense in controlling against the initial and persistent rise in inflation, say for instance for a range of 2.5% to 4%. This can be implemented by shifting the allocation to inflation-sensitive assets. For example, it is well established that outright long positions in traditional asset classes such as stocks and nominal bonds, in aggregate, deliver lackluster nominal returns under inflationary scenarios over short horizons (particular combinations, such as yield curve steepeners, implemented by going short longer-dated bonds while hedging the duration risk with shorter-dated bonds allow one to benefit from rising inflation expectations). To be clear, outright long positions in equities show a mixed sensitivity to inflation as the horizon changes. While in the short term rising inflation risk premia (which accompany rising inflation) lead to an inverse relationship of equity returns and inflation, over longer horizons, high earnings yield (accompanied with high interest rates) may lead to positive correlations between inflation and equity expected returns. Alternative asset classes, such as Treasury Inflation-Protected Securities (TIPS), gold, commodities, real estate investment trusts (REITS), certain foreign currencies, timber, intellectual property, farmland and long-short baskets of inflation-sensitive equities, are other examples that can be utilized to create more robust portfolio mixes against moderate persistent rises in inflation (see Johnson, 2013). Of course, for practical implementation, we would need to estimate the potential transaction costs of switching from traditional assets to these alternatives, and liquidity concerns might prohibit the use of many of the less liquid categories.

In contrast, and importantly so, when we talk about inflation “spikes” or “tails,” the dynamic asset allocation approach to managing inflation needs to be complemented with more convex inflation-hedging strategies. There are three main reasons for this. First, as mentioned earlier, large inflation shocks can result in larger unanticipated movements in asset prices. The linear instruments mentioned above cannot capture the convexity inherent in market responses to inflation spikes. This issue is especially important when initial inflation levels and inflation expectations are very low. Second, hedging against inflation spikes by rebalancing alone would require a much higher than normal allocation to inflation-sensitive assets in the steady state. Since this portfolio rebalance has an implicit cost (in terms of lower nominal yields from underweighting nominal bonds, for example), the cost of a static inflation-sensitive portfolio using asset allocation alone would be a permanent drag if inflation spikes are not realized or inflation is not persistent. Inflation spikes are by definition low-probability events, thus the expected implicit cost of hedging from asset allocation should be compared with the potential benefits. Finally, and this follows from both the inherent instability of correlations of assets to inflation and the trouble with forecasting inflation with accuracy, if instead of inflation rising we actually realize deflation, a static asset allocation approach would prove to create large portfolio underperformance or at least additional tracking error.

For these reasons, if the object of hedging is to control against inflation tail risks, we believe a powerful approach is to directly use an extension of the conceptual framework developed for equity/risky asset/deflationary scenario tail risk hedging that commits a finite amount of hedge “premium” to potentially inflation-sensitive derivatives (see Bhansali, 2008). It relies heavily on applying an option theoretic framework to hedging inflation risk. To achieve these objectives in practice, the framework relies on both linear and non-linear option-like instruments across multiple asset classes. Surprisingly, volatility across macroeconomic markets has been driven to historical lows on the back of the supply of central bank liquidity, and creates the opportunity, albeit short-lived, to hold assets and hedge them too. So in our view today’s macroeconomic environment creates an attractive set of conditions for implementing inflation tail hedging relatively inexpensively.

**Tail hedging realized inflation versus inflation expectations **

The simplest metric for inflation risk is the percentage gain or loss on a price index. In the U.S., the CPI is such a metric. Inflation-linked securities such at TIPS derive their value from the change in the CPI. Thus, to target inflation risk management, the simplest dynamic approach would be to create a portfolio of securities that are sensitive to the CPI.

For instance, a direct linear hedge against inflation would consist of inflation swaps, where one receives the inflation rate measured by such an index in exchange for a contractually fixed payment (these trade with a fair amount of liquidity). Another alternative is to purchase an inflation option, which typically trades in the form of inflation caps (for rising inflation risk mitigation) and offers more convexity if inflation starts to rise quickly. The caps come in the form of year-over-year caps and zero-coupon caps. Quite simply, year-over-year caps pay off if the inflation measured over a yearly period exceeds a pre-defined strike. Zero-coupon caps pay off if the actual value of the inflation index (e.g., the CPI) exceeds a pre-defined strike level of the index.

