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PLANTRONICS, INC

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PLANTRONICS, INC. IMPROVING P&L PREDICTABILITY THROUGH EFFICIENT FX MANAGEMENT “ We didn’t have confidence in the static Oracle reports. Besides the fact that they weren’t very functional and we couldn’t easily export them to Excel to run scenarios, they got outdated in a hurry. Identifying exposures was like trying to find a needle in a haystack, but we didn’t know what it looked like or if it was even there in the first place.” Stuart White Senior Director, Tax and Treasury Plantronics, Inc. CUSTOMER SUCCESS STORIES Plantronics, Inc. and FiREapps “ IMPROVING P&L PREDICTABILITY THROUGH EFFICIENT FX MANAGEMENT We’re confident in our processes and in our numbers. Predictable results speak for themselves. Company & Case Study Quick Facts Industry Audio communications Annual Revenues $713 million (FY2012) Business Landscape Offices in more than 20 countries with over 40 percent of revenue generated internationally Stuart White Senior Director, Tax & Treasury Plantronics, Inc. Executive Summary When a combination of rapid international growth, currency volatility, and a lack of visibility into exposure data began to negatively impact the Other Income line on the P&L statement, the treasury team at Plantronics acted quickly to implement a new process for managing foreign exchange. A cross-functional group from treasury, accounting, tax, and IT worked together to correct multicurrency accounting and remeasurement issues and establish a systematic process for accurately identifying, understanding, and managing the company’s true FX exposure and risk. Today, using the cloud-based FiREapps FX management platform, Plantronics has dramatically reduced FX gain/loss variance and improved predictability in the P&L statement by: Achieving and maintaining better than 90 percent balance sheet forecast accuracy. ERP Environment Oracle Improving the effectiveness of the hedge program to keep forecast-to-actual FX gain/loss variance below $250k per quarter. FiREapps Solution FiREapps Enterprise Reducing the impact of foreign exchange on earnings from several cents per share to nearly zero. EUR Estimated vs Actual Exposure Rapid dampening of FX gain/loss volatility Thousands of Dollars $30,000 25,000 20,000 15,000 10,000 5,000 0 2 Actual Exposure Estimated Exposure Implementation Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep 2008 2009 Since implementing FiREapps, treasury has consistently met its goal of achieving 90-percent balance sheet forecast accuracy using euro and pound hedging results as the metric. “ CHALLENGE NEUTRALIZING FX GAIN/LOSS VOLATILITY IN A DYNAMIC BUSINESS ENVIRONMENT Identifying exposures was like trying to find a needle in a haystack, but we didn’t know what it looked like or if it was even there in the first place. Stuart White Senior Director, Tax & Treasury Plantronics, Inc. Plantronics is headquartered in Santa Cruz, California with offices in 20 countries, including major facilities in China, England, Mexico, and the Netherlands. The company’s products are sold and supported through a worldwide network of resellers, systems integrators, retailers, and mobile carriers. And, with nearly 50 years of innovation and customer commitment under its belt, Plantronics enjoys a leading market share in its core business, a history of strong comparative profit margins, and growing cash flow from operations. Despite all of these positive factors, quarterly earnings reports were being impacted by large fluctuations in below-the-line expense. The main culprit? Foreign exchange volatility. “The business was changing internationally, creating larger exposures to currencies in growing and emerging markets,” says Barbara Scherer, Senior Vice President and CFO at Plantronics. “At the same time, the volatility of those currencies was ramping up, causing quarterly FX gain/loss to become increasingly unpredictable as international revenues became a larger percentage of total revenues.” While the company’s formal hedging program had worked well for over a decade, Treasury was no longer able to hedge the appropriate FX exposures because they weren’t getting accurate or complete data from the Oracle-based ERP system. “We didn’t have confidence in the static Oracle reports,” recalls Stuart White, Senior Director, Tax and Treasury. “Besides the fact that they weren’t very functional and we couldn’t easily export them to Excel to run scenarios, they got outdated in a hurry. Identifying exposures was like trying to find a needle in a haystack, but we didn’t know what it looked like or if it was even there in the first place.” The result was extreme volatility in FX gain/loss, which negatively impacted the Other Income line in the P&L statement. “We quickly learned how significant the risk of not properly identifying exposure can be,” Stuart says. “When you have an FX gain, analysts, management, and the board don’t take much notice. But if FX hurts, it will come back to haunt you because it shows an inability to effectively manage a certain aspect of your expenses. We needed to fix the problem, and we needed to do it fast.” “The business was changing internationally, creating larger exposures to currencies in growing and emerging markets. At the same time, the volatility of those currencies was ramping up, causing quarterly FX gain/loss to become increasingly unpredictable as international revenues became a larger percentage of total revenues.” Barbara Scherer Senior Vice President & CFO 3 SOLUTION “ CONNECTING THE DOTS BETWEEN ACCOUNTING AND TREASURY FiREapps gives us ease of use, and the ability to get data into a manageable format. We can really delve into the details to see what the exposures are, and then decide how to deal with them either internally through intercompany clean up, or externally through forward hedges. Stuart White Senior Director, Tax & Treasury Plantronics, Inc. After evaluating the functionality and capabilities of the FiREapps exposure management platform and determining that it was a good fit to help them improve the forecasting process and hedge results in a dynamic business environment, a cross-functional Plantronics team began working closely with FiREapps professional services to put a new FX management process in place. The first order of business was to analyze the accounting issues that were preventing treasury from accurately forecasting the exposures that needed to be hedged. “It wasn’t just a ‘treasury problem’ or an ‘accounting problem’,” Stuart says. “We needed to be able to connect the dots between the two in order to hedge appropriately. To do that, we needed the support of accounting, tax, and IT, and we had management buy-in to get it done.” The team’s systematic approach included: Identifying all sources of exposure, marking accounts for remeasurement, and going back into Oracle to better align intercompany accounting data. Configuring the FiREapps software to automatically