Real Estate Volatility Index and Its Economic Significance

Transcription

1 Real Estate Volatility Index and Its Economic Significance ABSTRACT We develop a 30-day forward-looking real estate volatility index (RVX) based on the fundamental Arrow (1964) and Debreu (1959) state preference approach using US Real Estate Investment Trusts (REITs). It is shown that RVX is a useful predictor of future REIT realized volatility, as reflected by an explanatory power (measured by the adjusted R-squared in the regression of future REIT realized volatility on RVX) of 71.61%. In comparison, the Chicago Board Options Exchange (CBOE) volatility index (VIX) has an explanatory power (measured by adjusted R-squared) of 58.86% on the next 30-day S&P 500 realized volatility. We further explore a pair of economic applications of RVX. First, by uncovering the asymmetric relation between the daily changes in RVX and the REIT index, we show that RVX plays a counterpart role to the VIX relative to the stock market, in the sense that RVX serves as the investor fear gauge for the real estate market. Second, we develop a natural trading strategy based on the probability distributions of historical volatility and the last trading day s RVX. We then provide a direct test of this strategy using the REIT exchange traded funds (ETFs) as instruments and show that it is effective and profitable. These analyses provide evidence that RVX contains important information regarding future REIT volatility and returns and, therefore, the value added by RVX is economically significant. JEL classification: G12; G14; R33 Keywords: REITs; Volatility index; State preference approach; Economic significance; Fear gauge; Timing 1

2 1. Introduction In this paper, we develop a real estate volatility index (RVX) and explore a pair of its economic applications. Market volatility is no doubt an important factor in investment and risk management. The Chicago Board Options Exchange (CBOE) Volatility Index (VIX) launched in January 1993 measures the market s expectation of the next 30-day volatility implied by S&P 500 Index option prices 1 and is the premier benchmark for the stock market volatility. Developed even earlier in April 1988, the Merrill Option Volatility Expectations (MOVE) index, calculated as a weighted average implied volatility on 1-month Treasury options, 2 is usually deemed as the equivalent of the VIX for the US Treasury bond market. In addition, applying the VIX methodology, CBOE has developed a number of volatility indexes for other financial/commodity markets such as the Crude Oil Volatility Index (OVX), Gold Volatility Index (GVZ), EuroCurrency Volatility Index (EVZ), Emerging Markets ETF Volatility Index (VXEEM), China ETF Volatility Index (VXFXI), Energy Sector ETF Volatility Index (VXXLE), etc. 3 As one of the major investment asset classes, 4 surprisingly, to date real estate does not have its own market risk measure. The urgency of developing a real estate volatility index is further supported by the rapid expansion of the REIT industry over the last two decades. Figure 1 shows the market capitalization trend of all US REITs. With a market capitalisation of $15.91 billion at the end of 1992, the REIT market grew to $ billion at the end of 2002 and further to $ billion by the end of 2012, representing an average growth of 185% per year. 5 This speedy growth is not unreasonable. While investment into direct property assets requires a large outlay of funds and these assets generally lack liquidity, REITs offer an indirect avenue for average investors to gain access to the real estate market. More importantly, REITs possess economic importance because they offer considerable investment benefits including diversification, liquidity, continuous outperformance, high dividend yields, growth 1 On 22 September 2003, CBOE updated the computational methodology of VIX and changed the underlying index from S&P 100 Index to S&P 500 Index. 2 These 1-month Treasury options are across 2-year, 5-year, 10-year and 30-year US Treasury securities, with a weight of 20%, 20%, 40% and 20%, respectively. 3 For a complete list and detailed information of the volatility indexes at CBOE, refer to 4 Investment or asset groups may be classified into equity, cash, fixed interest and property (Brailsford, Heaney and Bilson, 2011). 5 Also note the decrease of market capitalisation of REITs during

3 in real value and transparency. 6 Today, almost all aspects of the economy are closely linked to REITs, for example, apartments, hospitals, hotels, industrial facilities, nursing homes, offices, shopping malls, storage centres, student housing and timberlands (NAREIT, 2013). Consequently, we rely on REITs to develop the real estate volatility index. Consistent with the VIX and MOVE, the RVX, computed on a daily basis, is a 30-day forward-looking volatility measure. The method we employ is the time-state preference approach developed by Arrow (1964) and Debreu (1959). As one of the most general frameworks in financial economics for contingent claims pricing under uncertainty, this approach offers distinctive advantages. First, unlike the VIX and other CBOE volatility indexes whose computation process is rather complicated and not easy to understand, our model constructed based on the state preference approach possesses a strong theoretical underpinning, and therefore is intuitive and easy to implement. In simple words, the RVX at time t is just the sum of the expected real estate volatility in all the possible states at time t, where the real estate volatility in each state s is estimated as the state price (i.e. the price of a security that pays $1.00 at time t state s and zero otherwise) multiplied by the payoff (i.e. the expected real estate market volatility when the stock market is at time t state s). Second, as just mentioned in the brief explanation of the RVX construction, our model incorporates all the possible market states/conditions and hence captures the stochastic characteristic of volatility. One measure of market risk in the REIT literature is to use REIT market beta. However, as market beta is commonly assumed to be constant in many of the asset pricing models, it cannot capture stochastic volatility and hence different market states. In addition, the VIX, MOVE and other CBOE volatility indexes are all weighted averages of implied volatility on certain options. These weighted averages likewise fail to reflect this important attribute of volatility and this could possibly explain the outperformance of the RVX over the VIX in terms of predicting the next 30-day volatility in each respective market. After the construction of the RVX, we first examine its statistical significance. Our results show that RVX is a useful predictor of REIT realized volatility, as reflected by an explanatory power (measured by the adjusted R-squared in the regression of REIT realized volatility on RVX) of 71.61%. 6 See Appendix A for a brief discussion of each of these benefits. 3

