Time Diversification under Loss Aversion: A Bootstrap Analysis

Time Diversification under Loss Aversion: A Bootstrap Analysis Wai Mun Fong Department of Finance NUS Business School National University of Singapore Kent Ridge Crescent Singapore 119245 2011 Abstract We examine time diversification from the viewpoint of prospect theory investors. We use a block bootstrap approach to generate returns of U.S. stocks and Treasury bills for time horizons from 1 year to 20 years. On average, bootstrapped value functions are mainly positive and increase monotonically with the time horizon. In contrast, mean-variance optimal portfolios are more conservative, with the optimal proportion of the portfolio invested in stocks declining with time horizon. Our results suggest that time diversification ought to be viewed more favorably by prospect theory investors than by mean-variance investors. Key words: time diversification, bootstrap, prospect theory JEL codes: C15, G11

1. Introduction The question of whether an investor s holding period affects his or her optimal asset allocation is an important one for theory and practice. This issue, which is commonly known as time diversification, has been vigorously debated by academics since Samuelson (1963) s paper which argues that time diversification is a fallacy. Subsequent debate on time diversification has been framed mostly within mean-variance (MV) analysis or expected utility theory. Empirical research, however, indicates that people systematically violate expected utility theory (Starmer, 2000). One important reason why people violate expected utility is that they are loss averse. Loss aversion is central to prospect theory, a descriptive theory based on extensive experimental evidence of how people actually make decisions under risk (Kahneman and Tversky, 1979, Tversky and Kahneman, 1992). In this paper, we examine time diversification from the viewpoint of prospect theory investors. We do so by computing value functions for portfolios of U.S. stocks and Treasury bills (T-bills) across different holding periods. Since historical data is not long enough to provide large numbers of independent long horizon returns, we compute value functions using a block bootstrap approach, which also allows us to capture possible general forms of time dependencies in the data. We show that the optimal proportion of stocks increase monotonically with the time horizon in contrast to results based on MV analysis. This indicates that time diversification should to be viewed more favorably from the viewpoint of prospect theory investors than MV investors. The next section reviews the main ideas behind prospect theory. Section 3 describes our data and elaborates on the bootstrap methodology. Section 4 reports our findings and Section 5 concludes. 1

2. Prospect Theory Prospect theory posits that investors have value functions defined over deviations in wealth from a reference point. The value function is S-shaped, i.e., investors are risk averse for gains and risk seeking for losses. In addition, it has a steeper slope for losses than gains, implying that individuals are loss averse. Loss aversion has emerged as one of the most important properties for understanding economic behavior (Rabin, 2000, Barberis and Huang, 2001). In this paper, we use the original functional form of the value function proposed by Tversky and Kahneman (1992). This can be written as: v x x if x 0 v x x if x 0 where v is the value function defined over outcome X where X > 0 denote gains relative to a reference point. For stocks, a reasonable reference point is the return that could be earned from a risk-free asset. Hence, we define X as excess stock returns over the one-month T-bill rate. The two parameters in the value function are which measures the curvature of the value function and which is the loss aversion parameter. Tversky and Kahneman estimate to be 0.88 and to be 2.25. We use these parameter values to compute our bootstrap value functions. 3. Data and Methodology 3.1 Data Our data comprise monthly returns for U.S. stocks and one-month Treasury bills from January 1927 to December 2010. The data for stocks are total returns from CRSP. Data for Treasury bills are from Ibbotson Associates. Since investors are concerned with the 2

purchasing power of money, we convert the two series of returns into real returns using the Consumer Price Index (CPI). CPI data is from CRSP. 3.2 Bootstrap Methodology The block bootstrap was introduced by Knusch (1989). The idea is to divide the data into overlapping blocks and generate new data by resampling the blocks rather than the individual data so that the blocks mimic the same dependence structure as the original data, whatever that structure is. Large sample properties of the block bootstrap have been extensively studied in the statistics literature (Lahiri 1991, 1992). Jin (1997) shows that the block bootstrap performs well both at the center and tails of distributions. We form overlapping blocks of returns of length n from the original data. We set n to 60 months which is reasonably long enough to capture mean reversion 1. We then randomly resample the overlapping blocks with replacement 1,000 times and stitch them to compute long-horizon returns of 10, 15 and 20 years. We resample stocks and T-bills jointly so that they belong to the same time period as the original data. 3. Results Table 1 reports mean and standard deviation of bootstrap returns for various stocks-bills combinations. Panel B shows that the volatility of returns increases with time horizon but at slower rate than if annual returns were i.i.d. This is consistent with mean reversion. More conservative portfolios exhibit stronger mean-reversion than portfolios where stocks dominate. As a result, portfolios with high Sharpe ratios tend to be those with a high proportion of bills (Table 2). The optimal portfolio (highest Sharpe ratio) at the 5-year horizon is one that has 60% in stocks. The optimal proportion in stocks declines to 50% for 1 We obtain very similar results using n = 120 months. These results are available on request. 3

