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A Portfolio Insurance Strategy for Volatility Index (VIX) Futures

Young Cheol Jung*

This paper proposes a dynamic asset allocation strategy for portfolio insurance of which underlying risky asset is the Volatility Index (VIX) Futures and evaluates the effectiveness of the strategy. For this purpose, we build Option-Based Portfolio Insurance (OBPI) and Constant Proportion Portfolio Insurance (CPPI) for VIX futures. From the simulation for eight sample periods between Feb.2007 and April 2014, this paper finds that practitioners can apply the dynamic asset allocation strategy to VIX Futures. In particular, CPPI can secure the floor more tightly compared to OBPI and well capture the upside potential of the market under the highly unpredictable volatility of VIX futures.

Volatility derivatives on the stock index have attracted much attention to both academics and practitioners over the past two decades. Since the Chicago Board Options Exchange (CBOE) introduced Volatility Index (known by its ticker symbol VIX) in 1993, there had been a growing demand for instruments to hedge volatility risk. To address this demand, in March 2004, CBOE created the exchange-traded VIX futures and in Feb. 2006, launched VIX options. In particular, the volatility shock that hit the financial markets globally in autumn 2008 had pronounced effects on the trading volume of volatility derivatives.1

VIX, often known as “fear index”, is the standard measure of volatility risk for investors in the U.S. stock market. VIX is currently based on the S&P500 Index (SPX) and is devised to estimate the expected volatility (i.e., standard deviation) over the next thirty calendar days by averaging the weighted prices of SPX Index options over a wide range of strike prices (CBOE[2009]). The negative correlation of VIX to stock market returns and a mean-reverting feature of VIX have been well documented.

Exhibit 1 illustrates the historical behavior of SPX and VIX from Jan 1, 1990 through June 30, 2013. From the figure, we can observe mean-reversions after several spikes in VIX and a strong negative correlation between the daily movements of SPX and VIX. In particular, the negative correlation intensifies in a market downturn. Moreover, VIX has been almost six times more volatile than SPX for the period. 2 VIX futures, of which level represents a forward volatility over thirty calendar day that begins at the expiration date of the futures contract, also has a peculiar characteristic. Because VIX itself is not tradable, VIX futures has a very different property from other futures on tradable assets, which is implied by the standard cost-of-carry model. Due to the mean-reversion of the volatility, VIX futures tends to be lower than VIX when VIX is high and vice versa. Even at the expiration date, VIX futures is not exactly converge to VIX due to the non-tradability of VIX.

                                                            * Assistant Professor, PhD, CFA, Department of Policy Studies (Economics), Mount Royal University, 4825 Mount Royal Gate, Calgary, Alberta, Canada T3E 6K6, 403-440-5079, yjung@mtroyal.ca. 1 The trading volume of volatility derivatives has grown to its 11-fold for VIX futures and 4-fold for VIX options between 2007 and 2011. The data source for S&P500 Index, VIX and VIX futures is the CBOE Futures Exchange (CFE). 2 For the period, the correlation between the daily movements of SPX and VIX is -0.60 in days with a negative return on SPX and -0.47 in days with a positive return. The asymmetric relation between market volatility and stock market return has been frequently observed (Schwert [1990], Fleming et al. [1995], Whaley [2000], Daigler and Rossi [2006] and Sarwar [2012]). In addition, VIX ranges from 9.3 percent (Dec.22, 1993) to 80.9 percent (Nov.20, 2008) and averages to 20.3 percent; the annualized volatility of SPX and VIX are 18.6 percent and 107.7 percent, respectively, for the period. 

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Since Brenner and Galai [1989] and Whaley [1993] first introduced the concept on the volatility derivatives, there has been extensive literature on VIX Futures. The literature has mainly focused on building and testing pricing models (Grünbichler and Longstaff [1996], Zhang and Zhu [2006]), assessing the forecastability (Nossman and Wilhelmsson [2009], Lu and Zhu [2010], Konstantinidi and Skiadopoulos [2011]), modeling a term structure (Zhang et al. [2010], Huskaj and Nossman [2012]) and examining a causality between VIX and VIX futures prices (Shu and Zhang [2012]). In particular, several scholars have noticed the diversification benefit from adding VIX futures to the existing portfolio as one of hedging tools (Moran and Dash [2007], Szado [2009], Chen et al.[2011]). EXHIBIT 1 VIX (right axis) vs. SPX (left axis): Jan 1, 1990 - June 30, 2013

EXHIBIT 2 3-Month Buy-and-Hold Return of VIX Futures: Feb.2007 – April 2014

However, in spite of the diversification benefit that previous studies presented, VIX futures as an investment asset has been hardly justified due to its negative expected return.

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Jan/08

Jan/09

Jan/10

Jan/11

Jan/12

Jan/13

SPX

VIX

‐100%

‐50%

0%

50%

100%

150%

200%

G (Feb 07)

K (May 07)

Q (Aug 07)

X (Nov 07)

G (Feb 08)

K (May 08)

Q (Aug 08)

X (Nov 08)

G (Feb 09)

K (May 09)

Q (Aug 09)

X (Nov 09)

G (Feb 10)

K (May 10)

Q (Aug 10)

X (Nov 10)

G (Feb 11)

K (May 11)

Q (Aug 11)

X (Nov 11)

G (Feb 12)

K (May 12)

Q (Aug 12)

X (Nov 12)

G (Feb 13)

K (May 13)

Q (Aug 13)

X (Nov 13)

G (Feb 14)

3‐month return (VIX Futures)

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Particularly, the risk premium is negative for short-term VIX futures, which works as an insurance against drops in the stock market (Nossman and Wilhelmsson [2009], Huskaj and Nossman [2012]). For instance, Exhibit 2 depicts the 3-month buy-and-hold return of VIX futures until the expiration date during Feb.2007 – April 2014. As the chart shows, VIX futures contracts have revealed a negative return for 72 percent of the contracts and the average loss of the contracts was 27.6 percent. Moreover, the frequency of the loss was more intensified recently. Given this reality, the role of VIX futures as a hedging tool is limited because adding VIX futures to an investment portfolio may drag the overall return of the portfolio.3

