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Hedge funds face many limits on leverage they can employLIMITS ON HEDGE FUND LEVERAGE Margin requirements limit the amount of leverage hedge funds can create with futures, options, or margin loans. Initial Margin Hedge funds face many limits on leverage they can employ. Most investors know Regulation T or at least know something about margin that is required by Reg T. To read the regulation, see www.access.gpo.gov/nara/cfr /waisidx_03/12cfr220_03.html. A series of regulations was promulgated by the Federal Reserve Bank. Reg T limits lending by a broker or dealer to customers secured by securities. Regulation U extends to provisions to banks and other lenders; to read the regulation, see www.access.gpo.gov/nara /cfr/waisidx_03/12cfr221_03.html. Regulation X extends the provisions to U.S. entities that seek such loans from financial institutions exempt from Reg U and Reg T; to read the regulation, see www.access.gpo.gov/nara/cfr /waisidx_03/12cfr224_03.html. The regulations impose initial margin requirements on these securities loans. Margin is the part of the security value that is funded by capital (i.e., the value of the position minus the debt extended). The portion that may not be financed with borrowings is formally one of the tools the Federal Reserve Bank may use to implement monetary policy. In practice, the margin amounts have not changed for decades, as the central bank has focused on other policy tools. Different initial margin requirements apply to different assets. A margin percentage of 50 percent applies to most exchangetraded common stock. Smaller initial margin requirements exist for fixed income securities. Certain types of assets (options owned and nonmarginable stock) may not be used as collateral. For example, a hedge fund with $1 million could buy no more than $2 million of common stock to comply with a 50 percent initial margin requirement. It is important to note that once this initial margin requirement is met the requirement does not apply to subsequent market values. The hedge fund would not be in violation of the initial margin requirement even if the value of the position dropped to $1 million and the value of the securities just matched the loan amount. As noted in the next subsection, other limitations would require the hedge fund to post additional margin if this happened, but Reg T (and Reg U and Reg X) would not require adjustments. Maintenance Margin The Federal Reserve imposes only initial margin requirements. However, the major stock exchanges require their members to hold minimum maintenance margin (the minimum margin required based on updated values). Maintenance margin is recalculated frequently to account for the updated market value of the positions securing the lending. These requirements establish only the lowest margin that members must require. Brokerdealers are free to require maintenance margin in excess of exchange minima. If the broker carrying the $2 million in common stocks required maintenance margin of 35 percent, the fund would get a margin call if the value of the positions fell below $1,538,461.54. The remaining value of the margin is $538,461.54 after subtracting the $1 million loan balance from the total value. This margin exactly equals 35 percent of the position. If the value of the position fell below $1,538,461.54, the hedge fund would need to either post additional margin in cash or sell part of the position until the maintenance requirement is met. If the fund failed to meet this margin maintenance voluntarily, the broker would sell positions to satisfy the requirement. More generally, suppose the margin maintenance requirement was Maintain%. The current loan amount is Loan, and the minimum margin for that loan amount is Margin. The maintenance margin imposes a limitation on the amount of leverage available to a hedge fund if the hedge fund holds assets subject to margin requirements and if the hedge fund must observe the requirements (many offshore hedge funds bypass margin requirements by financing positions with dealers or subsidiaries of dealers located outside the United States). Maximum leverage is equal to the total value of the positions that can be carried divided by the margin required to carry those positions. The margin maintenance percentage can be readily converted into the leverage possible under that margin requirement Of course, funds generally run leverage below this amount to avoid margin calls for routine changes in market value of the positions carried. Equity Option Margin Exchanges collect margin for equity options. Buyers of puts and calls on individual stocks must fully pay for all options and cannot use the value of the options as collateral on a loan. Short option margin is calculated two ways and the seller is charged the higher of the two amounts. First, the proceeds of the sale plus 20 percent of the underlying stock price are reduced by the amount (if any) that the option is out of the money. Second, the proceeds of the sale are combined with 10 percent of the share price. Margin is somewhat lower for index options than the margin for individual options. Options offer a way to participate in the price of a stock but options, generally move less than the underlying asset. The delta of an option measures the sensitivity of the option price relative to