*This article was originally published in The Option Strategist Newsletter Volume 1, No. 12 on June 11, 1992. *

With myriad investment advisors and the media trumpeting the fact that the market is overvalued, and with scary comparisons to the summer of 1987 abounding, an owner of stocks might justifiably be concerned with how he can safeguard his portfolio. He may not want to sell out his portfolio and go into an all cash position, but he would like to have some "insurance" in case the market takes a nosedive. Most investors in today's markets are familiar with the fact that index futures or index options can be used to protect one's portfolio. However, few know exactly how to adequately and correctly protect their portfolio of stocks. In this week's feature article, we'll describe the way in which one can compute the number of futures or options that would be needed to properly hedge his portfolio.

In addition, the advantages and disadvantages of different hedges will be demonstrated.

If one owned a portfolio of stocks that were the exact makeup of the S&P 500 or the S&P 100 (OEX), then he could easily compute the number of options or futures that would be required to hedge his position. However, no individual investors and few institutional investors are in this position. Rather, one usually has a portfolio of stocks that bear little resemblance to the indices themselves. However, he is forced to use an option or future that is based on one of those indices in order to hedge his portfolio. How does he equate the two? He computes how his portfolio relates to the overall stock market.

The correct way of computing the portfolio's correlation to the stock market is to use the Beta of each stock. Beta is a statistical measure of how the stock moves in relationship to the overall market. Simply stated, if a stock has a beta of 2.0, then it will rise or fall twice as fast as the overall stock market. Many options traders don't have ready access to Beta, however, but they do have access to implied or historical volatility. This can be used as a substitute for Beta in many cases: just divide the stock's volatility by the market's volatility.

Suppose one owns a portfolio that consists of only three stocks: a volatile biotech stock, a stodgy utility, and an industrial company. These are three diverse holdings whose combination may not relate directly to the OEX Index or the S&P 500 Index. As long as one knows each stock's Beta, he can calculate the Adjusted Capitalization of his portfolio by multiplying the quantity of each stock owned by it's closing price and Beta.

*Example:* The following table shows the closing price, quantity owned, and Beta for each of the three fictional stocks. The Adjusted Capitalization is the product of the three and is the right-hand column.

Stock Name Price Quantity Value Beta Capitalization Biotech Inc. 6,000 25 $150,000 4.0 $600,000 Steel Corp. 9,000 50 450,000 2.0 900,000 Elec. & Power 10,000 40 400,000 0.8 320,000 $1,000,000 $1,820,000

The Adjusted Capitalization of this portfolio is $1,820,000. Note that the actual worth of this portfolio is $1,000,000. But after adjusting for volatility, it is clear that the portfolio is more volatile than the stock market in general. In fact, the portfolio behaves more like $1.8 million dollars worth of the "stock market". Stated in another way, this portfolio is about 1.8 times as volatile as the stock market.

This illustrates the first and most important step in evaluating how to hedge one's portfolio: adjusting the portfolio for volatility so one can see how it compares to the market. Once this step has been accomplished, it is merely a simple matter of arithmetic to define the available alternatives. The investor can then chose the one that suits him best.

**Selling Index Futures Against The Portfolio: The Complete Hedge**

If one sells futures against his portfolio, he will be removing nearly all of the risk as well as the reward potential of his stock holdings. This is a relatively extreme measure, but it might be used by someone who does not want to liquidate his stocks because of the taxes he might owe; such stocks might be long-term holdings with a greatly reduced cost basis. In this case, one could sell futures against the portfolio and virtually remove all risk and reward potential. The only profits or losses that would occur would be those generated by the fact that the portfolio does not exactly track the market index underlying the futures.

Example: The owner of the above portfolio of stocks is very concerned about the overall stock market and would like to get out. However, he does not want to sell the stocks because of the taxes that would be due on the long-term gains. Rather, he decides that he will sell S&P 500 futures against the portfolio. The proper quantity of futures to sell is determined by dividing the Adjusted Capitalization of the portfolio by the value of one futures contract. Since the S&P futures are worth $500 per point, the value of the futures contract is $500 times the price of the contract.

S&P 500 September futures contract: 412.00 Value of the Futures Contract at $500 per point: 412.00 × $500 = $206,000 Quantity of futures to sell Adjusted Capitalization of Portfolio to hedge this portfolio = Value of 1 futures contract = $1,820,000 = 8.83 contracts 206,000

Thus one would sell 8 or 9 September S&P 500 futures contracts to completely hedge this theoretical portfolio. Once the sale was made, there would be very little profit potential or risk potential remaining.