However, limiting to the observed CPI for inflation risk management could be a rather restrictive way to manage inflation tail risks, especially in today’s market environment. There are two main reasons for this. First, a cursory look at the components that make up the CPI shows that a dominant portion (over 40%) is related to housing and another major portion to food and energy. Since the housing market is recovering from the excesses of the crisis, and there is ample supply of housing to keep inflation in this sector moderated, focusing on CPI inflation could miss the impact of inflation in other sectors of the economy. From a forward-looking and broader perspective, macroeconomic considerations such as low rates of capacity utilization can keep the short-term inflation rate low, while longer-term inflation and inflation expectations can rise due to policy, expectations and uncertainty. Second, for longer-term investment decisions, sensitivity to inflation expectations is likely to be a key input. There are, of course, many metrics for inflation expectations. One could use directly observable market metrics such as the pricing of inflation swaps (though they also carry exposure to counterparty risk and illiquidity). As another alternative we could look at surveys (for instance the Philadelphia Fed Survey of Professional Forecasters, or the Thomson Reuters/University of Michigan Survey). As a third option we could use implicit market metrics such as the steepness of the Treasury yield curve or inflation break-even rates. This last metric – that is, the inflation break-even – is simply the difference between a nominal government bond yield and the corresponding maturity real yield.^{2}

As can be observed in Figure 2, the 10-year break-even is currently hovering around 2.5%.

If instead of using the spot yield differentials we use the differential of forward yields, the result is the forward break-even rate. For example, the five-year forward, five-year break-even is the difference between the nominal five-year forward, five-year yield and the real five-year forward, five-year yield (see Figure 3). Each five-year forward yield can be computed by observing the 10-year spot yield and the five-year spot yield and using traditional compounding formulas for the yield curve. This forward breakeven rate has also gained some notoriety as the gauge that reflects in real time the inflation-fighting credibility of the Fed (Gurkaynak, 2006). So as an added benefit this metric and the tail hedges on it are directly related to the liquidity/inflation risk tradeoff that central bankers are also paying close attention to.

So to recap: Using a linear instrument to protect against realized inflation we can use CPI swaps on the observed inflation rate, or to protect against expected inflation we can use the “four-legged” break-even (sell a 10-year nominal and five-year real bonds and buy the five-year nominal and 10-year real bonds) or their nonlinear option counterparts.

These linear instruments like swaps and break-evens are always “delta-one” – that is, they will respond one-for-one to changing inflation rates or inflation expectations. As mentioned above, since both the swaps and the break-even trade can result in unlimited mark-to-market losses if inflation falls, the risk budget allocated to the hedge is hard to quantify ex-ante, as compared with that for their option counterparts. To enter into such swaps counterparties would require both maintenance and variation margins against the mark-to-market volatility of the positions.

The linear strategies are also always “at-the-money,” while there is more flexibility in picking the “strike” for the options-based strategies. Also note that we can combine options-based strategies to reconstruct the linear strategies, but not vice versa. In this sense the options-based strategies allow for more customization of inflation-protection portfolios.

**Inflation dynamics and inflation spikes
**Before we can dive deep into the quantitative aspects of options-based inflation tail hedging, it makes sense to explore briefly the theory and empirical properties of rising inflation. We should note that both the economics and empirical characteristics of inflation have been well studied over multiple decades

^{3}, but forecasting bouts of high or hyperinflation remains difficult as ever. In the interest of focusing on practical methods for inflation tail hedging, we will not discuss the fundamental economics of inflation, but summarize the data on the time-series and cross-sectional properties.

The two canonical examples of inflation super-spikes (with episodes of high inflation present essentially in all developed and developing economies over the last three hundred years) are from the experiences of the Weimar Republic (early 1900s Germany) and Zimbabwe (late 1900s early 2000s).^{4}

While prices shot up 1 trillion times higher in terms of marks, they remained stable in terms of gold, widely considered to be an inflation hedge (source: http://inflation.us/charts.html). In Zimbabwe the inflation rate in August 2008 went up to 6.5 sextillion (10^21). In effect, this meant the currency had no residual exchange value.

While these anecdotes are interesting in their own right, any currently popular inflation modeling paradigm does not allow for the possibility of rapidly rising inflation, let alone hyperinflation.

In the figures below we show inflation episodes in the U.S. over the last 100 years. As is obvious, though there are sustained periods of higher-than-normal inflation, the actual experience in the developed markets recently has not supported fears of very high inflation. Thus, any empirically calibrated model for inflation to recent experience would find the probability of hyperinflation to be negligible.

The distribution of inflation over this super-secular period has also shown no signs of being normal (see Figure 6). As a matter of fact, if one computes the raw probability of CPI inflation exceeding 5% per annum, the long history suggests that year-over-year inflation measured monthly exceeded 5% more than 20% of the time. Note, however, that the low levels of realized inflation and low inflation volatility as embedded in the pricing of inflation options (CPI options) suggest that the cumulative probability of exceeding 5% inflation in one year is only 2.2%, in five years is 6.4%, and in 10 years is 9.5%.^{5} So clearly the “risk-neutral” pricing of inflation risk is not compatible with the very long history of realized inflation, unless we make the heroic assumption that credible monetary policy has the power to quash inflation and inflation expectations unconditionally to keep them within a short range of current levels.

