PUT OPTION PORTFOLIO INSURANCE VS. ASSET ALLOCATION

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1 Advances in Quantitative Analysis of Finance and Accounting, Volume 7(2009), pp Center for PBBEFR and Airiti Press PUT OPTION PORTFOLIO INSURANCE VS. ASSET ALLOCATION Ken Johnston Department of Accounting and Finance, Campbell School of Business, Berry College John Hatem Department of Finance and Quantitative Analysis, College of Business, Georgia Southern University ABSTRACT The purpose of this study is to develop a model that uses index put options to replace fixed income securities in an individual investor s portfolio. Such a portfolio would allow the investor to reduce downside risk, consistent with the rationale for the fixed income allocation, while also allowing the investor to participate to a greater degree, in any potential gains from a market upturn. Results indicate that in order for the model to be superior to a stock/bond portfolio, substantial movement in stock returns is necessary. Implications of results are discussed. Keywords: Portfolio insurance, Investments, Options 253

2 254 Johnston and Hatem I. INTRODUCTION There has been considerable debate regarding the optimal percentages of equity and fixed income securities in an individual investor s portfolio. There is both support for and against higher bond allocations ( Leibowitz & Kogelman, 1991; Samuelson, 1994; Siegel,1994; Asness, 1996; Campbell & Viceira, 2002; Brennan & Xia, 2002; Campbell, Chan, & Viceira, 2003). The main issues of this debate are a combination of terminal wealth, investment horizon, and risk. The primary risk factors discussed are individual risk aversion, risk of shortfall, and real interest rate risk. The rationale for the increased fixed income allocation is to reduce risk. But there is a cost to this reduction in risk, a reduction in the amount of upside capture. As discussed in Leland (1980), if an investor desires to maximize their expected return while insuring against any losses they may want to purchase portfolio insurance. Portfolio insurance s purpose it to eliminate or reduce downside risk while allowing the investor to participate to a greater degree in any potential gains from a market upturn. Pozen (1978) discusses portfolio insurance in its most basic form, requiring the investor to purchase protective puts. Using the underlying stock and stock index futures, put options can also be created synthetically. Another way to create synthetic put options on a portfolio is by dynamically varying the stock-bond composition of the option-replicating portfolio (Fortune, 1995). Prior studies investigating portfolio insurance, using either historical data or employing theoretic or stochastic modeling, have focused primarily on the investment tradeoffs involved with the insurance exclusively (costs vs. potential benefits). This paper makes two contributions to the literature. First, it examines how portfolio insurance may work in place of asset allocation changes. Specifically, the purpose of this study is to develop a model that uses index put options in place of bonds to reduce the risk of an individual s portfolio, while allowing for the potential of higher valuations when compared to an asset allocation portfolio (stocks and bonds). Second, the potential pitfalls of using dynamic Black and Scholes (1973) modeling are avoided. Most theoretical models of portfolio insurance are created in a Black and Scholes (1973) world with complete markets. These complete markets allow for continuous trading with the returns on risky assets conforming to Weiner processes. The Black and Scholes model could be used to determine put premiums or put options can be created synthetically. In this world, synthetic put options can be created through the continuous revision of portfolios with long positions in the risk free asset and short positions in a stock or stock index. As discussed above, there are other ways to synthetically create

3 Put Option Portfolio Insurance vs. Asset Allocation 255 put options. In the context of this study, due to the synthetic nature of these put options, the quantities of puts purchased would vary continuously with the stock index price. In addition, to use the Black and Scholes model, estimates of implied volatility and the risk free rate have to be determined. Why introduce potential error into the model, when actual put option premiums are available from the market? Even when less than complete markets are incorporated into the theoretical modeling, assumptions are made that significantly weaken the conclusions. For example when Benninga and Blume (1985) model incomplete markets the investor is permitted to purchase put options on the risky asset with any strike price that the investor sets. Therefore a continuous distribution exists for option strike prices. In the real world this does not exist, in reality exercise prices are set mechanically by the exchange. For example S&P 500 leaps have a minimum tick for options trading below $3.00 of.05 ($5.00) and for all other series of.10 ($10.00). Also, the option writers choose which available exercise prices they want to offer. The strongest argument against using a dynamic world like Black and Scholes to model portfolio insurance is that these synthetic strategies are found not to function well when the stock market crashes. Tian (1996) and Fortune (1993) found that investors were not able to protect their positions due to several reasons, one being suspended trading. Therefore, this study will not examine the potential use of synthetic options. This model will not attempt to time the market, i.e. buy index put options when market downturns are predicted to be more likely. The current literature on market timing, overall finds little evidence of timing ability 1 ( Daniel, Grinblatt, Titman, & Wermers, 1997; Brinson, Hood, & Beebower, 1986; Graham, & Harvey, 1997; Blake, Lehmann, & Timmermann, 1999; Goetzmann, Ingersoll, & Ivkovi, 2000; Dellva, Demaskey, & Smith, 2001; Bollen & Busse, 2001). II. PORTFOLIO INSURANCE TO REDUCE RISK The insurance portfolio includes shares in a stock index and put options on that index. 2 The value of this portfolio at t = 0 without insurance is given by: 1 Although the most recent study Bollen and Busse (2001), using daily tests instead of monthly, showed that mutual funds may possess more timing ability that previously thought. 2 Index options differ from individual stock options in that, index options are settled in cash, since it would be costly to duplicate. Therefore instead of each index option contract demanding a number of shares to be transferred, index put option have a multiplier.