4 In comparison, the VIX has an explanatory power (measured by adjusted R-squared) of 58.86% on the next 30-day S&P 500 realized volatility. As such, the statistical importance of RVX is supported. We further examine the economic significance of RVX by exploring a pair of its economic applications. 7 First, as the VIX is often referred to as the investor fear gauge for the stock market (Whaley, 2000; Whaley, 2009), we assess if RVX plays a counterpart role to the VIX and serves as an investor fear gauge for the real estate market. Our analysis supports this role by showing that RVX captures the asymmetry of real estate investors fear. That is, RVX increases at a higher rate when the real estate market goes down than it decreases when the real estate market is bullish. Second, we examine whether RVX can be used for market timing for investors. We develop a simple and natural trading strategy based on the probability distributions of historical volatility and the last trading day s RVX and provide a direct test of this strategy using the REIT exchange traded funds (ETFs) as instruments. Results show that our trading strategy is highly effective and profitable. These analyses taken together provide evidence that RVX contains important information regarding future REIT volatility and returns and, therefore, the value added by RVX is also economically significant. The contributions of this study are four-fold. First, building on a solid theoretical framework, RVX is the first standardized ex-ante volatility measure for the real estate market. Embracing the stochastic characteristic and all the possible states in RVX, our aim is that RVX will be well-accepted and become the core systematic risk measure for this specific market. Second, we provide evidence to support the economic importance of RVX. Specifically, we show that RVX fills the void of the real estate investor fear gauge and is also a useful indicator for market timing strategies. In addition, RVX has far more potential applications (see Section 5.3 Other Potential Applications for a brief discussion). In this sense, we contribute to not only academia but also industry. Third, the trading strategy we propose using RVX as a timing signal is innovative and natural. It can be simply described as: If the probability of getting the same return under the conditional distribution (i.e. the probability distribution conditional on the most recent RVX observed) is greater than that under the 7 The usefulness of RVX is not limited to the two applications examined in this paper. See Section 5.3 Other Potential Applications for more potential applications of RVX. 4

5 unconditional distribution (i.e. the probability distribution of historical volatility), we shun the REIT market by selling or short selling the REIT exchange traded fund (ETF). This timing approach based on the conditional and unconditional probability distributions is new to the literature. Giot (2005) develops a trading strategy to examine the relation between the VIX and future stock index returns. Although he utilizes the probability distribution of historical VIX, his probability distribution is only based on a rolling two-year history of VIX and therefore not an unconditional distribution. Moreover, he does not make use of the probability distribution conditional on VIX (i.e. conditional distribution). The intuition of our approach is straight forward. Using information extracted from the conditional and unconditional distributions, we are essentially comparing the RVX with the historical volatility while keeping the return constant. Comparison with the methods employed in the prior studies suggests at least two advantages of our approach. On one hand, our approach of timing strategy takes into account both return and volatility, whereas Copeland and Copeland s (1999) method only considers volatility (their trading rule is based on the percentage deviation of today s VIX from its 75-day historical average). On the other hand, unlike Breen, Glosten and Jagannathan (1989) and Chen and Liang (2007), our approach does not rely on any model and thus avoids any potential modeling errors. Last but not least, following the idea of this study and applying the fundamental and powerful state preference approach, similar ex-ante volatility indexes can be developed for any individual REIT, any other asset classes and countries if needed, as long as the options data for computing the state prices are available. The remainder of this paper proceeds as follows. Section 2 details the intuition of the state preference approach and the process of developing RVX. Section 3 describes the data, sample filtering criteria and presents the sample descriptives. Section 4 documents the results of RVX and evaluates its statistical significance. The economic applications of RVX are explored in Section 5. Section 6 concludes. 5

6 2. Developing the Real Estate Volatility Index 2.1 Intuition of the State Preference Model The fundamental pricing equation for any asset is PP tt = EE tt (MM tt+1 XX tt+1 ) (1) where PP tt is the price of the asset at time t, EE tt is the conditional expectations at time t, MM tt+1 is the stochastic discount factor, and XX tt+1 is the asset s payoff at time tt + 1. The stochastic discount factor MM tt+1 represents the marginal rate of substitution between consumption in the current period and consumption at time t, state s. In other words, MM tt+1 (a.k.a. state price) is the y in equation (2) below, denoting the number of units of consumption that investor k would be willing to give up today in exchange for 1 unit of consumption at time t, state s. As this makes no difference to the investor s utility, δuu = 0 yyuu CC kk ππ kk tttt UU CC kk tttt = 0 (2) where y is the units of consumption that investor k would be willing to give up today, U is investor s kk kk utility, CC 0 is investor k s consumption in the current period, ππ tttt is investor k s estimate of probability of time t state s occurring, and CC tttt kk is investor k s consumption at time t state s. Rearranging equation (2), we get yy = ππ tttt kk UU CC kk tttt UU CC kk 0 (3) In general, the state price MM tt+1 may differ across investors, as individual investors may have different quantities for y. However, in complete markets, asset prices PP tt are the same for all investors, resulting in the same state prices MM tt+1. Hence, the Arrow (1964)-Debreu (1959) state preference model is expressed as SS PP 0 = φφ tttt dd tttt ss=1 (4) where PP 0 is the price of any asset or portfolio, φφ tttt is the state price, and dd tttt is the payoff at time t state s. 6

