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H. Kevin Byun
H. Kevin Byun

The Risk-Reward Relationship Revisited

October 31, 2007

The Risk-Reward Relationship Revisited

Let’s say you have $100 in your pocket. Someone offers you two choices for a bet: 1) you win $10 or lose $10 or 2) you win $20 or lose $20. Both wining and losing have equal probability and the outcome is determined by a coin flip. Which choice is more risky? Clearly, the second choice is perceived as more risky and this is the basis of the commonly accepted risk/reward graph: up and to the right, meaning the greater the risk, the greater the reward.

The risk-reward relationship is assumed to have a positive correlation such that the greater the risk (beta) taken, then the greater the required reward (returns) and the greater the potential for reward (return), then the greater the assumed risk (beta). We have seen this relationship in countless finance books with risk (beta) on the x-axis and reward (return) on the y-axis and their relationship plotted as a line that extends up and to the right. This relationship tells us that to achieve excess returns, one must be exposed to excess risk. This is the classic framework that has goes largely unquestioned.

I submit that this relationship is wrong.

Before carting the author away to the asylum, a fundamental assumption in academia is that markets are efficient with perfect information so no unusual investment opportunities exist outside of this relationship. As such, the downside risk must be compensated for in increase upside such that the upside is approximately equal to the downside.

Well, let’s consider this further. Let’s divide the universe of stocks into ten groups (with an equal number of stocks) from cheapest to the most expensive (on various fundamental metrics such as Price/Book Value, Price/Sales, Price/Earnings, EV/EBIT, EV/EBITDA, etc.).

Given this division, what are the expected returns for each group?

Well, numerous studies have shown the relationship looks something like this: down and to the right, meaning the more expensive a stock, the lower the expected return.

This is very interesting. The data is telling us that the less one pays for a stock (based on fundamental metrics), the higher the returns one should expect. Hence, we can say:

Take Away 1: Cheaper stocks have higher expected returns than expensive stocks.

Okay, then what about Beta?

Now, what if we look at the relationship to beta (the traditional definition of risk)? What type of relationship does beta have with this grouping of stocks? Interestingly, the data shows that the relationship looks like this: up and to the right, meaning the more expensive a stock, the greater the beta.

This is very interesting as well. The data is telling us that the less one pays for a stock (based on fundamental metrics), the lower the risk (beta) one should expect. Hence, we can say:

Take Away 2: Cheaper stocks have lower risk (beta) than expensive stocks.

Both Take Away 1 and 2 are very interesting. Now, if you can accept both Take Away 1 and 2, then we are ready for the next step. The next step is to combine these two take aways. From the combination, we can logically conclude that:

Conclusion: Low risk stocks offer higher expected returns and high risk stocks offer lower expected returns

This is really astounding! Another way to phrase this relationship is:

Conclusion: Cheap stocks have lower risk (beta) and higher returns and expensive stocks have higher risk (beta) and lower returns.

Is it possible to have your cake and eat it too? Looks like it.

So then why does the traditional relationship tell us the opposite?

Well, let us divide the entire universe of stocks into ten groups according to market cap this time, which is how it is usually done. Would you then expect a positive or negative correlation?

It turns out that beta is weighted to market cap such that larger cap stocks are less volatile than smaller market caps.

So when you look at all the stocks together, it appears there is a positive correlation. However, when looking a bit deeper, we see that it is a positive correlation of a relative size effect and a liquidity effect.

The author I credit with having done very thorough work on this analysis is Robert A. Haugen. I consider his work very valuable, and although the work and conclusions themselves have been around for a long time, rarely do the conclusions get the attention they deserve.

So why does this matter and how does this affect us?

It matters because the data shows us that the traditional mindset about risk-reward is not necessarily correct. It affects us because we can use this new framework to recognize which opportunities are inherently more attractive. What have we determined today? Simply that:

Conclusion: Cheap stocks have lower risk (beta) and higher returns and expensive stocks have higher risk (beta) and lower returns.

However, now that we have established an alternative framework for the risk-reward relationship, let’s add another dimension to the analysis. One concept that Haugen does not consider in his book is intrinsic value. Haugen necessarily needed to remain in an area with quantifiable and available historic data (metrics that can be counted - betas, ratios, etc.) whereas intrinsic value is an estimate. By considering this dimension, I believe we have a distinct advantage to Haugen’s already invaluable alternative framework. With our advantage, we can discover an exception to Haugen’s work in which we can find great investment opportunities.

Rating: 3.4/5 (12 votes)


Billytickets - 10 years ago    Report SPAM
This article is very good and is the"premise" of capital allocation . Beta is an ASININE way to measure risk.

Web says it BEST when he explains about how it "BETA" explains it is "riskier" to buy a stock at 40 dollars than it is at 80. ( or something to the effect when he was explaining the WASHINGTON POST dropping in value) ONLY A FOOL would think that it is riskier to buy a stock at 40 than 80

Which makes me LAUGH is when my friend who bought CFC at 39 tells me now it is RISKY to buy it at 13. NEWSFLASH its a LOT LESS RISKY to buy it at 13 than 39.Anyone who does NOT believe that should choose another profession. Ironically Charles Brandes one of the most 150 richest men in the world recently bought CFC.This is not a "recommendation" of CFC ( ihate Mozilo and his selling of stock with a passion ) but ONLY afool would say that the"risk" of buying it now has INCREASED now that the stock has dropped from 45 to 16

EVERY GREAT INVESTOR KNOWS what Mr BYUN has "explained" . PRICE PAID IS PARAMOUNT to SUCCESS . WEll done.Many will not"understand" what you have said Mr Byun but trust me ,this forum has MANY bright people that will"embrace" your theory.Well written this is worth 5 stars easily.peace
John.sturges - 10 years ago    Report SPAM
This article is a great read. I would add that the term "Risk" refers to a iyr avg deviation to the expected return and is how in part Beta is calculated. This reveals that the market's problem is that most investors are thinking of investment returns in a time frame of 1yr or less. When you think of an investment cycle of 5yr+, then "Risk" becomes Standard Deviation(STD) in the long cycle and your goal becomes applying patience and analysis in identifying where the lower part of the cycle is and the pricing of the stock relative to the underlying business. By thinking STD in a cycle, "Risk" becomes your ally as it permits you to buy cheaply a management whose business is simply in an economic slow period. You capture the "RECOVERY" a couple of years of normal earnings and then are left to decide whether to sell or hold throught the next downturn. This approach is very, very profitable. Tweedy Brown has on their site a group of market studies that support the value approach. I do think they do not sell when stocks are at high valuations because of tax implications. Being a little more sensitive to overvaluations goes a long way towards redeploying capital to better returns.

The net/net is that "Risk" becomes STD in a 5yr+ investment timeframe and becomes your ally if you know how to use it wisely.

Tweedy Brown-"What has worked in Investing"


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