4 256 Johnston and Hatem V I0 = n T S 0 (1) Where: V I0 is the value of the insured portfolio at time zero, S 0 is the value of the index, n T is the number of shares of the stock index owned before purchasing any index put options. The value of this stock portfolio with insurance (put options) at time zero equals: V I0 = n S1 S 0 +n P MP 0 (2) n S1 = n TS 0 n P MP 0 S 0 Where: n S1 is the number of shares of the index held after purchasing the put options, n p is the number of put options purchased, M is the index option multiplier, and P 0 is the cost of each put option at time 0. 3 For the model developed, the assumption is made that the investor can only exercise the options on the maturity date (European style). The value of the insured portfolio at time T (when the set of options purchased expires) is given by: (3) V IT = n S1 S T +n P M(max[X S T, 0]) (4) Where: M(max[X S T, 0]) is the value of each put option at expiration, X is the strike price on the put options, S T is the value of the index at time T and S T = S 0 (1 +r s ), where r s is the return on the stock index. III. ASSET ALLOCATION TO REDUCE RISK The asset allocation portfolio includes shares in a stock index and shares in a bond index. The value of the asset allocation portfolio at time zero is the weighted average of the value of the stock and bond indexes: V A0 = n S2 S 0 +n B B 0 (5) Where: V A0 is the value of the asset allocation portfolio at time zero, which equals the initial value of the insurance portfolio V I0. n S2 is the number of shares of the index owned, B 0 is the value of the bond index at time zero, n B is the number of shares of the bond index owned. The value of the asset allocation portfolio at time T is: VAT = ns2st+nbbt (6) 3 It is assumed that the investor can obtain fractional amounts of all securities.

5 Put Option Portfolio Insurance vs. Asset Allocation 257 Where: B T = B 0 (1+r b ) and r b is the return on the bonds. The purpose of the insurance is to provide downside protection. If there is a stock market stock market downturn, theoretically portfolio insurance is achieved if: V IT = V AT (7) That is, at a specific expiration date in future, if the stock market declines, the value of the put option insurance portfolio (V IT ) equals to the value of the asset allocation portfolio (V AT ). This is different from 100% portfolio insurance in that the insured portfolio is allowed to fall but not to a level lower than the asset allocation portfolio. If the options expire out of the money and the stock market increases, the higher weighting of stock in the insurance portfolio compared to the asset allocation portfolio, may result in the insurance portfolio outperforming the asset allocation portfolio. In order to determine the minimum number of put option needed to satisfy equation 7, it is restated using equations 4 and 6: n S1 S T +n p M(max[X S T, 0]) = n S2 S T +n B B T (8) Given that S T = S 0 (1+r s ) and B T = B 0 (1+r b ), substituting for n S1 from equation (3), equation (8) can be written as: n T S 0 (1+r s ) n P MP 0 (1+r s )+n P M(max[X S 0 (1+r s ),0]) = n S2 S 0 (1+r s )+n B B 0 (1+r b ) Where: n T S 0 (1+r s ) is the potential value of the stock portfolio with no insurance, n P MP 0 (1+r s ) is the potential cost of the insurance, n P M(max[X S 0 (1+r s ),0]) is what you can potentially make on the insurance and n S2 S 0 (1+r s )+n B B 0 (1+r b ) is the potential value of the asset allocation portfolio at time T. The discussion is in terms of potential here since at time 0, r s and r b are estimates. 4 The purpose of the insurance is to provide downside protection. The insurance is activated when the intrinsic value of the option is positive. This occurs when r s is sufficiently negative, resulting in a positive insurance payout. Therefore the potential payout of the insurance, in equation (9), becomes n P M(X S 0 (1+r s )). Subsequently, solving equation (9) for the number of options: (9) 4 The impact of this will be addressed later in the paper.

6 258 Johnston and Hatem n P = (n S2 n T )S 0 (1+r s )+n B B 0 (1+r b ) M[X (S 0 +P 0 )(1+r s )] (10) Equation (10) gives us the minimum number of puts needed to provide the insurance. By design n S2 n T < 0, more stock index shares are held in the insurance portfolio (before options purchase) than in the asset allocation portfolio. Given that n B B 0 (1+r b ) and S 0 (1+r s ) are positive and (n S2 n T ) is negative by design, the numerator is positive if: (n S2 n T )S 0 (1+r s ) < n B B T (11) Divide both sides by B T = B 0 (1+r b ) and (n S2 n T ) which is negative: S 0 (1+r s ) n B > B 0 (1+r b ) (n S2 n T ) (12) Equation (12) will always hold since the right hand side is negative and the left hand side of the equation will be positive. Therefore the numerator of equation (10) will always be positive. For n P to be positive, the denominator has to be positive. What you potentially make on each option M(X S 0 (1+r s )) is greater than the potential cost of each option M(P 0 (1+r s )). In the model a negative denominator, and hence a negative n P, would not imply that the investor should write index put options. The exercise price (X) and the premium (P 0 ) are chosen from the available set of index put options in the market. Given the option information, the investor chooses r s, this allows you to solve for the number of options needed such that if the actual return (r sa ) r s, the insurance payoff will result in the value of the insurance portfolio being greater than or equal to the value of the asset allocation portfolio (V IT V AT ). 5 If the investor chooses r s such that r s > (X S 0 P 0 )/S 0, the investor wants the insurance to kick in before the available options give any positive economic payoff. In this case the number of options (n p ), in equation (10), is forced to be negative for the equality to hold. This problem is avoided by requiring the stock index return satisfy the following condition: r s < (X 0 S 0 P 0 ) S 0 (13) 5 This proof is shown later in the paper.

