Chapter 11 Assessing Risk

11.1 Risk and Return Basics

If you are given a choice between these two following options, which would you pick (assume you only get to pick once):

1. You receive $1,000.
2. You flip a coin. If it lands heads, you receive $2,000, but if it lands tails, you get nothing.

Option A is guaranteed, while option B has some uncertainty: we don’t know for certain what our outcome will be. Thus, we say that option A is risk free, while option B entails a certain degree of riskUncertainty in outcome., or uncertainty in outcome.

The option that you prefer will depend upon your level of risk aversionThe tendency for most people to prefer less risk to more risk, all else being equal.. This is the tendency for most people to prefer less risk to more risk, all else being equal. Most people presented with this problem (with these dollar amounts) prefer option A. Some people might prefer to take the risk and prefer option B, in which case we say they are risk seekingAn entity who prefers a more risky alternative to a less risky one, all else being equal. (an entity who prefers a more risky alternative to a less risky one, all else being equal). And some might not care which of the two they pick, in which case we say they are risk neutralAn entity who is ambivalent between choices with different levels of risk, all else being equal. (an entity who is ambivalent between choices with different levels of risk, all else being equal).

Different people have different levels of risk aversion, and even the same person can exhibit different amounts, depending upon the specifics of the question! For example, many more are willing to take the gamble if the amounts of money are smaller (say $10 in option A vs. $0 or $20 in option B), or if the wording of the questions are framed differently.

Much of finance involves changing risk exposure, especially the transfer of risk from those who are more risk averse to those who are less. Usually, the more risk averse party rewards, directly or indirectly, the bearer of more risk. The classic example is buying insurance: a policy holder will pay the insurance provider to bear the risk that something unfortunate will happen (like a fire or car accident).

When we talk about financial risk, the outcomes we compare are solely the monetary returnsFor an investment, the amount of money we end with, less our invested money. that are possible in the different outcomes. These returns are the amount of money we end with, less our invested money. They can be positive or negative (which would occur if the money received is less than our investment). When figuring returns, we need to include any payments received or made (such as interest or dividends) as well of the selling price of the asset. To compare investments of different sizes easily, we will typically discuss the rate of returnThe ratio of our return to our investment., which is the ratio of our return to our investment.

11.2 Portfolios

Determining the risk of holding a single asset is a difficult task in itself. In reality, investors will typically hold many different assets of potentially different types. Another common alternative is that investors hand their money over to a manager that pools all the investors’ contributions to purchase such, which we call a portfolioA collection of financial assets. (a collection of financial assets).

While individual stocks or bonds might rise or fall in value, ultimately an investor cares about the performance of the whole portfolio. If we make one bad investment but five good ones, we might come out ahead and be satisfied. Since investors tend to hold stocks as parts of their portfolios, managers need to understand how their company’s performance will affect the risk and return of the portfolio.

Determining the return of the portfolio is a simple matter of taking the average of the returns of the components, weighted by the portion of the portfolio initially invested in each. Thus, if we invest $20 thousand in ABC stock and $80 thousand in FEG stock, our weights would be 20/(20+80) = 20/100 = .2 and 80/(20+80) = 80/100 = .8, respectively.

If the return on ABC was 10% and the return on FEG was −2%, the return of our portfolio would be (.2 × .10) + (.8 × −.02) = .004 or 0.4%. We can confirm that this is correct: we earn 10% on our $20 thousand investment for a return of $2 thousand, and we lose 2% on our $80 thousand investment for a return of −$1,600. Adding the returns together gives a positive $400, which is precisely 0.4% of our original $100 thousand investment.

Notice that our portfolio return lies somewhere between the two extremes. Almost always, some investments will do better than others, so that our portfolio’s returns won’t be as good as our best investments. But the benefit is that our portfolio’s returns won’t be as bad as our worst investments. This is the key to diversificationHolding a portfolio of different assets so as not to concentrate exposure on one particular asset. (holding a portfolio of different assets so as not to concentrate exposure on one particular asset). This is embodied by the old saying, “Don’t put all your eggs in one basket.” Diversification can lower risk to a point, and is the closest thing to a “free lunch” in finance! To see this more concretely, we need to have a specific measure for our risk, which we will introduce later in this chapter.

11.3 Market Efficiency

Talking about risk only makes sense if we are dealing with uncertainty. If we truly are superior investors, able to always “pick the winners”, then we have no need to diversify significantly. For example, Bill Gates earned most of his fortune by owning a significant stake in one company (Microsoft).

A key difference, however, is that Bill Gates wasn’t just an investor, he was also the CEO for a significant portion of Microsoft’s history (and thus much more in control of the investment). And his tenure certainly wasn’t without uncertainty: for every Microsoft, Google, or Facebook there are numerous other companies that have failed, losing much of their investors’ capital. Unfortunately, many investors and managers only remember the success stories and forget those that lost their gamble. We constantly hear of these companies that were successful, but rarely hear of the former high-fliers of the internet bubble.

