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.
* 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: $0 \leq \alpha \leq 1$
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.
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.