7 Put Option Portfolio Insurance vs. Asset Allocation 259 The investor is unable to obtain insurance for drops in the market that are smaller than the negative stock return needed to make the available chosen index put options profitable. This restriction results in equation (10) having a positive denominator. An operational concern can occur when the number of options is positive but too large in terms of the objective of the portfolio. The more options in your portfolio the larger the insurance payment you receive if the stock market falls significantly. Higher potential insurance payments result from choosing a negative stock index return (r s ) closer to zero. The closer r s is to zero the lower the potential economic profit M(X S 0 (1+r s ) P 0 ) on each option, the more options are needed to substitute for the bonds. This increased insurance comes at a cost and that cost is a reduction in upside potential due to the cost of the greater number of options. To have realistic upside potential the stock portion of the insured portfolio, at time zero, has to dominate in terms of value over the option component. 6 Although we show mathematically that we can solve for n P that will eliminate downside risk in the sense that the value of the insurance portfolio will not fall below the value of the asset allocation portfolio, in reality this risk can not be completely eliminated. n P is calculated based on a known r s and r b. Using a portfolio of zero coupon bonds the actual return on bonds can be known at time zero. That is not the case for the return on the stock index. An initial return (r s ) is required to calculate the number of options purchased, it is very unlikely that this value will turn out to be the actual stock index return (r sa ), since the stock index return is a random variable. Therefore equation (7) (V IT = V AT ) which states that at a specific expiration date in the future, and given a stock market down turn, the value of the insurance portfolio is always equal to the value of the asset allocation portfolio, will not hold. Equations (4) and (6) need to be reexamined distinguishing between the initial stock index return (r s ) and the actual stock index return (r sa ). At time zero the actual stock index return (r sa ) is unknown, choosing an initial r s, for equation (10), tells us the actual level of loss in the stock index where the insurance payoff, results in the insurance portfolio value matching the value of the asset allocation portfolio. Solving equation (10) tells the investor the minimum amount of options needed to achieve that goal. With the number of options determined the value of the two portfolios will depend on the actual return on the stock market index. Therefore equation (4) and (6) are restated as: 6 Simulations shown later in the paper indicate that in order to have realistic upside potential the stock portion of the insurance portfolio must be over 90% of the portfolios value. Given the trade off between the potential insurance payment and the opportunity cost of the insurance, investor preference will determine the exact allocation between stock index shares and index put options.

8 260 Johnston and Hatem V IT = n S1 S 0 (1+r sa )+n P M(max[X S 0 (1+r sa ), 0]) V AT = n S2 S 0 (1+r sa )+n B B 0 (1+r b ) (14) (15) The difference in value at time t between the insurance portfolio and the asset allocation portfolio is: V IT V AT = (n S1 n S2 )S 0 (1+r sa )+n P M(max[X S 0 (1+r sa ), 0]) n B B 0 (1+r b ) (16) There are two possible outcomes for the actual stock index return: the return on the stock index moves such that the options end up out of the money and two, the return on the stock index moves such that the options end up in the money. 7 If the options in the insurance portfolio end up in the money, equation (16) becomes: V IT V AT = (n S1 n S2 n P M)S 0 (1+r sa )+n P MX n B B 0 (1+r b ) (17) The value of the insurance portfolio is greater than the value of the asset allocation portfolio if: (n S1 n S2 n P M)S 0 (1+r sa )+n P MX > n B B 0 (1+r b ) (18) Dividing both sides by S 0 and (n S1 n S2 n P M) which is negative: n B B < 0 (1+r b ) n P MX r sa 1 (n S1 n S2 n P M)S 0 (19) Substitute the value of n T from equation (3) into equation (10) and solve for r s : r s = n B B 0 (1+r b ) n P MX (n S1 n S2 n P M)S 0 1 (20) Therefore: r sa < r s (21) At time T the value of the insurance portfolio will be greater than the asset allocation portfolio if the actual stock market index return r sa is less than the initial stock market index return r s. If the options in the insurance portfolio end up out of the money, equation (16) becomes: V IT V AT = (n S1 n S2 )S 0 (1+r sa ) n B B 0 (1+r b ) (22) 7 At the money has the same result as out of the money.

9 Put Option Portfolio Insurance vs. Asset Allocation 261 In this case the value of the insurance portfolio is greater than the value of the asset allocation portfolio if: or (n S1 n S2 )S 0 (1+r sa ) > n B B 0 (1+r b ) r sa n > B B 0 (1+r b ) 1 (n S1 n S2 )S 0 (23) (24) Solving equation (20): n B B 0 (1+r b ) r s (n S1 n S2 n P ) n P MX n S1 n S2 n P M = + + (n S1 n S2 )S 0 (n S1 n S2 ) (n S1 n S2 )S 0 (n S1 n S2 ) (25) Substituting equation (25) into equation (24) and solving, if the index options ends up out the money at time T, the value of the insurance portfolio will be greater than the asset allocation portfolio if: r sa > r s + n PM[X/S 0 r P 1] (n S1 n S2 ) (26) The net result of the actual stock index return being different from the initial stock index return is that the insurance portfolio ends up having a payoff structure that requires substantial movement in stock index returns up or down in order to be superior to a stock/bond portfolio. To attain further insight into the operational characteristics of equation (10), partial derivatives are taken and simulations are conducted with respect to the initial stock index return, the exercise price, the percentage of stocks in the asset allocation portfolio, and the return on the bond portfolio. For all partial derivatives the denominator is positive due to the operational constraint imposed by equation (13). The partial derivative of n P with respect to the initial stock market return chosen is: n P (n S2 n T )S 0 X+n B B 0 (1+r b )(S 0 +P 0 ) = (27) r M(X (S 0 +P 0 )(1+r s )) 2 s Given that n S2 n T < 0, the numerator is positive if: (n S2 n T )S 0 X < n B B 0 (1+r b )(S 0 +P 0 ) (28) Divide both sides by (n S2 n T )S 0 which is negative: X > n BB 0 (1+r b )(S 0 +P 0 ) (n S2 n T )S 0 (29)