the underlying common stock. An option with a delta of .5 moves only half as fast as the stock. Accounting for this delta, options can create leverage of about 2:1 relative to an equally responsive position in the underlying stock. The amount of leverage is also subject to change, depending on the level of the stock relative to the strike, the time to expiration, and other factors.2 Futures Margin Futures exchanges also impose margin requirements on hedge funds. Futures margin differs somewhat from margining of stocks and bonds. When a new trade is created, both the buyer and the seller must post initial margin. This amount is a goodfaith deposit and makes it possible for a third party, a clearing corporation, to guarantee performance to both the buyer and seller. The initial margin is set by the exchanges and varies from time to time. The exchange may raise initial margin requirements when a futures contract becomes volatile. This margin may be satisfied by depositing cash but the exchange pays no interest on the balances. Most futures traders deposit shortterm Treasury bills, which are also accepted as initial margin and earn a return. Futures exchanges also impose a maintenance margin on both the buyer and seller. Futures are revalued daily and customers must post cash if the margin value falls below the maintenance level. The maintenance margin must be in cash because much of this money will be deposited in the accounts of traders who are making money, and those traders have the right to withdraw the cash immediately. Because futures margin is considerably less than the value of the assets underlying the futures contract, the hedge fund creates leverage when the fund buys or sells a future. For example, the S&P futures contract (based on the Standard & Poors 500 index) is valued at 250 times the level of the index. If the S&P index is 1,120, the futures contract represents $280,000 worth of stock. This amount is 77 larger than the initial margin of $3,625. Although this future creates considerable leverage, it does demonstrate that futures margins place limits on hedge fund leverage. SPAN Margin Futures margin has historically been calculated individually for various futures positions (and options on futures). Futures exchanges allow brokers to collect lower margin for positions with less risk than an outright long or short position. As the variety of futures products has risen, exchanges have adopted a more comprehensive way to allow for the combined riskiness of a futures portfolio. The margin method is called standardized portfolio analysis of risk (SPAN). SPAN margin equals the largest likely loss on the entire position for a oneday horizon. For hedge strategies, this can be substantially less than the margin calculated in the traditional way. Brokers may require more margin that the minimum SPAN margin. See the futures exchanges for more information about how their SPAN margin is calculated. IMPACT OF LEVERAGE ON RISK OF HEDGE FUND PORTFOLIOS Leverage may increase the risk of a portfolio compared to a longonly unleveraged portfolio of assets. A leveraged long position can lose more than 100 percent of a hedge funds capital, while an unleveraged long position can lose only 100 percent. Similarly, any short position can appear to be more risky than a long position even if neither position is larger than the capital base of the hedge fund because short positions can lose more than 100 percent of capital. There is no absolute limit on the losses because an asset can double (losing 100 percent of capital), triple (losing 200 percent of capital), or more. Hedge funds are structured so that individual investors cannot lose more than 100 percent of their capital, so potential losses on the portfolio that exceed 100 percent dont necessarily affect hedge fund investors. However, the use of leverage may increase the probability of a loss of a certain magnitude. Investors are concerned about the impact leverage has on the probability of these losses (of all sizes). Creditors, in turn, are interested in the probability of losing more than the hedge funds capital, so creditors monitor hedge fund leverage. Risk in Unlevered Portfolio The normal distribution provides a convenient way to study the impact of leverage on hedge fund risk. The standard bell curve is actually a probability curve and displays the probability of a range of outcomes. In Figure 6.1, the normal distribution is displayed for an asset having an expected return of 15 percent and a standard deviation of return equal to 18 percent. The area to the left of 0 percent on the xaxis shows the times when this asset produces a loss. In this case, the investment would lose money 20.2 percent of the time.3 Risk in Leveraged , Unhedged Portfolio A hedge fund that borrows money to buy more of the same asset can increase the expected return of the fund if the expected return of the asset is higher than the borrowing rate. This kind of levered trading can increase risk. Risk is often described as the standard deviation of return. Figure 6.2 shows the impact of leverage when the borrowed money is used to buy more of the asset held in the fund. The leftmost point on the line is an unleveraged position. A portfolio containing this asset has a standard deviation of return equal to 18 percent. As the fund borrows money to buy more of the same asset, leverage