This is called the maxmin expected utility, and is often used to derive robust strategies. However, the strategy given by this approach often implies that the investor should not participate in the market. Although robust, this strategy is meaningless for investors who wish to make a profit by entering the market.
Finally, we present a numerical example using market data of stocks listed on the Tokyo Stock Exchange. We choose 20 names of stocks with relatively low PERs (Price Earnings Ratio) around 1 to 5, noting that the average PER of stocks on the Tokyo Stock Exchange is about 15; i.e., Fullcast Holdings Co., Ltd. (code: 4848), Daiichi Commodities Co., Ltd. (8746), Fuji Oil Co., Ltd. (5017), FIDEA Holdings Co., Ltd. (8713), Yoshicon Co., Ltd. (5280), PADO Corporation (4833), Sado Steam Ship Co., Ltd. (9176), Joban Kaihatsu Co., Ltd. (1782), Meiwa Estate Company Limited (8869), Oi Electric Co., Ltd. (6822), Takata Corporation (7312), Toei Reefer Line Ltd. (9133), Nihon House Holdings Co., Ltd. (1873), Sanei Architecture Planning Co., Ltd. (3228), Shinhoku Steel Corporation (5542), Daiko Denshi Tsushin Ltd. (8023), Shinnihon Corporation (1879), Asahi Industries Co., Ltd. (5456), Seiwa Electric MFG. Co., Ltd. (6748), and Daisue Construction Co., Ltd. (1814).
We sampled historical data of these names from 26 March 2015 to 25 May 2015 and applied the Phillips–Ouliaris cointegration test with a p-value 0.05. We then found six pairs of cointegration from 190 () pairs; i.e., (Daiichi Commodities Co., Ltd. (8746), Asahi Industries Co., Ltd. (5456)), (Fuji Oil Co., Ltd. (5017), Sado Steam Ship Co., Ltd. (9176)), (Fuji Oil Co., Ltd. (5017), Takata Corporation (7312)), (PADO Corporation (4833), Oi Electric Co., Ltd. (6822)), (PADO Corporation (4833), Seiwa Electric MFG. Co., Ltd. (6748)), and (Sado Steam Ship Co., Ltd. (9176), Daiko Denshi Tsushin Ltd (8023)).
Having found the pairs, we can estimate parameters of Ornstein–Uhlenbeck processes,
, via maximum likelihood estimation; e.g., the parameter of the pair value of Daiichi Commodities Co., Ltd. (8746) and Asahi Industries Co., Ltd. (5456) is given by
To calculate the optimal boundary, we further need parameters and . Here we set , according to the Monthly Report of the Bank of Japan issued in May 2015, where it was reported that yields on 10-year government bonds were moving in the range of 0.40–0.45 percent.
depends on the investors’ trustfulness for the model. Low
implies low confidence while high
implies high confidence. Here, we consider four cases; i.e.,
. Note that the case
is consistent with the case of the boundary
. Using these parameters, we calculate boundaries
and show them in Figure 1
together with pair values of stocks from 25 May 2015 to 26 June 2015 that are outside the sample. The figure reveals that the boundary
; i.e., to
. Furthermore, the distance from
decreases. This is consistent with the theory.
Furthermore, we consider the performance of our pair trading strategy using real market data. The trading policy is set as follows. The position is set when the pair value touches either the boundary
or the mean-reverted point
. If the position is set for the case that the pair value touches
, the position is liquidated when the pair value touches the mean-reverted point, and vice versa. After the liquidation, the next position is set when the pair value touches either
and this position is liquidated by the same rule, and so forth. According to this trading policy, we simulate the rate of return for pair trading, using a sample from 25 May 2015 to 2 September 2015. The rate of return of this trading policy is presented in Table 1
] pointed out, an important obstacle for the pair trading to make a profit is the transaction fee. One transaction of pair trading requires transaction fees to be charged twice, owing to the requirement of two stocks. Furthermore, there may be frequent transactions for pair trading if the investor often determines profits for the movement of the pair value. We thus discounted the loss due to the transaction fee from the rate of return. The results given in Table 1
are these discounted returns. Although the transaction fee depends on the market maker, 0.1% for one transaction may be used in the Japanese market. We discounted 0.2% for one transaction because our trading policy is based on pair trading.
For the evaluation of the above result, we compare the result with the rate of return of another trading policy. We here consider the buy-and-hold strategy. Because all stocks used in this numerical example have small PERs, they may be good candidates for the buy-and-hold strategy.
Similarly for the trading policy of pair trading discussed above, we used the sample ranging from 25 May 2015 to 2 September 2015. The buy-and-hold strategy requires only the buying of a stock on 25 May 2015 and the liquidation of the stock on 2 September 2015. For this simple strategy, the rate of return is summarized in Table 2
Except for a few stocks, almost all stocks make a loss for this strategy, at least in the term considered. We note that even if we choose the group of stocks using criteria other than the PER and the buy-and-hold strategy makes a big profit, it may not be stable. Table 1
gives the stability of pair trading. This is why we suggest the trading strategy of pair trading while taking model uncertainty into account.