10 262 Johnston and Hatem Since the right hand side of the equation is negative, equation (29) will hold, therefore the numerator and the derivative (equation (27)) are positive. As the initial negative stock return (r s ) moves closer to zero, the number of options needed increases. In Table 1, a simulation is shown that examines this issue. To simplify the simulations, actual stock market index values and index put option prices are used. Using actual market data removes the difficulties related to the use of the Black and Scholes options pricing model, for example determining the appropriate volatility of the stock market index. The premium and exercise prices for put option leaps on the S&P 500 (expiration date 12/16/2006) and the index value are taken from Market Data Express on the Chicago Board Options Exchange website ( com) for the trading date of December 16, This gives us a 12 month period. All other data is simulated. The following information is used for the simulations in Tables 1 through 5: Value of portfolio at time zero = $100,000 1 period leap put option on the stock market index: premium = 18.5, exercise price = 1100 December 16, 2005 S&P 500 stock index value = Annual return on bond index = 6% Current bond index value = 200 Asset allocation = 60/40 stocks/bonds S&P index Multiplier = 100 The simulation in Table 1, examines how changing the initial stock index return r s, effects the comparative valuations between the insured portfolio and the asset allocation portfolio. At t = 0, the initial stock index return (r s ) is used to determining the number of options needed to provide the insurance. The less negative r s is set at, the less you potentially make on each option, the more options needed to substitute for bonds. For example if the initial r s is 15%, the potential economic profit on each option is $ (M(X S 0 (1+r s ) P 0 )), if r s is set at 16% the potential economic profit is $1,695.12, therefore you would require less options. In Table 1 the initial stock index return can take one of five values 14.67%, 14.85%, 15.68%, 16.45% and 19.81%. Smaller negative values for r s are excluded from the simulation due to violation of the constraint imposed by equation (13), that the investor is unable to obtain insurance for drops in the market that are smaller than the negative stock return needed to make the available chosen index put options intrinsic value positive. The first case where this constraint is satisfied occurs with an r s of 14.67%. With this initial r s, the insured portfolio requires options and the portfolio is only 45.56% stocks. 8 As you can see from Table 1, the cost 8 The inability to obtain fractional shares of the index put options could easily be integrated into the model, having little overall effect on the investor s potential payoff structure and hence results of this study.

11 Put Option Portfolio Insurance vs. Asset Allocation 263 of the options is so high that in reality there would be no upside potential. Even if the actual annual index return is 40% the assets allocation portfolio is worth $62, more than the insured portfolio. This allocation is inappropriate for our goal to develop a strategy that uses put options in place of bonds to reduce the risk of an individual s portfolio while allowing for the potential of higher valuations when compared to an asset allocation portfolio. Similar results are found if the investor uses an initial return (r s ) of 14.85% and 15.68%. Initial returns of 16.45% and 19.81%, give us stock ownership levels greater than 93%. The numbers of options needed are and respectively. In these cases there are both upside potential and downside gain. The more options in your portfolio the larger the insurance payment you receive if the actual stock market index return is lower than the initial. To get a higher potential insurance payment results from choosing a smaller negative initial return. For example if your initial return is 16.45% and the actual return (r sa ) is 20% the insured portfolio s value is $14, greater than the asset allocation portfolio. If your initial return is 19.81% and the actual return is 20% the insured portfolio s value is only $ greater than the asset allocation portfolio. The increased insurance comes at a cost and that cost is a reduction in upside potential due to the cost of the greater number of options. Again using equation 26, if your initial return is 16.45% the actual stock index return would have to greater than or equal to 26.44% for the insurance portfolio to be greater in value than the asset allocation portfolio. If your initial return is 19.81% the actual stock index return would only have to be greater than or equal to 13.90%. To discuss the effects of choice of exercise price on the model the partial derivative of n P with respect to the exercise price is determined: n P [(n S2 n T )S 0 (1+r s )+n B B 0 (1+r b )] X = M(X (S 0 +P 0 )(1+r s )) 2 (30) Given the necessary conditions that n S2 n T < 0, and it has previously been shown that (n S2 n T )S 0 (1+r s ) < n B B 0 (1+r b ), the numerator is positive, making the partial derivative negative. Options involve financial leverage increasing the strike price increases what you potentially make on the options less their cost, reducing the number of options needed to provide the insurance. Therefore the benefit of an increased strike price is the increase in the amount of upside capture and it occurs at a lower positive interest rate. Fewer options result in lower payoffs when the market decline is greater than the initial r s.

