**FX Quant > **Numerical Simulations and Evidence of Profitability

Below are given return-leverage graphs for the **
Fx Quant 11** strategy (V.10/2008), which
started live trading on November 1, 2008. Actual trading results after this date
(until August 2011, when v.10/2008 was replaced by v.08/2011)
can be found on the **FXQ Real Performance** page.

**Figure 1.** A very important feature of
the Fx Quant 11 system is stationarity, or mean-reversion, of the system leverage. We define
the system leverage as a ratio of the **total long currency exposure** (all long
portfolio components, USD denominated) to **Net Asset Value** **(NAV)**.
Numerical experiments show that the leverage oscillates/ reverts
to its long term average of 0.77. This can help us
determine the optimal timing to start trading (or NOT to start
trading).
The system either: 1.) makes a new profit (the leverage decreases), or 2.) the leverage
increases (the equity curve temporarily retraces).

As time passes, the leverage oscillates around its mean, BUT the return makes a new high - see the diagram above.

**Figure 2. **We define the **
crossover VAMI **(the **red line** in diagram
above) as a **value of VAMI when the leverage crosses over the mean leverage**
of 0.77. Note that the crossover VAMI monotonically rises on every
leverage crossover. Although the leverage goes up and down all the time,
it eventually reverts and crosses over its mean (0.77), making a new VAMI high
on each crossover. This is a strong evidence of profitability of the system; as
long as the leverage reverts to mean (at least it did so in the past), the
system makes new profit. The above graph shows that, however large the drawdown
and leverage are, the system recovers from losses and sets a new VAMI peak.
Sometimes it takes months (and large losses in account), but patience is needed
to see a new profit.

**Figure 3.** There is a very strong negative
correlation between the monthly rate of return (ROR) and the month-to-month (MTM)
leverage change. An increase in monthly leverage, however, produces not as
sharp drop in monthly ROR. The opposite also holds true: a positive monthly ROR
does not decrease the leverage as much as a negative ROR increases the leverage (see the graph below).

**Figure 4.** The correlation between the monthly ROR
and leverage change can mathematically be described
as:

**Predicted Monthly ROR = 2.115% - 2.44% x Leverage Change**

Let us sum the above formula across all months from the testing interval. Leverage changes will cancel each other, i.e. their sum will be zero (remember the leverage change has a zero mean) and we come to conclusion that

**Cumulative ROR = Sum(Monthly RORs) = N x
2.115% = 118 x 2.115% = 249.6% **(non-compounded)

where N=118 is the number of months in the testing interval

Hence the **Average
Predicted Monthly** **ROR = 2.115%** (look at the intercept on the Monthly ROR axis). This
correlation was persistent and the intercept was always positive, regardless the
choice of system parameters. This brings us to a conclusion that, in long run,*
the system should be profitable, if a reasonably large leverage is used.*
**Using too large a leverage can deplete the trading account, despite the long
term system
profitability.**

You can open this
**Excel table** and click on the
**Drawdown & Corel. Analysis** and** ROR vs. Lever. Change **tabs to see calculations
for the latest version of the system (V.10/2008) .

**Figure 5.** 5-Day
ROR Histogram

The above histogram shows that:

1. There is very regular,
bell-shaped, thin tailed distribution of 5-day rolling rates of return . Robust systems, which are *not curve-fitted* usually have
this type of distribution.

2. The 5-day rolling rate of
return (ROR) mean is *positive* and equals 0.37%. There is a 50%
probability that the 5-day rate of return will be greater than 0.37% and 50%
probability the return will be less than 0.37%.

3. 5-Day RORs are more clustered in the positive territory in the histogram above. There is a 28.5% probability that a 5-day ROR will be negative and a 71.5% probability the 5-day ROR will be positive.

4. The 5-day
Value-at-Risk (VaR)
for the 95% confidence level is 1.01%. This means that the risk of loss in a
5-day period is 1.01%, with a 95% probability. The probability of loss of more
than 1.01% in a 5 day period is 5%. Similarly, the 5-day VaR for the 99%
confidence level is 2.20%. **Excessive negative RORs do and will happen, although
with decreasing probability.** To see these calculations open the **5 Day ROR
Percentile** tab in this
Excel workbook.

Detailed actual and backtesting performance reports can be found in the
**FXQ Real Performance** and**
FXQ Back testing** section of our web site.