As a value investor I have often found myself highly dependent on thinking in terms of “margin of safety” when it comes to value. But Michael Mauboussin, Chief Investment Strategist at Legg Mason Capital Management and author of *More Than You Know,* suggests there is a better way to quantify value and opportunity: **Expected Value**.

It may have been Buffett and Graham who popularized the renowned principal of “margin of safety” (or discount to fair value), but it was also Buffett who defined Mauboussin’s definition of expected value:

“Take the probability of loss times the amount of possible loss from the probability of gain times the amount of possible gain. That is what we’re trying to do. It’s imperfect but that’s what it’s all about”

#### Margin of Safety vs. Expected Value?

To define margin of safety you only need market price and fair value estimate. The margin of safety is simply the difference between the two. But seasoned investors know that **investing is much more complex than this**. Expected value looks at value and opportunity in terms of the weighted average of the probability of the bear scenario plus the bull scenario (or as many scenarios as you would like to include in the weighted average). This type of thinking can help you assess risk more effectively.

#### Magnitude of Correctness over Frequency of Correctness

Yet another bonus is that expected value highlights the fact that what matters in investing is the ** magnitude** with which you are right . . .

**no****the frequency. While this lesson may seem elementary, it is often hard to actually internalize this concept when making investment decisions. Expected value helps you quantify this concept, which makes it easier to internalize.**

*t*In the words of Michael Mauboussin:

“Focus not on the frequency of correctness but on the magnitude of correctness.”

(click to tweet the above quote)

#### Expected Value in Action

Consider this example: You are working at an investment firm and your team is considering a large investment in one of two stocks. Let’s call one Growth and the other Boring.

Your firm is 60% certain Growth will obliterate earnings estimates and appreciate 40%. But your firm isn’t quite as certain about the upside for Boring. You are also 60% certain that Boring will beat earnings estimates but it will only result in an appreciation of 25%. This must mean Growth is a better investment, right? Not at all. We need to also take into consideration the bear scenario for each stock.

#### Don’t Forget the Bear Scenario!

If your team’s bull scenario estimates are wrong, investors may be deeply disappointed and the stock could plummet as much as 35%. Your firm sees a 40% chance for such a situation occurring. For Boring, however, your team believes that there is a 40% probability of a downside after earnings of only -5%.

#### Weighted Expected Value Calculation

So, despite a high probability of a massive 40% gain for Growth, Boring offers a higher expected value of 13%. Furthermore, Boring offers greater downside protection.

#### Expected Value Makes “Boring” Seem More Interesting

While this is only a simplified scenario of expected value thinking, it does make two very good points:

**Magnitude of correctness needs to be the focus, not the frequency of correctness.****Downside protection is very important to expected value.**

So . . . that “hot” stock that your brother-in-law is sure to rocket upward is not always a better investment than that “boring” stock your grandpa suggested that will most likely go nowhere. Remember: consider the expected value . . . consider the possible magnitude of gains and losses!

In other words, being right more than you are wrong is not what makes a good investor. What makes a good investor is the *magnitude of your correctness.*

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Good traders know this and practice it religiously.

Expected return =(P(W)*Avg W)-(P(L)*Avg L).

Simple to understand but hard because our egos get in the way. Being wrong is tough because it is potentially embarrassing and uncomfortable. It is not a coincidence that the title of Ned Davis’ best book was “Being right or making money.”

However, I take issue with the concept of expected value being superior to that of a margin of safety. The two concepts are related: without a MOS, your P(W) is reduced dramatically.

Trading too big is what tends to break vlue investors, since we tend to overestimate our ability to assess probabilities. The solution is to trade smaller, diversify across sectors and asset classes and use stops.

H

It would be nice if things were that simple and predictable. The problem is that I don’t know what I don’t know. How do you predict the probability of being right or wrong on a stock?

Moreover, how do you predict the expected increase or decrease of a stock. In practice if you could compute either of these factors then plugging the variables into a simple probability formula would make you rich fairly quickly.

However, in my experience, stock investing is not an exercise in objectivity. In most cases, its totally subjective. There are literally an infinite number of ways of looking at a situation. How does one assign probability to one’s investment style, unless one is using a mechanical system approach?