extract, aggregate, harmonize, and validate current and historical balance sheet data from the Oracle system. Providing detailed analytical views in FiREapps that could also be exported to Excel as needed. Reports show currency exposures across key dimensions including corporate and legal entity views, monetary asset and liability class, general ledger account in reporting, and local and transaction currency detail. Automated data capture and management gives treasury a complete picture of FX exposure whenever they need it. With access to data they can trust, they can uncover new or previously hidden exposures, find and exploit natural hedging opportunities, track actual-versus-predicted trends to improve forecast accuracy, and increase the scope of the hedge program to achieve earnings and EPS predictability on an ongoing basis. “FiREapps gives us ease of use, and the ability to get data into a manageable format,” Stuart says. “We can really delve into the details to see what the exposures are, and then decide how to deal with them either internally through intercompany clean up, or externally through forward hedges.” In the last 12 months, Barbara has made only one adjustment to the treasury team’s recommendations. “I don’t have to think about the monthly balance sheet exposure forecast; I don’t expect surprises, and I don’t get surprises,” she says. “FiREapps takes what can be a very time-consuming issue off the table for the team, which allows me to focus on things a CFO should focus on, like revenue growth, OPEX control, and gross margins.” 4 Treasury is also working with FiREapps and the Plantronics planning team to prepare for what might happen in the future. NATURAL EXPOSURE For example, what would dollar/euro parity mean? Or, conversely, what if the euro collapses? “In the current environment, FX is something we need to watch very closely,” Stuart says. “We need a system and processes in place that will allow us to react quickly to changes in the currency market so new exposures don’t creep up on us. To manually gather the kind of data we get from FiREapps could take weeks or even months. As long as exchange rates are erratic, and there is no indication that they will stabilize any time soon, we don’t have that kind of time.” December 2011 Final 12/31/2011 Reporting Currency USD Value at Risk Confidence 99% confidence Value at Risk Horizon (days) 30 days Entity Group Currencies FX Rate Exposure Currency 1 Currency 2 Net VaR EUR-USD 1.2969 27,122,566 20,913,383 (27,122,566) 1,545,973 GBP-USD 1.5534 8,119,433 5,226,878 (8,119,433) 429,958 USD-CNY 6.2949 7,891,897 (7,891,897) 49,678,705 147,822 AUD-USD 1.0247 5,834,623 5,693,982 (5,834,623) 442,312 Plantronics, Inc. Group USD-MXN 13.9787 2,426,369 2,426,369 (33,917,484) 171,058 Plantronics, Inc. Group USD-BRL 1.8751 2,004,762 (2,004,762) 3,759,129 168,022 Plantronics, Inc. Group USD-JPY 76.98 1,778,373 (1,778,373) 136,899,187 55,750 Plantronics, Inc. Group EUR-GBP 0.8349 736,642 568,002 (474,225) 25,744 Plantronics, Inc. Group USD-CAD 1.0171 665,819 665,819 (677,204) 33,394 Plantronics, Inc. Group USD-SEK 6.8733 631,010 631,010 (4,337,123) 45,824 Plantronics, Inc. Group USD-INR 53.06 480,063 480,063 (25,472,167) 42,615 Plantronics, Inc. Group GBP-SEK 10.677 278,186 (179,082) 1,912,061 14,566 Plantronics, Inc. Group NZD-USD 0.779 238,752 306,485 (238,752) 18,708 Plantronics, Inc. Group USD-HKD 7.7665 172,803 (172,803) 1,342,074 755 Plantronics, Inc. Group USD-SGD 1.296 152,341 152,341 (197,434) 6,271 Plantronics, Inc. Group DKK-SEK 1.1995 131,373 (752,807) 902,992 4,325 Plantronics, Inc. Group USD-RUB 32.094 93,932 (93,932) 3,014,654 13,282 Plantronics, Inc. Group EUR-SEK 8.914 88,412 (68,171) 607,680 2,913 Plantronics, Inc. Group USD-DKK 5.7303 82,952 (82,952) 475,341 4,626 Plantronics, Inc. Group USD-MYR 3.167 59,146 (59,146) 187,314 3,263 Plantronics, Inc. Group USD-CHF 0.9374 50,817 50,817 (47,636) 3,251 Plantronics, Inc. Group USD-ZAR 8.0562 42,802 42,802 (344,823) 3,911 Plantronics, Inc. Group USD-CZK 19.94 37,519 37,519 (748,137) 3,344 Plantronics, Inc. Group USD-PHP 43.82 30,038 (30,038) 1,316,244 1,588 Plantronics, Inc. Group USD-KRW 1,152.20 26,749 26,749 (30,819,938) 2,014 Plantronics, Inc. Group GBP-CHF 1.4562 18,828 12,121 (17,650) 879 Plantronics, Inc. Group USD-TWD 30.268 16,918 (16,918) 512,060 1,964 Plantronics, Inc. Group EUR-CHF 1.2157 1,600 (1,234) 1,500 58 Plantronics, Inc. Group Plantronics, Inc. Group Plantronics, Inc. Group Plantronics, Inc. Group » » » Group Totals: 59,214,727 Grand Totals: 59,214,727 Percent Exposed: 100.0% Treasury uses natural exposure data from FiREapps to identify which entities have the highest net Value at Risk (VaR), a calculation of the potential loss from unhedged exposures due to currency volatility. Because VaR takes into account both gross exposure and volatility per currency pair, the team can clearly see which currencies to hedge (highlighted in green). For example, even though the Chinese yuan represents the third-largest exposure, the Australian dollar represents a higher VaR due to higher volatility, and is therefore a more favorable hedging opportunity for Plantronics and this analysis led us to begin hedging the Australian dollar. The magnitude of prehedge exposure relative to realized FX gain/loss illustrates the positive impact of improved data and analytics on forecast accuracy on hedge results. 5 RESULTS “ IMPROVED CONFIDENCE, COLLABORATION, AND PREDICTABILITY With a level of certainty we didn’t have before, we have a lot less stress, and we can focus on other areas of our work that need our attention more. Stuart White Senior Director, Tax & Treasury Plantronics, Inc. When the new FX exposure management system was first implemented three years ago, process improvements across accounting and treasury was almost immediate. “In the beginning, FiREapps allowed us to quickly discover and deal with the big issues, and within the first few months, we were able to gain confidence in our data,” Stuart explains. Eight months ago, FiREapps worked with Plantronics to do a follow-up deep dive into the system, as FX is always a high risk area. The goal was to take a fresh look after the original implementation to see if any exposures were missed the first time. They didn’t find any. “Now, we’re focused on forecasting more accurately and making better hedging decisions rather doubting our numbers and hoping for a positive outcome,” he says. “We’re confident in our processes and in our numbers. Predictable results speak for themselves.” New-found confidence leaves time for value-added analysis and planning Because treasury doesn’t have to manually hunt down and manipulate questionable, outdated, and incomplete data each month, they can spend their time improving forecast accuracy and hedging results, even as company growth introduces new currency pairs and ERP complexity to the FX management process. “We trust the data we see in the FiREapps reporting, and we have great confidence that we are doing the right things to manage our exposure and