Fig 1.
a: The trend of Berkshire A stock price, b: Berkshire A stock price returns. Fig 1A shows Berkshire A stock price trend from 1980/3/17-2018/09/28. The figure shows that Berkshire A stock price rises very quickly. It also shows that the return volatility for Berkshire A is also relatively high. Fig 1A shows that the stock price fell significantly during the period of the recent financial crisis, but the stock price also rises faster after the financial crisis. Fig 1B shows the return of Berkshire A stock price from 1980/3/17-2018/09/28. Fig 1B shows that the volatility of stock price return was significantly higher in 1987 and 2008 than in other periods.
Fig 2.
a: Buffett-factor model returns, b: five-factor model returns, c: Buffett-factor model returns and five-factor model returns for the sub-period (2008–2009). Fig 2A shows the returns of the replicating asset constructed by using the Buffett-factor model. Fig 2B shows the returns of the replicating asset constructed by using the five-factor model. We can find that the overall behavior patterns of the two replicating assets are similar to those of the original Berkshire A stock. Fig 2C overlays the information presented in Fig 2A and 2B in one plot using a shorter time frame, 2008–2009. This figure shows that the information presented in Fig 2A and 2B, which depict the returns of the replicating assets obtained from using the Buffett- and five-factor models are not identical to each other. In the figure, the maximum return of the Buffett-factor model is 7.2%; the minimum return is -0.068%. The maximum return of the five-factor model is 6.26%; the minimum return is -0.0733.
Fig 3.
a: Replicating asset price of Buffett-factor model, b: Replicating asset price of five-factor model, c: Replicating asset price of Buffett-factor model and replicating asset price of five-factor model for the sub-period (2008–2009). Fig 3A shows the replicating asset price when the replicating asset is constructed using the Buffet model. Fig 3B shows the replicating asset price when the replicating asset is constructed using the five-factor model. Overall price behavior closely matches the price behavior of the original Berkshire A price. Fig 3C overlays the information presented in Fig 3A and 3B in one plot using a shorter time frame, 2008–2009. This figure shows that the information presented in Fig 3A and 3B, which depict replicating asset price when the replicating asset is constructed using the Buffet- and five-factor models, are not identical to each other. The maximum value of replicating assets price of Buffett- factor model is $141,109; the minimum value is $73,088. The maximum value of the five-factor model is $140,284; the minimum value is $73,375.
Fig 4.
The relationship of expected return, a, and c when the replicating asset is constructed by the Buffet-factor model. This figure presents the relationship between the expected return, a, and transaction cost when the replicating asset is constructed by the Buffett-factor model. This example uses parameters α = 0.006439, η = 0.05401, c = 0.001~0.007.
Fig 5.
The relationship of expected return, a, and c when the replicating asset is constructed by the five-factor model. This figure presents the relationship between the expected return, a, and transaction cost when the replicating asset is constructed by the five-factor model. This example uses parameters α = 0.01702, η = 0.05748, c = 0.001~0.007.
Table 1.
The expected return of optimal statistical arbitrage of Berkshire A and replicating asset pair.
Fig 6.
a: c vs a for Buffet-factor model, b: c vs expected return for Buffet-factor model. Fig 6A shows the relationship between transaction cost and Fig 6B shows the relationship between transaction cost and the expected return from the optimal trading strategy. The replicating asset for Fig 6 is constructed using the Buffett-factor model.
Fig 7.
a: c vs a for five-factor model, b: c vs expected return for five-factor model. Fig 7A shows the relationship between transaction cost and a. Fig 7B shows the relationship between transaction cost and the expected return from the optimal trading strategy. The replicating asset for Fig 7 is constructed using the five-factor model.
Fig 8.
a: The behavior of the synthetic asset formed from Berkshire A stock price and the replicating asset BerkA* when the replicating asset is constructed using the Buffett-factor model, b: The behavior of the synthetic asset formed from Berkshire A stock price and the replicating asset BerkA* when the replicating asset is constructed using the five-factor model. Fig 8A presents the relationship between the synthetic asset price and the entry level "a" and exit level "m." In Fig 8A, the results are computed using the Buffett-factor model to construct the replicating asset. In computing Fig 8A, α = 0.006439, η = 0.05401, and the "a" and "m" are obtained assuming c = 0.001. "c" represents the transaction cost; "a" represents the entry level; "m" is the exit level. Fig 8B presents the relationship between the synthetic asset price and the entry level "a" and exit level "m." In Fig 8B, the results are computed using the five-factor model to construct the replicating asset. In computing Fig 8B, α = 0.01702, η = 0.05748, and the "a" and "m" are obtained assuming c = 0.001. "c" represents the transaction cost; "a" represents the entry level; "m" is the exit level.
Fig 9.
The relationship of expected return, a, and c for the optimal statistical arbitrage of the S&P500 using the replicating asset constructed by the Buffet-factor model. This figure presents the relationship of expected return, a, and transaction cost for the optimal statistical arbitrage of the S&P500 using the replicating asset constructed by the Buffett-factor model. This example uses parameters α = 0.008735, η = 0.007170, c = 0.001~0.007.
Fig 10.
The relationship of expected return, a, and c for the optimal statistical arbitrage of the S&P500 using the replicating asset constructed by the five-factor model. This figure presents the relationship of expected return, a, and transaction cost for the optimal statistical arbitrage of the S&P500 using the replicating asset constructed by the five-factor model. This example uses parameters α = 0.009195, η = 0.007369, c = 0.001~0.007.
Table 2.
The expected return of optimal statistical arbitrage of S&P 500 and replicating asset pair.
Table 3.
The expected Sharpe ratio of optimal statistical arbitrage of Berkshire A and replicating asset pair.
Table 4.
The expected Sharpe ratio of optimal statistical arbitrage of S&P500 and replicating asset pair.