## Estimation of Expected Returns, Time Consistency of A Stock Return Model, and Their Application to Portfolio Selection

##### Abstract

Longer horizon returns are modeled by two approaches, which have different impact on skewness and excess kurtosis. The Levy approach, which considers the random variable at longer horizon as the cumulants of i.i.d random variables from shorter horizons, tends to decrease skewness and excess kurtosis in a faster rate along the time horizon than the real data implies. On the other side, the scaling approach keeps skewness and excess kurtosis constant along the time horizon. The combination of these two approaches may have a better performance than each one of them. This empirical work employs the mixed approach to study the returns at five time scales, from one-hour to two-week. At all time scales, the mixed model outperforms the other two in terms of the KS test and numerous statistical distances.
Traditionally, the expected return is estimated from the historical data through the classic asset pricing models and their variations. However, because the realized returns are so volatile, it requires decades or even longer time period of data to attain relatively accurate estimates. Furthermore, it is questionable to extrapolate the expected return from the historical data because the return is determined by future uncertainty. Therefore, instead of using the historical data, the expected return should be estimated from data representing future uncertainty, such as the option prices which are used in our method. A numeraire portfolio links the option prices to the expected return by its striking feature, which states that any contingent claim's price, if denominated by this portfolio, is the conditional expectation of its denominated future payoffs under the physical measure. It contains the information of the expected return. Therefore, in this study, the expected returns are estimated from the option calibration through the numeraire portfolio pricing method. The results are compared to the realized returns through a linear regression model, which shows that the difference of the two returns is indifferent to the major risk factors. This demonstrates that the numeraire portfolio pricing method provides a good estimator for the expected return.
The modern portfolio theory is well developed. However, various aspects are questioned in the implementation, e.g., the expected return is not properly estimated using historical data, the return distribution is assumed to be Gaussian, which does not reflect the empirical facts. The results from the first two studies can be applied to this problem. The constructed portfolio using this estimated expected return is superior to the reference portfolios with expected return estimated from historical data. Furthermore, this portfolio also outperforms the market index, SPX.

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