risk,” Stuart says. “With a level of certainty we didn’t have before, we have a lot less stress, and we can focus on other areas of our work that need our attention more.” Cross-functional collaboration improves efficiency and results FiREapps helps the treasury team at Plantronics work holistically and collaboratively with the accounting, tax, and IT departments to maximize efficiency across the organization. As Stuart sees it, “If you don’t understand and appreciate what each department’s challenges are, you can’t work together toward a common goal. And if you don’t have an efficient and reliable process in place to facilitate working together, you have a couple of risks that are almost impossible to mitigate: exhausting your employees and getting hit with a big problem you didn’t see coming.” Volatility is down, predictability is up Depending on currency movements, Plantronics was experiencing FX surprises equating to as much as three or four cents per share. Now, the FX line is rarely over a penny. “Our FX gain/loss used to be erratic and significant in the quarterly earnings releases. Now, because of the forecasting and hedging we’re able to do, it’s immaterial,” Stuart reports. “I don’t hear from the board, which is a good thing.” Treasury has met its goal of achieving 90-percent balance sheet forecast accuracy using euro and pound hedging results as the metric. The FP & A group has exceeded its original goal of keeping total FX gain/loss below $500k per quarter. And improved forecast accuracy has bolstered the effectiveness of the hedge program to consistently keep forecast-to-actual variance below $250k per quarter. 6 RESULTS “ Our FX gain/loss used to be erratic and significant in the quarterly earnings releases. Now, because of the forecasting and hedging we’re able to do, it’s immaterial. Stuart White Senior Director, Tax & Treasury Plantronics, Inc. “Given the volatility we saw in the past, this is a major, major improvement,” Stuart says. “We don’t hedge all of our currencies, so there is always some chance of FX variability, but it’s negligible in terms of variance from what we expect. We haven’t had to explain any kind of volatility in our Other Income line for quite a while now. When the street, the board, and the executive team are happy, we’re all happy.” “FiREapps represents a quantum leap in terms of how we identify currency exposures and their associated risks,” Barbara adds. “With a clear picture of our FX landscape, we have lowered our net gain/loss after the hedge. And, because we’re never behind the 8-ball when it comes to developing exposures, we have also increased the percentage of exposure and currency pairs we hedge, further reducing the FX impact on net earnings each quarter. I am proud of the team for identifying and implementing the appropriate technology to improve processes and outcomes, and doing so in less than three months with excellent results.” “FiREapps represents a quantum leap in terms of how we identify currency exposures and their associated risks. With a clear picture of our FX landscape, we have lowered our net gain/loss after the hedge.” Barbara Scherer Senior Vice President & CFO To learn more, visit us at www.fireapps.com 7 Phone: +1.866.928.FIRE (3473) | Fax: +1.928.223.0133 www.fireapps.com 1. Consider one of the Greeks, "rho," in answering this question. (I can't insert the actual greek letter here in Sakai.) Now, note that at one point, Buffett's Berkshire Hathaway sold what is probably the world's largest short position in S&P 500 put options http://www.cboeoptionshub.com/2014/08/04/checking-warren-buffetts-equity-indexput-trade/. Update 5/8/2021: The link in question 1 of the exam redirects to the CBOE site and the article referenced in the question is now unavailable. (http://www.cboeoptionshub.com/2014/08/04/checking-warren-buffetts-equity-index-puttrade/) I guess the CBOE did not archive it :-( Use these: How Warren Buffett Use Put Options to Increase Berkshire's Returns (re-thinkwealth.com) Buffett's S&P 500 Puts: Big Blunder? | Seeking Alpha ______________________________ What would happen to this type of short position if interest rates rise dramatically? In other words, will Berkshire's possible liability to the counterparty go up or down? Remember the company is the put writer (also known as the seller), not the buyer. Stated another way, if you "marked Berkshire Hathaway to market," would Berkshire's position (short) make money or lose money if rates rose...(holding everything else constant, i.e., ceteris paribus)? In other words, don't spend any time predicting the stock market, concentrate on the question, which only addresses interest rates an their effect on put values. In addition to stating your result (your prediction), explain why this is true. In other words, is rho positive or negative? 400 words 2. Open the attachment from FIREAPPS, a consulting/software firm whose products address FX exposure and management. Then write about a two-page answer*, in "memo" format. Address it to your company's treasurer. Assume that you are an MBA-trained treasury analyst. Tell the treasurer if your company should adopt (or not adopt) FIREAPPS solutions for forex exposure. You don't necessarily have to agree with FIREAPP's point of view. It's just s starting point for thought. In this memo, GRAMMAR COUNTS. And you must not resort to short superficial answers. On this question, your answer should be at least 400 words (about 1 1/2 typed pages double-spaced). *Please NO SHORT SUPERFICIAL ANSWERS. 3. Go to CMEGROUP.COM We are going to hedge oil, for a bank that has made huge loans to oil producing companies. Choose the July 2020 contract as the underlying future. State if you will go long or short oil. Explain CAREFULLY, not briefly, why you are long or short. What would ten contracts cost you, using the actual CMEGROUP last trade as your price? Show your math, i.e., #gallons times price per futures contract. Is this contract in contango, or normal backwardation? 4. Some analysts favor the use of VIX-related options, such as UVXY, as a hedge with respect to net income reporting by corporations. I think of this as a "VIX beta." See attached article by WElch. Write, carefully, the equivalent of about two typed pages (400 or more words ) about how this would work. State pros and cons, give some examples. No short superficial answers, please. 5. Explain the main thesis and main points. Show the pros and cons of what (the idea, the concept or the strategy). Aim for 400 words. 1.Breif information about cryptocurrencies 2. risk Investment on cryptocurrencies and future contract 3. How to hedge crypto and strategies. For example, Short-Selling and futures contract Multiple Choice: 1. Institutional portfolios can take advantage of what is called "portfolio margining" under the rules of the CBOE and other options exchanges. What IS portfolio margining? Choose only one answer. A. You only pay attention to the bottom margin of the portfolio. • • B. Great leverage is possible, as the intercorrelations and natural offsets (or what we call in English a "wash") are taken into consideration. • C. No margin is required, because these are not individual investors. 