rises from 1:1 to 2:1. The standard deviation of return rises proportionally to 36 percent. The expected return on the portfolio is just the weighted average of the returns of the assets in the portfolio, adjusted for borrowing costs if the portfolio is leveraged. Note that the weights need not sum to 1 (or 100 percent of the portfolio) and will exceed 1 for a levered portfolio. Financing applies only to the portion that exceeds the capital. It is possible to derive the portfolio standard deviation of return using standard statistical formulas found in many statistics books. The impact of this leverage can be seen in Figure 6.1. The expected return rises to 25 percent (15 percent on the unleveraged part of the portfolio and 15 percent  5 percent borrowing rate on the leveraged portion). Notice that the higher expected return means (all other things being equal) that a loss is less likely. Of course, things are not equal, but the doubling of portfolio risk as defined by the standard deviation of return leads to a probability of loss equal to 24.4 percent.4 Under this particular set of assumptions, leverage significantly increases the expected return but increases the chance of losing money less significantly. The reader should not conclude that risk of loss is a superior measure of risk than the standard deviation. Hedge fund investors measure risk in a variety of ways. Also, the chance of losing greater amounts (say 25 percent or 50 percent) may be higher in the levered portfolio. Without leverage, the portfolio cannot lose more than 100 percent of the capital, but the levered portfolio in Figure 6.1 can lose up to twice the capital (and perhaps a little bit more because of financing costs and transactions costs). In going from an unlevered to a levered portfolio, a hedge fund may use the borrowed funds to buy a different asset. If the assets added to the unlevered portfolio provide diversification to the portfolio, the increase in risk caused by the leverage is mitigated by the riskreducing impact of diversification. Correlation measures the degree to which two assets move together. Suppose a second asset was bought that was only 40 percent correlated with the unleveraged portfolio. That is, for every dollar in capital, the fund buys $1 of one asset and $1 of a second asset that is correlated 40 percent to the first (leverage 2:1). Although the portfolio contains as much leverage as the examples shown in Figures 6.1 and 6.2, the diversification reduces the standard deviation of the portfolio to 30.12 percent, down from 36 percent. As can be seen in Figure 6.3, the portfolio has about the same probability of loss (20.3 percent)5 as an unleveraged portfolio (20.2 percent). It is clear, however, that the chance of large losses is higher on the levered portfolio. In return, this levered portfolio also has a considerably higher probability of large gains. Risk in a Levered , Hedged Portfolio Hedge funds commonly construct portfolios so that the risk of the long positions is significantly mitigated by the risk of the short positions. (This kind of position is called a hedge or arbitrage.) Two nearly identical positions behave almost exactly opposite each other if one position is held long and the other is held short. When analyzing the returns of the assets, these are generally described as being highly correlated (near 100 percent). The assets in Figure 6.4 are constructed to be equal in value and 90 percent correlated. However, the first asset is held long and the second asset is short. The combined portfolio has an expected return of 6 percent (in part by design because the expected return on the long is assumed to be 15 percent and the expected return on the asset held short is 14 percent).6 The standard deviation of the combined returns is 8.05 percent. While the standard deviation is much lower than the unhedged portfolio, the risk of loss is 22.8 percent,7 higher than the unhedged portfolio. Nevertheless, the probability of a large loss is much smaller for the levered, hedged portfolio than for the outright long, unlevered, and unhedged portfolio. Because it is possible to remove much of the aggregate market risk from a portfolio, it is not accurate to presume that more leverage leads to more risk. In fact, the least volatile hedge funds are the funds following arbitrage strategies. These funds also have the highest leverage of all hedge fund investment styles. A highly levered, wellhedged portfolio has other risks that are not easy to measure with standard measures of risk. These risks, including the risk of losing financing capacity, are discussed in Chapter 11, which deals with risk management. CONCLUSIONS Hedge funds use leverage to construct portfolios that differ significantly from conventional long, unlevered portfolios. While leverage often creates a riskier portfolio than the assets that make up the portfolio, there is no simple relationship between leverage and risk. Investors and creditors need to understand the nature of the positions to understand the impact of the leverage on the risk in a hedge fund portfolio. QUESTIONS AND PROBLEMS A hedge fund has $40 million in common stock long and $50 million in common stock short. It has stock loan agreements of $60 million as assets and $30 million as liabilities. The fund has $40 million is capital (including both limited partner and general partner capital). Answer questions 6.1 and 6.2 about this hedge funds positions: 6.1How do the stock loan amounts relate to the stock positions held asassets and liabilities?6.2What is the leverage in this hedge fund?6.3A hedge fund is reviewing a stock or bond. It appears that the asset should have a return equal to 3 percent. The fund can borrow or lend the security at the rate of 5 percent. Why might a hedge fund use leverage with this instrument? 