In that sense, this study begins with an inquiry, “Can we use VIX futures as an investment asset with a downside protection?” To answer the inquiry, this study examines whether a dynamic asset allocation strategy for the portfolio insurance (‘the PI strategy’) works for VIX futures. Currently Option-Based Portfolio Insurance (OBPI) and Constant Proportion Portfolio Insurance (CPPI) are two most prominent dynamic asset allocation strategies that are designed to guarantee a minimum value (called the ‘floor’) of the portfolio at investment horizon while retaining some exposure to a rising market. Although these strategies have their applications to most of financial products, they have not been used to VIX futures yet. Thus the main purpose of this paper is to implement OBPI and CPPI on VIX futures and to evaluate their effectiveness.

To the best of our knowledge, this paper is the first study on the dynamic asset allocation strategy for the portfolio insurance applied to VIX futures. The simulation results indicate that practitioners can apply OBPI and CPPI to VIX futures in order to reduce its downside risk; in particular, CPPI secures the floor more tightly compared to OBPI and well follows up the market under the high unpredictable volatility of VIX futures.

OBPI on VIX Futures

Rubinstein and Leland [1981] originally introduce OBPI by showing that there exists a

replicating portfolio strategy, involving stock and cash only that creates payoff identical to that of a call option on stock. Using the same framework, we can implement OBPI on the market of VIX futures by replicating volatility options. The idea on volatility option originates from Brenner and Galai [1989]. They construct the volatility index from historical standard deviation and price the volatility option with the Cox-Ross-Rubinstein binominal technique. Whaley [1993] advances the idea further using the Black-Scholes [1973] option pricing model (the B-S model) and values European style options on the volatility index futures. Grünbichler and Longstaff [1996] derive a formula for the volatility option by assuming that the volatility index follows a mean-reversion process. For the sake of brevity and feasibility, however, this paper uses Whaley’s model to construct OBPI.4 According to Whaley, the price of a European style call and put option on the volatility index futures is respectively,

                                                            3 Alexander and Korovilas [2012] find that volatility diversification with VIX futures was only optimal during the periods of the recent credit and banking crises, even after accounting for skewness preference. 4 Grünbichler and Longstaff [1996] provide a more delicate framework for pricing volatility option because their model allows a mean-reversion of volatility which is frequently observed in the financial markets. However, to build the portfolio insurance strategy using their model, it needs to estimate more parameter values. In that case, parametric mis-specification may more likely affect the outcome. On the contrary, Whaley’s model suffers relatively less from the estimation errors because volatility of volatility is the only parameter to estimate. Related to the issue, Wang and Daigler [2010] examine the pricing performance of VIX option models and find that Whaley’s model produces the best results for in-the-money VIX option.  

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)()( 21 dXNdVNec rtv (1)

)()( 12 dVNdXNep rtv (2)

where ,5.0)/ln( 2

1t

tXVd

v

v

tdd v 12

X : exercise volatility

V : price of VIX futures r : annualized risk-free interest rate t : the remaining life of option

v : volatility of volatility

)(N : the cumulative standard normal distribution function

OBPI can also be implemented by combining VIX futures and VIX put option traded in market. However, it is not easy to find market traded VIX options of which maturity and exercise price exactly match with those of the insured portfolio that has a pre-determined floor. Even if suitable options are found, the portfolio insurers may find buying them in the market is too expensive. Therefore, in practice, replicating VIX option by the dynamic asset allocation can be a better (albeit imprecise) solution. In this section, we build three strategies of OBPI: Protective Put, Volatility Cap and Resetting. (1) Protective Put

Protective Put is the basic strategy of OBPI. When we add the price of VIX futures, V , to both side of equation (2), we obtain:

)()(1 21 dNXedNeVpV rtrtv

(3)

Equation (3) provides the key to create OBPI. Protective Put is created synthetically by selling a proportion of VIX futures equal to its put option delta, )( 1dN , and placing the proceeds in the risk-free asset. OBPI typically considers the put option cost as an initial out-of-pocket investment so that it sets the exercise price of the put option equal to the desired floor. However, this paper will implement the strategy with a little revision. The payment for the put options, i.e., the insurance cost, is included in the investment principal to calculate the floor. Thus to replicate the pay-off of a call option on VIX futures or equivalently the payoff of holding VIX futures plus its put option, the portfolio insurer needs to invest v

rt pVdNeV /)(1 1 of the total portfolio

value on VIX futures and to hold the remains as the risk-free asset whenever rebalancing occurs. As vpV , and )( 1dN change, the asset allocation between VIX futures and the risk-free asset

must be rebalanced so as to maintain the floor. Note that the portion invested in VIX futures is always less than one implying that no leverage is required. The floor of Protective Put is the minimum return that the insured portfolio must protect at the maturity and can be calculated as

)/( 0 vpVX where :0V initial price of VIX futures.

(2) Volatility Cap

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Protective Put can be revised to a strategy, say, Volatility Cap that is composed of VIX futures, put options (its exercise price is X ) and written out-of-money call options (its exercise price isX ). To replicate Volatility Cap, equation (1) is deducted from both sides of equation (3) to obtain:

)()()()(1 2211 dNXdXNedNdNeVcpV rtrtvv , .XX (4)

Volatility Cap is created synthetically by investing vvtTr cpVdNdNeV /)()(1 11)(

of the total portfolio value on VIX futures and holding the remains as the risk-free asset at each rebalancing. The floor of Volatility Cap is calculated as )./( 0 vv cpVX

An advantage of Volatility Cap is that it can lessen the adverse impact of the volatility reversal on the portfolio value while maintaining an upside capture on market. As a rising VIX futures approaches the exercise price of its synthetic VIX call option, the exposure to VIX futures begins to decline rapidly. If an abrupt fall of VIX futures is followed, the portfolio is less affected by the reversal. However, Volatility Cap must give up additional gains above the exercise price of the call option. Therefore, the strategy will be suitable for investors which expect a moderately rising market with mean-reversals. Setting up the level of volatility cap is arbitrary and depends on the investor’s risk tolerance. This paper sets the exercise price of the out-of-money call option at 1.5 times of the initial price of VIX futures.