2. What does the symbol ^VVIX mean? • A. Apple's volatility. • B. An exchange-traded (ETF) volatility fund. • C. The implied volatility of the VIX options themselves. 3. An example of purchasing a "real" option is a company's opening of a Shanghai office, in order to have the capacity to pursue business in China. True False 4. A hedge fund is "short" a portfolio of VIX call options. It's "theta" is $16,788 per day. This means, barring some big market or political or terrorist event, the fund's net asset value will increase by about $16,788 per day. In other words, let's assume that the VIX does not vary much during this fund's portfolio's timeframe. True False 5. Long options positions(long does not mean long-dated) have positive gamma, and short positions have negative gamma. True False 6. In valuing credit default swaps, one must estimate default probabilities. A start would be to obtain historical data for default rates from rating agencies such as Standard & Poor's. But a more complete risk profile, or risk prediction, limiting ourselves to the US market (US-based companies) would be to [pick only one choice, the best one, in your opinion]: • A. Also consider the risk of another earthquake in Japan. • B. Also consider the systematic risk associated with an economic downturn in the US, accompanied by a downturn in US stocks. • C. Also consider the possibility that Facebook options are trading at an implied volatility that is not as high as it should be, considering the lofty level of its P/E ratio. 7. A total return swap can be a form of financing vehicle to allow an investor to buy a specific bond, because: (there are two correct answers, you must pick two) A. The payer (a financial institution) could agree to pay the bond's capital appreciation or depreciation, plus the interest payments, in return for receiving LIBOR plus some basis points. B. The payer (a financial institution) could agree to receive the bond's capital appreciation or depreciation, plus the interest payments, in return for receiving LIBOR plus some basis points. C. The payer retains ownership of the bond for the life of the contract. 8. Imagine an equity-oriented state pension fund (such as the State of Maryland Pension Funds). Suppose their equity portfolio is NOT identical to the S&P 500 index. Further, suppose that due to a state budget shortfall, the year's additions to the pension fund are likely to be less than actuarially sound. Thus, the fund is becoming very risk-averse. And it needs to implement this feeling by hedging. CHOOSE THE BEST ANSWER, even though there might be more than one true answer.This is actually a question about hedge ratios. • A. If the equity portfolio's beta is close to 0, then they could decide to match the overall exposure, via S&P Index futures contracts, to the fund's value. That is, # of contracts times the contract multiplier times the index would approximately equal the value of the fund. • B. If the equity portfolio's beta is close to .4, then they would have a tough time using S&P 500 contracts, so they would instead use foreign exchange contracts. • C. If the equity portfolio's beta is close to 1, then they could decide to match the overall exposure, via S&P Index futures contracts, to the fund's value. That is, # of contracts times the contract multiplier times the index would approximately equal the value of the fund. Critical Finance Review, 2016, 5: 399–415 How Should Firms Hedge Market Risk? Bhagwan Chowdhry Eduardo Schwartz 1 University of California, Los Angeles, Anderson School of Management, USA; bhagwan@anderson.ucla.edu 2 University of California, Los Angeles, Anderson School of Management, USA; eschwart@anderson.ucla.edu ABSTRACT Consider a firm whose stock returns are affected by market returns and an idiosyncratic market-orthogonal factor. The level of the firm’s cash flows depends on the level of the market and the level of the idiosyncratic factor multiplicatively because of compounding. Although a large hedge against the market index minimizes the variance of cash flows, such a hedge does not minimize the costs of financial distress associated with low cash flow realizations below a debt threshold. A hedge ratio based on asset-rate-of-return regression estimates is then incorrect. This holds even in continuous time and with dynamic hedging policies. Our paper provides a simple heuristic for corporations wishing to hedge out the adverse consequences of market risk. Keywords: Finance JEL Codes: G30 We thank Jeremy Stein, René Stulz (who also told us about Fischer Black’s critique), and Ivo Welch for many insightful conversations. We also thank seminar participants at UCLA Anderson brown-bag seminar series and at the UCLA-Lugano Finance Conference. We thank anonymous referees for many useful comments, and are deeply grateful to one who helped provide the analytical proof for the main result in the paper. ISSN 2164-5744; DOI 10.1561/104.00000023 ©2016 B. Chowdhry and E. Schwartz 400 1 Bhagwan Chowdhry and Eduardo Schwartz Introduction There is an extensive literature that shows that firms can, under some circumstances, increase shareholder wealth by reducing the volatility of their cash flows. In particular, firms that face significant costs of financial distress if they experience abnormally low cash flows can decrease the present value of financial distress through hedging. In a seminal paper, Froot et al. (1993) show that firms that have to finance their investments out of their cash flows are forced to give up positive net present value projects if they experience poor cash flows. Such firms benefit from hedging because it enables them to take advantage of investment opportunities they would have to forsake or give up otherwise. A number of other reasons for why firms can benefit from decreasing total cash flow volatility have been discussed in the literature.1 Total cash flow volatility is a function both of firm idiosyncratic volatility and of volatility induced by systematic risk. Consequently, it would seem that firms could also create shareholder wealth by reducing their exposure to systematic risk. However, we do not observe firms hedging their exposure to the market either by shorting a market index or by using financial derivatives on the market index. Nor do many academics2 or practitioners recommend that firms do that. Fischer Black pointed out this embarrassing fact many years ago. We show that the simple intuition which would suggest that a firm with positive exposure to market movements (i.e., a positive beta) hedge by taking an offsetting position in the overall market requires careful consideration when more than one source of uncertainty affects the variability of the firm’s cash flow. This is because the effects of different sources of uncertainty on a firm’s level of cash flows at a distant date in the future are