6.4 One fund holds assets equal to its capital and is using no margin or other techniques to create leverage. A second fund has no long positions but carries short positions in a margin account equal to its capital. Which fund has higher leverage? 6.5 A fund has assets equal to $100 million, including $25 million com mon stock carried in a cash margin account. The fund has $50 million in capital. What is the leverage on the fund? 6.6 Another fund has assets equal to $100 million, including $2 mil lion in margin at futures exchanges. The fund has long positions in futures contracts representing an additional $50 million in assets. The fund has $50 million in capital. What is the leverage of this fund? 6.7 During the day, an entity carries up to $20 million in securities. All positions are liquidated by the close of day each day. The entity has $5 million in capital. What are the limits on leverage for this company? 6.8 A hedge fund has positions (long and short) in stocks, index futures, commodity futures, and currencies. The funds prime broker finances all the cash positions and carries all the futures positions. To what extent can SPAN margin rules reduce the margin required for the positions? 6.9How can a hedge fund exceed the limit of 50 percent margin requiredon cash stock positions? 6.10 Your hedge fund has long positions in U.S. Treasury securities totaling $50 million and short positions worth $35 million. You pay an average repo rate of 4.5 percent to finance your longs and receive an average reverse repo rate of 4 percent on your cash collateral covering your short positions. You post haircuts averaging 0.25 percent on the long positions and 0.5 percent on the short positions. What amount of capital is tied up in haircuts? 6.11How levered is the Treasury part of the portfolio in question 6.12 A hedge fund has $100 million in capital levered approximately 20:1. It maintains a position of long notes and bonds roughly equal to its position in short bonds. The financing rate on the long positions averages about 0.5 percent (50 basis points) higher than the rate on the short positions. How much does the fund need to make trading (annually) to break even after financing costs? 6.13 You borrow 25,000 shares of XYZ common and deliver the shares to satisfy a short sale. While you are carrying the short, the stock pays a dividend of $1 per share. Who receives the dividend? 6.14 You borrow 25,000 shares of XYZ common and deliver the shares to satisfy a short sale. While you are carrying the short, the stock splits 2:1. How does this affect the stock loan transaction? 6.15 You borrow 25,000 shares of XYZ common and deliver the shares to satisfy a short sale. While you are carrying the short, a proxy fight develops over control of the company. How do you restore the vote to the lender of the shares? .16 What is the tax treatment of a Treasury coupon or stock dividend received as a replacement payment from a securities borrower? 6.17 Why might it be reasonable to allow a hedge fund greater leverage for risky positions held as outright futures (long or short) than for levered positions in the underlying cash instruments? 6.18 Why would the U.S. Federal Reserve Bank want to limit the amount of leverage possible on securities loans? 6.19 A hedge fund has $10 million in marginal positions and has loans totaling $8 million. The fund is subject to maintenance margin of at least 30 percent. How much new cash would the hedge fund need to deposit to satisfy a margin call? 6.20If the hedge fund in question 6.19 chose to liquidate positions ratherthan deposit additional margin, how much would need to be liquidated? 6.21 A hedge fund buys a sixmonth call on a common stock at a price of $5. The strike of the option is $102. The current price of the stock is $100. The shortterm rate of interest is 5 percent. The current value of the call is $5.25. How much margin must the hedge fund put up to carry the call? .22 What margin would the hedge fund need to post if it wrote (sold) an option like that described in question 6.21? Assume the sale is a naked short sale (i.e., you have no position in the stock). 6.23 What is the standard deviation of the return on a hedge fund portfolio that is 100 percent invested in stock A and also carries an equal amount of stock B on leverage. Stock A has a standard deviation of return equal to 22 percent. Stock B has a standard deviation of return equal to 25 percent. The two stocks have a correlation of 75 percent. 6.24 What is the expected return on the leveraged portfolio in Question 6.23 if the expected return on each stock is 20 percent (unleveraged) and the riskfree rate is 5 percent? 6.25What is the probability of loss for the portfolio in questions 6.23 and6.24?6.26What is the probability of losing 25 percent or more for the portfolioin questions 6.23 and 6.24? 




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