One of the reasons why rapidly rising inflation is out of the zone of common expectations of current theoretical models is that inflation modeling has relied generally on the same technology of continuous time finance that is used in the pricing of options on equities (where adjustments have to be made to explain tail behavior). In one such popular model inflation rises to a certain level, but the further it deviates from a long-term average, the more likely it is to revert back to the longer-term mean. This dynamic is ultimately based on the credibility of the central bank not being challenged. If long-term inflation expectations are anchored in this way and remain fixed, this class of models predicts that the mean and the variance of inflation are always well behaved and converge to a finite value; thus they cannot reach values that are out of the ordinary with any large probability. Note that while this approach is reasonable for modeling inflation under well-behaved economic conditions, it might not be appropriate under the presently observed economic conditions, low realized and expected inflation and policymaker activism, not to speak of the increasing challenges to the notion of central bank independence and credibility.

Another similar approach, more econometrically motivated but still backward-looking and rooted in empirical data analysis, is to model year-over-year inflation as an autoregressive process. The economic rationale behind this assumption of “inertia” is that rising inflation results in a recalibration of expectations, and hence inflation rate changes show momentum; that is, rising or falling inflation is likely to be followed by rising or falling inflation.^{6} Thus discontinuous changes, especially reversals, cannot be accounted for within this approach either.

Abandoning the assumptions of continuity, at the other extreme we can model the inflation process as a compound jump process. A compound jump process combines randomly occurring jumps with randomly distributed magnitudes of jumps. The easiest approach to simulate assumes that inflationary jumps occur randomly, following, for instance, a Poisson process, and the size of the jumps is also random (for instance the size of the shocks can be normally distributed about a mean jump size).^{7}

The modeling of hyperinflation from a purely macroeconomics viewpoint is based on the classical quantity theory of money, which relates money times the velocity (circulation) of money to price times economic output (i.e., **M×V=P×Y**). Under this theory, holding all else constant, if the supply of money increases, then the price level has to increase. The expected increase of the money supply (i.e., under the promise of continued central bank quantitative easing, so-called **QE ∞**) would result in such an outcome for inflation expectations.^{8}

We do not believe that any of these simplified dynamics are the complete and unique answer to modeling inflation, let alone to forecasting its future evolution. Indeed, none of these theoretical approaches does very well in fitting the actual distribution, especially the tails of the inflation distribution. Given the inability of these models to fit actual experience both in the U.S. and elsewhere, pinning too much faith on inflation hedging strategies on such foundations is likely to prove erroneous for practical portfolio construction.

As Figure 7 shows, the empirical distribution of realized inflation is much fatter-tailed than an auto-regressive model, a normal distribution or even a calibrated jump process. What this suggests is that while modeling inflation dynamics with an eye toward tail hedging, we might have to make a larger leap than any of the traditional models would allow.

**Framework for inflation tail hedging
**Given that there are severe limitations in forecasting the dynamics of inflation, we take the view that investors ought to be less concerned about forecasting the probability of inflation spikes and more concerned about the severity of such events and their impact on portfolios. This “Pascal’s wager” approach to tail risk hedging necessarily forces us to reflect on the exposures within the portfolio to construct the hedging program.

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We also believe that any tail-hedging program is best implemented with reference to the underlying asset and liability portfolio being hedged. This holistic picture allows for active management, proper accounting of hedging costs and the ability to rebalance opportunistically. This asset allocation perspective also makes it possible for the tail-hedged portfolio to perform better over long-term horizons, since inflation tails, by definition, are low-frequency but high-severity events, and excessive reliance on our ability to forecast the timing of such events is fraught with danger.

Three essential ingredients are critical to the understanding of any tail-hedging program, whether for the equity type of risk, or more relevant to the current discussion, for interest rate or inflation sensitivity. First, we need to understand portfolio risk exposures. As is well known, for fixed income portfolios, we can condense and encapsulate the exposures in terms of a handful of key risk factors. The most important ones are duration, curve duration (sensitivity to yield curve steepening and flattening), spread duration (sensitivity to OAS changes) and convexity. Of these, the primary risk factor is duration. For equity-heavy portfolios, it is various equity factors, such as betas (overall market betas, growth, value, etc.). For a blended portfolio, a look-through into both these factor exposures is important. Just as high-yield bonds carry a substantial amount of equity beta, many risky assets thought of as pure equities carry a large amount of inflation or duration exposure. For example, think of utility sector equities that are exposed to economic cycles and interest rate levels, or even the financial sector, which is exposed to the level of interest rates as well as the shape of the yield curve (since financial sector companies derive profits primarily from borrowing short and lending long – the “carry” trade).

Second, we need to specify tail risk “attachment levels.” For hedging purposes, the attachment level is simply the level of inflation or interest rates beyond which we want the portfolio to be protected. If we estimate the duration of a portfolio to be five years (e.g., for a portfolio similar to the Barclays U.S. Aggregate Bond Index), then a 100-basis-point rise in yields (say from inflation) would result in roughly a 5% fall in the value of such a portfolio. So we can translate easily from the portfolio loss tolerance to the attachment level in terms of inflation if we assume a pass-through from inflation to interest rates.