multiplicative even when these effects appear to be separable in stock returns over short horizons. The intuition for the main insight in the paper can be illustrated by the following simple example. Suppose a firm’s cash flow in 5 years is $100 on average. Suppose that the realized cash flow is determined by a factor that is idiosyncratic to the firm and also by overall market conditions. Assume that the idiosyncratic factor alone can make the realized cash flow go up or down by 60% with equal probability, and that the market factor alone can make the realized cash flow go up or down by 50% with equal probability. When both factors are present and are uncorrelated, the realized cash flows can take four different values: 1 For instance, Smith and Stulz (1985) show that lower cash flow volatility can reduce the present value of taxes; Stulz (1984) makes the case that high cash flow volatility can make the firm’s insiders more risk-averse; using different mechanisms, Breeden and Viswanathan (1998), and DeMarzo and Duffe (1995) show that lower cash flow volatility can help outsiders in assessing the performance of firms. 2 See Bolton et al. (2011) for an exception. How Should Firms Hedge Market Risk? Idiosyncratic factor up by 60% down by 60% 401 Market up by 50% Market down by 50% $100 × 1.60 × 1.50 = $240 $100 × 0.40 × 1.50 = $60 $100 × 1.60 × 0.50 = $80 $100 × 0.40 × 0.50 = $20 Suppose now that the firm shorts $100 of the market to hedge market risk, so that when the market goes up, it loses $50; but when the market goes down, it gains $50. The hedged cash flows are shown as follows. Idiosyncratic factor Market up by 50% Market down by 50% up by 60% down by 60% $240 − $50 = $190 $60 − $50 = $10 $80 + $50 = $130 $20 + $50 = $70 Hedging has reduced the range of cash flows from $80–$240 to $130–$190 when the idiosyncratic factor is up. However, the worst cash flow realization has deteriorated from $20 to $10 when the idiosyncratic factor is down. If the firm had debt of $15, it would be bankrupt with the market hedge but not without. Our paper shows that the optimal market hedge is much more conservative than hedging $100 of market risk. In our example, with $15 worth of debt, the optimal hedge is $15. The intuition is as follows. Suppose the firm takes a short position in the market assuming the average realization of cash flow. Then, if the realized cash flow turns out to be low, the short market hedge would have been excessive and in fact may lead to significant losses if the market goes up. This could be devastating. On the other hand, if the realized cash flow turns out to be high, the market hedge based on the average cash flow would have been inadequate, but this is not so critical. Thus, the asymmetric payoff should induce the firm to be more (though not completely) conservative in hedging its market risk.3 If the firm’s objective were to minimize the total variance of the cash flow, then the appropriate market hedge would be large. This large hedge would reduce variance both when cash-flow realizations are large as well as when they are small. However, because bankruptcy deadweight costs are relevant only when the cash-flow realizations are low, the firm’s objective should be to minimize the total variance of the cash flow only when its cash flows are likely to be low. It does not matter that this low a hedge increases the unconditional variance or the variance when cash flows tend to be high. This intuition also suggests that Shiller’s 2004 suggestion that individuals hedge more (market) risk requires the caveat that their optimal hedge ratio is 3 That there is a tradeoff between financing and risk management is identified in Holmstrom and Tirole (2000) and Mello and Parsons (2000), and Rampini and Viswanathan (2010, 2013). Dynamic models of such tradeoffs are developed in Bolton et al. (2011) and Rampini et al. (2013). 402 Bhagwan Chowdhry and Eduardo Schwartz not the sensitivity of their welfare with respect to the market, but potentially considerably lower. Interestingly, the optimal market-risk hedge has an easy heuristic. Managers should choose a hedge ratio equal to the firm’s contractual obligations as a fraction of expected cash flows, multiplied by the firm’s market beta. The hedge ratio does not depend on the firm’s own volatility and the market volatility. 2 2.1 The Model The Firm Consider a firm and a market index whose short-run return dynamics are r t = α + β · r tM + ν t , r tM = α M + ν M t , where r t is a stock return, r tM is the contemporaneous return on the market index, β is the exposure to systematic risk, and ν t and ν M t are mean-zero idiosyncratic components. When β is positive and the variance of ν t is large, this is a canonical example in which exposure to r tM is significant and therefore an offsetting market hedge would appear to be helpful. Textbooks often suggest rate-of-return regressions to estimate a coefficient β, which is then argued to be the optimal hedge ratio, because this hedge ratio minimizes the variance of returns. We will show that this is incorrect if the goal is to avoid financial distress. To simplify, assume that β = 1. The return dynamics over a finite period of, say, one year, R t and R M t , have to be exponentiated, (r t ) (1 + R M , t )=e M (1 + R t ) = e(r t ) . The economy is risk-neutral,4 so f E(R t ) = E(R M t )=R , where R f is the rate for a risk-free investment. Then, it follows that f M e(r t ) = (1 + R M t ) = (1 + R ) · (1 + ε t ) M e(rt ) = (1 + R t ) = (1 + R f ) · (1 + ε M t ) · (1 + ε t ), M M and E(ε t ) = E(ε M t ) = 0. If r t , r t , ν t and ν t are normally distributed, then R t , M M R t , ε t and ε t are log-normally distributed. 4 The firm’s incentive to hedge in our model will arise from its desire to avoid financial distress, and not from risk-aversion. How Should Firms Hedge Market Risk? 403 Now, consider a two-date model in which a firm generates one cash flow at date 1, C1 . Because we are assuming risk neutrality, the value of the firm at date 0 is E0 ( C1 ) S0 = . 1 + Rf Because the gross rate of return over one period (1 + R1 ) = C1 /S0 , C1 = S0 · (1 + R f ) · (1 + ε1M ) · (1 + ε1 ). Notice that the cash flow depends on the market risk (1+ε1M ) and the idiosyncratic risk (1 + ε1 ) multiplicatively. 2.2 The Hedge Now consider hedging the firm’s cash flow by shorting forward contracts on a market index. Consider a market index with current value Z0 . Its value at date 1 is Z1 = Z0 · (1 + R1M ) = Z0 · (1 + R f ) · (1 + ε1M ). The forward price at date 0 of the market index is F = Z0 · (1 + R f ). If the firm goes short one forward contract on the market index (or, equivalently, shorts the market index and invests the proceeds to earn the risk-free rate of return on the proceeds), then the cash flow on date 1 will be H1 = F − Z1 = −Z0 · (1 + R f ) · ε1M . The expected value of the market hedge is zero. H1 is positive if the market falls and negative if the market rises. Define y0 = (S0 /Z0 ) · h0 to be the number of market hedges (short the forward contracts) and h0 the fraction of the market-risk hedged. Then the hedged cash flow for the firm is C1 + y0 · H1 = S0 · (1 + R f ) · [(1 + ε1M ) · (1 + ε1 ) − h0 · ε1M ] = E0 ( C1 ) · [1 + ε1 + ε1 · ε1M + (1 − h0 ) · ε1M ]. If B1 denotes the contractual obligations to firm’s creditors or bondholders, then the firm will be bankrupt if its hedged cash flow at date 1 is C 1 + y0 · H 1 < B 1 . The bankruptcy condition above can be rewritten as ε1 + ε1 · ε1M + (1 − h0 ) · ε1M < −(1 − φ), (BC) 404 Bhagwan Chowdhry and Eduardo Schwartz where φ≡ B1 (< 1) E0 (C1 ) is the contractual obligations of the firm as a fraction of its expected cash flow at date 1. 