the 10 and 15-year horizon and to 20% for the 20-year horizon. Hence, from the viewpoint of mean-variance investors, there is no time diversification. [Table 1 and 2 about here] The goal of prospect theory is not to choose the strategy that yields the highest Sharpe ratio but one that maximizes the prospect theory value function. Table 3 reports the average value function for each stock-bill combination. Two results are noteworthy. First, for each horizon, the value function increases monotonically with the percentage of stocks in the portfolio. Second, for a given portfolio, the value function increases monotonically with the time horizon. Overall, the strategy that gives the highest value function is to buy and hold an all-stock portfolio for 20 years. Thus, time diversification should be viewed favorably by prospect theory investors than MV investors. [Table 3 about here] Table 4 reports the lower tails of the simulated value function distribution. The 10 th percentiles (Panel A) are negative except for portfolios with at least 70% stocks held for 20 years. Thus, extremely loss-averse investors should simply hold Treasury bills and not invest in stocks unless they have a long time horizon. The 20 th percentile value functions (Panel B) are more encouraging for stock investors. In particular, the critical values are positive for portfolios with 50% or more in stocks held for at least 10 years. Similar to the average results, the tail values increase significantly with the time horizon. For example, for a portfolio with 70% in stocks, the 20 th percentile goes from marginally positive at the 10-year 4

horizon (0.08) to 0.58 at the 20-year horizon. Thus, given a sufficiently long holding period, even investors who are extremely loss averse can still consider investing most of their wealth in stocks. [Table 4 about here] As a sensitivity check, Table 5 reports results for an investor with loss aversion parameter of 5.25. As expected, the value functions are smaller in every case compared to the individual with 2.25. The other results, however, are qualitatively similar. This can also be seen from Figure 1 which shows that the value function is positive and increasing with the time horizon for all cases except the 1-year holding period. [Table 5 about here] [Figure 1 about here] 4. Conclusion We have used a bootstrap procedure to examine whether time diversification holds for prospect theory investors. Our results support time diversification in the sense that average value functions are positive and increasing with the time horizon. While there is the risk of negative value functions at the extreme left tail, investors who are able to bear such tail risks should still find stocks to be an attractive asset class over the long run compared to bills. References: Barberis, N., and M. Huang (2001), Prospect theory and asset prices, Quarterly Journal of Economics, 116, 1-53. 5

Jing. B.Y (1997), On the relative performance of the block bootstrap for dependent data, Communications in Statistics: Theory and Methods, 26, 1313-1328. Kahneman, D., and A. Tverksy (1979), Prospect theory: an analysis of decision making under risk, Econometrica, 47, 263-291. Knusch, H.R. (1989), The jackknife and the bootstrap for general stationary observations, Annals of Statistics, 17, 1217-1241. Lahiri, S.N. (1991), Second order optimality of stationary bootstrap, Statistics and Probability Letters, 11, 335-341. Lahiri, S.N. (1992), Edgeworth correction by moving block bootstrap for stationary and nonstationary data, in Lepage, R., and L. Billard (eds.), Exploring the Limits of the Bootstrap, Wiley: New York. Rabin, M. (2000), Risk aversion and expected utility theory: a calibration theorem, Econometrica, 68, 1281-1292. Samuelson, P.A. (1963), Risk and uncertainty: a fallacy of large numbers, Scientia, 98, 108-113. Starmer, C. (2000), Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk, Journal of Economic Literature, 28, 332-382. Tversky, A., and D. Kahneman (1992), Advances in prospect theory: cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5, 297-323. 6