This investor would have achieved a nearly full hedge of his portfolio without incurring any tax liability on his stocks, and also at a greatly reduced commission expense (the commission for selling 8 or 9 futures contracts is assuredly less than that for selling 25,000 shares of stock). At a later time, if his opinion of the overall market become more sanguine, he could cover the futures contracts that he sold, thereby restoring his holdings to their original state.

**Insurance: Buying Index Puts**

This technique of selling futures against the entire portfolio is a little extreme for many investors. Rather, most would like to have some insurance that their stock holdings won't be devastated by a steep or sharp market decline, but they do not want to completely eliminate the chance for further rewards, either. For this type of investor, the purchase of index puts is warranted for they act as an insurance policy against disaster. Since there are many choices of striking prices and expiration dates possible for the puts used in the hedge, the investor will have to make a decision regarding 1) how much he wants to spend on insurance and 2) how much price protection he needs.

*Example:* First, let's look at one example of how this same stockholder might use a particular OEX put option to hedge his portfolio. The number of puts needed is determined by dividing the Adjusted Capitalization of the portfolio by the dollar value of the striking price of the put. The dollar value of the striking price is easily determined by multiplying the striking price times the value of a one point move. For OEX options, the value of a one point move is $100 (for S&P 500 futuresoptions, it would be $500).

OEX Current Price: 390.00 OEX Aug 370 Put Price: 4 Value of the OEX 370 strike = $100 × 370 = $37,000 Quantity of Aug 370 puts to buy Adjusted Capitalization of Portfolio to insure this portfolio = Value of 370 strike = $1,820,000 = 49.19 puts 37,000

If the stockholder decided to purchase these OEX Aug 370 puts as insurance, he would need to buy 49 or 50 of them to provide insurance. The cost of 50 of these puts would be $20,000 at current prices. He can view this cost as an insurance premium. The insurance does not fully "kick in", however, until OEX is below 370 -- a 5% decline from current prices. Thus, his overall "insurance policy" could be stated something like this: to insure this portfolio for two months against a general market decline of greater than 5% costs $20,000.

The above case is one example of using puts to insure a portfolio, in which the striking price of the puts is relatively close to the current market price -- only about 5% below. This type of insurance is going to be more expensive than would be insurance using puts that are farther out-of-the-money. For example, to insure the portfolio with the Aug 360 puts, one would have to buy 51 of them at $225 each, for a total cost of $11,475, but this cheaper insurance policy would not offer full protection until the OEX Index had fallen below 360, a drop of closer to 8%. As with any insurance, it costs less to insure against a more remote occurrence of loss.

There is no lower strike than 360 for OEX August puts. However, if one goes out farther in time -- perhaps even using the Index LEAPS puts -- he could buy insurance that is much farther out-of-the-money. The following table shows the results of the previous example as well as some other choices of protection that this theoretical stockholder could use (in this table, S&P refers to the puts on the futures contracts):

Option Percent Cost of Cost as Annual Length of Out-of-money Insurance Pct of Holdings Coverage OEX Aug 370 5% $20,000 7% 2 months OEX Aug 360 8% 11,475 4% 2 months OEX Sep 360 8% 20,400 4% 3 months S&P Dec 350 15% 20,800 2% 6 months S&P Mar 360 13% 34,400 3% 9 months OAX Dec('93) 32½ 17% 70,000 3% 1½ years OAX Dec('93) 37½ 4% 127,000 5% 1½ years

The OAX Index is merely the OEX Index divided by 10. Smaller investors would probably want to use OAX puts as insurance since they can be bought in smaller dollar amounts than their OEX or S&P counterparts. In general, the farther out-of-the-money one wants the insurance to be, the less will be his annual cost. Similarly, buying longer-term insurance is usually less expensive on an annualized basis. There are both benefits and detriments to the various time periods. If one buys insurance for a long period of time, the market may move much higher, thereby rendering his insurance policy so far out-of-the-money that it is virtually worthless well before its expiration. On the other hand, if one buys very short-term insurance, expecting to re-purchase it in two months or so, he may find that its cost has increased quite a bit if the market drops in the meantime, thereby making the striking price much closer to the then-current market price.

If the stock market begins to decline immediately, any puts purchased as insurance would surely increase in price, thereby providing some protection even while the puts were out-of-the-money. The exact amount of that instant protection can be calculated by using the delta of the put options. However, it is usually the case that one computes his insurance needs as if they begin at the striking price of the put, since that is where he will have "full" coverage at expiration. Finally, note that if one buys OEX puts as insurance, his portfolio will not perform exactly like the OEX, so the insurance will not necessarily cover 100% of his losses below the striking price. However, it should cover a substantial portion of them. The more diversified that his portfolio is, the better the insurance will be from using a broad-based index such as the OEX or S&P.

*This article was originally published in The Option Strategist Newsletter Volume 1, No. 12 on June 11, 1992. *

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