(3) Resetting

Resetting is another revised strategy of Protective Put. The strategy is adjusting the floor of the insured portfolio according to subsequent portfolio return.5 When the price of risky asset is substantially increased, investors are concerned about protecting today’s value from a reversal of the price. If the price of VIX futures rises substantially, the exposure to VIX futures will increase up to 100 percent. The portfolio that fully consists of VIX futures will lose all gains if the price is abruptly mean-reverting. Therefore, an appropriate strategy for the investors who expects the reversal is to reset the floor whenever the price of VIX futures increases by a certain percentage point. Then the exposure to VIX futures would be scaled down and consequently the sharp fall of volatility would affect the portfolio value less. Resetting is a similar strategy to Volatility Cap. Once its floor is reset, however, Resetting has a better ability to secure gain in a type of ‘whipsaw’ market - a market that involves an upward movement in price, which is then followed by a drastic downward move. A rule of resetting is arbitrary. In this paper, the floor is reset by 1 percent point up for every 10 percent increase in the price of VIX futures. Resetting is more effective for the portfolio with a low floor rather than a high floor because the portfolio with a low floor can maintain relatively high exposure to the risky asset.

CPPI on VIX Futures

OBPI is often compared to CPPI. While OBPI is based on the B-S option pricing model, CPPI is based on the theory of expected utility maximization.6 CPPI has its theoretical origin on

                                                            5 Perold and Sharpe [1988] initially provided the basic concept of resetting. 6 For a comparative study of portfolio insurance between OBPI and CPPI, see Bertrand and Prigent (2005) and Pain and Rand (2008).

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the optimal portfolio rules in a continuous-time model by Merton [1971], and was modified practically by Black and Jones [1987] and Black and Perold [1992]. Under CPPI, the mixture of VIX futures and the risk-free asset in the portfolio satisfies the relationship:

)( rtttt eFWmcmE (5)

tE : exposure to VIX futures

m : multiple

tc : cushion (portfolio value – floor)

tW : portfolio value at time t

F : the insured amount at the maturity (i.e., floor)

For CPPI, the multiple is a key parameter that importantly affects a trade-off between performance and risk. The multiple is typically chosen to reflect the risk preference of the investor, the expected volatility of risk assets, the risk-free interest rate, the floor and the investment horizon. A ‘conventional’ method that chooses the multiple of CPPI arbitrarily, say at 4 or 8, is not appropriate for the purpose of comparing its payoff to OBPI’s. Thus to evaluate its payoff more comparably at an identical initial condition, this paper chooses a multiplier that equals the initial risk exposure of CPPI to that of an equivalent Protective Put for each period.

When CPPI adjusts portfolio positions between VIX futures and the risk-free asset, the portion of VIX futures can be greater than 100 percent (i.e., leverage is required) in a strong and persistently rising market. This paper will assume that leverage is not allowed for CPPI to maintain the consistency with OBPI and also to follow the common practice in financial applications. Along with its simplicity, CPPI has several advantages over OBPI. First of all, CPPI does not need to estimate volatility of underlying risky asset explicitly and as a result can reduce the volatility misspecification risk. In addition, CPPI does not require a definite maturity date. Thus the possible loss from liquidity-oriented trades at the maturity date can be avoided. More importantly, its relative solidarity to secure the floor has been frequently cited. However, differently from OBPI, CPPI is path-dependent: i.e., the final payoff depends on the path of underlying asset prices during the investment horizon. Once the allocation to VIX futures falls to zero, CPPI has no chance to recover its risk exposure while OBPI can restore it. 7

Portfolio Simulations

(a) Design

For simplification, this paper pursues a portfolio insurance strategy free from estimation. Therefore, all data in this paper is easily obtainable from financial markets. In the CBOE, there are nine contracts traded on each trading day. The daily trading volume of VIX futures is estimated to be reduced by half beyond 150 days (Lu and Zhu [2010]). Thus to secure a proper liquidity for trading, the portfolio in this paper sets its maturity at 3-month and invests VIX futures of which expiration month is immediately after the maturity date. For example, if the investment period of a portfolio covers from June 30, 2009 through Oct.1, 2009, the portfolio

                                                            7 A simple simulation can confirm the result.

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uses VIX futures expired in Oct. 2009 as its underlying asset8. Since VIX is not tradable, the settlement price of VIX futures is used as the portfolio benchmark. The data set that is employed to determine the optimal exposure to risky assets consists of the settlement price of VIX futures, the volatility of the settlement price, and 4-week US T-bill secondary market rate.9

As Rendleman and O’Brien [1990] mentioned, the replication of synthetic options depends critically on accurate estimation of the volatility of the underlying risky asset. However, as shown by numerous researches, the estimated volatility is far from accurate.10 Moreover, the volatility of volatility is even less predictable. Thus to simplify the procedure, this paper uses an annualized (252 days) rolling 22-day historical volatility of VIX futures as a proxy for the volatility of volatility.