12 264 Johnston and Hatem Table 1. Value of Insured Portfolio - Value of Asset Allocation Portfolio Changing the Initial r s 14.67% 14.85% 15.68% 16.45% 19.81% Number of options Actual r sa 40% $948, $516, $161, $96, $30, % $798, $434, $135, $80, $24, % $648, $352, $108, $63, $18, % $499, $269, $82, $47, $12, % $349, $187, $55, $30, $6, % $199, $105, $28, $14, $ % $49, $23, $2, $1, $5, % $55, $33, $16, $12, $9, % $55, $33, $14, $11, $8, % $56, $32, $13, $10, $6, % $56, $32, $12, $8, $5, % $57, $32, $11, $7, $3, % $57, $31, $10, $6, $2, % $58, $31, $8, $4, $ % $59, $30, $7, $3, $ % $59, $30, $6, $2, $2, % $60, $30, $5, $ $3, % $60, $29, $4, $ $5, % $61, $29, $3, $1, $6, % $62, $28, $1, $3, $8, % $62, $28, $ $4, $9, Stock % in Insured Portfolio 45.56% 69.89% 89.84% 93.54% 97.23% Lowest r sa where V IT > V AT % 42.08% 26.44% 13.90% Note: Value of insured portfolio and asset allocation portfolio at time zero = $100,000 Value of insured portfolio - value of asset allocation portfolio at time t = (n S1 n S2 )S 0 (1+r sa )+n P M(max[X S 0 (1+r sa ), 0]) n B B 0 (1+r b ) Where: n S1 is the number of shares of the index held after purchasing the put options = n T S 0 n P MP 0 S 0 n S2 is the number of shares of the index owned in the asset allocation portfolio M is the stock index multiplier = 100 S 0 is the value of the stock index at time 0 = X is the strike price on the stock index put options = 1100 r sa is the actual stock index return n P is the number of put options purchased = (n S2 n T )S 0 (1+r s )+n B B 0 (1+r b ) M[X (S 0 +P 0 )(1+r s )]

13 Put Option Portfolio Insurance vs. Asset Allocation 265 n T is the number of shares of the stock index owned in the insurance portfolio before purchasing any put options r s is the initial return on the stock index. It is the negative return required, where the insurance payoff results in the value of the insurance portfolio being equal to the value of the asset allocation portfolio n B is the number of shares of the bond index owned B 0 is the value of the bond index at time zero = 200 r b is the annual return on the bonds = 6% P 0 is the cost of each index put option at time 0 = 18.5 Asset allocation = 60/40 stocks/bonds The simulation in Table 2, examines how changing the exercise price of the option affects the difference between the value of the insured portfolio and the value of the asset allocation portfolio. A second leap option on the S&P 500 with the same expiration date, an exercise price of 1,175 and a premium of 31.5 is examined. The simulation information from Table 1 is used and the initial stock index return is set at 16.45%. Again, increasing the strike price from 1,100 to 1,175 increases what you potentially make on each option, the more you potentially make on each option, the fewer options needed to substitute for bonds (3.495 vs ). Fewer options result in lower payoffs when the market declines more than the initial return. If the stock market falls 24% the insurance portfolio value is $30, greater than the asset allocation portfolio when then exercise price is 1,100. If the exercise price is 1,175, the insurance portfolio s value is $6, greater than the asset allocation. If actual stock returns are better than the initial, with a strike price of 1,100, the insured portfolio is not superior to the asset allocation portfolio until the actual index return is over 26.43%. If options with a strike price of 1,175 are used, the insured portfolio value is greater than the asset allocation portfolio value for actual stock index returns above 15.05%. To examine the effects of asset allocation changes on the model, the partial derivative of n P with respect to the number of shares in the asset allocation portfolio is derived: n P n S2 = S 0 (r s r b ) M(X (S 0 +P 0 )(1+r s )) (31) The initial stock return has to be sufficiently negative to satisfy constraints and make equation (10) operational. The bond portfolio contains zero coupon bonds. Therefore the return on the bonds will be greater than the initial return on the stock index used to determine n p. Thus the numerator and derivative will be

14 266 Johnston and Hatem negative. Increasing the amount of stock in the asset allocation reduces the number of options needed. If you have fewer bonds in your portfolio less insurance is required, since options are substituting for bonds in the insurance portfolio. Table 2. Value of Insured Portfolio - Value of Asset Allocation Portfolio Changing the Exercise Price Exercise Price Premium Number of options Actual r sa 40% $96, $21, % $80, $17, % $63, $13, % $47, $10, % $30, $6, % $14, $3, % $1, $ % $12, $3, % $11, $7, % $10, $7, % $8, $5, % $7, $4, % $6, $2, % $4, $1, % $3, $ % $2, $1, % $ $3, % $ $4, % $1, $6, % $3, $7, % $4, $9, Stock % in insured Portfolio 93.54% 96.85% Lowest r sa where V IT > V AT 26.44% 15.06% Note: Value of insured portfolio and asset allocation portfolio at time zero = $100,000 Value of insured portfolio - value of asset allocation portfolio at time t = (n S1 n S2 )S 0 (1+r sa )+n P M(max[X S 0 (1+r sa ), 0]) n B B 0 (1+r b ) Where: n S1 is the number of shares of the index held after purchasing the put options = n T S 0 n P MP 0 S 0 n S2 is the number of shares of the index owned in the asset allocation portfolio

15 Put Option Portfolio Insurance vs. Asset Allocation 267 M is the stock index multiplier = 100 S 0 is the value of the stock index at time 0 = X is the strike price on the stock index put options r sa is the actual stock index return n P is the number of put options purchased = (n S2 n T )S 0 (1+r s )+n B B 0 (1+r b ) M[X (S 0 +P 0 )(1+ r s )] n T is the number of shares of the stock index owned in the insurance portfolio before purchasing any put options r s is the initial return on the stock index. It is the negative return required, where the insurance payoff results in the value of the insurance portfolio being equal to the value of the asset allocation portfolio = 16.45% n B is the number of shares of the bond index owned B 0 is the value of the bond index at time zero = 200 r b is the annual return on the bonds = 6% P 0 is the cost of each index put option at time 0 Asset allocation = 60/40 stocks/bonds Table 3 shows the results of the simulation where the percentage of stocks in the asset allocation portfolio is varied. The initial stock index return is set at 16.45% and the actual stock index return is varied. The asset allocations examined are 80/20, 60/40, and 40/60 stocks/bonds. Increasing the percentage of bonds in the portfolio from 20% to 60% increases the number of options needed to provide the insurance (1.747 vs ). Again the increase in options results in higher values for the insured portfolio when the actual stock index return is less than the initial stock index return. When the stock market index return is 20% the insurance portfolio value is $21, greater than the 40/60 asset allocation portfolio but only $7, greater than the 80/20 asset allocation portfolio. Although the point at which the actual index returns result in the insurance portfolio being greater than the asset allocation portfolio is the same for all allocations (26.44%), the allocation with the most options has more dollar upside potential. At first this may seem counterintuitive given the previous findings and discussions about the cost of the options. What drives this result is the size of the bond allocation is increasing significantly, causing much smaller allocations of the stock index in the asset allocation portfolio. The upside potential from holding more stock in the insurance portfolio vs. the asset allocation portfolio dominates the cost of the options. To determine how changing the bond return affects the model, the partial derivative of n P with respect to the bond return is derived: n P n B B 0 r = (32) B M(X (S 0 +P 0 )(1+r s ))