Third, we need to specify the horizon of the tail hedge. Is the hedge for a “cyclical” six-month horizon, or for a “secular,” say five-year or even longer horizon? Since inflation is much lower frequency and longer lasting than equity market shocks, we believe that inflation tail hedging horizons necessarily have to be longer than those for equity market tail hedging. As we will discuss further below, the longer the horizon, the more confidence we can have in forecasting correlations with indirect hedging assets, which we can also use in our favor for building more efficient hedges.

Finally, we need to have some estimate of the cost of the hedge and whether the budget will be used up front or on an ongoing basis.

Obviously these inputs have to be iterated to find appropriately priced hedges in the market, or the parameters have to be varied until a suitable exposure, attachment and cost can be found. For instance, if the attachment level is too close to the current level of rates, and the cost budget is not large enough, then either the duration exposure has to be reduced by portfolio repositioning, or cheaper, indirect hedges have to be utilized.

**Benchmarking inflation tail hedges
**When we discuss tail hedging for typical equity-like risks, we find it important to define a benchmark hedge that we can call the “direct” hedge as reference. For instance, let us look to hedge the equity component of a 60% equity/40% bond portfolio at an attachment level of 15% of the portfolio value (i.e., when the portfolio falls 15% or more it is hedged) for one year. The reference hedge is approximately a 25% out-of-the-money equity put on 60% of the notional for the same one-year maturity (since the 60% equity component would have to fall by 15%/0.60 = 25% to reach the attachment level if the rest of the portfolio was unchanged). We can directly price this option using the price of an index option that replicates the equity component. Of course if volatilities are high, then the hedge might be too expensive, and one might make the decision to reduce the total size of the hedge, look at correlated “indirect” hedges or actively manage the quality (strikes, maturities, mixes) of the hedges. For inflation hedges, we can take the same approach. We can either benchmark the hedge in terms of the total expected payoff if the CPI (or the five-year forward, five-year break-even as an alternative) rises beyond a particular level. Or we can price the theoretical CPI cap or break-even option and then try to make the actual hedge more attractive relative to the direct inflation hedge. This exercise in benchmarking leads us to a brief discussion of the pricing of inflation options.

**Pricing of inflation options
**While options on both realized and expected inflation currently do not have liquid markets, we find it instructive to use this pricing for setting a theoretical “benchmark.” To this end, we go through the steps of pricing a CPI option and a break-even option.

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**Options on the CPI
**To set the stage, we first compute the pricing of options on the CPI index (CPURNSA on Bloomberg). These options come in two forms. The first form is the zero-coupon (ZC) option on the cumulative CPI. The second form is an option on year-over-year CPI inflation (YOY). To price these options, the market convention is to assume that the CPI index itself is a lognormal variable. Sample term sheets for these options are provided below.

As of December 2012, the index had a value of 230.221. The forward CPI for one year, using data from the survey of professional forecasters, was 238.28 (using a CPI YOY forecast of 3.5%). If we assume that the last 12 months’ volatility of the CPI is a good estimate of volatility for option pricing, we can compute an annualized volatility of approximately 1.5% as an input (although note this volatility has been quite variable, and some practitioners use the volatility implied from the implicit floor in TIPS prices to price such options).

Using these inputs, an at-the-money forward CPI option for one year would cost 1.24% (of notional). Similarly, a 1% out-of-the-money option would cost 62 bps, and a 2% out-of-the-money option would cost 24 bps. From 1968 onward, the average annualized volatility of the CPI has been approximately 1.15%, so the average price of an at-the-money option at the theoretical volatility would be approximately 0.5%. Similarly, the average theoretical price of a 1% out-of-the-money CPI option would be 19 bps, and that of a 2% out-of-the-money CPI option would be approximately 8 bps.

Just to be clear, market participants also quote the volatility in terms of bps/day for inflation option pricing. If at-the-money volatility is 1.5% for the CPI, this translates roughly to volatility of 10 bps per day. This increases to 2 bps per day more (or 180 bps per annum) for a 5% strike cap, reflecting the “skew” in the pricing of inflation options (Mirani et. al., 2013). Of course we do not doubt that the pricing will be variable based on models and demand and supply considerations, but having these estimates allows us to do basic analysis in terms of the relative richness or cheapness of these “straight” hedging alternatives.

**Options on the break-even inflation rate**^{13}** **

In a similar vein, as discussed above, if we are concerned that the rise in inflation expectations needs to be hedged, we need to price options not on the current CPI or inflation rate, but on longer-term inflation expectations. We believe the best metric (of many such metrics, none of which is really perfect), as displayed above (and watched by the Fed), is the forward inflation rate as embedded in the pricing of nominal and inflation-linked bonds. The theoretical benchmark pricing of options on these break-evens is slightly more involved and, to be clear, these options do not currently trade.