This paper chooses the following assumptions for simulation. The portfolio is rebalanced on a daily basis for a precise replication of options.11 The optimal exposure to VIX futures is determined based on the settlement price of each trading day. The required rebalancing amount is completely settled at the settlement price of the previous trading day in the next trading day. This assumption is generally taken for granted in the literature on the portfolio insurance. For reference, the tables in this paper include an implementation shortfall (Shortfall) cost, i.e., the gain or loss caused by the difference in the settlement price between the two adjacent trading days. The return is calculated without deducting the transactions cost. To help estimating approximate transaction costs, however, ‘turnover’ of each strategy is disclosed. For example, if the turnover is 4, the transaction cost is estimated to be 0.6 ~ 1.2 percent of the principal for the investment horizon. 12 The portfolio is self-financing, i.e., money is neither injected nor withdrawn during the investment horizon. The margin requirement for VIX futures is assumed to be 100 percent.

(b) Results

Each strategy is tested for eight sample periods between Feb. 2007 and April 2014.13 The floor is aggressively set at 90 (maximum loss of 10 percent) and 95 (maximum loss of 5 percent) over the 3-month investment horizon.

Exhibit 3 and Exhibit 7 (Appendix) present a summary result of each strategy for the sample periods. First of all, CPPI protects its floor in all sample periods regardless of its floor. For instance, in period 8 (Sept.30, 2010- Jan.3, 2011) that revealed the worst return on VIX futures (-37.4 percent) among the sample periods, CPPI ends up with 10 percent loss at floor 90 and 5 percent loss at floor 95, which equal to each floor. As mentioned previously, CPPI’s solidarity to secure the floor is frequently observed in other research. CPPI also well captures bull markets. For example, in period 1 (July 31- Oct.31, 2008: Exhibit 3), CPPI achieves 110

                                                            8 We assume that the fund is cashed at the maturity date. 9 The settlement price of VIX futures is more appropriate to measure the return because VIX futures is marked to market according to the settlement price defined by the CFE. The T-bill rate is obtained from Board of Governors of the Federal Reserve System. 10 See Poon and Granger [2003] for a comprehensive survey on the issue. 11 The frequency of rebalancing provides a trade-off between transactions cost and protection. Since the volatility of VIX futures is extremely high, the daily rebalancing may be acceptable to secure solid protection in spite of the increased transactions cost. 12 It was challenging to measure the average transaction cost including market impact cost for this simulation. According to Alexander and Korovilas [2012], the average bid-ask spread for VIX futures quarterly declined from 0.58% (Oct.2006 – March 2009) to 0.31% (April 2009- Dec.2011). The transaction cost due to a bid-ask spread is generally calculated as half the bid-ask spread.  Based on the information, this paper estimates the average transactions cost per unit to be in the range of 0.15~0.3 percent. 13 The eight periods are arbitrarily selected to reflect various market situations.

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percent gain at floor 90 while VIX futures gains 125.8 percent. In period 3 (July 29- Oct.31, 2011: Exhibit 7), CPPI shows relatively lower return (16.6 percent) compared to VIX futures return (36.4 percent) at floor 95 but it is still higher than the return of Protective Put (14.8 percent). Interestingly, the multiple of CPPI is even higher at floor 95 than at floor 90 (e.g., in period 8, the multiple is 11.8 at floor 95 while it is 8.4 at floor 90). It implies that the multiple of CPPI must be higher as its floor increases in order to achieve an equivalent payoff to OBPI.

Protective Put also shows satisfying performance in bull markets. For example, in period 1, in which the collapse of Lehman Brothers provoked the volatility shock, Protective Put accomplishes the highest return (114.3 percent at floor 90 and 95.1 percent at floor 95) among others, which is a little below VIX futures’ return (125.8 percent). However, Protective Put mostly fails to protect its floor in bear market or ‘whipsaw’ market. For instance, in period 8, Protective Put incurs 13.9 percent loss at floor 90 and 7.3 percent loss at floor 95.

One main culprit for the failure of Protective Put to secure its floor is a misspecification of volatility. The volatility misspecification causes mispricing of Protective Put and consequently results in a misallocation between VIX Futures and the risk-free asset. However, if we could know in advance the actual volatility of VIX futures between each rebalancing and maturity day of the insured portfolio, Protective Put would secure floor more tightly in a deep weak market. As illustrated in Exhibit 4, if the actual volatility had been used, Protective Put could almost have secured its floor: e.g., in period 8, the 10.2 percent loss at floor 90 and 5.1 percent loss at floor 95; in period 7, 10.2 percent loss at floor 90 and 5.2 percent loss at floor 95. It is also interesting to see that Protective Put would have a lower turnover in the deep weak market had the actual volatility been used. This indicates that a certain portion of transactions in Protective Put is caused by the mis-forecasting of volatility.

Resetting and Volatility Cap appear to be the least effective in a strong bull market. For example, in period 1 in which the volatility displays a strong up-trend consistently, Resetting ends up with the lowest return (44 percent at floor 90 and 15.2 percent at floor 95) owing to a pre-mature reduction of its risk exposure. In the same period, Volatility Cap also reveals a low return due to a similar reason. In severe bear markets (period 5, period 7 and period 8), both strategies show the same or similar results as Protective Put because there was no chance to reset the floor or to exercise the out-of-money call option. However, in ‘whipsaw’ market such as period 4 and period 6, Resetting could capture the chance of volatility reversal in part and consequently comes out with a relatively better outcome than others. Once Resetting works, the return to average risk exposure is high. Its higher Sharpe ratios for the periods verify the results. Exhibit 8 and Exhibit 9 in Appendix provide graphical illustrations how the risk exposures of each strategy are dynamically adjusted.

When the PI strategy is applied to real situations, implementation shortfall (Shortfall) should also be an integral part of the transaction cost to consider: e.g., for Protective Put at floor 90 in period 6 (Exhibit 3), actual loss would be 8.6 percent after adding Shortfall cost of 2 percent.