16 268 Johnston and Hatem Table 3. Value of Insured Portfolio - Value of Asset Allocation Portfolio Changing the Percentage of Stocks/Bonds in Asset Allocation Portfolio Asset Allocation 40/60 60/40 80/20 stocks/bonds Number of options Actual r sa 40% $144, $96, $48, % $120, $80, $40, % $95, $63, $31, % $70, $47, $23, % $46, $30, $15, % $21, $14, $7, % -$2, $1, $ % $19, $12, $6, % $17, $11, $5, % $15, $10, $5, % $13, $8, $4, % $11, $7, $3, % $9, $6, $3, % $7, $4, $2, % $5, $4, $1, % $3, $2, $1, % $1, $ $ % $ $ $ % $2, $1, $ % $4, $3, $1, % $6, $4, $2, Stock % in insured Portfolio Lowest r sa where V IT > V AT 90.30% 93.54% 96.77% 26.44% 26.44% 26.44% Note: Value of insured portfolio and asset allocation portfolio at time zero = $100,000 Value of insured portfolio - value of asset allocation portfolio at time t = (n S1 n S2 )S0(1+r sa )+n P M(max[X S 0 (1+r sa ), 0]) n B B 0 (1+r b ) Where: n S1 is the number of shares of the index held after purchasing the put options = n T S 0 n P MP 0 S 0 n S2 is the number of shares of the index owned in the asset allocation portfolio M is the stock index multiplier = 100 S 0 is the value of the stock index at time 0 = X is the strike price on the stock index put options = 1100 r sa is the actual stock index return n P is the number of put options purchased = (n S2 n T )S 0 (1+r s )+n B B 0 (1+r b ) M[X (S 0 +P 0 )(1+r s )]

17 Put Option Portfolio Insurance vs. Asset Allocation 269 n T is the number of shares of the stock index owned in the insurance portfolio before purchasing any put options r s is the initial return on the stock index. It is the negative return required, where the insurance payoff results in the value of the insurance portfolio being equal to the value of the asset allocation portfolio = 16.45% n B is the number of shares of the bond index owned B 0 is the value of the bond index at time zero = 200 r b is the annual return on the bonds = 6% P 0 is the cost of each index put option at time 0 = 18.5 Since the asset allocation portfolio contains bonds, the numerator is positive. Therefore the derivative is positive, the higher the bond return the more options are required, since options are substituting for bonds in the insurance portfolio. Table 4 shows the results of the simulation where the bond index return is altered. The initial stock index return is set at 16.45%, the asset allocation is 60/40 stocks/ bonds and the actual stock index return is varied. If the bond index annual return is 3%, options are required whereas options are required if the bond index annual return is 9%. More options result in higher insurance payments if the actual return is less than the initial return. If the actual return is 20% and the bond index annual return is 9%, the insured portfolio s value is $16, greater than the asset allocation portfolio. If the bond index annual return is 3%, the insured portfolio s value is only $12, greater than the asset allocation portfolio. Like previous simulations increasing the number of options reduces the upside potential. When the bond return is 3% (9%), the stock index return would have to be greater than or equal to 19.78% (33.46%) for the insurance portfolio to be greater in value than the asset allocation portfolio. Table 4. Value of Insured Portfolio - Value of Asset Allocation Portfolio Changing the Bond Index Return Bond Index Annual Return 3% 6% 9% Number of options Actual r sa 40% $82, $96, $110, % $68, $80, $91, % $54, $63, $72, % $40, $47, $54, % $26, $30, $35, % $12, $14, $16, % $1, $1, $2, % $10, $12, $14, % $9, $11, $13, % $8, $10, $9, % $6, $8, $10,928.76

18 270 Johnston and Hatem Table 4. Value of Insured Portfolio - Value of Asset Allocation Portfolio Changing the Bond Index Return (Continued) Bond Index Annual Return Number of options 3% % % Actual r sa 4% $5, $7, $3, % $4, $6, $8, % $2, $4, $7, % $1, $3, $5, % $78.85 $2, $4, % $1, $ $3, % $2, $ $1, % $4, $1, $ % $5, $3, $ % $6, $4, $2, Stock % in insured Portfolio 94.40% 93.54% 92.67% Lowest r sa where V IT > V AT 19.78% 26.44% 33.46% Note: Value of insured portfolio and asset allocation portfolio at time zero = $100,000 Value of insured portfolio - value of asset allocation portfolio at time t = (n S1 n S2 )S 0 (1+r sa )+n P M(max[X S 0 (1+r sa ), 0]) n B B 0 (1+r b ) Where: n S1 is the number of shares of the index held after purchasing the put options = n T S 0 n P MP 0 S 0 n S2 is the number of shares of the index owned in the asset allocation portfolio M is the stock index multiplier = 100 S 0 is the value of the stock index at time 0 = X is the strike price on the stock index put options = 1100 r sa is the actual stock index return n P is the number of put options purchased = (n S2 n T )S 0 (1+r s )+n B B 0 (1+r b ) M[X (S 0 +P 0 )(1+r s )] n T is the number of shares of the stock index owned in the insurance portfolio before purchasing any put options r s is the initial return on the stock index. It is the negative return required, where the insurance payoff results in the value of the insurance portfolio being equal to the value of the asset allocation portfolio = 16.45% n B is the number of shares of the bond index owned B 0 is the value of the bond index at time zero = 200 r b is the annual return on the bonds = 6% P 0 is the cost of each index put option at time 0 = 18.5 Asset allocation = 60/40 stocks/bonds

