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Defining and Evaluating Portfolio Insurance StrategiesAuthor(s): Harold Bierman, Jr.Source: Financial Analysts Journal, Vol. 44, No. 3 (May - Jun., 1988), pp. 85-87Published by: CFA InstituteStable URL: http://www.jstor.org/stable/4479116 .

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Defining and Evaluating Portfolio Insurance Strategies

by Harold Bierman, Jr., The Nicholas Noyes Professor of Business Administration, Cornell University

The word "insurance" implies that it is possible to buy a "policy" that will reduce the probability of loss to zero. In portfolio management, the term "portfolio insurance" has been applied to a wide range of techniques that modify risk but are not insurance in the same sense as defined in the first sentence. This note defines and evaluates three strategies that have been described as being portfolio insurance.

To understand the underlying theory of portfolio insurance, consider Figure A, which shows a conven- tional Capital Market Line and Efficient Frontier. Point A defines a zero-risk portfolio consisting of risk- free securities. Point B is a mixture of risk-free and risky securities (common stocks). Point C represents a portfolio of only common stock. Portfolio insurance strategies build on the concepts of Figure A.

Strategy One Consider the portfolio strategy represented by

points A and B. A risk-free security's return is per- fectly predictable. Assume a $1,000 risk-free (more exactly, default-free) security maturing in 10 years can be purchased for $385.54 (a 10 per cent yield to maturity). Assume the investor has $1,000 to invest and needs to be assured that the portfolio will be worth $1,000 in 10 years.

The portfolio manager can invest $385.54 in the risk-free security (investment A) and $614.46 in secu- rities with risk. This is analogous to point B of Figure A, but the magnitude of the risk-free component has been defined by the desire to have $1,000 in 10 years (time 10) with certainty (probability one). The inves- tor cannot lose (if we define loss as not having $1,000 at time 10). There is an opportunity cost because $1,000 could grow to be $2,593.74 at time 10, given 100 per cent investment in the risk-free security.

This type of portfolio insurance (Strategy One) consists of investing initially in a risk-free security that will provide the guaranteed amount. The re- mainder of the initial investment can be placed in risky securities. The problem with this strategy, rep- resented by point B, is that many alternative strate- gies offer a higher expected return.

Strategy Two A second strategy invests a larger percentage of the

initial amount in common stock; it thus offers a higher expected return than Strategy One (i.e., the portfolio is to the right of point B). In fact, if one believed in the strategy completely, one would invest the entire initial amount in common stock (point C). If stock prices go up the investor wins. If stock prices go down, the insurance strategy dictates that stock index futures be sold. Now if the market goes down further, the investor wins on the futures and loses in the stock portfolio. Portfolio insurance has been pur- chased.

The first problem is determining when and what

Figure A Capital Market Line and Efficient Frontier

Expected Rate of Return | Capital Market Line

Efficient Frontier * (without risk-free asset)

Risk-Free Rate A

O Standard Deviation

FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1988 0 85

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amount of futures should be purchased. Because there are transaction costs and because upside poten- tial is reduced (the investor loses on the futures if stock prices increase), the purchase of futures is limited to a level that supplies adequate but not complete protection.

Now consider what can happen (and what may have happened during the week of October 19, 1987). Stock prices go down because of a variety of reasons (e.g., higher interest rates, increased trade deficit, unpopular tax proposals), and portfolio insurers sell massive amounts of stock index futures, driving the price of the futures down. Arbitrageurs buy the cheap futures and sell stocks, further driving down stock prices. Portfolio insurers then sell more stock index futures, and the cycle is repeated. As a result, the portfolio is much less well protected from loss than was originally thought.

At any time, an insurer can sell a large enough amount of futures so that the value of the portfolio is increased by a stock price decrease. But then the portfolio would be harmed by a stock price increase. Also, the transaction costs and the loss in expected profits from stock price increases would be large. The amount of protection that should be purchased de- pends on the risk preferences of the investor.

Strategy Two has problems because it requires the selling of a large amount of futures, and this action triggers further stock price decreases. The third port- folio insurane strategy avoids the use of stock index futures.

Strategy Three Strategy Three was recommended in a Goldman

Sachs research report authored by Fischer Black and Robert Jones. The objective of the Black-Jones insur- ance technique is "to meet the needs of pension funds that do not want the value of their pension fund assets to fall below the floor defined by the present value of their liabilities."'