2.3 Hedged Firm Risk The variance of the hedged cash flows is proportional to the variance of ε1 + ε1 · ε1M + (1 − h0 ) · ε1M . Lemma 1 states that a hedge that offsets 100% of the variability in cash flows minimizes the variance of the hedged cash flows. Lemma 1. The hedge h0 = 1 minimizes the variance of ε1 + ε1 · ε1M + (1 − h0 ) · ε1M . Proof. ε1 and ε1M have means equal to zero and are independent of each other. Therefore Var ε1 + ε1 · ε1M + (1 − h0 ) · ε1M = Var(ε1 ) + Var(ε1 ) · Var(ε1M ) + (1 − h0 )2 · Var(ε1M ). Setting h0 = 1 minimizes the right-hand side (RHS). However, the firm would want to minimize the variance of its cash flows only if minimizing the variance also minimizes the likelihood that its hedged cash flow will fall below a certain threshold. This is the case if the firm has no idiosyncratic risk, but not usually otherwise. Lemma 2. A hedge h0 where φ ≤ h0 ≤ 1 can eliminate financial distress if the firm has no idiosyncratic risk (ε1 = 0). Proof. Zero idiosyncratic risk implies that ε1 = 0. Setting ε1 = 0 in the bankruptcy condition (BC), and then noting that φ ≤ h0 ≤ 1 ⇒ (1 − h0 ) · ε1M > (1 − h0 ) · (−1) = −(1 − h0 ) ≥ −(1 − φ) proves that such a firm will avoid bankruptcy in all states of the world. The firm can minimize the likelihood of financial distress by taking an offsetting short position in the market. However, this result does not generalize when its own idiosyncratic risk is positive and significant. Instead, a more conservative market hedge, i.e., h0 < 1, can increase the conditional variance when ε1 is positive and reduce it when ε1 is negative. This increases the overall variance but reduces the likelihood that the firm will face financial distress. How Should Firms Hedge Market Risk? 405 Assumption 1. The firm minimizes a cost associated with financial distress that is proportional to the difference between its contractual obligation and its hedged cash flow in states in which it is bankrupt.5 The firm minimizes Z B1 min K · y0 (B1 − x) · f (x) d x, −∞ where x ≡ C1 + y0 · H1 . f (x) is the density function of the hedged cash flow x, and K is a scaling constant that parameterizes the cost of financial distress.6 Normalizing all cash flow numbers by E0 (C1 ), and setting K = 1, the firms’ objective function can be rewritten as Z ∞Z ∞ Γ ≡ min max − (1 − φ) − {ε1 + (ε1 + 1 − h0 ) · ε1M }, 0 h0 −1 −1 × f (ε1M ) · f (ε1 ) dε1M dε1 . Notice that when the hedged cash flow is higher than the threshold, the maximum in the integrand sets bankruptcy cost to zero. Theorem 1. If the firm has contractual obligations of B1 , expected cash flows of E0 ( C1 ), sensitivity of cash flows to market returns of β = 1, and the distress cost is proportional to the shortfall in cash flows to the contractual obligations, then the firm minimizes the expected cost of bankruptcy for h0 = φ ≡ B1 . E0 ( C1 ) Proof. See Appendix. 5 We do not posit that the firm minimizes the probability of bankruptcy for two reasons. First, minimizing the probability of bankruptcy introduces a discontinuity when the firm is just at the boundary of bankruptcy. Second, in some states of the world when the firm’s unhedged cash flow C1 < B1 , the firm may have a perverse incentive to have a speculative short position on the market index. 6 Notice that because we allow the hedged cash flow to become negative, we are in effect assuming that the firm has unlimited liability and thus it will honor its obligations on the short market hedge. Thus the pricing of the forward contract that assumed no default is appropriate. Assuming limited liability complicates the analysis - the derivative short position must be priced to account for default and an additional perverse incentive to hold a speculative position. This additional complexity does not lead to any additional insights that have not already been analyzed in the related papers mentioned in the introduction. Therefore we stay with the simpler formulation of unlimited liability. 406 Bhagwan Chowdhry and Eduardo Schwartz This is a simple yet remarkable result. The optimal market hedge (when the firm’s return sensitivity to market factor is one-to-one) is equal to the level of the firm’s contractual obligations as a fraction of expected cash flow. The optimal hedge does not depend on idiosyncratic volatility, market volatility, and exact distribution of returns as long as the shocks have zero means. This means that corporate managers can estimate and implement this hedge relatively easily. 3 Numerical Simulations We now plot firm’s hedged cash flow for various parameter values. We assume log-normal distributions for the idiosyncratic shock ε1 and the systematic shock ε1M with means equal to zero. We vary the bankruptcy threshold φ. Figure 1 shows a case in which the firm’s contractual obligations are 30%, 40%, or 50% of its expected cash flows. The curvature of the functions is based on σ(ε1 ) = σ(ε1M ) = 40%, which are plausible estimates over a four-year horizon. The top panel shows the probabilities of bankruptcy. These are plausibly low. For a small hedge, the probability is rapidly declining. At least a little market-risk hedging is clearly superior. For φ = 0.3, it is fairly flat (very close to zero) from 25% to 50%. For φ = 0.5, it is minimized at 60%. The bottom panel shows the expected cost of bankruptcy as a function of the hedge ratio. Figure 1 confirms that the optimal hedge that minimizes the expected cost of bankruptcy h0 is equal to φ for each of the three values of φ. In general, minimizing the probability of bankruptcy leads to a higher hedge ratio than the hedge that minimizes the expected cost of bankruptcy. This is because minimizing the probability of bankruptcy, in some states of the world when the firm’s unhedged cash flow C1 < B1 , may provide a perverse incentive to have a speculative short position on the market index. This is seen more clearly for φ = 0.5 where the hedge that minimizes the probability of bankruptcy is 60% instead of the optimal 50%. Figure 2 shows the distribution of the hedged cash flow with a 100% hedge and the optimal hedge of 30%. Notice that the firm is bankrupt for values less than −(1 − φ) = −0.70. The optimal hedge creates more values in the upper tail and reduces values in the lower tail. The standard deviation, however, is higher with the optimal hedge (51%) than with a 100% hedge (43%). This confirms that the