19 Put Option Portfolio Insurance vs. Asset Allocation 271 The simulation in Table 5, examines how changing the term of the put options affects the difference between the value of the insured portfolio and the value of the asset allocation portfolio. 9 A second leap option on the S&P 500 is chosen with the same exercise price as the options used in Table 1 (1100). This second option has a premium of $36.9 and an expiration date that is approximately 1 year longer (Dec. 22, 2007 vs. Dec. 16, 2006). The focus of this study is on wealth changes over time, therefore compounded geometric mean returns have to be incorporated into the model. Given a two year expiration date equation (10) becomes: n P = (n S2 n T )S 0 (1+r s ) 2 +n B B 0 (1+r b ) 2 M[X (S 0 +P 0 )(1+r s ) 2 ] (33) The difference in value at time t between the insurance portfolio and the asset allocation portfolio in equation (16) becomes: V IT V AT = (n S1 n S2 )S 0 (1+r sa ) 2 +n P M(max[X S 0 (1+r sa ) 2,0]) n B B 0 (1+r b ) 2 Table 5. Value of Insured Portfolio - Value of Asset Allocation Portfolio One Year vs. Two Year Leap, Changing the Initial r s One Year Leaps Two Year Leaps Initial r s 15.68% 19.81% 15.68% 19.81% Number of options years Acual (34) 1 year Actual r sa Holding Period R sa Annual GM return 40% $161, $30, % 22.54% $9, $2, % $135, $24, % 20.00% $6, $ % $108, $18, % 17.54% $2, $2, % $82, $12, % 15.15% $ $4, % $55, $6, % 12.82% $4, $6, % $28, $ % 10.55% $7, $8, % $16, $5, % 8.35% $10, $11, % $16, $9, % 6.19% $12, $12, % $14, $8, % 4.08% $11, $10, % $13, $6, % 2.02% $9, $9, The partial derivative of np with respect to the expiration date of the options is not shown, since the length of the options are not part of the original equation (equation (10)). It could be incorporated by using compound returns. For example equation (33) adjusts equation (10) for a two year compounding period (the use of two year leap options).

20 272 Johnston and Hatem Table 5. Value of Insured Portfolio - Value of Asset Allocation Portfolio One Year vs. Two Year Leap, Changing the Initial r s (Continued) Initial r s Number of options One Year Leaps 15.68% % years Acual Two Year Leaps 15.68% % year Actual r sa Holding Period R sa Annual GM return 0% $12, $5, % 0.00% $8, $7, % $11, $3, % 1.98% $7, $6, % $10, $2, % 3.92% $5, $4, % $8, $ % 5.83% $4, $3, % $7, $ % 7.70% $2, $1, % $6, $2, % 9.54% $1, $ % $5, $3, % 11.36% $ $1, % $4, $5, % 13.14% $1, $2, % $3, $6, % 14.89% $3, $4, % $1, $8, % 16.62% $4, $5, % $ $9, % 18.32% $6, $7, Stock % in Insured Portfolio 89.84% 97.23% 96.47% 97.29% Lowest r sa where V IT > V AT 42.08% 13.90% Lowest annual GM r sa where V IT > V AT 11.01% 9.79% Note: Value of insured portfolio and asset allocation portfolio at time zero = $100,000, n S2 is the number of shares of the index owned in the asset allocation portfolio, M is the stock index multiplier = 100, S 0 is the value of the stock index at time 0 = , X is the strike price on the stock index put options = 1100, r sa is the actual stock index return, n T is the number of shares of the stock index owned in the insurance portfolio before purchasing any put options, r s is the initial return on the stock index. It is the negative return required, where the insurance payoff results in the value of the insurance portfolio being equal to the value of the asset allocation portfolio, n B is the number of shares of the bond index owned, B 0 is the value of the bond index at time zero = 200, r b is the annual return on the bonds = 6%, Asset allocation = 60/40 stocks/bonds. n S1 is the number of shares of the index held after purchasing the put options = n TS 0 n P MP 0 S 0 One Year Leap Options Value of insured portfolio - value of asset allocation portfolio at time t = (n S1 n S2 )S 0 (1+r sa )+n P M(max[X S 0 (1+r sa ), 0]) n B B 0 (1+r b ) n P is the number of put options purchased = (n S2 n T )S 0 (1+r s )+n B B 0 (1+r b ) M[X (S 0 +P 0 )(1+r s )]

