The traditional valuation model, whether it is the model from DCF or wall street’s popular P/E multiple method, requires estimating a fair value of a stock, and either use that value to see the upside potential (potential return of investment), or apply a discount as “margin of safety”.

This model, if used simply as it is, is inherently flawed. The reason is that it doesn’t consider the possible variations of the fair value. Or using the language of statistics, it doesn’t consider the “variance” of the expected mean value.

Some people may argue that the “riskiness” of the investment is already compensated when we change the discount rate (or cost of capital) for different situations. If it is a highly leveraged company, we will use higher cost of capital rate, or high discount rate, or lower P/E multiple.

However, this kind of “compensating” method is pretty ambiguous. Because the compensated higher margin of safety really consists of two parts:

1. The lower expected mean value:

If a company has high financial leverage or operating leverage, it may have 20% chance to go bankrupt in the next 5 years, so 80% chance we have $100 share price, and 20% chance we have $0 share price, the expected mean value is $80 share price. In this case, the additional discount required is 20%

2. The additional return required to compensate the additional risk observed:

This is also a part of the high discount we needed, but it is hard to quantify this either from intuition or from math, since it really depends on personal risk tolerance and the position size.

So if a company has debt to equity ratio of 2 to 1, should we use 12% discount rate? or 10%? or 15%?

What is the justification for that percentage? And how much of that percentage is the part 1 of the premium, or the part 2 of the premium?

If a company has a clearly bad CEO who is very likely to make a very expensive acquisition to waste all the cash on the balance sheet, how much discount should you apply to that fair value?

That is why I call the traditional method very “ambiguous”.

In the modern finance (quant finance), the situation is much clearer, at least in the theory. We just estimate a mean value with different scenarios assuming a probability for each scenario, and then estimate the variance of that mean value given those scenarios. Then we will apply a discount based on the variance.

Despite the additional clarity, there are still two challenges:

1. How much additional discount we need for a given variance still depends on personal risk tolerance.

This problem is not really an issue in two cases:

a. If the position size is very small, and the individual company’s risk can be fully diversified away, we can simply ignore the variance completely. In practical cases though, manual selection of stocks requires a lot of research and follow-up, plus best opportunity is very rare, so this kind of massive diversification is not practical. Still, if position is smaller, variance is not that bad any more, therefore we can require less discount on the risky stock.

b. There is a well defined upper bound of max position size for each stock given its variance and expected return.

The beauty here is that this upper bound (as defined in “Kelly bet”) does not depend on individual’s risk tolerance. No matter how aggressive you are, once you risk more than this limit, you are simply wrong!

“Kelly bet” defines the upper bound, but it doesn’t tell you the “right” bet size, since practical situation is quite different from a theoretical setup, where you don’t have a large number of identical bets waiting for you, and you don’t know the predefined risk-reward ratio. So the “right” bet size still depends on personal risk appetite, but at least we have some theoretical ground work here, and some people just choose “half kelly bet” as their choice.

In Kelly bet, the right position size should be inversely proportional to variance, and proportional to expected return. Remember variance is square of standard deviation, this means if a stock is twice risker, we need to put 25% of position size, or for the same position size we have to require 4 times more expected return.

For example, if buying BRKB has 15% standard deviation, and buying sears has 60% standard deviation, and I am willing to put 60% of my net worth in BRKB if it is 33% discounted (50% expected return), it means I can only put 60% / (4 ^2) = 4% of my net worth to sears if it is expected return is also 50%. Or I have to require 50% * (4 ^ 2) = 800% expected return to put 60% of my net worth into sears.

This is why I found it wrong when many value investors estimated sears’ real estate asset value and bet big on it. Yes, it does have a lot of asset value, but given its deeply troubled retailed business, the uncertainty of how many years of continued bleeding, the cost of liquidation, and the illiquid nature of real estate asset (it is hard to sell a lot of them in short term), the variance is very big. Therefore, applying 33% discount or even 50% discount may be not enough, especially when someone tries to bet big on it.

2. It is already very hard to estimate fair value, it would be even harder to estimate variance.

This is true, but a general sense would still help here. Plus, if we can list out a few scenarios and its probability, we can have a very rough estimate on the variance.

For quite some time, I was always puzzled about how much certainty I would need before making an investment. Since “certainty” is what Buffett cares most about. If I require too much certainty, it will bring very few opportunities. Too less, I am risking too much.

Later I figured out that, the certainty is important, but we can’t expect too much. There is always risk in an investment. What we should do is to adjust the position size and/or the margin of safety to compensate it.

A word of caution is that impact of risk goes quadratically, not linearly. So twice of risk (standard deviation) would require 4 times more potential return or 4 times less position size. Since it is very unlikely to get that much potential return, what usually ends up is a much smaller position size.

Apparently, this makes a very risky investment not worthwhile at some point, since we don’t want to diversify too much and we have limited time to do the research and follow-ups.

In conclusion, I think we should not neglect “variance” when we evaluate a stock, as it is as critical for evaluation. We should also remember, higher variance doesn’t always require a much higher expected return or higher discount, it really depends on how diversified we are. This concept of quant finance can help to refine the traditional valuation model.

Just so everyone knows, i’ve been in touch with Mingran for a few days. I think two of my investing mistakes support adjusting the margin of safety for risk and the size of the investment. They were both in London (U.K.) listed companies. One was a catastrophe insurer called Catlin, which was cheap by price / book value and yield when I invested, not long after the financial crisis. They got cheaper after new catastrophes affected results, and I wanted to buy more, but didn’t because I’d invested too much to start with. Management seemed to be OK but not excellent, so it wasn’t like adding to a large Berkshire Hathaway holding on a dip. In other words, there wasn’t much margin of safety, taking the size into account. Catlin have done well since, but I lightened up when the share price recovered. I’ll probably get another chance when catastrophes send rates up and the share price down.

Ath Resources plc were a U.K. coal miner. The story was that they’d sold coal forwards at low pices, the price of coal had risen, and profits would rise when the forward selling was worked off. They were cheap, but having to sell forwards at low prices indicates either a bad business, bad management, or both. The share price fell until they went into administration, due partly to hitting hot coal. With hindsight, they were a completely stupid investment, due to the high risk and limited reward, and the quote attributed to Mark Twain that a mine is a hole in the ground with a liar at the top. Fortunately I only risked a small ‘investment’. Anyone tempted by the supposed value story could have avoided loss by adjusting the required margin of safety for the high risk.

IMO, both mistakes could be identified without using value investing concepts, but they also fit the margin of safety concept when the margin is adjusted for the size and risk of the investment.

Many of the suggested value investments I read about seem too risky, and I usually scan the comments before reading an article on Seeking Alpha, because the stock’s followers seem to know the risks better than an author who probably found the stock with a screen.

Thanks. Those are good examples, I have many of them myself, when I invested in financial companies myself during 2008.