7 2.2 Developing RVX using State Preference Model As any security or portfolio of securities can be priced as the state price multiplied by the payoff, we propose the following theoretical model for RVX SS RRRRRR tt = φφ mm,tttt EE[RR RRRR,tttt RR mm,tttt ] ss=1 (5) 2 2 where RRRRXX tt is the real estate volatility index, φφ mm,tttt is the state price, and EE RR RRRR,tttt RR mm,tttt is the expected real estate market volatility when the stock market (proxy for the general market) is at time t state s Estimation of State Price As the state of the overall economy is most commonly reflected by the stock market, we estimate the state prices based on the S&P 500 Index options. Furthermore, if one wanted to estimate the state prices based on the REIT market states, it would be infeasible since the options on the REIT indexes are not actively traded and according to Whaley (2009), to create similar volatility indexes to the VIX, the only important requisite is that the underlying index option market has deep and active trading across a broad range of exercise prices. Following Breeden and Litzenberger (1978), the state prices in equation (5), φφ mm,tttt, are estimated using the delta securities approach. That is, φφ KK ii, KK ii+1 = ee rrrr [NN(dd 2 (KK ii ) NN(dd 2 (KK ii+1 )] (KK ii < KK ii+1 ) (6) NN dd 2 (KK ii ) = llll(ss 0/KK ii )+ rr dd σσ 2 /2 TT σσ TT (7) where φφ KK ii, KK ii+1 is the delta securities, measuring the prices of the securities with a payoff of $1 if the market at time t is between KK ii and KK ii+1, rr is the continuously compounded risk free rate, NN(dd 2 ) is the standard normal cumulative distribution function, SS 0 is the current S&P 500 Index level, KK ii is originally the option price in the Black and Scholes (1973) model and herein the prespecified market level in state i, d is the continuously compounded dividend yield, σσ is the implied volatility, and T is time to maturity in years. With this method, the increment of KK ii is discretionary depending upon 7

8 different research objectives. In this paper, we choose an increment of 0.10 for KK ii to create states ranging from 300 to 2400 (i.e. approximately 50% of the minimum to 150% of the maximum of the S&P 500 Index levels during 3 January August 2012). 8 This fairly wide range with a fine grid of increment ensures that all the possible states (i.e states at each time t) are taken into account Estimation of the Implied Volatility For equation (7), the challenging part is the input of implied volatility, σσ. We estimate it as the average of four implied volatilities on the near-the-money S&P 500 options with time to maturity closest to 30 days (i.e. one call and one put with maturity less than 30 days, one call and one put with maturity more than 30 days). 9 Ideally, an at-the-money series should be used since they are most actively traded and consequently their prices contain timely information. However, when at-themoney options are not available, we use the next best alternative, which is near-the-money options. The 30-day timespan is chosen for our RVX to keep pace with VIX and MOVE. In addition, Fleming, Ostdiek and Whaley (1996) point out that implied volatilities on short-term options tend to exhibit greater variability than those on longer-term options. By keeping the constant 30-day to expiration, we eliminate the impact of this concern on RVX (Whaley, 1993). In line with Carr and Wu (2006), the following filtering rules are applied: 1) Call (put) options with strike price smaller (greater) than the forward index level are excluded, because in-the-money options are less liquid than at-the-money and out-of-the-money options. This is consistent with the construction of VIX, which is computed based on selected out-of-themoney calls and puts on S&P 500 centered around an at-the-money strike price. The forward 30 (rr index level is calculated as SS 0 ee 365 ), where SS 0 is the current S&P 500 Index level and r is the continuously compounded risk free rate. 8 The minimum and maximum of the S&P 500 Index levels during 3 January August 2012 are and , respectively. The mean is with a standard deviation of (total observations of 3925). 9 Occasionally when the call and put with maturity less than 30 days are not available, we estimate the implied volatility as the average of the two implied volatilities on the call and put with maturity more than 30 days. 8

9 2) Options with the best bid price less than $0.05 are removed, ensuring that all the bids are strictly positive. 3) Options with time to maturity less than 7 days are further removed to mitigate the microstructure effect, as Whaley (1993) and Fleming, Ostdiek and Whaley (1996) note that implied volatilities on options become a lot more volatile during the last week of trading. 4) Options with an implied volatility of greater than 1 or smaller than 0 are dropped Estimation of the Payoff 2 2 The payoff in equation (5), EE RR RRRR,tttt RR mm,tttt, is the expected variance of the REIT returns at time t state s conditional on the stock market being at time t state s. It is estimated as 2 2 RR RRRR,tttt = αα tt + ββ tt RR mm,tttt + εε tttt (8) where RR RRRR,tttt is the REIT market variance at time t state s and RR mm,tttt is the stock market variance at time t state s (i.e. the prespecified states), measured as llll [(KK ii + KK ii+1 )/2SS 0 ]. The estimates of αα tt and ββ tt used for all the states at time t are generated using a rolling regression of prior 365 calendar days. 11 After estimating the state prices and payoffs respectively, the sum of their products across all the states at time t is the RVX squared at time t. We then take the square root of this number and divide it by 22/252 to get the annualized RVX. 10 This is just one of the methods to estimate the expected payoff. See Hastie, Tibshirani and Friedman (2009) for other nonlinear methods. 11 We also try rolling regressions with a window of prior 60 and 180 calendar days as well as growing regressions. The rolling regressions with a window of prior 365 calendar days achieve the best outcome in terms of the beta coefficient estimate and adjusted R-squared. 9