Table 1. Mean and Volatility of Simulated Holding Period Returns Panel A reports average (across 1,000 simulation runs) of 1, 5, 10, 15 and 20-year holding period returns for various combinations of stocks and treasury bills indicated in the first column. The column Relative Returns displays the ratio of 5, 10, 15 and 20-year holding period returns to the 1-year holding period returns. Panel B reports in a similar format the standard deviation of holding period returns. All returns are generated using moving block bootstraps with block size of 5 years (60 months). The original data are monthly real returns for the CRSP value-weighted stock portfolio and 1-month Treasury bills from January 1927 to December 2010. Panel A. Mean Holding Period Returns Relative Returns % Stocks 1y 5y 10y 15y 20y 5:1 10:1 15:1 20:1 0.0 0.005 0.041 0.073 0.108 0.133 7.7 13.8 20.5 25.2 0.1 0.014 0.072 0.170 0.288 0.472 5.2 12.4 21.0 34.4 0.2 0.022 0.123 0.263 0.480 0.833 5.5 11.8 21.5 37.3 0.3 0.029 0.160 0.359 0.658 1.106 5.4 12.3 22.5 37.7 0.4 0.033 0.197 0.488 0.862 1.450 5.9 14.6 25.8 43.4 0.5 0.043 0.232 0.580 1.000 1.703 5.4 13.5 23.3 39.6 0.6 0.047 0.283 0.671 1.243 2.054 6.0 14.2 26.2 43.4 0.7 0.068 0.327 0.807 1.427 2.484 4.9 12.0 21.1 36.8 0.8 0.069 0.375 0.916 1.519 2.723 5.4 13.3 22.0 39.4 0.9 0.077 0.358 0.979 1.706 3.136 4.7 12.8 22.3 41.0 1.0 0.101 0.451 1.073 1.912 3.298 4.4 10.6 18.9 32.5 Panel B. Standard Deviation of Holding Period Returns Relative Volatility % Stocks 1y 5y 10y 15y 20y 5:1 10:1 15:1 20:1 0 0.04 0.15 0.22 0.27 0.34 3.9 5.7 7.0 8.7 0.1 0.04 0.15 0.22 0.31 0.45 3.4 5.1 7.1 10.5 0.2 0.05 0.16 0.29 0.45 0.77 2.9 5.3 8.1 13.9 0.3 0.08 0.18 0.36 0.64 1.10 2.3 4.5 8.0 13.9 0.4 0.09 0.22 0.49 0.80 1.52 2.5 5.4 8.9 16.7 0.5 0.11 0.27 0.55 0.91 1.77 2.4 4.8 7.9 15.5 0.6 0.13 0.31 0.67 1.21 2.16 2.4 5.1 9.2 16.5 0.7 0.16 0.39 0.81 1.42 2.77 2.4 5.0 8.8 17.1 0.8 0.16 0.43 0.91 1.57 2.97 2.7 5.6 9.7 18.3 0.9 0.20 0.45 0.98 1.80 3.36 2.3 5.0 9.1 17.0 1.0 0.22 0.53 1.17 1.93 3.46 2.5 5.4 8.9 16.0 7

Table 2. Sharpe Ratios This table reports Sharpe ratios of portfolios comprising stocks and treasury bills for various holding periods. The Sharpe ratio is computed as the average return (across 1,000 simulation runs) divided by the corresponding standard deviation. All returns are generated using the moving block bootstrap described in Table 1 Sharpe Ratios % Stocks 1y 5y 10y 15y 20y 0.0 0.14 0.27 0.33 0.40 0.40 0.1 0.32 0.50 0.77 0.94 1.04 0.2 0.41 0.78 0.91 1.07 1.09 0.3 0.37 0.87 1.00 1.03 1.00 0.4 0.37 0.89 1.00 1.07 0.95 0.5 0.37 0.86 1.06 1.10 0.96 0.6 0.36 0.90 1.00 1.03 0.95 0.7 0.42 0.85 1.00 1.00 0.90 0.8 0.43 0.87 1.01 0.97 0.92 0.9 0.39 0.79 1.00 0.95 0.93 1.0 0.47 0.85 0.91 0.99 0.95 8

Table 3. Value Functions This table reports average prospect theory value functions across 1,000 simulation runs for portfolios of stocks and treasury bills held over various holding periods. The assumed parameter values for the value function are 0.88 and 2.25 based on Tversky and Kahneman (1992). The reference asset is one-month Treasury bill. All returns are generated using the moving block bootstrap as described in Table 1. Holding Periods % stocks 1y 5y 10y 15y 20y 10 1 0.1 0.02 0.07 0.01 0.07 0.22 0.24 0.2 0.01 0.01 0.11 0.30 0.61 0.61 0.3 0.01 0.06 0.22 0.48 0.85 0.85 0.4 0.01 0.10 0.35 0.67 1.14 1.15 0.5 0.00 0.14 0.46 0.80 1.35 1.35 0.6 0.00 0.19 0.53 0.99 1.62 1.62 0.7 0.02 0.23 0.65 1.14 1.96 1.94 0.8 0.02 0.27 0.74 1.21 2.12 2.10 0.9 0.01 0.24 0.79 1.35 2.44 2.42 1.0 0.04 0.33 0.84 1.53 2.56 2.52 9

