A Cash is King Theorem

You’ve probably heard the street adage “cash is king”. But how much cash ? What is the best ratio of cash that one should hold in a portfolio ? Here in this note I’ll give a quantitative answer.

Assumptions:

* The portfolio consists of cash and a risky asset S (S can be a composite asset, i.e. made of many securities)

* For simplicity, assume that the interest rate is zero (the theorem can easily be modified for the case of non-zero interest, by substracting  the interest rate from the expected return)

* S has expected return $\mu$ and volatility $\sigma$

* Call $\alpha$ the weight of S, and $1- \alpha$ the weight of cash, in the portfolio

* No short-selling: $0 \leq \alpha \leq 1$

Theorem:

Under the above asumptions, we have:

* If $\mu \leq 0$ then the best asset allocation strategy is $\alpha = 0$ (don’t buy any asset with negative expected return)

* If $\mu > \sigma^2$ then the best allocation strategy is $\alpha = 1$ (fully invest when the risky asset has expected return greater than volatility square)

* If $0 < \mu < \sigma^2$ then the best allocation strategy is $\alpha = \mu / \sigma^2$, i.e. one should keep $1 – \mu/\sigma^2$ of portfolio in cash.

Proof. The above value of $\alpha$ is the solution of an optimization problem for the Bernoulli utility function. It’s not difficult. Maybe I’ll write down the proof later.

Practical application:

Let’s say you have an asset (say wonderstock) whose expected 1-year return is 50%, but which has volatility 100%. Then you should buy that asset with only 50% of your money, and keep the remaining 50% in reserve. (If your wonderasset falls down significantly or rise up, you can rebalance your portfolio so that the ratio will become 50/50 again).

If your asset is expected to grow  6% more than the interest rate, and volatility = 20%, then it’s still OK to be fully invested (20%^2 = 4% < 6%). But when the volatility shoots up to 30%, then it’s time to keep 1/3 of your portfolio in cash.