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`

.