Suppose we take the current five-year forward, five-year break-even to be 3%, and we want to price an option on the break-even struck at 4%. To price this option, we will require the volatility of the break-even rate itself. The difficulty is that options on the break-even do not trade in the market currently, so there is no direct way of inferring the break-even volatility. One alternative is to use historically realized break-even volatility. The historical estimate (since 2007) of the volatility of the five-year forward, five-year break-even is approximately 100 bps a year.

The other alternative is to use the fact that the break-even rate is basically the difference between the nominal and the real yields. The variance of the break-even can be estimated if we can use the volatility of the nominal rates and the real rates and the correlation between them. The nominal rate volatility can be read off from the actively traded swaption market (it’s of the order of 80 bps to 100 bps per year). The volatility of the real rate can also be implied out of options on TIPs (called TIPtions), but these options again don’t trade with much volume or liquidity. To make things simple, we assume that we can forecast real rate volatility (which should ultimately be related to real growth rate variability).

As an illustrative example, if we assume that the volatility of the real yield is 75 bps per annum, and the correlation between the real yield and the nominal yield is 70%, then we obtain an estimate of the volatility of the forward break-even of approximately 65 bps per year using the formula relating the variances. We can now use this volatility as an input in the pricing of a break-even call option. An option struck 100 bps out of the money (say at a forward break-even rate of 4%) would cost approximately 96 basis points up front. If we annualize this number, the cost is approximately one-fifth or 20 bps per year. In practice, there is likely to be a substantial skew to the volatility surface; that is, since the volatility of inflation is expected to be higher for higher inflation, the input volatility for a deeply out-of-the-money call option on the break-even is likely to be closer to 100 bps.^{14}

As mentioned above, these computations for both the CPI and the break-even option are largely theoretical as of this writing, since break-even options really do not trade in any real volume. But the real value of benchmarking the price of these options is to set a reference for indirect hedging of inflation tail risk. To replicate the payoff of such direct options we have to create proxy or “indirect” hedges, whose relative attractiveness might be graded with reference to the theoretical “direct” options. We will discuss this approach now.

**Indirect inflation tail risk hedging and basis risk
**Given that many direct inflation tail options are practically not implementable, we now turn our focus to approximations. We will do this systematically, but starting with underlying assets that are fundamentally most correlated to inflation, and then to assets that have less predictable co-movements with inflation under normal situations but high co-movements especially under inflation shocks.

While it may seem that our task would be done if we can find assets that show a high positive beta to inflation by doing a simple correlation analysis, it is important as mentioned above to note that the potency of an inflation hedge also depends on the persistence of inflation. Since inflation itself is a low frequency time process, incorporating inflation persistence in estimation of the hedge betas is critical before we select hedges. We will briefly discuss the role of inflation persistence in determining the potency of hedges.

Following the framework of Schotman and Schweitzer (2000), if inflation evolves as an autoregressive process (with the caveats discussed previously on making this assumption),

and asset returns are a function of expected and unexpected inflation

(where *α β, Φ, σ_ε, σ_η* are inflation persistence, inflation beta of asset, beta of returns to unexpected inflation, return volatility and inflation volatility), then the estimate of the long-term beta (the “hedge ratio”) of the hedge is

which requires ((1 - *α*) *Φ* + *αβ*) > 0 for the hedge to be a bona-fide hedge (i.e., for the beta to be positive to inflation).

Equally important is the case when inflation becomes a random walk, that is

Then for any time-horizon *k*, the estimate of the hedge beta is

If we were to plot this hedge ratio as a function of the time horizon *k* we would find that the hedge potency increases monotonically but at a decreasing rate as a function of the horizon. What this tells us is that even though the short-term behavior of indirect inflation tail hedges might be open to much debate, the potential longer-term benefits of a diversified basket of inflation tail hedges, purchased at an attractive price, are quite predictable. We invite the curious reader to plug in some sample parameters into the equations above to see the time behavior of inflation hedges. This also cautions us that the traditional approach of using contemporaneous correlations between asset returns and inflation leaves us short of extracting all the value we can from the role of these assets as inexpensive longer term hedging vehicles.

**Pricing of tail interest rate swaptions
**Our first step toward deriving an indirect hedge is to assume that the effect of rising inflation is reflected in rising nominal interest rates while real rates remain relatively static. If this assumption is borne out, then options on the nominal yield curve (i.e., swaptions) would be the simplest replication of the option on the break-even inflation. Indeed, as a matter of practice, we have recently observed that many inflation tail hedgers have been using “out-of-the-money” swaption hedges (which explains the “payer” skew for swaption volatilities).

Here is an example of the pricing of a payer swaption (the right to pay fixed swap rates at a pre-defined strike). Assume that the current five-year forward, five-year swap rate is 2.97%. Suppose we fix the premium budget at 50 bps per year, which equals 2.50% for the next five years. Using a Black model for swaptions and an implied volatility of 95 bps, we can price the option and solve for the strike that this cumulative premium gets us (the strike turns out to be 3.64%). The distance of this strike from the current value of the forward is 0.67%. Based on a Black model the delta of the swaption turns out to be 0.38. To evaluate the potency of this hedge in a portfolio and perform risk analysis, we can shock the interest rate curves and re-evaluate the new value for the portfolio with and without the hedges. This exercise is quite standard and most of the computations can be done with ease.