hedge that minimizes the bankruptcy costs or the probability of bankruptcy is less than the hedge that minimizes the variance. The analytical results in our paper were derived assuming beta was equal to 1. When the beta is different from one, the optimal hedge that minimizes the expected cost of bankruptcy is approximately equal to the product of β and φ. This is seen in Figure 3 which plots firm’s hedged cash flow for beta of 0.5, 1 and 1.5 when φ = 0.5. The optimal hedge ratios are 25%, 50% and 70% respectively. How Should Firms Hedge Market Risk? 407 Panel A: Probability 0.18 0.16 σ(ε) = σ(εM) = 40% Probability of Bankruptcy 0.14 0.12 0.10 0.08 0.06 50% 0.04 =40% 0.02 =30% 0.00 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% Hedge Ratio Panel B: ExpectedCost 200 180 σ(ε) = σ(εM) = 40% Cost of Bankruptcy (x104) 160 140 120 100 80 ? = 50% 60 40 ? = 40% 20 ? = 30% 0 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% Hedge Ratio Figure 1: Probability and Expected Cost of Bankruptcy as a function of the Hedge Ratio Description: The firm’s contractual obligations are 30%, 40%, or 50% of its expected cash flows. The curvature of the functions is based on σ(ε1 ) = σ(ε1M ) = 40%. Interpretation: The optimal hedge that minimizes the expected cost of bankruptcy h0 is equal to φ for each of the three values of φ. Minimizing the probability of bankruptcy leads to a higher hedge ratio than the hedge that minimizes the expected cost of bankruptcy. 408 Bhagwan Chowdhry and Eduardo Schwartz 100% Hedge sd=43% 30% Hedge (Optimal) sd=51% Figure 2: Distribution of Hedged Cash Flow Description: The figure shows the probability distribution with a 100% hedge and with optimal hedge for σ(ε1 ) = σ(ε1M ) = 40% and φ = 0.3. Interpretation: The optimal hedge creates more values in the upper tail and reduces values in the lower tail. 400 350 Cost of Bankruptcy (x104) 300 σ(ε) = σ(εM) = 40% ? = 50% 250 200 β = 0.5 150 100 β = 1.0 50 β = 1.5 0 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% Hedge Ratio Figure 3: Optimal Hedge Ratios for different Betas Interpretation: The optimal hedge that minimizes the expected cost of bankruptcy is approximately equal to the product of β and φ. How Should Firms Hedge Market Risk? 409 To summarize our results from the numerical simulations, we find that (a) when beta is equal to one the optimal hedge that minimizes the cost of financial distress is equal to the fraction of contractual debt obligations to expected cash flows which is often smaller than 100%, a variance minimizing hedge, (b) a hedge that minimizes the probability of bankruptcy is typically higher than the hedge that minimizes the expected cost of bankruptcy, and (c) the optimal hedge when beta is different from one is approximately equal to φ · β. 4 Continuous-Time Hedging Our theoretical analysis proved that because date 1 cash flow C1 = S0 · (1 + R f ) · (1 + ε1M ) · (1 + ε1 ) is affected by the idiosyncratic shock ε1 and the market-related shock ε1M in a multiplicative fashion, a 100% market hedge is not optimal. One might wonder if this result arises because we have imposed a requirement that the hedge be put in place at the beginning of date 0 and have not allowed the hedge to change at more frequent intervals. We now show that allowing the hedge to change dynamically does not alter the conclusion that a 100% hedge is not optimal. Suppose we were to subdivide the period from 0 to 1 into N sub-periods. As N approaches infinity, the approximation turns into continuous time. A 100% hedge at the beginning of sub-period t y t−1 = S t−1 Z t−1 results in a hedge profit at time t of H t = −S t−1 · (1 + r 0 ) · ε M t , where (1 + r 0 ) = (1 + R f )1/N and ε M t is the (much smaller) shock to the market for one sub-period. If H t is invested in the risk free asset until date 1, its future value will be H t · (1 + r 0 )N −t = −S t−1 · (1 + r 0 )N −t+1 · ε M t ? t−1 ? Y 0 t−1 M = −S0 · (1 + r ) · (1 + εi ) · (1 + εi ) · (1 + r 0 )N −t+1 · ε M t i=1 ? t−1 ? Y f M = −S0 · (1 + R ) · (1 + εi ) · (1 + εi ) · ε M t , i=1 410 Bhagwan Chowdhry and Eduardo Schwartz where ε t is the (much smaller) idiosyncratic shock for one sub-period. The total hedged cash flow at date 1 then is ? t−1 ? N X Y f M C1 − S0 · (1 + R ) · (1 + εi ) · (1 + εi ) · ε M (1) t . t=1 Because C1 = S0 · (1 + R f ) · i=1 N Y (1 + ε t ) · (1 + ε M t ), i=1 substituting in the hedged cash flow from equation (1), simplifying, and keeping only terms that have the product of at most two ε terms, the hedged cash flow is equal to ? ? N t X X f M S0 · (1 + R ) · εt εi . t=1 i=1 Therefore, even though each second-order product term is small in the above expression, the number of these terms is of the order of N · (N + 1)/2. Our simulations confirm that the standard deviation of the hedged cash flow is of a similar order of magnitude as the yearly standard deviation of return on the market. We also considered a second scenario in which the cash flow is also generated continuously, but the firm needs to hedge its accumulated cash flow at some date in the future. Even in this case, we confirm that even though the market sensitivity of the firm’s cash flow at any given instant could be perfectly hedged by the market hedge, the fact that the hedge for both near and distant cash flows must be determined in advance at date 0 precludes the possibility of completely eliminating the sensitivity to market movements. 5 Discussion It has been a long standing puzzle in the risk-management literature that firms do not seem to hedge many important risks to their cash flows. The most obvious such risk is the exposure to market conditions. Our paper resolves a part of this puzzle. It shows that the naive suggestion of a full variance-minimizing hedge overstates the optimal hedge ratio. This is because exposure to market risks interacts multiplicatively with other idiosyncratic risks that firms face. If a firm were to take a short position in the market index and if the firm’s realized cash flow were to be low because of its idiosyncratic factors, then it could face much larger net losses if the market turned out to be high. Instead, firms should be rather conservative in hedging their exposure to the market. Our analysis clarifies that a hedge that minimizes the variance of the cash flow is not equivalent to a hedge that minimizes the costs associated with financial How Should Firms Hedge Market Risk? 411 distress. Textbooks often recommend stock return regressions on various risk factors to determine the exposure to risks and then equate this with an optimal hedge ratio to avoid such risks. This approach has several problems: 1. Hedges that minimize cash-flow variance do not minimize the costs of financial distress. Typically, the optimal hedge is smaller. This was the key point of our paper. 2. Stock return dynamics may already anticipate that managers hedge the firms’ risk exposures (and thus a regression coefficient would tend to underestimate the true exposure). 