Good thoughts. Just as “margin of safety” is not as simple and predictable as it is in calculation, neither is expected value. The source of power behind expected value is that it provokes investors to think about different scenarios and the “magnitude” of correctness over the “frequency” of correctness. Furthermore it highlights the importance of downside protection. Finally, it causes us to question analysts “target prices” in terms of weighted probabilities and multiple scenarios.

By no means is expected value simple. Just as with any model it is extremely hard to predict anything. It’s just a lattice for complex situations that have multiple possible outcomes.

H, I’d say that margin of safety should be a prerequisite to any potential investment. Expected value should never be used in its place. However I’d say it might be superior in the fact that it forces you to look at the stock from different angles and the total, weighted probability of expected value. It’s strength lies in the process of implementation. While many seasoned investors might internalize expected value, many new investors could use it as a helpful tool to add a devil’s advocate scenario to their prediction.

Daniel, a few comments.

1) I would think your weights should add to 1. So I’m guessing you intended to put either 60/40 or 80/20.

2) Calculating standard deviation would also be useful here. It would show why in your example (modified) Boring looks more attractive than Growth. To do this we’ll make some modifications. First we’ll replace 80% (in both cases) with 60%. This way the weights will add to 1. Then we’ll change the expected return in the bull case for Boring to +20% instead of +25%. This will make the expected value to be the same in both cases. Here’s what it looks like.

Growth:

Expected Value = E(X) = 10%

Variance = V(X) = .135

Standard Deviation = SD(X) = 36.7%

Boring:

E(X) = 10%

V(X) = .015

SD = 12.2%

Standard deviation for Boring is much lower than Growth which indicates it’s more attractive given that they have the same expected values.

3) I think comparing expected value to margin of safety probably isn’t appropriate here. If anything, margin of safety is comparable with standard deviation (if we are going to do a comparison) than it is with expected value. After all, when you value a stock you shouldn’t get one value but a range of values (e.g. $100 +/- $20). Standard deviation is related more to the range. Purchasing at a margin of safety makes it more likely that even if your valuation is off, you’re still buying at a fair price.

I think margin of safety is more comparable to standard deviation than it is to expected value. Expected value is like valuing the stock. When you value a stock you shouldn’t get one value but a range of values (e.g. $100 +/- $20). The point of buying at a margin of safety is that the “correct” value might not be what you expected it to be ($100) but could be lower (say $100- $20 = $80). If you buy at a margin of safety you reduce your odds of overpaying. But there are some differences as well. (For example, the standard deviation we calculated above is not the same as the one we’d be interested in for margin of safety.)

somrh,

Thanks for pointing out the need for the modification in the numbers there! I had changed it at the last minute and I forgot to adjust the bull scenario accordingly! I’ll get that fixed.

I find your combination of standard deviation with expected value very useful. But If the downside deviation is very small I wouldn’t mind an overall high standard deviation score if it was simply a result of large upside potential (but I doubt an investor woud encounter a situation like this very often!).

Thank you for the post. Often times, the difficulty in the EV model lies in the probability estimation. in your experience, how have you come up with the probability assignment? it may be useful if Jae can build excel tools for this as well :).

Daniel,

It took me a bit to construct the kind of scenario you might be thinking of. Here are the conditions:

1) Scenario A and Scenario B have the same expected value.

2) Scenario A has a larger standard deviation than B.

3) Scenario A, however, is still preferable to B.

Here’s the proposal:

Scenario A

p=.5 ~ -10% return

p=.4 ~ +20% return

p=.1 ~ +70% return

E(A) = +10%

SD(A) = 24.5%

Scenario B

p=.5 ~ -12%

p=.5 ~ +32%

E(B) = +10%

SD(B) = 22%

E(A) = E(B), so condition (1) is satisfied.

SD(A) > SD(B), so condition (2) is satisfied.

I suppose A looks preferable to B given that the magnitude of the downside risk is only 10% instead of 12% (the probability of downside is the same in both cases).

Somrh,

Yes, this is what I was referring to. Thanks for constructing the scenario! Overall, however, standard deviation seems like it would make a good partner to expected value.

“It would be nice if things were that simple and predictable. The problem is that I don’t know what I don’t know. How do you predict the probability of being right or wrong on a stock?”

That’s definitely a very real challenge.

With insurance and banking, you have people who do this every day. For example – if you wrote an auto loan, it will only take a few years to know how you truly did. If you write a policy for medical malpractice though, it could be 20 years before you get the surprise gifts that keep on taking, because a doctor may have tied up a blood vessel that wasn’t discovered till recently.