Note that swaptions allow the customization of both the option expiry and the rate (underlying) of the hedge. If one is worried about inflation rising in the short term and the short end of the nominal curve reacting to it, shorter-dated swaptions (say one-year expiry into the five-year swap curve) can be used. On the other hand, if we think that the rise of inflation will be a gradual process and really happen through expectations changing, then longer options (say 10-year options on the 10-year swap) might be warranted. In doing so, swaptions allow for an implicit view on the impact of rising inflation or inflation expectations on the shape of the yield curve to be simultaneously expressed with the most attractive point to buy volatility. In the current environment, the swaption volatility curve illustrates a “hump”; that is, it peaks around the two-to five-year point and declines both for shorter- and longer-dated swaptions. This allows for efficient volatility positioning as well. For instance, the purchase of a long-dated swaption results in volatility “roll-up”; that is, as time passes the longer-dated swaption becomes a shorter-dated swaption with a higher implied volatility. In doing so, the natural time decay incurred as part of buying an option is reduced. Any active inflation tail risk management paradigm should pay attention to such opportunities in the volatility term structure.

As we move further afield from interest rate options such as swaptions, we are faced with the problem of forecasting the correlations between these instruments and the rise in inflation and especially the robustness of the correlations. Theoretically, we can try to anticipate which assets are likely to have positive inflation sensitivity by noting that the value of any asset is the expected NPV of its future cash flows. If the NPV is calculated using the discount factor given by , where is the real rate, is the inflation rate and is the risk premium or credit risk, then we can see that the lower the sum of the three, the higher the discount factor. We can empirically partition the returns of any asset into their exposure to these risk factors and keep the inflation-sensitive component as a hedge. If we linearize the above equation by expanding the exponential, essentially what we are looking for are asset classes and combinations of securities that will show a positive relationship with inflation. To evaluate this, we looked at various inflationary periods and some core inflation-sensitive asset classes. Over most periods, it is clear that amongst other proxy instruments, gold and oil have tracked both realized and expected inflation fairly well, especially when we adjust for the persistence effect mentioned above.

While some assets like gold clearly show persistent positive relationships to rising inflation, especially if we condition our analysis on large rises in inflation, the paucity of data for periods similar to the current era makes dependence on backward-looking data analysis dangerous. To address this issue, we take a three-step approach. First we forecast which assets qualitatively are expected to respond to inflation in a dependable manner. Second, we estimate the relative cost of inflation-protection options on these assets against the cost of the more direct hedges. Finally, we iterate different mixes of such options in the portfolio to arrive at a probability distribution of possible outcomes that captures the basis risk versus the relative cheapening of the hedges compared with the most direct hedge. The spectrum of combinations measured through the lens of the cost versus basis risk allows a practical hedge program to select what point of the spectrum is appropriate.

**Gold and oil as proxy hedges
**As an example of indirect hedges that we can build theoretical and empirical support for, we can look at gold, crude oil and some foreign currencies as proxy hedges.

Even a cursory look at the historical returns on the CPI (YOY) and year-over-year returns on gold shows that the two series are highly correlated, especially in the extremes. Other than the empirical correlations (which happen to be close to 0.5 for inflationary episodes), there are good fundamental reasons to believe that gold and oil are robust inflation tail hedges. Simplistically put, if central banks are printing money to devalue their currencies, then gold, which is in limited supply and “the currency of no one central bank,” is clearly the beneficiary. In addition, there is anecdotal evidence that central banks are generally underweight gold as an asset, and a rebalancing toward reducing this underweight could easily be correlated with an inflation shock (see Erb and Harvey, 2012). While the empirical beta and the correlation are highly variable, we can easily see for fundamental reasons why options on gold at proper valuation levels can prove to be potent indirect hedges for both realized and expected inflation. For crude oil, the relationship might indeed be even more fundamental than it is for gold. First, a rise in oil prices has a direct flow-through into the pricing of finished products, both due to the energy cost of production and also because many oil derivatives are actually components of consumer products. Shocks to energy supply are also naturally positively correlated to rising inflation and inflation expectations. If we follow the supply shock argument as the precursor to inflation, then the case for energy-related assets, especially oil, as tail hedges becomes stronger.

**Example of gold options as a proxy tail hedge
**For this section, we will sketch the indirect tail hedging exercise with gold. We can easily extend the analysis to options on oil in a parallel fashion.

Let us return to the example discussed above, where we priced a CPI call option for one year at a theoretical price of 24 bps for a 2% out-of-the-money attachment point. We also priced an option on the five-year forward, five-year break-even for approximately 20 basis points a year.

We will focus on comparing the break-even option with the gold option as an indirect hedge.