3. The sensitivity of firms’ stock returns to overall market returns may arise not because cash flows are particularly sensitive to market movements, but because discount rates have a large common component. This would make a simple regression of stock returns on market returns indicate significant sensitivity, but an attempt to hedge cash flows by shorting the market would turn out to be misguided.7 Our arguments also shed some light on discussions in the risk management literature in which it is argued that firms should attempt to hedge their total economic exposure8 rather than focusing only on transaction exposure. Our analysis suggests that economic exposures are likely to be multiplicative and identification of these exposures using regression methods, as is often advocated, and then determining optimal hedges based on regression coefficients is likely to lead to incorrect results. Survey evidence presented in Bodnar et al. (2011) indicates that the most common risks that are managed using financial instruments are interest rate risk, foreign exchange risk, energy price risk, commodity price risk and credit risk. The evidence also suggests that foreign exchange risk that is managed arises largely from transaction exposures caused by contractual commitments. We suspect that interest rate, energy price, commodity and credit risks also arise largely from known transaction exposure whose cash flow value is known in advance and therefore hedging them using financial instruments is straightforward. Although market risk is considered to be the most important concern for firms surveyed, markets risks are rarely hedged using financial instruments. Finally, our analysis offers a more conservative perspective to the suggestions that people hedge too little and should use the financial markets to hedge many different types of risks, such as risks of housing price declines and unemployment risks (see Shiller, 2004).9 The problem with these recommendations is that 7 We thank René Stulz for discussing this insight with us. Total economic exposure includes transaction exposure, which is caused by contracts denominated in foreign currencies, and competitive operating exposure induced by relative price changes caused by exchange rate movements. See Shapiro (2009). 9 We thank Jeremy Stein for discussing these ideas with us. 8 412 Bhagwan Chowdhry and Eduardo Schwartz they do not appreciate that risks that affect people’s lives arise interactively and multiplicatively. Simple financial instruments that hedge these risks may in fact leave people more vulnerable, if they then have to fund a large cash outflow precisely when their own idiosyncratic capacity to pay has diminished, too. Instead, individuals should likely hedge only very modestly.10 6 Conclusion Hedging market risk depends crucially on why the firm wants to hedge. If the goal of the firm is to minimize the variance of its cash flows, perhaps because the owner-manager is risk averse, then fully hedging market risk is appropriate. However, if the goal is to minimize the costs associated with financial distress, then more moderate hedging of market risk is prudent. A key determinant of how much market risk should be hedged is the level of a firm’s contractual obligations that may trigger financial distress. For instance, a firm with only 25% debt in its capital structure (a typical U.S. manufacturing firm) should hedge market risk roughly half as much as a firm that has 50% debt in its capital structure (for instance, an airline company). Of course, the amount of market risk hedging also depends on the sensitivity of a firm’s cash flows to overall market conditions. For example firms in industries such as automobiles, retail, telecommunications, tend to have higher cash flow sensitivity to the market conditions, and therefore should hedge more than firms in industries such as food, tobacco, oil and gas, which have lower cash flow sensitivity to the market conditions. Our paper has suggested a simple rule for hedging market risk when the goal is to avoid financial distress. First, a firm should estimate the sensitivity of its cash flows to market movements. Second, the firm should estimate its fixed obligations as a proportion of expected cash flows. Then, its optimal market hedge as a proportion of expected cash flows is the product of these two estimates. Although our analysis has suggested that there are potentially large gains to the first dollar hedged, the optimal hedge is likely to be far more conservative than the more naïve full-variance hedge. We can help explain why firms do not fully hedge their market exposure, although it remains a puzzle why most firms do not hedge their market exposure, at all. 10 If one could write contracts that are simultaneously contingent on several risk factors, significant risk reductions may be possible. However, the feasibility of such complex instruments is doubtful. Not only would this require an accurate quantification of risk exposures, which are likely to be different for each individual, but also the instruments’ contingencies would have to involve variables that can be easily measured and cannot be manipulated. The possibility of misusing financial instruments to speculate rather than hedge, for personal profit at the risk of putting the organization in peril, and thus the costs of instituting internal controls and systems that can minimize or prevent such abuse, make the case for hedging with financial instruments even more tenuous. How Should Firms Hedge Market Risk? A 413 Appendix: Proof of Theorem 1 The optimization problem is Z ∞Z ∞ max − (1 − φ) − {ε1 + (ε1 + 1 − h0 ) · ε1M }, 0 Γ ≡ min h0 −1 −1 × f (ε1M ) · f (ε1 ) dε1M dε1 . The maximum function inside the integral is convex and the optimization problem is strictly convex and has a unique minimum, and therefore a local minimum is also the global minimum. We can rewrite the optimization problem as ? Γ ≡ min [−(1 − φ) − {ε1 + (ε1 + 1 − h0 ) · ε1M }] · f ( ε1M ) · f (ε1 ) dε1M dε1 , h0 Σ( h0 ) where Σ( h0 ) is the region of integration. This region for φ ≤ h0 < 1 is ε1 ≤ −(1 − h0 ) and ε1M ≥ (1 − φ) + ε1 . ε1 + 1 − h0 For 0 < h0 < φ, ε1 ≥ −(1 − h0 ) ⇒ ε1M ≥ [(1 − φ) + ε1 ]/(ε1 + 1 − h0 ) ε1 ≤ −(1 − h0 ) ⇒ ε1M ≥ −1. The Leibniz-Reynolds theorem implies that the derivative with respect to h0 is ? ∂ · −(1 − φ) − {ε1 + (ε1 + 1 − h0 ) · ε1M } · f (ε1M ) · f (ε1 ) dε1M dε1 , ∂ h0 Σ(h0 ) because [−(1 − φ) − {ε1 + (ε1 + 1 − h0 ) · ε1M }] along the boundary of the region of integration is zero. For φ ≤ h0 < 1 the integral above is Z Z ε1 ≤−(1−h0 ) 1−φ+ε1 1 +1−h0 ε1M ≥ ε ε1M f (ε1 ) · f (ε1M ) dε1M dε1 . This derivative is strictly positive except at φ = h0 where it is 0. This is because ε1M has a mean of zero and ∂ ε1M ∂ ε1 = h0 − φ >0 (ε1 + 1 − h0 )2 for h0 > φ. 414 Bhagwan Chowdhry and Eduardo Schwartz For 0 < h0 < φ, the integral is Z Z ε1 ≤−(1−h0 ) + ε1M ≥−1 Z ε1M · f (ε1 ) · f (ε1M ) dε1M dε1 Z ε1 ≥−(1−h0 ) 1−φ+ε ε1M ≤ ε +1−h1 1 0 ε1M f (ε1 ) · f (ε1M ) dε1M dε1 . The first term is zero. 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