10 3. Data and Descriptive Statistics 3.1 Data The options data on the S&P 500 Index used to estimate the state prices, including the continuously compounded risk free rate r, time to maturity T, continuously compounded dividend yield d and implied volatility σσ (see equations (6) and (7)), are sourced from OptionMetrics. The risk free rate used in the option pricing models in OptionMetrics is the zero curve, which is a series of continuously compounded zero-coupon interest rates with various maturities, derived from British Bankers Association (BBA) LIBOR rates and settlement prices of Chicago Mercantile Exchange (CME) Eurodollar futures. 12 Since there is no convexity adjustment on this zero curve, when the 30- day zero-coupon interest rate on a specific date is not available, we derive it by either linearly interpolating or extrapolating on the zero curve. The current dividend yield defined in OptionMetrics is the most recently announced dividend payment divided by the most recent closing price for the security. The implied volatility calculated by OptionMetrics is by setting the theoretical price of the European-style options using the Black-Scholes model equal to the market price, which is the midpoint of the option s best closing bid price and best closing offer price. The current index level of S&P 500, i.e. SS 0 in equation (7), is sourced from CRSP. The returns on the S&P US REIT Index 13 used as a proxy for the REIT market, i.e. RR RRRR,tt used in the rolling regressions to generate the estimates of αα tt and ββ tt in equation (8), are obtained from Datastream. Our sample period covers an extensive period of 3 January August 2012, 14 including different market conditions, special economic events and across the Global Financial Crisis (GFC), which is usually deemed to be initiated from the real estate market. 12 The detailed derivation of the zero curve is demonstrated in Chapter 3 Calculations, Ivy DB File and Data Reference Manual, version 3.0, rev. 19/05/2011, pages The S&P US REIT Index covers the investable universe of the U.S. publicly traded REITs and updates its constituents timely to reflect the market s overall composition (First Trust S&P REIT Index Fund Fact Sheet, 2014). 14 The options data on the S&P 500 index are available from 4 January When estimating the payoff in the state pricing model (i.e. equation (8)), we use a rolling regression with a window of the prior 365 calendar days. Therefore, the RVX starts from 3 January

11 3.2 Descriptive Statistics Recall that the input of implied volatility σσ in equation (7) is estimated as the average of four implied volatilities on the near-the-money options with time to maturity closest to 30 days. Table 1 provides the descriptive statistics of all the selected near-the-money options and the average implied volatility calculated therein. The Date difference shows that the selected options have a mean expiration of calendar days with a minimum of 7 and maximum of 80 calendar days. The Price difference, calculated as the absolute value of strike minus spot price for all the selected calls and puts, shows a mean of $5.49 with a minimum of $0 and maximum of $ The average implied volatility estimated from these options is 20.41% p.a. with a standard deviation of 8.17% p.a.. Table 2 presents the descriptive statistics of the S&P US REIT Index and S&P 500 Index returns used in the rolling regressions to generate the estimates of αα tt and ββ tt in equation (8). It can be seen that the S&P US REIT Index returns have a higher mean but also a higher standard deviation than the S&P 500 Index returns for the period of 4 January August The correlation between the pair is reasonably high at 66.81%. The summary statistics of the αα tt and ββ tt estimates from the rolling regressions are reported in Table 3. These estimates are then used in the estimation of the payoff in equation (8). The mean of the intercept estimates is , significant at the 1% level, indicating that the stock market variance cannot fully represent the real estate market variance and hence the necessity of creating a specialized volatility index for the real estate market. The mean of the beta coefficients is 0.94, significant at the 1% level, supporting that the stock market volatility is a significant factor in explaining real estate market volatility. The average adjusted R-squared of 30.23% reinforces that the stock market volatility has some explanatory power for the real estate volatility but cannot substitute RVX. 15 Only 44 out of selected options have a price difference between strike and spot price that exceeds $30 (i.e. 0.32%). 16 The rolling regressions have a window of prior 365 calendar days. Therefore, the descriptive statistics start from 4 January

12 4. Results and Performance Evaluation 4.1 Results of RVX The summary statistics of RVX and VIX are presented in the first two rows of Table Over the sample period, RVX has a mean of 17% p.a., which is less than that of VIX (22.49%), indicating that REIT investments on average are less risky than equities. This result is consistent with Chan, Hendershott and Sanders (1990) who conclude that real estate, represented by the equity REITs, is less risky than stocks. The standard deviation of RVX (15.29%) is much greater than that of VIX (8.64%). The range of RVX is between 1% and %. 4.2 Effectiveness of RVX While Figure 2 shows that RVX coincides closely with the REIT realized volatility, to assess the predictive power of the RVX on the subsequent 30-day REIT realized volatility, we run the following regression RRRRRRll RRRR,tt,tt+22 = αα 1 + ββ 1 RRRRXX tt + εε tt 18P (9) where RRRRRRll RRRR,tt,tt+22 is the REIT realized volatility for the period of time t to t+22 (measured in trading days) and RRRRXX tt is the ex-ante REIT volatility for the same period, available at time t. The REIT realized volatility is computed as: RRRRRRRR RRRR,tt,tt+22 = ln SS&PP UUUU RRRRRRRR IIIIIIIIII LLLLLLLLLL tt+ii 2 22 ii=1 (10) &PP UUUU RRRRRRRR IIIIIIIIII LLLLLLLLLL tt+ii 1 The summary statistics of the REIT realized volatility is shown in the third row in Table 4. All the statistics on the REIT realized volatility are higher than those of RVX. The correlation between RVX and REIT realized volatility is 84.63%. The regression results for equation (9) are reported in Table 5. The coefficient estimate on RVX is 1.25 and statistically significant at the 1% level. It is also significantly different from unity thereby indicating that the realized volatility of REITs is on average higher than RVX, consistent with the statistics reported in Table 4. The estimated intercept is not significantly different from zero indicating 17 The daily figures for RVX are available from the authors upon request and can be made available to public. 18 As the realized volatility is ex-post whereas RVX is ex-ante, we have ensured that they represent the same period. 12