To show how the computation of gold as a proxy hedge works, we first estimate the comparable attachment point in gold terms. If, for instance, we believe the conditional tail correlation between gold and the CPI to be 0.8, and assume that the volatility of gold for one year is 15% (we could read this from option-implied volatilities on gold, for instance), then we can approximate the beta of gold to CPI to be approximately the volatility ratio of gold to the CPI times the correlation. With this predicted beta of eight, we can see that a theoretical call option on gold will require a strike price that is scaled to be proportionately further away. In other words, in percentage terms we want the strike of the gold call option to be eight times away from the equivalent strike for the direct inflation option. The price of this indirect option is easy to obtain from the market. However, we still need to figure out how much of the indirect option to buy.

Thus, the second step is to compute the future value of both the inflation option and the gold option conditional on the inflation variable crossing the targeted inflation attachment point and scaling the size of the gold option such that when such an event happens their future dollar values are equal. This step is necessary to make sure that the indirect option can be exchanged with a high degree of certainty against the theoretical inflation option at that unknown (random) future date. This step also allows us to measure the potential basis risk from selecting an indirect hedge. The final step in the analysis is to compare the present price of this appropriately scaled indirect gold option against the price of the direct inflation option.

Continuing with our example, suppose the break-even option crosses the threshold of a 4% inflation rate in six months. Assuming that the volatility of the 4.5-year forward now increases to 100 bps/year (see the discussion of skew above), the new price of the theoretical break-even option is 384 bps; that is, the initial 96 bps has realized a mark-to-market gain of 3x and is now worth 4x its original value. So the proper way to scale the gold call option is such that its dollar value increases to the same amount. Of course, ultimately we need to apply judgment on how much of the ultimate hedging portfolio should consist of indirect hedges. To achieve this, we include the mix of direct and indirect options in the portfolio at different weights and compute the distribution of scenario payoffs versus costs. This tradeoff of basis risk versus cost savings is standard for any tail-hedging practice, and ultimately is one that the end user has to make based on considerations of the desire for certainty in hedge performance against the budget available.

The longer the time for crossing the threshold, the more certain we can be that the indirect hedges will actually realize the tail correlation that we theoretically expect them to. In other words, the basis risk of indirect hedges is effectively reduced as the horizon of hedging increases.

**Conclusions**

The unprecedented creation of liquidity in the aftermath of the credit crisis, the massive debt burden emanating from this liquidity and the well-advertised and potent threat of inflation-loving central banks raise the risk of an unanticipated inflationary tail event. Whereas a spike in actual inflation is not impossible, the magnitude of adverse impact to investment portfolios is likely to be tied to an unanticipated increase in inflationary expectations. In this paper, we discussed how to systematically address inflation-tail hedging via the framework that we have been able to apply to equity market risks. To do so, we first introduced direct, non-linear options on both realized and expected inflation. Unfortunately, since these options do not have a liquid market at the time of this writing, we must work with approximations to the direct hedges. To do so we first laid out the key elements for building an inflation tail hedging program, using benchmarks, exposures, attachment points, cost and basis risks as parameters. Finally, we illustrated, by simple examples, how to incorporate indirect hedges from other asset classes.

In summary, there is macroeconomic need today for crude yet potent inflation hedging solutions. Whereas the actions of central banks have possibly mitigated short-term deflation risk and risk to equity markets, in their wake inflation and duration tail risk are not far behind. We believe that the framework of this paper can be used to create a portfolio of potent hedges against such an adverse tail event.

**Acknowledgements **

I would like to thank my colleagues at PIMCO for their collaboration and critical comments on the draft of this paper. Chris Dialynas read an earlier version of this paper and made many helpful comments and pointed to other historical research. Mitch Handa and Qingxi Wang helped with the pricing of inflation options. Josh Davis and Bruce Brittain have assisted in developing our tail risk hedging approach, of which this is the latest iteration. I would also like to thank many clients who have asked for thought and discussion on the topic of tail risk hedging of inflation risk.

**References**

Bhansali, Vineer (2008). “Tail Risk Management,” *The Journal of Portfolio Management*, Summer, Vol. 34, No. 4, pp 68-75.

Erb, Claude, and C.R. Harvey (2012). “The Golden Dilemma,” available on SSRN website.

Gordon, Robert J., and vanGoethem, Todd (2007). “Downward Bias in the Most Important CPI Component: The Case of Rental Shelter, 1914-2003,” *Hard-to-Measure Goods and Services: Essays in Honor of Zvi Griliches*, University of Chicago.

Gurkaynak, Refet S., Sack, Brian, and Wright, Jonathan H. (2008). “The TIPS Yield Curve and Inflation Compensation,” *Finance and Economics Discussion Series*. FRB, 2008-05.

Johnson, Nicholas, Walny, Ronit (2013). “Responding to Inflation: A Multi-Asset Approach,*” PIMCO Strategy Spotlight,* January.