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EXHIBIT 3 Performance Comparison (Floor 90)

Sample periods (SPX Return)

VIX Futures

Protective Put

Resetting

Volatility Cap

CPPI

1  

July31,2008/ Nov.3,2008 

 (‐23.8%) 

Return Avg. risk exposure 

Shortfall* Turnover 

Sharpe Ratio Final floor 

125.8% ‐ ‐ ‐ 

0.29 ‐ 

114.3% 0.832 0.1%          4.2 0.282 90 

44% 0.52 3.9%         4.1 0.276 105 

51.9% 0.562 ‐1.4%          3.7 0.28 90 

110% 0.797 ‐0.5%        5.0 0.276 90 (5.8) 

2  

Apr.30,2007/ Aug.1,2007 

 (‐1.1%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

54.6% ‐ ‐ ‐ 

0.294 

53.3% 0.972 ‐0.0%         2.8 0.291 90 

48.2% 0.906 0.6%         4.7 0.29 94 

52.9% 0.967 0.6%           3.2 0.295 90 

53.8% 0.99 ‐0.2%      2.8 0.291 90 (7.8) 

3  

July29,2011/ Oct.31,2011 

 (‐3.0%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

36.4% ‐ ‐ ‐ 

0.086 ‐ 

23.3% 0.918 ‐2.9% 2.8 0.062 90 

23.5% 0.261 ‐0.6%        3.5 0.196 98 

37.8% 0.501 1.2%         7.5 0.141 90 

31.1% 0.978 ‐1.3%          2.5 0.077 90 (6.5) 

4  

April 30,2010/ Aug.2, 2010 

 (‐5.1%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

‐3.8% ‐ ‐ ‐ 

‐0.016 ‐ 

‐8.7% 0.913 0.8%          3.0 

‐0.041 90 

16.4% 0.225 1.7%         3.3 0.148 93 

‐5.8% 0.763 ‐2.2%          3.8 

‐0.032 90 

‐4.1% 0.98 0.5%         2.2 

‐0.018 90 (7.5) 

5  

June30,2009/ Oct.1,2009 

 (+12.0%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

‐7.5% ‐ ‐ ‐ 

‐0.057 ‐ 

‐10.3% 0.685 0.4%          3.4 

‐0.115 90 

    ‐7.1% 0.578 1.7%       4.0 

‐0.093 90 (7.4) 

6  

April 30,2013/ Aug.1,2013 

 (+6.8%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

‐18.8% ‐ ‐ ‐ 

‐0.153 ‐ 

‐6.6% 0.647 2.0%          2.9 

‐0.066 90 

6.2% 0.505 3.2%         2.8 0.073 92 

‐6.8% 0.651 1.9%          2.7 

‐0.066 90 

‐7.8% 0.604 2.3%       3.3 

‐0.086 90 (5.7) 

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Jan.31,2012/ May 1,2012 

 (+7.1%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

‐29.0% ‐ ‐ ‐ 

‐0.147 ‐ 

‐16.9% 0.47 ‐1.0%         4.0 

‐0.189 90 

    ‐10.0% 0.326 ‐1.3%      4.2 

‐0.149 90 (8.3) 

8  

Sept.30,2010/ Jan.3,2011 

 (+11.5%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

‐37.4% ‐ ‐ ‐ 

‐0.303 ‐ 

‐13.9% 0.236 0.1%          2.8 

‐0.384 90 

    ‐10.0% 0.181 ‐0.2%          2.2 

‐0.351 90 (8.4) 

* If Shortfall>0 (<0), it is cost (gain). Notes: (1) For reference, the return on S&P500 Index for the sample period is shown in the parenthesis of the first column. (2) The number in the parenthesis of the seventh column is the multiple of CPPI. (3) Turnover = Total transactions for VIX futures during the investment horizon (including the last day trades for cashing VIX futures at the maturity date) / Principal (4) The empty column means that the results are identical to those of Protective Put.

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Exhibit 5 compares the statistics of daily return on the four PI strategies to the benchmark (VIX futures) after aggregating the eight sample periods at floor 90 (518 days). The daily return is calculated as the logarithm of the price relative on two consecutive end prices. First of all, as Exhibit 5 presents, the daily mean returns of all PI strategies are greater than the benchmark’s. In particular, the daily mean return of CPPI (0.002) is much higher than VIX Futures (0.0008).14 The standard deviation is lower in the PI strategy than in the benchmark. Consequently, the Sharpe ratios of all PI strategies are much higher than VIX Futures.15 The average turnover ranges from 3.2 to 3.9, which seems not prohibitively be high to prevent implementation of the PI strategy. The returns of VIX futures and the PI strategy are clearly non-normally distributed with high kurtosis and positive skewness. Kurtosis of all PI strategies is well over that of benchmark, which means to exhibit “fat tails” relative to the benchmark as well as to the normal distribution. In particular, the positive skewness is relatively great in both Resetting and Volatility Cap. It implies that some risk-averse investors may prefer the two strategies to others.

EXHIBIT 4 Portfolio Return and Turnover of Protective Put Using Actual Volatility

EXHIBIT 5 Statistics of Daily Returns and Average Turnover for Full In-Sample Period (518 Observations) (Floor 90)

VIX Futures Protective

Put Resetting Volatility Cap CPPI

Mean 0.0008 0.0015 0.0014 0.0011 0.0020

Standard Dev. 0.0350 0.0294 0.0160 0.0221 0.0299

Sharpe Ratio 0.0228 0.0515 0.0876 0.0508 0.0663

Kurtosis 2.2167 4.5 5.5899 5.9471 4.9354

Skewness 0.3645 0.5336 1.3337 0.8445 0.6084

Average Turnover - 3.23 3.58 3.89 3.28

It is interesting to see what if the PI strategy of this paper would be applied to the

October 1987 Crash. To examine the effectiveness of the strategy for the best case scenario of

                                                            14 However, a t-test for the sample daily returns (518 observations) at floor 90 shows that the outperformance of the four PI strategies against the benchmark is all statistically insignificant at 95 percent confidence level. 15 The Sharpe ratio may not a good indicator to measure the risk-adjusted return of the portfolio insurance because the strategy is designed to provide an upside potential combined with a downside protection. As emphasized by Leland (1999), the strategy limiting downside risk with positively skewed returns, is incorrectly underrated. In that sense, Annaert et al. (2009) introduce the stochastic dominance framework to evaluate the portfolio insurance strategy that shows non-normality of return.