13 that there is no information that RVX fails to capture in terms of predicting future REIT volatility. The adjusted RR 2 of the model is 71.61%. To infer the strength of this goodness-of-fit, we compare it to the adjusted RR 2 from the regression of realized volatility on S&P 500 on VIX: RRRRRRll mm,tt,tt+22 = αα 2 + ββ 2 VVVVXX tt + ηη tt (11) where RRRRRRll mm,tt,tt+22 is the S&P 500 realized volatility from time t to t+22 (measured in trading days) and VVVVXX tt is the ex-ante stock market volatility for the same period, available at time t. The S&P 500 realized volatility is calculated in the same way as the REIT realized volatility, i.e. RRVVVVll mm,tt,tt+22 = llll SS&PP 500 LLLLLLLLll tt+ii 2 22 ii=1 (12) SS&PP 500 LLLLLLLLll tt+ii 1 The summary statistics of the S&P 500 realized volatility can be found in the last row in Table 4. In contrast to the REIT market, the S&P 500 realized volatility has a lower mean than VIX. Their pairwise correlation is 76.73%, smaller than the 84.63% for RVX and REIT realized volatility. Regression results for equation (11) in Table 6 show a statistically significant coefficient estimate on VIX of The estimated intercept is also significantly different from zero, implying that VIX fails to fully capture the information contained in the future market volatility. Remarkably, the explanatory power (measured by adjusted RR 2 ) of VIX on S&P 500 realized volatility is only 58.86%, significantly weaker than that (i.e %) of RVX on REIT realized volatility. In summary, it is shown that RVX outperforms the VIX in terms of predicting its own market realized volatility, and hence the statistical significance of RVX is supported. 5. Economic Significance of RVX In this section, we demonstrate the usefulness of RVX through a pair of its economic applications. We first assess if RVX is competent at gauging real estate investors fear. By developing a novel and natural trading strategy, we then test if RVX can be used as a market timing indicator. A series of other potential applications of RVX are also briefly discussed. 13

14 5.1 Real Estate Investor Fear Gauge It is fitting that the VIX is called the investor fear gauge. Whaley (2000) explains that investor indicates that the VIX is set by investors and conveys their consensus view about the next 30-day stock market volatility, fear reflects the risk-averse attribute of investors and gauge means a measure. The reason why the VIX is called the investor fear gauge is because it spikes during periods of market turmoil, regardless of whether the turmoil is due to stock market decline, the threat of war, an unexpected change in interest rates or any other newsworthy event (Whaley, 2009). Whaley (2009) further shows that the VIX rises more when the stock market is retreating than it falls when the market is advancing. In addition, Giot (2005) also offers an examination of the linkage between the old VIX in use before 2003 and its contemporaneous S&P 100 index (OEX) returns. In line with Whaley (2000), he not only finds that a significantly inverse relation exists between the VIX and S&P 100 index, but that the relation is asymmetric as described above. Similarly, we not only expect that RVX reflects the investors consensus forecast of the next 30- day real estate market volatility, but more importantly, it can capture the asymmetry of real estate investors fear. That is, when the real estate market is not doing well, investors become more concerned and, hence, the RVX increases at a higher rate than it decreases when the real estate market is bullish. To formally test this proposition, following Whaley (2009), we run the following regression: ΔΔΔΔΔΔXX tt = γγ 0 + γγ 1 ΔΔSSSSSSSS tt + γγ 2 ΔΔSSSSSSSS tt + uu tt (13) where RRRRXX tt is the daily rate of change in RVX, measured as llll (RRRRXX tt /RRRRXX tt 1 ), SSSSSSEE tt is the daily rate of change in the S&P US REIT Index, measured as llll (SSSSSSSS LLLLLLLLll tt /SSSSSSSS LLLLLLLLll tt 1 ), SSSSSSEE tt is the daily rate of change in the S&P US REIT Index conditional on the change being negative, i.e. SSSSSSEE tt = SSSSSSEE tt if SSSSSSEE tt < 0 and SSSSSSEE tt = 0, otherwise. If the proposition is true, we expect the intercept to be insignificantly different from zero and both coefficients γγ 1 and γγ 2 to be significantly negative. Results are reported in Table 7. Newey and West (1987) standard errors and covariance matrix with a lag length of 8 are employed to account for residual heteroscedasticity and autocorrelation. 14

