Posted by CANbike on Thu, 20 Feb 2014

[Finance] Coach Potato Rebalancing with New Cash

The Canadian Coach Potato is a simple passive investment strategy using indexed funds/ETFs. It’s aim is to match the average market performance which outperforms the majority of active portfolios.

There are variances but the most common portfolio would consist of four indexed funds/ETFs allocated with the following ratios:

  • 40% Bonds
  • 20% Canadian Equity
  • 20% US Equity
  • 20% International Equity

Over time, the ratio will change and the portfolio needs to be rebalanced to maintain the asset allocation. Typical rebalancing is once a year if necessary.

Another option, though, is to add new cash and rebalance it in the process. No funds are to be sold. The problem is then to solve for the optimal distribution of cash into the four funds/ETFs.

New Cash + Ideal Rebalancing

Value = Bonds + Canadian Equity + US Equity + Foreign Equity

Let’s represent this as variables

V = B+C+U+F

Ideally,

B = 0.4V
C = 0.2V
U = 0.2V
F = 0.2V

Investing new cash (N) the equation is

V+N = B+C+U+F+N

Let Xi, where i = {b,c,u,f}, represent the coefficient ratios to be calculated. N*Xi is then the amount of cash to add to asset i.

B+N*Xb = (V+N)0.4
C+N*Xc = (V+N)0.2
U+N*Xu = (V+N)0.2
F+N*Xf = (V+N)0.2

Solving for N*Xi gives the values to rebalance the portfolio. These values may be positive or negative. Negative values mean funds/ETFs need to be sold.

New Cash Additions Only

Given the stipulation that no funds may be sold, focus will be on finding an optimal distribution of cash.

Additional constraints are:

  • Xi is now range bound as funds/ETFs cannot be sold. Xi>=0 as it cannot be negative.
  • Since ΣN*Xi=N, it can be derived that ΣXi=1 and Xi<=1, given Xi>=0.

The new cash (N) may not be enough to rebalance the portfolio. But, it should be distributed as close as possible. Let Zi represent the ideal coefficient ratios to be calculated. Where N*Zi represents the amount of cash to add to asset i.

Thus,

B+N*Zb = (V+N)0.4
C+N*Zc = (V+N)0.2
U+N*Zu = (V+N)0.2
F+N*Zf = (V+N)0.2

Rearranging the above results in the ideal unbound solution.

Zb = [(V+N)0.4-B]/N
Zc = [(V+N)0.2-C]/N
Zu = [(V+N)0.2-U]/N
Zf = [(V+N)0.2-F]/N

Now let

~Xi = {Zi ,if Zi>=0
      {0  ,if Zi<0

to satisfy the first constraint.

However, ~Xi could be greater than 1, and may not satisfy Σ~Xi=1. The simple solution is to normalize the coefficients to satisfy ΣXi=1.

Thus, Xi = ~Xi/(Σ~Xi), the normalization of ~Xi.

The amount of cash to add to asset i is N*Xi.


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