Period  1  2  3  4  5  6  7  8 

VIX Future Return 

125.8%  54.6%  36.4%  ‐3.8%  ‐7.5%  ‐18.8%  ‐29.0%  ‐37.4% 

Floor 90 Return  Turnover 

 96.4%       3.6 

 48.9%      3.1 

 7.2%       3.3 

 ‐10.8%     3.6 

 ‐10.0%     3.2 

 ‐10.4%     2.9 

 ‐10.2%     2.1 

 ‐10.2%     1.8 

Floor 95 Return  Turnover 

 84.7%       3.5 

 42.5%      3.5 

 ‐0.6%       3.1 

 ‐5.2% 4.0 

 ‐5.2% 2.5 

 ‐5.3% 2.8 

 ‐5.2% 1.3 

 ‐5.1%      1.0 

11  

the volatility futures (or equivalently the worst case scenario of stock), Exhibit 6 displays a simulation result for the 3-month period from Sept.30, 1987 to Dec.31, 1988. Since VIX Futures did not exist at the time, the VXO Index is used as a proxy of an investment asset and its benchmark.16 As expected, the return of most PI strategies is remarkable. In particular, Volatility Cap shows an outstanding return (142.3 percent at floor 90 and 104.7 percent at floor 95 vs. 76.3 percent for the VXO Index). The main reason for the outstanding performance of Volatility Cap is that the VXO Index return peaked at 406 percent in Oct.26, 1987 and thereafter abruptly declined. Protective Put also reasonably follows up the strong market: the return is 60.9 percent at floor 90 and 43.6 percent at floor 95. On the contrary, CPPI is relatively sluggish. CPPI reveals 54.9 percent at floor 90 and 32.1 percent at floor 95. Resetting shows a contrary result depending on the level of floor: the return is 96.3 percent at floor 90 and 12.6 percent at floor 95. It is observed that the period exhibits an extremely high kurtosis and positive skewness both for the VXO Index and strategy. Also it is notable that the extreme rate of implementation shortfall for the periods distorts actual return. For instance, the actual payoff after reflecting Shortfall is 36.5 percent for Resetting and 6 percent for CPPI at floor 95.

EXHIBIT 6 Performance Comparison: 3-Month Period (1987 Market Crash- Sept.30, 1987~Dec.31, 1988)

VXO Index

Protective Put

Resetting Volatility Cap CPPI

Return Avg. risk exposure

Shortfall Turnover

Sharpe Ratio Kurtosis

Skewness Final floor

76.3% - - -

0.041 28.667 4.018

-

60.9% 0.836 1.8% 3.8

0.037 31.049 4.312

90

96.3% 0.101

-24.9% 2.6

0.178 50.563 6.814 147

142.3% 0.223

-14.9% 7.9

0.156 48.629 6.629

90

54.9% 0.92 5.3% 3.1

0.032 29.620 4.115

90 (4.7)

Return Avg. risk exposure

Shortfall Turnover

Sharpe Ratio Kurtosis

Skewness Final floor

76.3% - - -

0.041 28.667 4.018

43.6% 0.784 6.0% 2.4

0.029 30.938 4.262

95

12.6% 0.064

-23.9% 1.4

0.174 10.585 1.049 152

104.7% 0.176

-10.6% 5.6

0.152 49

6.673 95

32.1% 0.872 26.1%

2 0.02

30.917 4.235

95 (6.6) Notes: S&P500 return for the period was (-)23.2%. The average 1-month T-bill rate was 6 percent for the period.

Concluding Remarks

The main purpose of this paper is to propose a methodology using VIX futures as an investment asset while controlling downside risk. For this purpose, this paper built four PI strategies using OBPI and CPPI for VIX futures and evaluated the effectiveness of each strategy. The simulation results indicate that the dynamic asset allocation strategy can be applied to the market of VIX Futures. Since VIX futures is a high-risk investment tool, investors who trade VIX futures tend to be the least risk-averse in nature. In that sense, the PI strategy for VIX futures might be able to attract more risk-averse investors and consequently contributes to

                                                            16 The VXO is the original CBOE volatility index designed to measure the market’s expectation of 30-day volatility implied by at-the-money S&P100 index option prices.

12  

broadening the market for VIX futures. However, the results of this paper must be accepted with some caveats. Obviously, the price of VIX futures is highly volatile. Thus in order to practically apply the PI strategy that this paper has presented, more detailed research on liquidity and transactions cost (e.g., bid-ask spread, market impact cost) of VIX futures is required beforehand. In addition, this paper tested only eight sample periods with a 3-month maturity. Testing more sample periods with diverse maturities would help securing the robustness of the PI strategy. Lastly, to assess more thoroughly the effectiveness of the PI strategy as a hedging tool for the stock index fund, the analysis for the risk-adjusted performance of the mixed portfolio with various hedge ratios must be preceded.

Several scholars have pointed out that portfolio insurance affects financial markets in destabilizing ways because its trading rule is ‘trend-following’. In that sense, it has been generally recognized that the October 1987 Crash was exacerbated by the trading rule of the portfolio insurance that sell stocks in down-markets. Furthermore, Jacobs [2004] reports that the collapse of Long Term Capital Management (LTCM) in 1998 and the ensuring market instability was, in part, linked to portfolio insurance. However, it must be noted that the trading rule of the PI strategy for VIX futures impacts the stock market in the opposite direction. In the presence of a strong negative correlation between the stock index and its volatility, the trading rule that sells VIX futures as the stock market volatility declines can intensify the up-trend of the stock market. By contrast, the rule that buys more VIX futures as the volatility increases may affect the stock market adversely. Particularly bearing in mind that the negative correlation is greater in a negative return on stock market, the impact of the PI strategy on the stock market may be greater in a bear stock market. In that case, Resetting or Volatility Cap can lessen the adverse effect on the stock market by some degree. Also once the risk exposure of the PI strategy attains to 100 percent in a strong bull market of volatility, the PI strategy may not destabilize the stock market any further.

