I read a lot these days about folks who believe there is a "correction" coming, and they are "holding cash" in order to be able to buy shares of stock at reduced prices.

Suppose I want to buy IBM at reduced prices, so I hold cash in case IBM goes down.

Is holding cash a good idea? Will it pay off?

It depends.

Either I hold cash, or I don't hold cash.

Either IBM goes up, or IBM goes down.

There are 4 outcomes:

- I hold cash, IBM goes down: I buy shares of stock at reduced prices.
- I hold cash, IBM goes up: I reap little return on held cash, and I lost out on a (paper) capital gain.
- I don't hold cash, IBM goes down: I have a (paper) capital loss.
- I don't hold cash, IBM goes up: I have a (paper) capital gain.

I can control whether or not I hold cash, but I cannot control Mr. Market.

What "should" I do?

Before I can answer that question, I will discuss probability and expected return.

**"****Everybody talks about the weather, but nobody does anything about it."**

Wikipedia defines "probability" as "a measure or estimation of likelihood of occurrence of an event."

What does it mean when the weather report says, "There is a 27% probability of rain"? Either it will rain or it won't rain, so what does probability mean in this context?

It means that in the past, on days like today, it rained an average of 27 times out of 100.

What does it mean to be a day "like" today? How similar does another day have to be to today, in order to be considered as "like" today? Must they have the same temperature? Same pressure? Same humidity? Perhaps it rained on 5% of days with a temperature "like" today, it rained on 22% of days with a temperature and pressure "like" today, and it rained on 27% of days with a temperature, pressure, and humidity "like" today.

Perhaps if we use more dimensions, we can achieve a higher probability, which would enable us to make more accurate forecasts.

**What is "Expected Return"?**

Wikipedia defines "expected return" as "the weighted-average outcome in gambling, probability theory, economics or finance". Expected return is computed as "the probability of each possible return outcome … multipl[ied] by the return outcome itself".

Here's an example. I could play the following game. I pick a card from a standard deck of cards. If I pick a 2, 3, or 4, I win $2. If I pick a 5, 6, 7, 8, or 9, I win $1. If I pick a 10, Jack, Queen, King, or Ace, I lose $3.

My expected return is (3/13 * $2) + (5/13 * $1) + (5/13 * -$3), which is $0.4615 + $0.3846 - $1.1538, which is -$0.3077. In any one play of the game, I could be lucky and win $2, or I could be unlucky and lose $3, but over time, if I play the game repeatedly, on average I will lose over $0.30 each time I play.

I wouldn't choose to play that game.

**"The Envelope, Please"**

Here's another game. I must choose one of two envelopes, one of which is green and the other is yellow. One envelope contains a note that says "You win $1", the other envelope contains a note that says "You lose $1", but I can't tell which is which.

My expected return is (the probability of choosing the envelope that contains a note that says "You win $1" * $1) + (the probability of choosing the envelope that contains a note that says "You lose $1" * -$1).

Observe that **without knowing the probabilities, there is no way to compute the expected return.**

What if I learn that there is a 75% probability that the green envelope contains a note that says "You win $1"?

I might reason, "I will choose the green envelope 75% of the time, and the yellow envelope 25% of the time". 75% of the time I choose the green envelope, and 75% of those times the green envelope contains a note that says "You win $1", but 25% of those times the green envelope contains a note that says "You lose $1"; 25% of the time I choose the yellow envelope, and 25% of those times the yellow envelope contains a note that says "You win $1", but 75% of those times the yellow envelope contains a note that says "You lose $1". My expected return is (0.75 * 0.75 * $1) + (0.75 * 0.25 * -$1) + (0.25 * 0.25 * $1) + (0.25 * 0.75 * -$1), which is $0.25. In any one play of the game, I could be lucky and win $1, or I could be unlucky and lose $1, but over time, if I play the game repeatedly, on average I will win $0.25 each time I play.

Before I congratulate myself, I should realize that have a better strategy open to me.