15 The regression results in Table 7 are fully consistent with our expectations. The estimated intercept is not significantly different from zero. This indicates that if the real estate market (represented by the S&P US REIT Index) does not change over the day, then RVX should also exhibit negligible change. The estimated coefficient on ΔΔSSSSSSSS tt is , statistically significant at the 1% level. This supports the inverse relation between movements of expected volatility and prices documented in Whaley (2009) and Giot (2005). The estimated coefficient on ΔΔSSSSSSSS tt is , statistically significant at the 5% level. This negative coefficient reflects the asymmetry in the relation between REIT returns and change in RVX and supports the view that RVX can be taken as a real estate investor fear gauge. To see this more clearly, if the S&P US REIT Index rises by 100 basis points, the RVX will fall by: ΔΔΔΔΔΔXX tt = (1.00) = % However, if the S&P US REIT Index falls by 100 basis points, the RVX will increase by: ΔΔΔΔΔΔXX tt = ( 1.00) ( 1.00) = % Since the magnitude of the RVX increase when the real estate market goes down exceeds the counterpart magnitude of the RVX decline when the RE market goes up, we conclude that the RVX is a plausible gauge of real estate investors fear. 5.2 Market Timing with RVX The exploration of RVX as a market timing indicator is primarily motivated by Copeland and Copeland (1999). Using the deviation between today s VIX and the 75-day historical average, all divided by the 75-day historical average as a market timing signal, Copeland and Copeland (1999) find that following the VIX timing signal to switch between portfolio strategies style (value versus growth) or size (large cap versus small cap), produces positive future excess returns. Specifically, they find that when the VIX increases, value and large-cap portfolios outperform their counterparts whereas when VIX decreases, growth and small-cap portfolios outperform. Accordingly, the trading rule that they propose is that whenever today s VIX is x percent higher than its prior 75-day moving average, they switch into a portfolio that is long value (large cap) and short growth (small cap); 15

16 conversely, whenever today s VIX is x percent lower than its prior 75-day moving average, they long growth (small cap) and short value (large cap). The implementation of this strategy is by changing positions in futures contracts (to avoid high transaction costs and the need for a stand-alone hedge fund strategy) representing each of these four portfolios. Their results support the ability of the VIX to effectively identify good performers in different time periods (e.g. value (growth) would outperform growth (value) following an increase (decrease) in the VIX) leading to enhanced portfolio returns. As RVX functions as the counterpart of the VIX in the real estate market, it is reasonable to expect it to possess some timing ability. The extant literature on timing using implied volatility is scarce. Giot (2005) investigates whether the VIX is a leading indicator of stock index returns. Building on the hypothesis that (very) high implied volatility indicates an oversold market and, hence, underpriced security 19, he develops a trading strategy to test whether a (very) high level of implied volatility would bring positive future index returns. Specifically, he first calculates the 20 equally spaced percentiles (i.e. 5, 10,, 95 percentiles) based on a rolling two-year history of VIX. For example, at time t, he generates the 20 percentiles using all the historical VIX values from the prior two years until the last trading day. Then today s VIX is compared to these 20 values and assigned a rank. Other things equal, the higher is today s VIX, the higher its rank. If today s VIX is higher than the maximum of the historical VIX levels, it will be assigned a rank of 21. Then for each of these 21 ranks, assuming a long position in the S&P 100 index, he computes 1-, 5-, 20- and 60-day forward-looking returns. His hypothesis predicts that the higher ranks would generate significantly positive returns. Giot s finding is that only rank 21 (extremely high VIX) produces significantly positive returns, whereas rank 1 (extremely low VIX) produces significantly negative returns and all the other ranks have insignificant results. Fleming, Kirby and Ostdiek (2001) also focus on the economic value of volatility timing. The economic value measured in their approach is embodied through the asset allocation strategies (specifically, for short-horizon risk-averse investors) using predictable changes in volatility. In a 19 As explained in Giot (2005), during (very) high implied volatility periods, as investors fear increases, they would indiscriminantly sell the assets to try to mitigate potential losses. This is believed by some market practitioners to be overreacting and indicate an oversold market. As too much selling would drive the asset prices down, these market participants regard this to be a good signal to buy in the hope of earning positive future asset returns. 16

17 mean-variance optimization framework and assuming four asset classes (i.e. stocks, bonds, gold and cash, proxied by corresponding futures contracts), the two trading strategies that they propose are: either aiming to maximize expected return for a given level of volatility (the maximum return strategy) or to minimize volatility for a given level of expected return (the minimum volatility strategy), an investor would dynamically adjust the optimal asset weights in his portfolio based on the one-stepahead estimates of the conditional covariance matrix. By comparing these two strategies with the unconditional mean-variance efficient static strategies with the same target expected return and volatility, their results reveal significant outperformance of the former and support the economic value added by volatility timing. Another study underpinning the worthiness of examining economic importance can be traced back to 1989 when Breen, Glosten and Jagannathan evaluated the economic significance of predictability of nominal interest rates on nominal stock excess returns. Based on the documented statistically significant negative correlation between nominal stock excess returns and nominal interest rates in the literature, Breen, Glosten and Jagannathan (1989) build a forecasting model of xx tt+1 = ββ 0 + ββ 1 rr ffff + εε tt+1 where xx tt+1 is the monthly excess market return, calculated as the CRSP valueweighted or equally-weighted market index return minus the one-month treasury bill rates; and rr ffff is risk-free interest rate, proxied by the one-month treasury bill rate. Assuming only two types of assets (i.e. stocks and T-bills), they then develop a trading rule, as follows. At the end of month t (when the following month interest rate is known), if the model predicts a positive excess market return in month t+1, the portfolio manager would invest all the funds in the stock index; otherwise, all funds would be invested in the T-bills. The beta coefficients used to forecast the month t+1 excess stock returns are obtained from the linear projection of the prior 36 months. Breen, Glosten and Jagannathan (1989) find that while this timing portfolio produces a mean return of 2 basis points higher than that of the value-weighted index, its standard deviation is about 22% lower. The estimated economic value of using the negative correlation between the value-weighted (but not equally-weighted) index return and risk-free interest rate to time the market and shift funds according to the trading rule would be 17

