Once the assets to be traded have been selected, and all possible pairs have been created as well as the forecasts for each of the pairs, we can go to the next stage, to wit, the portfolio construction. We calculate the portfolio weights for separate assets and answer the question “How much are we going to buy or sell and when?”. In order to calculate the portfolio weights, it is necessary to determine the single asset forecasts.
Let us say that at the identification point, our pair forecast for the next day equals 1. In that case the relative single asset forecast equals:
1, for the asset in the numerator, and
•
–1, for the asset in the denominator.
•
The opposite holds if the pair forecast equals –1. The forecast for a single asset is then the sum of all the relative forecasts calculated based on the pair forecasts for all pairs that particular asset appears in.
Here we should bear in mind that the same asset can appear in more than one pair and therefore it is possible that it is undervalued in one pair while it is overvalued in the other. In case of the contradictory relative forecasts (that sum up to 0), the final single asset forecast for that particular asset will be defined as neutral and the proposed weight for that asset will be zero.
The portfolio weight of stock j at the time t is calculated as
, ,
,
( | and Selected) . ( | Selected)
k
i t k
j t
i I i t
p i i I i
W p i i
Ik designates the industry to which the asset belongs, pj, t represents the sum of the relative forecasts for the period t for the asset xj.
Since we are constructing a cash neutral (as well as market neutral) portfolio, we have j Ikpj t, 0, then it also holds jWj t, 0. Based on the previous two equations, it can be easily shown that jWj t, 1.
58 Integration of Forecasting and Portfolio Revision
Finally, under the assumption that the whole disposable capital is invested in stocks, according to the previously calculated portfolio weights, and based on the assumption that the orders have been executed at the opening prices, the portfolio value at the time t equals
1 , , , , 1
1
1 k ( | |)
t t j t j t j t j t
j
PF PF W r c W W
Where xj, t represents the opening price of a stock j at the time , ,
, 1
, j t j t 1
j t
t r x (return of the jth stock), and c represents transaction costs. x
In practice, the calculation of the proposed weights is done every working day after the closing of the markets and the proposals are sent to the traders the following working day before the opening of the markets. The transactions are supposed to be executed at the opening prices on that day, but the execution strategy and the choice of the submission of the orders (whether market or limit orders) are left to be decided by the trader. In the proposal sent to the trader, there is no information on the time-distribution of the trading orders or the time distribution of the quantities to be sold or bought, but only the information on the portfolio weights for the assets to be traded.
Figure 5 shows the total relative value portfolio development (black line) over the period from January 2001 through December 2005, as well as the development of the industry portfolios (grey lines) in the same period. The portfolio rebalancing is done on a daily basis, while the selection of the pairs of assets is done once a year.
The performance of the total relative value portfolio, as well as the performances of the separate relative value industry portfolios shown in figure 5, is calculated using the opening prices with the transaction costs included in the calculation. In practice, the actual trading prices, which are not necessarily equal to the opening prices, should be used for the portfolio evaluation. However, this will be discussed in more detail in section 3.
Table 2 shows the relative value portfolio statistics for the years 2001 to 2005. For the purposes of this calculation, a risk free rate of 2.5% per annum was used.
Although some industry portfolios show a moderate performance, total portfolio performance has been relatively high in the whole period and shows an upward trend.
59 Integration of Forecasting and Portfolio Revision
Figure 5: Performances of the portfolio and the industry portfolios
Table 2: Portfolio Statistics
Return Vola (%) Sharpe Ratio+ Correlation between Portfolio Value and DJ Stoxx 600
2001 19.2 5.8 2.9 –0.7
2002 8.0 6.4 0.9 –0.9
2003 9.2 5.4 1.2 0.9
2004 3.9 2.7 0.5 0.4
2005 6.1 2.0 1.8 0.9
+ Risk free rate = 2.5% p.a.
The total relative value portfolio in figure 5 (black line) is depicted again in figure 6, but this time together with the DJ Stoxx 600 Index. In comparison with the underlying index, total relative value portfolio performance shows less volatility than the Index and it is also evident that at the times where the
60 Integration of Forecasting and Portfolio Revision
Index experienced a downward trend, there is a strong negative correlation between the two (see table 2).
Figure 6: Relative Value Portfolio vs. DJ Stoxx 600
80 90 100 110 120 130 140 150 160 170 180
01-2001 04-2001 07-2001 10-2001 01-2002 04-2002 07-2002 10-2002 01-2003 04-2003 07-2003 10-2003 01-2004 04-2004 07-2004 10-2004 01-2005 04-2005 07-2005 10-2005 150
200 250 300 350 400 Relative Value Portfolio (left)
DJ Stoxx 600 (right)
The main conclusion of this section is that the Relative Value Strategy applied on the subset of the DJ Stoxx 600 shares shows a consistent and good performance. Also, the co movement of the Relative Value portfolio performance and the DJ Stoxx 600 Index is weakening in the “bad times”
(i.e. in the year 2001 and 2002, where the Index experienced a downward trend) and where the correlation coefficient of the two series proves that there is a highly negative correlation. On the other hand, the correlation of the two is highly positive in the periods in which the underlying index shows a consistent upward trend (year 2003 and 2005).
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