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# Optimal f

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## TOO much risk.

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I firmly believe that the key to effective money management over long periods of time is controlling risk. In fact, most investors and money managers ignore risk and concentrate on return, balancing a portfolio by weighting it highly toward those assets they believe will "perform" the best. In other words they balance a portfolio too heavily toward "risky" assets.

This ignores the possibility of "what can go wrong" and the benefit in such a case of having a certain part of the portfolio in "less risky" assets. This prevents the investor from committing financial anathema: selling assets at a loss solely because too much risk has been assumed.

The average equity mutual fund holds only 4% of its assets in cash, a level that hasn't changed much in the last few years and one of the lowest levels in history. Make no mistake, investing in equity mutual funds is not an allocation process between risky stocks and cash based on valuation; they are a stock market surrogate, pure and simple.

So when investing in a mutual fund, treat it like any other asset or asset class: allocate a portion to it among other assets to maximize risk-adjusted returns. So how do we do this?

Optimal f is a method, a guideline, of managing a portfolio by maintaining a constant proportion called "f" in risky assets. This requires a constant rebalancing based on expected risk-adjusted returns between risk-less and risky assets to keep this proportion the same. This is how it works for a gambler; it works the same for an investor.

Suppose a player is given \$1000 (or any finite amount) and can risk any amount repeatedly in a game where a player has an edge.

If she bets \$1 she wins W dollars with probability P or loses L dollars with probability (1-P) (the total probability is 1).

The best strategy for the player to maximize her "expected winnings" is to risk in each game a fixed fraction f of his total capital such that:

f = ((W X P) - (L X (1-P))) / W

For example, if the game is such that the player betting \$1 wins W=2 dollars with probability P=0.4 or 40% or loses L=1 dollar with probability 1-P = 0.6 or 60%, her optimal f is:

f = ((2 * 0.4) - (1 * 0.6)) / 2 = 10%

The player should always risk 10% of her current capital per game.

If the probabilities are not known for sure, it is possible to prove that for a player it is better to be conservative than aggressive: under-risking will result in the less than maximal winning amount, but over-risking can result in the loss of the whole capital before the probabilities are allowed to work. In portfolio trading practice, this rule translates into a multiple simulation of the portfolio looking at each trade as a probabilistic bet.

The most important point is that when probabilities are not certain, it is better to be more conservative: err by assuming higher risk (a lower probability) for risky assets. This will result in a less than optimal return, but a too aggressive approach may lead to unexpected and unacceptable losses.

In my experience, people gamble (and invest) in the exact opposite manner, assigning too high a probability to risky assets. If they did not, the equity markets would be much lower.

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