18 worth an annual management fee of 2% of the assets being managed. Therefore, the main conclusion drawn is that the forecast ability of the one-month Treasury bill rates is economically significant. A more recent article by Chen and Liang (2007) is also relevant to our study in the sense that they examine whether market timing ability exists in the hedge funds industry and if yes, the associated economic significance. The timing ability defined by them is the ability of a fund manager to effectively adjust portfolio beta (i.e. exposure to the equity market) when observing a signal about the future market. They further categorize the timing into three types, i.e. return timing, volatility timing and joint timing. Specifically, for the return timing, a fund manager would increase (decrease) the portfolio beta when the market return is expected to go up (down), whereas the volatility timing manager would increase (decrease) the portfolio beta when the market is expected to be less (more) volatile. The joint timing measures the manager s ability to forecast the market return and volatility simultaneously. Using Treynor and Mazuy s (1966) and Henriksson and Merton s (1981) return timing models, Busse s (1999) volatility timing model and Chen and Liang s (2007) own proposed joint timing model, the authors find the presence of all the three types of timing ability in the selfdescribed market timing hedge funds at both the aggregate and individual fund level. More importantly, they conclude that the timing ability delivers significant economic value to investors RVX and Expected REIT Returns In this section, we provide a foundation for the trading strategy developed in the next section. The literature documents evidence of the link between implied volatility and expected returns. Using a regression analysis, Banerjee, Doran and Peterson (2007) show that the current VIX level (but not VIX innovations) can predict future 30- and 60-day excess returns on the S&P 500 index. The significantly positive relation found by them is equivalent to saying that when VIX is increasing, the S&P 500 prices will decrease. Although the authors cannot conclude whether VIX is a priced risk factor, they ascertain that the VIX levels and innovations have strong predictive ability for future portfolio returns. In addition, from Giot s (2005) point of view, it is also not unreasonable to expect the predictability of implied volatility on forward-looking returns, since the implied volatility has been 18

19 shown to be a good predictor for future realized volatility (Christensen and Prabhala, 1998; Blair, Poon and Taylor, 2001). For the RVX, another way of interpreting it as the real estate investor fear gauge is that if the expected market volatility increases, investors become concerned and demand higher returns and hence the asset prices will fall (French, Schwert and Stambaugh, 1987; Whaley, 2000; Whaley, 2009). Figure 3 shows the daily movements of RVX and the S&P US REIT Index from 3 January August The inverse relation between changes in RVX and the S&P US REIT Index is particularly evident after January 2007 (see the green box). For example, during the period of the GFC (February December 2008), RVX spiked upward while the S&P US REIT Index tumbled downward. Subsequently, until 30 August 2012, the movements of RVX and the S&P US REIT Index are also always in the opposite direction, with RVX showing a decreasing trend and the S&P US REIT Index exhibiting an increasing trend. To formally assess the relation between daily changes of RVX and S&P US REIT Index levels, we perform the following regression for the period of 3 January August 2012: ΔΔSSSSSSSS tt = ββ 0 + ββ 1 ΔΔRRRRRR tt + εε tt (13) 20 where ΔΔSSSSSSSS tt is the daily rate of change in the S&P US REIT Index, measured as ln (SSSSSSSS LLLLLLLLll tt+1 /SSSSSSSS LLLLLLLLll tt ) and ΔΔRRRRRR tt is the daily rate of change in RVX, measured as ln (RRRRXX tt+1 /RRRRXX tt ). In light of the pattern observed in Figure 3, it is expected that the estimated intercept to be not significantly different from zero whereas the coefficient estimate β 1 to be significantly negative. The regression results in Table 8 broadly conform with the pattern observed in Figure 3. The estimated intercept is not significantly different from zero. This indicates that if RVX does not change over the day, the S&P United States REIT Index will remain roughly the same. This is consistent with the mean daily return of of the REIT index for the test period. More importantly, the slope coefficient estimate is negative and significant at the 1% level. This confirms that if at time t, RVX is expected to increase, the predicted REIT index level will fall from time t to t Note the difference from Section 5.1. While Section 5.1 examines the contemporaneous relation between change in RVX and REIT returns, this section focuses on the changes in RVX and future REIT returns. 19

