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.

4 comments to A Cash is King Theorem

• admin

OK, what about Vietnamese stock market ?

expected return = 16-17%/year, but only 4-5% above the interest rate.

volatility is rather high, say 40%

40%^2= 16%, 4/16 = 1/4

One should keep 70-75% cash in Vietnamese market and invest only 25-30%, unless one finds opportunies with better tan 16-17% expected return.

• Du

Utility function là maximizing wealth phải không ?
Thế thì chắc liên hệ tới Kelly’s criterion – ( bài sau nói Kelly và Bernoulli’s là tương đương)
http://en.wikipedia.org/wiki/Kelly_criterion
Bài của Kelly là binary distribution, nhưng chắc có thể mở rộng ra các distribution khác ( và nếu distribution là liên tục thì có thể xảy ra gambler’s ruin không nếu fully invest ?). Paul Samuelson rất chống lại Kelly, còn cụ Thorpe, tay tổ của Quant investing thì rất thích và viết nhiều bài về Kelly ở đây:
http://edwardothorp.com/id9.html (bài 21, 23 và 24)
và đây
http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/KellyCriterion2007.pdf
Cuốn Fortune’s Formula kể chuyện cụ đi đánh blackjack với Claude Shannon dùng cách đếm bài của cụ và cách đặt theo Kelly criterion.

• Du

À nếu giả sử log-normal thì chẳng bao giờ có gambler’s ruin cả – và tính toán có lẽ đơn giản hơn binary 1 chút thì phải.

• admin

Merci anh Du cho biết mấy thông tin đó ! Em cũng chưa biết Kelly criterion ra sao.