What if I choose the green envelope 100% of the time?

My expected return is (0.75 * $1) + (0.25 * -$1), which is $0.50. In any one play of the game, I could be lucky and win $1, or I could be unlucky and lose $1, but over time, if I play the game repeatedly, on average I will win $0.50 each time I play.

It might seem weird, strange, bizarre, counterintuitive, or silly to never choose the yellow envelope, but it's hard to argue with the math.

What if I learn that there is a 51% probability that the green envelope contains a note that says "You win $1"?

Any time there is a greater-than-50% probability that the green envelope contains a note that says "You win $1", I should choose the green envelope 100% of the time.

**Are the costs and/or benefits quantifiable?**

Sometimes the costs and/or benefits are difficult or impossible to quantify.

Here's a real-life example. When I leave my home this morning, should I take an umbrella or not?

There are 4 outcomes:

- I do take an umbrella and it does rain: I'm dry but my arm hurts from holding up the umbrella.
- I do take an umbrella and it doesn't rain: I'm dry but I've carried an umbrella for nothing.
- I don't take an umbrella and it does rain: I'm soaked.
- I don't take an umbrella and it doesn't rain: I'm dry and my arm feels fine.

How do we quantify the cost of my arm hurting, or the benefit of not being soaked?

**Should I hold cash in case IBM goes down?**

There are 4 outcomes:

- I hold cash, IBM goes down: I buy shares of stock at reduced prices.
- I hold cash, IBM goes up: I reap little return on held cash, and I lost out on a (paper) capital gain.
- I don't hold cash, IBM goes down: I have a (paper) capital loss.
- I don't hold cash, IBM goes up: I have a (paper) capital gain.

If I hold cash, my expected return is (the probability of IBM going down * the benefit of buying shares of stock at reduced prices) + (the probability of IBM going up * (the little return on held cash, and the psychic pain of a lost (paper) capital gain)).

If I don't hold cash, my expected return is (the probability of IBM going down * the cost of the (paper) capital loss) + (the probability of IBM going up * the benefit of the (paper) capital gain).

It is possible to quantify "the benefit of buying shares of stock at reduced prices" - if the shares I want cost $100 today, and IBM goes down by 10%, then the benefit is saving $10 per share.

It is possible to quantify "the little return on held cash" - it might be the current interest rate on a bank account.

It is possible to quantify "the cost of the (paper) capital loss" - if my shares are priced at $100 today, and IBM goes down by 10%, then my (paper) capital loss is $10 per share.

It is possible to quantify "the benefit of the (paper) capital gain" - if my shares are priced at $100 today, and IBM goes up by 10%, then my (paper) capital gain is $10 per share.

It is difficult to quantify "the psychic pain of a lost (paper) capital gain", although Wikipedia cites the work by Tversky and Kahneman that losses hurt twice as much as gains.

What is the probability of IBM going up or going down? Without knowing the probabilities, there is no way to compute the expected return.

What does it mean when someone says, "There is a 27% probability of IBM going up"? Either it will go up or it won't go up, so what does probability mean in this context?

It means that in the past, on days like today, it went up an average of 27 times out of 100.

What does it mean to be a day "like" today? How similar does another day have to be to today, in order to be considered as "like" today? Must they have the same EPS? Same P/E? Same ROE? Perhaps it went up on 5% of days with an EPS "like" today, it went up on 22% of days with an EPS and P/E "like" today, and it went up on 27% of days with an EPS, P/E, and ROE "like" today.

I don't know the probability of IBM going up or going down.

I doubt it anyone knows the probability.

Without knowing the probabilities, there is no way to compute the expected return.

If I can't compute the expected return of holding cash, and the expected return of not holding cash, then how will I know which to choose?

**Conclusion**

I believe it is not possible to know the probability of IBM, or any one company, or even the market itself, going up or going down.

Therefore I believe it is not possible to compute the expected return.

Therefore I believe it is not possible to know ahead of time if holding cash is a good idea, or if it will pay off.

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