20 Trading Strategy using RVX Based on the significant inverse relation between changes in RVX and the S&P US REIT Index demonstrated in Section 5.2.1, in this section we develop a natural and easily implemented trading rule using the probability density functions (pdfs) of historical volatility and the most recently observed trading day s RVX. We then perform direct tests of this trading rule using REIT ETFs as the instrument for the sample period of 10 May August 2012 (in total 1334 trading days). 21 To provide a clear illustration, we decompose the trading strategy into three steps. Step 1. Find the 5% 22 return cutoff value from the probability density function of historical volatility, i.e. the unconditional distribution of REIT returns. In line with the state price estimation for our RVX, we follow Black and Scholes (1973) and assume that the distribution of possible REIT prices is log normal. That is, by the law of large numbers, the average REIT returns are assumed to be normally distributed. Hence, mean and standard deviation are the only two moments required to determine the shape of the distribution. From the historical REIT returns, we generate the probability distribution for each trading day on a growing window basis. That is, we fix the first return observation at the start of the sample period (i.e. 3 January 1997) and augment the sample size by one trading day each time. With this method, the probability distribution incorporates all the available data at time t and best obeys the law of large numbers. From this unconditional distribution (see the distribution in red in Figure 4), we obtain the 5% return cutoff value (xx tt ) for each trading day t (see the 5% cutoff point in Figure 4). Step 2. Estimate the probability that a return is less than xx tt from the probability density function of RVX, i.e. the distribution of REIT returns conditional on last trading day s RVX. At time t, conditional on last trading day s RVX (RRRRXX tt 1 ), we generate the risk-neutral probability distribution with conditional mean of rrff tt 1 1 RRRRXX 2 2 tt 1 and conditional volatility of RRRRXX tt 1. From this conditional distribution (see the distribution in black in Figure 4), we find the probability (pp tt ) that a possible return is less than xx tt. As the pdf of historical volatility in Step 1 is estimated using daily 21 The First Trust S&P REIT Index Fund incepted on 8 May 2007 and its data are available since 10 May We also test other cutoff points including 6%, 7%, 8% and 9%. Results are reported in Table 9. 20

21 REIT returns, in this step the risk-free rate and RVX used in the estimation of the conditional pdf are both converted from annual to daily figures. Step 3. If the probability (pp tt ) that a return is less than xx tt is greater than 5%, which signals the increase of the RVX, we expect the REIT market return to decrease, so will the S&P US REIT Index level. Therefore, as risk-averse investors, today we sell or short sell the ETF tracking the S&P US REIT Index, which is the First Trust S&P REIT Index Fund (FRI) in the hope to mitigate potential losses in the predicted falling REIT market. 23 We calculate the average return for a period of 1 day, 1 week and 1 month after selling the REIT ETF and test the significance of these returns. The First Trust S&P REIT Index Fund daily prices are downloaded from Datastream. The reasons why we choose the ETF as the instrument to implement our trading strategy are threefold: First, Index ETFs are normally a good representative of the underlying index. The First Trust S&P REIT Index Fund aims to track the S&P US REIT Index prices and returns. Although the performance of the ETF is not guaranteed to be exactly the same as that of the underlying index, they should be fairly close. Second, trading ETFs incurs low trading costs. ETFs are traded on an exchange. Therefore, trading ETFs is more like trading stocks and involves brokerage fees. Figure 5 gives a snapshot of the online broker firms available for trading the First Trust S&P REIT Index Fund. As can be seen, the typical flat rate commission fee ranges from $4.50 to $10. For a trade with a large investment amount, this fee is negligible. Last but not least, for liquidity reasons. ETFs possess much higher liquidity than REIT index futures as evidenced by the large daily volume traded. The results of our trading strategy are reported Table 9. Panel A presents the results when 5% is used as the cutoff point. The first column indicates the period including 1 day, 1 week and 1 month after selling the REIT ETF. The second column records the number of days where the REIT ETF is sold. As can be seen, the trading rule is implemented around 40% 24 of the time during the sample period. The periodic return is calculated as ln (FFFFFF CCCCCCCCCCCCCC PPPPiiccee tt /FFFFFF CCCCCCCCCCCCCC PPPPPPPPee tt+ii ) where ii = 1, 5 aaaaaa 22 trading days, respectively. The average periodic returns are reported in the third 23 Chen and Liang (2007) also argue that it is more important for a volatility timer to get out of the market during volatile periods. 24 The total observations during the sample period of 10 May August 2012 are /1334=40.4%; 535/1334=40.1%; 518/1334=38.83%. 21

22 column with the corresponding standard deviation in column four. The results show that the strategy earns a positive return of 0.11% if the REIT ETF is purchased the next day. The return is even higher at 0.35% (1.36%) after one week (after one month). The significance becomes stronger with longer horizons with the average daily return significant at the 10% level, followed by the average weekly return at the 5% level, then by the average monthly return significant at the 1% level. This pattern is reasonable given that RVX is an estimate for the next 30-day volatility. The last three columns in the table report the number of cases where each single transaction produces positive, zero or negative returns. Following our trading rule, for any individual transaction with the 1 month horizon, the chance of an investor earning positive returns is 7.53% 25 higher than making a negative return. There is no theory operationally useful in guiding how to choose the optimal cutoff. Therefore, we also test our trading rule using various alternative cutoff points including 6%, 7%, 8% and 9%. Results are included in Panels B - E in Table 9. All of them strongly support the effectiveness and profitability of our trading rule at the weekly and monthly interval. First, as the cutoff return gets larger, the number of days where transactions take place should be decreasing. If the number of days does not change much, the signal to us would be that the riskneutral probability distribution conditional on last trading day s RVX has fat left tail, which would violate our assumption that REIT returns are normally distributed. Second, referring to the column of Mean Periodic Return, there is no strong evidence that the expected daily return from our trading rule is significant. This is not surprising given such a short time (i.e. one day) for the REIT index price to respond to the change in RVX. It is also consistent with the insignificant intercept estimate in equation (13). However, given a bit more time, at the weekly and monthly interval, all the mean periodic returns are statistically significant at the 1% level. More importantly, these average returns are all significant in an economic sense. For example, at the weekly interval, the descriptive statistics in Table 10 report a mean of REIT returns of 0.02% with a standard deviation of 5.45%. However, with our trading strategy, using a cutoff point of 6%, an investor could achieve a return of 0.55% with a lower standard deviation of 2.79%. The weekly higher return-lower / /518 = 7.53% 22