Table 4. Tails of the Value Functions This table reports lower tails (10 th and 20 th percentile) of prospect theory value functions based on 1,000 simulation runs for portfolios of stocks and treasury bills held over various holding periods. The assumed parameter values for the value function are 0.88 and 2.25 based on Tversky and Kahneman (1992). The reference asset is one-month Treasury bill. All returns are generated using the moving block bootstrap as described in Table 1. 10th Percentile % stocks 1y 5y 10y 15y 20y 10 1 0.1 0.20 0.67 0.79 0.84 0.87 0.67 0.2 0.19 0.56 0.71 0.58 0.49 0.30 0.3 0.25 0.51 0.61 0.49 0.42 0.17 0.4 0.27 0.50 0.55 0.48 0.24 0.03 0.5 0.33 0.45 0.40 0.26 0.03 0.30 0.6 0.36 0.50 0.45 0.31 0.04 0.31 0.7 0.41 0.59 0.46 0.32 0.12 0.53 0.8 0.41 0.57 0.48 0.29 0.02 0.43 0.9 0.49 0.68 0.50 0.44 0.16 0.66 1.0 0.50 0.71 0.67 0.19 0.15 0.65 % stocks 20th Percentile 0.1 0.11 0.38 0.47 0.42 0.34 0.23 0.2 0.11 0.30 0.35 0.10 0.11 0.22 0.3 0.14 0.30 0.24 0.02 0.21 0.35 0.4 0.16 0.24 0.02 0.13 0.27 0.43 0.5 0.18 0.23 0.05 0.19 0.35 0.53 0.6 0.20 0.23 0.07 0.19 0.39 0.59 0.7 0.22 0.21 0.08 0.22 0.58 0.79 0.8 0.23 0.21 0.10 0.25 0.46 0.69 0.9 0.27 0.27 0.10 0.26 0.66 0.93 1.0 0.26 0.26 0.03 0.34 0.67 0.92 10

Table 5. Value Functions (Highly Loss-Averse Investors) Panel A reports average prospect theory value functions across 1,000 simulation runs for portfolios of stocks and treasury bills held over various holding periods while Panel B reports the 10 th percentile of these simulated value functions. The assumed parameter values for the value function are 0.88 and 5.25 based on Tversky and Kahneman (1992). The reference asset is one-month Treasury bill. All returns are generated using the moving block bootstrap as described in Table 1. Average Value Function % stocks 1y 5y 10y 15y 20y 10 1 0.1 0.10 0.31 0.30 0.22 0.07 0.03 0.2 0.08 0.18 0.13 0.10 0.43 0.51 0.3 0.10 0.12 0.03 0.31 0.69 0.79 0.4 0.11 0.06 0.19 0.52 1.02 1.13 0.5 0.12 0.02 0.33 0.69 1.24 1.36 0.6 0.13 0.02 0.39 0.87 1.52 1.65 0.7 0.12 0.05 0.50 1.01 1.86 1.98 0.8 0.12 0.09 0.59 1.07 2.00 2.13 0.9 0.16 0.03 0.62 1.19 2.33 2.49 1.0 0.13 0.12 0.65 1.40 2.45 2.59 % stocks 10th Percentile 0.1 0.46 1.55 1.84 1.95 2.03 1.57 0.2 0.44 1.32 1.65 1.35 1.14 0.70 0.3 0.59 1.19 1.42 1.15 0.98 0.39 0.4 0.63 1.16 1.29 1.11 0.57 0.06 0.5 0.77 1.05 0.93 0.61 0.06 0.70 0.6 0.83 1.17 1.05 0.73 0.10 0.73 0.7 0.95 1.37 1.08 0.76 0.12 1.08 0.8 0.96 1.33 1.13 0.68 0.02 0.98 0.9 1.15 1.59 1.16 1.03 0.16 1.32 1.0 1.16 1.65 1.55 0.45 0.15 1.31 11

Figure 1. Value Function for Various Portfolios and Time Horizon ( 5.25 ) 2.5 2.0 1.5 1.0 0.5 0.0 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y 1 5y 10y 15y 20y 12