The PI strategy for VIX futures also can be used as an effective hedging tool for the portfolio insurance fund on the stock index. For example, despite that the insurance fund on the stock index fails to secure its floor in a strong bear stock market (tantamount to a strong bull market of volatility), the positive return of the PI strategy can partly compensate the loss. Oppositely, even though the PI strategy fails to secure its floor in a strong bear market of volatility, the high performance of the insurance fund on the stock index may make up the misfortune in part.

References

Alexander, C., D.Korovilas. “Diversification of Equity with VIX futures: Personal Views and Skewness Preference”, working paper, Social Science Research Network, 2012.

Annaert, J., S.V. Osselaer, B.Verstraete. “Performance Evaluation of Portfolio Insurance Strategies Using Stochastic Dominance Criteria”, Journal of Banking & Finance, 33(2009), pp.272-280.

Bertrand, P., J.L. Prigent. “Portfolio Insurance Strategies: OBPI versus CPPI”, Finance, 26(1) (2005), pp.5–32.

Black,F., M. J. Scholes. “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, 81 (1973), pp.637-59.

13  

Black, F., R. Jones. “Simplifying Portfolio Insurance”, Journal of Portfolio Management, 14 (1987), pp.48–51. Black, F., A. Perold. “Theory of Constant Proportion Portfolio Insurance”, Journal of Economic Dynamics and Control, 16 (1992), pp.403–427. Brenner, M., D. Galai. “New Financial Instruments for Hedging Changes in Volatility”, Financial Analysts Journal, (Jul. - Aug. 1989), pp.61-65. Chen, H., S.Chung, K.Ho. “The Diversification Effects of Volatility-Related Assets”, Journal of Banking & Finance, 35(2011), pp.1179-1189.

Chicago Board of Options Exchange. The CBOE Volatility Index – VIX, 2009.

Daigler, R.T., L. Rossi. “A Portfolio of Stocks and Volatility”, Journal of investing , 15(2006), pp.99-106.

Fleming, J., B. Ostdiek, R.E. Whaley. “Predicting Stock Market Volatility: a New Measure”, Journal of Futures Markets, 15(1995), pp.265-302. Grünbichler, A., F. A. Longstaff. “Valuing Futures and Options on Volatility”, Journal of Banking & Finance, 20(1996), pp.985–1001. Huskaj, B., M. Nossman. “A Term Structure Model for VIX Futures”, Journal of Futures Markets, Vol.0, No.0 (2012), pp.1-22. Jacobs, B. “Risk avoidance and market fragility”, Financial Analysts Journal, (Jan.-Feb. 2004), pp.26–30. Konstantinidi, E., G. Skiadopoulos. “Are VIX Futures Prices Predictable? An empirical investation”, International Journal of Forecasting, Vol. 27, No.2 (2011), pp.543-560. Leland, H.E. “Beyond Mean–Variance: Performance in a Nonsymmetrical World”, Financial Analysts Journal, (Jan.-Feb. 1999), pp.27–36. Lu, Z., Y. Zhu. “Volatility Components: the Term Structure Dynamics of VIX Futures”, Journal of Futures Markets, Vol.30, No.3 (2010), pp.230-256. Merton, R. C. "Optimum Consumption and Portfolio Rules in a Continuous-Time Model", Journal of Economic Theory, 3(1971), pp.374-413. Moran, M.T., S. Dash. “VIX Futures and Options; Pricing and Using Volatility Products to Manage Downside Risk and Improve Efficiency in Equity Portfolios”, Journal of Trading, 2(2007), pp.96-105.

Nossman, M., A. Wilhelmsson. “Is the VIX Futures Market Able to Predict the VIX Index? A Test of the Expectation Hypothesis”, Journal of Alternative Investments, Vol. 12, No. 2(Fall 2009), pp.54-67. Pain, D., J. Rand. “Recent Developments in Portfolio Insurance”, Bank of England Quarterly Bulletin, (Q1 2008), pp.37-46.

Perold, A.F., F. William, W.F. Sharpe. "Dynamic Strategies for Asset Allocation", Financial Analysts Journal, (Jan. - Feb. 1988), pp.16-27.

14  

Poon, S., C. Granger. “Forecasting Volatility in Financial Markets: A Review”, Journal of Economic Literature, Vol. XLI (June 2003), pp.478–539. Rendleman, R., T.J. O’Brien. “The Effects of Volatility Misestimation on Option-Replication Portfolio Insurance”, Financial Analyst Journal, (May-June 1990), pp.61-70. Rubinstein, M., H.E. Leland. “Replicating Options with Positions in Stock and Cash”, Financial Analysts Journal, (July-Aug. 1981), pp.63-72. Sarwar, G. “Is VIX an Investor Fear Gauge in BRIC Equity Market?”, Journal of Multinational Financial Management, 22(2012), pp.55-65. Shu, J., J.E. Zhang. “Causality in the VIX Futures Market”, Journal of Futures Markets, Vol.32, No.1 (2012), pp.24-46. Szado, E. “VIX Futures and Options – a Case Study of Portfolio Diversification during the 2008 Financial Crisis”, Journal of alternative investments, 12(2009), pp.68-85. Schwert, G.W. “Stock Volatility and Crash of ‘87”, Review of Financial Studies, 3 (1990), pp.77-102. Whaley, R. “Derivatives on Market Volatilities: Hedging Tools Long Overdue”, Journal of Derivatives, (Fall 1993), pp.71-84.