Johnson, Nicholas (2012). “Practical Models for Inflation Forecasting,”* Inflation Sensitive Assets,* Risk Books, pp 351.

C. Mirani et. al. (2013). “Inflation Volatility Digest,” Barclays Interest Rate Research, January 29.

Schotman, Peter C., and Schweitzer, Mark (2000). “Horizon Sensitivity of the Inflation Hedge of Stocks,” *Journal of Empirical Finance* , Vol. 7, pp 301-315.

**Notes**

^{1}*In this paper we will not address the question of whether the measurement of the CPI shows a persistent tendency for downward bias. For an interesting discussion please see Robert J. Gordon and Todd vanGoethem, 2007.*

^{2}*However, two qualifiers are important before one takes this metric seriously. First, the TIPS market is less liquid than the nominal treasury market, hence some of the break-even spread is really a liquidity differential. Second, given the purchase of both TIPS and nominal bonds by the Fed, care needs to be exercised in relying on the break-even rate as a pure metric of inflation expectations.*

^{3}*For instance, the time-tested Cagan model for hyperinflation is an elegant framework that dates back to the 1960s. One consequence of the model under a wide variety of assumptions is that expectations of an uncontrolled and rapid increase of the supply of money in the future will inevitably lead to an increase in inflationary expectations.*

^{4}*There are numerous books and websites that discuss these historic episodes. The website Shadow Government Stats has a compendium of such anecdotes and further references.*

^{5}*Please see *Barclays Inflation Volatility Digest*, January 2013.*

^{6}*If one naively fits the CPI YOY inflation series to an AR(1) process (i.e., with one lag), the dependence on the first lag is almost 0.98%, so there is indeed a lot of momentum in inflation as measured over short periods.*

^{7}*This is exactly the same as a Merton jump process that has been utilized for the modeling of fat tails and the volatility skew for equity markets.*

^{8}*Critics of the quantity theory of money would argue that hyperinflation only results if there is an accompanying supply shock, for example, shocks that arise from wars or famines, but these nuances are really irrelevant for our task at hand of creating portfolios robust to rapidly rising inflation or inflation expectations.*

^{9}*Pascal’s wager in this context simply means that since expected losses are the probability of loss times the severity of loss, we should behave as if we cannot forecast the probability, since the severity of the event can easily overwhelm any minor differences in the probability forecast error. Pascal used this logic in the context of whether one should believe in the existence of God.*

^{10}*For a robust hedge construction, we also need to manage the counterparty risk of the OTC options, and carefully select the currency in which the hedge payment will be made. In the extreme case, if the settlement for a dollar inflation option is in less valuable dollars, the price of the inflation hedge has to be appropriately adjusted. This quanto adjustment is standard.*

^{11}*Source: Deutsche Bank.*

^{12}*Source: Deutsche Bank.*

^{13}*I am grateful to my colleague Mitch Handa for the computations in this section.*

^{14}*The simplest way to use this in the pricing of break-even options is to use a class of models knows as SABR (Stochastic Alpha Beta Rho), where both the inflation variable, either the CPI or the break-even rate, and the respective volatility are stochastic and the inflation variable and the volatility have a positive correlation.*

##### The Author

# Vineer Bhansali

Portfolio Manager, Head of Quantitative Investing

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# Disclosures

**Past performance is not a guarantee or a reliable indicator of future results.** Investing in the **bond market** is subject to certain risks, including market, interest rate, issuer, credit and inflation risk. Investing in **foreign-denominated and/or -domiciled securities** may involve heightened risk due to currency fluctuations, and economic and political risks, which may be enhanced in emerging markets. **Inflation-linked bonds (ILBs)** issued by a government are fixed income securities whose principal value is periodically adjusted according to the rate of inflation; ILBs decline in value when real interest rates rise. **Treasury Inflation-Protected Securities (TIPS)** are ILBs issued by the U.S. government. **Commodities** contain heightened risk including market, political, regulatory and natural conditions, and may not be suitable for all investors. **REITs** are subject to risk, such as poor performance by the manager, adverse changes to tax laws or failure to qualify for tax-free pass-through of income. **Swaps** are a type of derivative; swaps are increasingly subject to central clearing and exchange-trading. Swaps that are not centrally cleared and exchange-traded may be less liquid than exchange-traded instruments. **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.

There is no guarantee that these investment strategies will work under all market conditions or are suitable for all investors and each investor should evaluate their ability to invest long-term, especially during periods of downturn in the market. 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 correlation of various indices or securities against one another or against inflation is based upon data over a certain time period. These correlations may vary substantially in the future or over different time periods that can result in greater volatility. The Consumer Price Index (CPI) is an unmanaged index representing the rate of inflation of the U.S. consumer prices as determined by the U.S. Department of Labor Statistics. There can be no guarantee that the CPI or other indexes will reflect the exact level of inflation at any given time. 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. This article is scheduled to appear as an invited editorial in the Winter 2014 issue of the Journal of Portfolio Management. 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. ©2013, PIMCO.