The investor defines a floor below which the port- folio is not to be allowed to fall. The difference between the amount to be invested and the floor is defined by Black-Jones to be the "cushion" (c). The initial exposure (e) is the amount invested in common stock and is defined by the portfolio manager. The ratio of the initial stock exposure and the initial cushion defines the multiple (m). We now have:

e e = mc or m =-

c

using Black-Jones symbols and results. Using Black and Jones' primary example, the initial

investment is $100 and the floor is defined to be $80. The initial stock exposure is $50 and the initial cush- ion is $100 minus $80, or $20, so the multiplier is:

m = 20= 2.5.

Fifty dollars is invested in stock and $50 in Treasury bills. (So far, this is analogous to point B of Figure A.) Assume the stock price index is initially 100.

While Black and Jones assume the floor grows through time because of interest, it will be assumed here that common stock price changes occur with very little lapse of time; in effect, the floor and the value of the Treasury bills do not change. We can then focus on the effect of changes in common stock prices. Assume the stock price index increases to 120. The value of the stock component is $60 ($50 X 1.2). The total value of the portfolio is now $110, and the cushion is $30 ($110-80). The cushion of $30 times the 2.5 multiplier gives the new exposure (stock portfolio) of $75. The amount invested in Treasury bills is $35 ($110-75). Because $60 was invested in stock, an additional $15 of stock must be purchased.

Note that the Black-Jones decision process results in buying more stock when the price index goes up. If stock prices were to go up even more, the $15 additional shift to stock would be better than staying with the original portfolio, but not as good as if the entire initial portfolio had been invested in stock.

If, after the index had gone to 120, it returned to 100, the $75 of common stock investment would have a value of $62.50 ($75/1.2) and the total value of the portfolio would be $97.50 (62.50 + 35.00). If the original common stock investment had merely been retained, the portfolio would have a value of $100. With a stock index increase and then a return to initial value the Black-Jones investment strategy will be inferior to a buy-and-hold strategy. Of course, Black and Jones are offering a procedure for insuring a portfolio, not for maximizing value.

Consider now what happens if, after the same initial situation, the stock price index goes dozvn to 80, and the value of the stock portion of the portfolio falls to $40. The total value of the portfolio is now $90, and the cushion is now $10. The new exposure (invest- ment in stock) is $25 ($10 x 2.5). The new amount invested in stock is $25 and the amount invested in Treasury bills is $65. Note that, with the decrease in the stock index, the portfolio manager sells $15 of common stock. The manager wins if the stock index goes down further. Obviously if a large amount of funds were invested following these decision rules, a price decrease would result in selling, and a further price decrease would result in further selling, thus exacerbating the decline.

If the stock index returns to 100, the value of the $25 investment in stock becomes $31.25 ($25 x 100/80) for a total portfolio value of $96.25 (31.25 + 65). If the initial portfolio had been retained when the index went to 80, the stock value would be $40; when the index went back to 100, the stock value would be $50 and the total portfolio would be worth $100. A

1. Footnotes appear at end of article.

FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1988 C 86

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buy-and-hold strategy beats the proposed insurance strategy if the stock index returns to its initial value after a price index decrease.

The Black-Jones strategy leads to buying stock when the price index increases and selling stock when prices decrease. This is not the conventional wisdom of old-time practices such as dollar-cost aver- aging. Also, if prices go up and then down, or down and then up, the strategy might not be as good from a profit-maximization viewpoint as a buy-and-hold strategy.

Conclusions Strategies using stock index futures have been the

most commonly used insurance strategy. It has been estimated that as much as $90 billion of stock was covered by this type of insurance immediately prior to October 19, 1987.2 When arbitrageurs begin selling stock, there have to be buyers or there will be a massive price drop. A stock price decrease will trigger the computers following Strategies Two and Three to

sell stock. There are fewer computers programmed to buy when the sell computers send their signals. The opportunity for wide swings in stock prices is created.

For many years, stop-loss orders have been used to insure the preservation of a given amount of invest- ments. But stop-losses could not always be executed at the defined price, hence were not always effective. Neither are portfolio insurance schemes.

The fact is that we live in an uncertain world and that it is extremely difficult to eliminate risk. If all risks are eliminated, then the return earned will probably approach the risk-free rate. When someone promises more (if tax considerations are not promi- nent in the analysis), beware of the catch. The proba- bility is close to one that there is a catch.

Footnotes 1. F. Black and R. Jones, "Simplifying Portfolio In-

surance" (Goldman Sachs Research, New York, August 1986), p. 1.

2. The New York Times, December 1, 1987, p. D6.

FINANCIAL ANALYSTS JOURNAL / MAY-JUNE 1988 U 87

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