21 Put Option Portfolio Insurance vs. Asset Allocation 273 P 0 is the cost of each index put option at time 0 = 18.5 Two Year Leap Options Value of insured portfolio - value of asset allocation portfolio at time t = (n S1 n S2 )S 0 (1+r sa ) 2 +n P M(max[X S 0 (1+r sa ) 2, 0]) n B B 0 (1+r b ) 2 n P is the number of put options purchased = (n S2 n T )S 0 (1+r s ) 2 +n B B 0 (1+r b ) 2 M[X (S 0 +P 0 )(1+r s ) 2 ] P 0 is the cost of each index put option at time 0 = 36.9 Apart from the additional option, all other simulation information from Table 1 is used. Increasing the expiration date one year increases the premium from $18.5 to $36.9. If the initial r s is 15.68%, the potential one year economic profit on each one year leap option is $1, (M(X S 0 (1+r s ) P 0 )) and the potential two year economic profit is $16, (M(X S 0 (1+r s ) 2 P 0 )) on each two year leap option. This substantial increase on what the investor potentially makes on the two year leap options less their costs compared to the one year leap options, reduces the number of options needed to provide the insurance. Again, if the initial stock index return is 15.68% the model uses one year leap options and two year options. Therefore the benefit of increasing the term to maturity is an increase in the amount of upside capture and it occurs at a lower positive interest rate. Given the initial r s of 15.68%, Table 5 shows that the insurance portfolio, using two year leap options, is greater than the asset allocation portfolio if the actual annual geometric mean is greater than 11.01%. This corresponds to a 2 year holding period return of 23.23% ((1.1101) 2 1). Using the same initial r s and the one year options, the value of the insurance portfolio is not greater than the value of the asset allocation portfolio until the actual one year return is greater than 42.08%. If the initial r s is changed to 19.81% the model uses one year leap options and two year options. Using the one year options, the value of the insurance portfolio is not greater than the value of the asset allocation portfolio until actual one year return is greater than 13.09%. For the two year leap options, the insurance portfolio is greater than the asset allocation portfolio if the actual annual geometric mean is greater than 9.79% (two year holding period return of 20.54% ((1.0979) 2 1)). The use of fewer options, when the term to maturity is increased, does result in lower payoffs when the market decline is greater than the initial return r s. With an initial return of 15.68%, if the stock market falls 20% over the two year leap option period ( 10.55% annual geometric mean) or 36% over the two year period

22 274 Johnston and Hatem ( 20.00% annual geometric mean), the insurance portfolio value is $7, less or $6, greater than the asset allocation portfolio, respectively. If the stock market falls 20% and the investor is holding the one year leap options, the insurance portfolio s value is $28, greater than the asset allocation. Findings are similar when using an initial stock return of 19.81%. Although the downside payoff is smaller due the use of fewer options, results indicate that using two year leap options enables the investor to choose smaller negative initial returns that have more realistic upside potential compared to the use of one year leap options. IV. CONCLUSIONS This paper develops a model that uses index put options in place of fixed income securities in an individual investor s portfolio. This model allows the investor to reduce downside risk, as they age, similar to increasing the bond allocation, while also permitting the investor to participate in any potential gains from a market upturn. For the model to be successful, the investor needs substantial movement in the underlying stock index. The model expands the investor s opportunity set, it allows for risk return combinations not otherwise available with a typical stock/bond asset allocation portfolio. This model is an improvement over just purchasing fixed amounts of index put options, given that it tells the investor what the potential payoff structure looks like in relation to a given asset allocation. It also allows the investor to make adjustments to the potential payoff structure by choice of put options purchased and the initial stock index return estimate used. If the investor believes the future stock index return distribution is more likely skewed in one direction versus the other, the investor can position their insurance portfolio to take advantage of the skewness, but still have some opportunity to beat the asset allocation portfolio if they are incorrect. Given the expected stock index return distribution and information on the available set of put options on that index, the model determines boundary positions. Given an expected index return distribution, the investor could conduct a Monte Carlo simulation to calculate the probability that the insurance portfolio will be greater than the asset allocation portfolio. There are significant problems with conducting simulations using index return probability distributions to calculate the average returns of the portfolios as well as the change in returns. How does the investor determine the expected stock index return distribution?

23 Put Option Portfolio Insurance vs. Asset Allocation 275 It would be incorrect to sample from an historical stock index return probability distribution, assuming independent and identically distributed (i.i.d.) returns, when conducting this simulation. Sampling under the assumption of i.i.d., from a historical stock index return probability distribution, ignores the fact that market expectations about the future probability distribution are incorporated into the options price and influences the available set of put options offered. In this study historical options data is used. If the investor then sampled under the assumption of i.i.d. from an historical stock index return distribution, this link would be severed, severely weaken any conclusions derived from the results. Given historical options pricing data, the Black and Scholes options pricing model could be used to back out implied volatility. However, Doran and Ronn (2006) find significant bias between the Black and Scholes (1973)/Black (1976) implied volatility and the realized volatility of the S&P 500 and S&P 100. Error would also be introduced when estimating the mean of the expected stock index return distribution. Preliminary results indicate that the use of longer term leaps is superior. Although the downside payoff is smaller due the use of fewer options, results indicate that using longer term leap options (2 year maturity) enables the investor to obtain downside protection for smaller drops in the stock market while also allowing for more realistic upside potential compared to the use of shorter term leap options. Empirical evidence indicates that the majority of options expire out of the money. This is consistent with the efficient market hypothesis. Information on expected volatility is used when the option writer chooses which exercise price to offer, and is a factor in determining the premiums of the available index put options. 10 Although market expectations are incorporated into the options market, market expectations are often incorrect. This error should increase with the option holding period and especially in periods with large market declines. This models use of small amounts of index leap put options in a stock portfolio has the potential to give the investor significant payoffs in these periods. The increased upside potential from a higher proportion of stock in the portfolio, along with the occasional options payout, may overcome the cost of the majority of periods where the options expire out of the money. Using long term leap options is desirable for a second reason. One of the potential issues in conducting a portfolio insurance program is the cost of rebalancing the portfolio when the horizon of the portfolio does not match the 10 Exercise prices are set mechanically by the exchange.