Buffett calls stocks to be like 100-year bonds and I think he’s mostly right, so we end up wanting a long runway over which things don’t change.

The challenge is that can I really say if the odds are 10% of an event or 15%? I’m not that smart and so I have to stick to really simple things. I prefer it when the odds are either: (1) Way out of line and thus, strongly in my favor relative to the price I pay, or (2) Close to 100% when looking out 10-20+ years.

The best way I have to determine this is to have studied the business and industry well to understand what protects a company from being competed against and losing profitability. Then, if a temporary issue arises, you’re “armed” with your own understanding and valuations that will be able to be put to use.

I’m not too good at playing the odds game and so I either need something close to a guaranteed return or a situation where I have an understanding of something the the market isn’t weighing. More and more, I get the feeling that the really good investment theses are ones that we won’t read about online, but they’re likely simpler than imagined. Right now, my process is to start with a specific industry and swim up and down the value chain to look at suppliers, distributors, retailers of the products, and everything. On occasion, there are just overlooked companies that aren’t too hard to bet on. They still come with risks, but there are opportunities.

Daniel, thanks for the post. One of the difficulties using this method is probability assessment. Can you please share some of the methods that you used for estimation? Thanks.

Ankit, Jason Y.

I think I could have done a better job with the title on this article. Perhaps I should have titled it: “Why magnitude of correctness matters more than frequency of correctness.” Because it is not the calculation that I meant to focus on. I meant to use the calculation to illustrate why stocks with little downside are favorable to riskier stocks that have huge upside potential.

In every investment we make I think we are all running probabilities through our head (if we are not writing them down.) Investing is ultimately a game of forecasting . . . right? Often we estimate some sort of future cash flow pattern. To arrive at those cash flows we had to think of some sort of probability. If the average growth over the last 5 years was 25% a year, we could assume that the growth will continue at that rate . . . but usually we assign a small probability to that happening so we take some sort of average of probabilities based on our analysis of the company to arrive at something more conservative like 15%. There is no way that anyone can be sure future cash flows will grow at 15% next year, it is simply an average of probabilities.

Expected value can easily be internalized. We do it every time we find ourselves saying “This stock has very little downside risk and great upside potential.” In this case we have highlighted the importance of minimizing our downside risk and focusing on the magnitude of correctness. EV calculations show favor toward stocks with low downside risk. Growth investors, for example, might really need to actually think through different scenarios with EV. Because growth stocks are often very difficult to forecast. Example: Perhaps a growth stock has been crushing estimates and for 20 out of the last 20 quarters. Everything seems to be going for it and you feel that it will continue to surprise over the next few years. If it does you see huge upside potential, but . . . if it disappoints it has a long way to fall. So with this growth stock we have a could chance (frequency) of the company to continue to surprise, but when we take into consideration the bear scenario, our magnitude (EV) is greatly reduced. On the other hand, as the price falls on a stock like this, EV will be greatly increased so it can help you keep an eye on growth stocks to see when they reach a bargain level with a risk level that is acceptable to you.

Let’s tie everything together: So I agree that it is very difficult to forecast probabilities, just as it is to difficult to forecast future cash flows (which are often forecasted by subconsciously averaging probabilities), but EV reminds us that it is very important to consider that small chance that things could go bad. Because a small probability of things going very bad can weigh down on our expected value. Ultimately it will most likely keep us interested in the “boring” or stocks or stocks in a unique situation where the underlying value is not given the respect it deserves instead of the “hot” stocks.

So, I agree with you guys: Estimating probability would be very hard. But you are essentially using EV when you find a stock like Ankit describes: “I’m not too good at playing the odds game and so I either need something close to guaranteed return or a situation where I have an understanding of something the market isn’t weight.” Ankit has essentially dramatically increased his EV by finding a stock with either a 0% chance of downside risk or only a small probability of a very small loss. In a situation like this, only a small amount of upside is needed.

In conclusion, I will not tell any investor how to estimate probability or to use EV at all. That is like me telling a skilled artist how to imagine. I’m only giving you a brush that you can use if you ever find a need for it. I ask only that you consider the implications highlighted by EV analysis.

By the way, thanks for all the great comments everyone!