Whaley, R. “The Investor Fear Gauge”, Journal of Portfolio Management, 26(2000), pp.12-17.

Zhang, J. E., Y. Zhu. “ VIX Futures”, Journal of Futures Markets ,26 (2006), pp.521–531. Zhang, J. E., Y. Zhu, M. Brenner. “The New Market for Volatility Trading”. Journal of Futures Markets, 30 (2010), pp.809-833.

15  

Appendix

EXHIBIT 7 Performance Comparison (Floor 95)

Sample periods (SPX Return)

VIX Futures

Protective Put

Resetting

Volatility Cap

CPPI

1  

July31,2008/ Nov.3,2008 

 (‐23.8%) 

Return Avg. risk exposure 

Shortfall* Turnover 

Sharpe Ratio Final floor 

125.8% ‐ ‐ ‐ 

0.29 ‐ 

95.1% 0.635 ‐0.5%         4.8 0.264 95 

15.2% 0.269 1.1%         3.1 0.185 110 

38.6%          0.372 ‐1.9%           3.7 0.281 95 

94.2% 0.642 ‐2.0%       5.0 0.262 95 (7.0) 

2  

Apr.30,2007/ Aug.1,2007 

 (‐1.1%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

54.6% ‐ ‐ ‐ 

0.294 

49.4% 0.879 ‐0.1%         3.6 0.284 95 

28.5%        0.676 0.1%         6.8 0.259 99 

49.0%          0.873 0.5%           4.1 0.288 95 

49.1% 0.897 0.1%       4.5 0.278 

95 (10.1) 

3  

July29,2011/ Oct.31,2011 

 (‐3.0%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

36.4% ‐ ‐ ‐ 

0.086 ‐ 

14.8% 0.863 ‐2.9%         3.0 0.043 95 

10.6% 0.075 ‐0.7%        1.5 0.178 103 

28.3% 0.446 1%          6.5 0.119 95 

16.6% 0.937 ‐2.8%         4.0 0.046 

95 (8.25) 

4  

April 30,2010/ Aug.2, 2010 

 (‐5.1%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

‐3.8% ‐ ‐ ‐ 

‐0.016 ‐ 

‐10% 0.815 2.3%          3.9 

‐0.053 95 

14.7% 0.098 0.1%        2.1 0.204 98 

‐7.2% 0.665 ‐0.5%          3.3 

‐0.045 95 

‐3.5% 0.921 4.1%        2.6 

‐0.016 95 (10.0) 

5  

June30,2009/ Oct.1,2009 

 (+12.0%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

‐7.5% ‐ ‐ ‐ 

‐0.057 ‐ 

‐6.7% 0.398 1.5%          3.4 

‐0.116 95 

    ‐4.3% 0.344 1.5%        3.3 

‐0.091 95 (9.7) 

6  

April 30,2013/ Aug.1,2013 

 (+6.8%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

‐18.8% ‐ ‐ ‐ 

‐0.153 ‐ 

‐4.7% 0.341 1.6%          3.1 

‐0.078 95 

4.0% 0.252 2.1%         2.8 0.086 97 

‐4.6% 0.357 1.5%          3.0 

‐0.072 95 

‐4.4% 0.385 1.6%        3.0 ‐0.07 

95 (7.1) 

7  

Jan.31,2012/ May 1,2012 

 (+7.1%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

‐29.0% ‐ ‐ ‐ 

‐0.147 ‐ 

‐11.5% 0.305 ‐0.3% 3.5 

‐0.188 95 

    ‐5.0% 0.166 ‐1.0%      3.1 

‐0.131 95 (11.6) 

8  

Sept.30,2010/ Jan.3,2011 

 (+11.5%) 

Return Avg. risk exposure 

Shortfall Turnover 

Sharpe Ratio Final floor 

‐37.4% ‐ ‐ ‐ 

‐0.303 ‐ 

‐7.3% 0.116 ‐0.3%         1.9 

‐0.373 95 

    ‐5.0% 0.086 ‐0.4%         1.5 

‐0.305 95 (11.8) 

16  

EXHIBIT 8 Graphical Illustration: Period 1 (July 31, 2008~ Oct.31, 2008) at Floor 90

a) Return

b) Risk exposure

‐20%

0%

20%

40%

60%

80%

100%

120%

140%

160%

180%

7/31/08

8/7/08

8/14/08

8/21/08

8/28/08

9/4/08

9/11/08

9/18/08

9/25/08

10/2/08

10/9/08

10/16/08

10/23/08

10/30/08

VIX futures

Protective Put

Resetting

Volatility Cap

CPPI

0%

20%

40%

60%

80%

100%

120%

7/31/08

8/7/08

8/14/08

8/21/08

8/28/08

9/4/08

9/11/08

9/18/08

9/25/08

10/2/08

10/9/08

10/16/08

10/23/08

10/30/08

Protective Put

Resetting

Volatility Cap

CPPI 

17  

EXHIBIT 9 Graphical Illustration: Period 3 (July 29, 2011~ Oct.31, 2011) at Floor 90

(a) Return

(b) Risk exposure

‐10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

7/29/11

8/5/11

8/12/11

8/19/11

8/26/11

9/2/11

9/9/11

9/16/11

9/23/11

9/30/11

10/7/11

10/14/11

10/21/11

10/28/11

VIX futures

Protective Put

Resetting

Volatility Cap

CPPI

0%

20%

40%

60%

80%

100%

120%

7/29/11

8/5/11

8/12/11

8/19/11

8/26/11

9/2/11

9/9/11

9/16/11

9/23/11

9/30/11

10/7/11

10/14/11

10/21/11

10/28/11

Protective Put

Resetting

Volatility Cap

CPPI