Calculating Leverage Space
So, we have three systems (N = 3) and n = 12. We want to determine our holding period return, or HPR, for a given set of f values — of which there are N, or 3 — so we seek the maximum geometric mean HPR, or GHPR( f1, f2, f3).
We could solve, say, for all values of f1, f2 and f3 and plot out the N + 1 – dimensional surface of leverage space (in this case, a four-dimensional surface), or we could apply an optimization algorithm, such as the genetic algorithm, to seek the maximum “altitude” of the curve. We will focus on the application part of the process that isn’t covered in more generalized texts on mathematical optimization. Although this discussion calls on some equations that may be intimidating for those whose algebra is a bit rusty, “Seeking HPRs” right, summarizes the calculations.
Notice that to determine the GHPR( f1, f2, f3), we must find nHPR( f1, f2, f3), or:
In other words, we go through each row in the joint probabilities table, calling each row “k,” and determine an HPR(k, f1, f2, f3) for each row as follows:
Notice that inside the HPR( f1 · · · fN)k formula, there is the iteration through each column, each of the N market systems, of which we discern the sum:
Assume we are solving for the f values of 0.1, 0.4 and 0.25 for MarketSysA, MarketSysB and MarketSysC, respectively. We would figure our HPR(0.1, 0.4, 0.25) at each row in our joint probabilities table, each k, as shown in the f columns of “Seeking HPRs.” By adding 1.00 to each of the f columns, we obtain the HPRs. We can sum these for each row, and obtain a net HPR for that row.
This example uses fixed f values for each system. In reality, we would optimize the f value based on given constraints and time horizons. Then, we would select those that meet our constraints.
A common set of constraints is to limit drawdowns. The joint probability tables with a given time horizon are used to calculate this. Say we want to find the optimal f and f dollar values to limit drawdown to 20% and have that occur less than 20% of the time. We will perform multiple horizon analysis, as we did for the multiple coin toss game, and calculate returns. We cap our total wealth relative (TWR) at 1.00. (This is our final stake after compounding.) We then look at HPR for the next time horizon. The difference between the TWRs of the previous and current horizons is the run up/down. For example, if TWR is 0.80 for the next horizon, that implies a 20% drawdown.