Furthermore, just as most people think they are “above-average drivers”, most investors are deluded into thinking they know more than everyone else in the market. Natural overconfidence, selective memory (we tend to remember our successes better than our failures), and a self-attribution bias (we tend to credit successes with skill but failures to “bad luck”) cause investors to believe they are better than they, in actuality, are. There is debate about whether Warren Buffett is a truly skilled investor, or whether he has “gotten lucky” because the market happened to fit his investment style during the years he was active. Given all of the numerous investors over the years, perhaps someone was bound to get as lucky as Buffett and make billions. Of course, many other investors were doomed to failure: many through lack of skill, but undoubtedly many also who were just “unlucky”. “It is better to be lucky than good” is a trading proverb that holds much truth.

Thus it is an open question whether skilled investors can reliably outperform the market. The argument for market efficiency goes as follows: investors sell or buy an asset based upon their valuation of the asset. This will cause the price to shift until the equilibrium price is reached (this process is called price discoveryThe process by which investors sell or buy an asset based upon their valuation of the asset. This will cause the price to shift until the equilibrium price is reached.). Once the proper price is reached, abnormal profits can no longer be earned. Furthermore, the market, in aggregate, has more information than any one investor could possibly have. The market price will be reached by this information, so an investor should not be able to abnormally profit from his or her private information. We call this the Efficient Market Hypothesis (EMH)The theory that the market, in aggregate, has more information than any one investor could possibly have. The market price will be reached by this information, so an investor should not be able to abnormally profit from his or her private information..

Studies have shown that those with insider knowledge (that is, material non-public financial information) about companies can profitably trade on this information, thus offering one refutation of complete market efficiency (in what we term the strong form of the EMHThe belief that all public and non-public information has been accounted for by the market. Those with insider knowledge (that is, material non-public financial information) about companies cannot profitably trade on the information if the market is strongly efficient.). Given that trading on insider information is typically against the law, this is of little benefit to the typical investor!

The semi-strong form of the EMHThe belief that all public information has been accounted for by the market (including financial analysis of released statements and analysts’ forecasts). If the semi-strong form is true, than investments based upon the fundamentals of a company shouldn’t be able to earn excess profits argues that all public information has been accounted for by the market (including financial analysis of released statements and analysts’ forecasts). If the semi-strong form is true, than investments based upon the fundamentals of a company shouldn’t be able to earn excess profits. Successful investors such as Warren Buffett and Peter Lynch are often put forth as counter-examples to the semi-strong form; proponents of the semi-strong form have put forth arguments that these investors were either lucky or had access to investment opportunities or information not available to the market as a whole. Other arguments against the semi-strong form include the existence of anomalies such as asset bubbles (such as the technology bubble and the real estate bubble) and post-earnings drift (the tendency for stock prices to not immediately jump to a new price based upon earnings surprise but slowly “drift” to their new level).

Most academics tend to embrace the semi-strong form, though many Wall Street traders reject it. One reasonable question to ask is, even if public information can be exploited for gain, can a typical investor access and make use of the data? Believers of the semi-strong form (or those who reject it but believe typical investors can’t benefit) advocate investing in “passive” broad market funds that don’t invest in particular companies based on analysis (unlike most “actively managed” funds).

11.4 Standard Deviation

There are many potential sources of risk, and not all methods of assessing risk are the same. One popular method (though certainly not the only one) for trying to quantify risk is to examine the standard deviation of the outcome returns. Standard deviation is a term from statistics that helps explain to what degree our results are expected to be different from the averge result. We are, in a sense, finding the “average” size of the “miss”. For a full treatment of standard deviation as a statistical measure, please consult a statistics text; for our purposes, a basic understanding will suffice.

Figure 11.1 Standard Deviation Comparison

Source: JRBrown / Wikimedia Commons / Public Domain

The blue investment has results that are less “clumped” around the mean than in the red investment. The blue investment has a correspondingly higher standard deviation.

Finding standard deviation involves a few steps; if we know all the outcomes and their probabilities, standard deviation is found as follows:

1. Find the expected rate of return.
2. For each outcome, find the difference between the outcome’s rate of return and the expected rate, and square the difference. So if the expected outcome is 2.5%, but the outcome’s return is −10%, then the difference is (−.10 − (.025) = −.125. This result squared is −.125² = .015625. Notice that the squaring effectively eliminates the negative numbers (so it doesn’t matter if we miss high or low, just how much we miss by).
3. Find the weighted mean of these: multiply each by the probability of the outcome and sum.
4. Take the square root of this weighted mean to find the standard deviation.

If we are using historical returns to estimate our expected return, then we need to make a slight adjustment to the formula. Typically, we assume each past observation (usually daily, weekly, monthly, or yearly returns) to have equal weight. Thus, the expected rate of return is just the arithmetic mean. But since we have just estimated the expected rate of return, we need to adjust our standard deviation calculation (for a detailed reason why, please see any standard statistics text, as it is beyond the scope of this text):

On the bright side, spreadsheet programs can quickly do this calculation for us.

=STDEV.S(outcome1, outcome2, outcome3…)

Thus, if our outcomes for the past 5 years are 5%, 2%, −6%, −2%, and 4%, we can enter:

=STDEV.S(.05,.02,−.06,−.02,.04)

and receive the correct result of 0.0456 or 4.56%.

Most financial calculators can also solve for standard deviation; please consult the instruction manual of your calculator for details.

There are some downsides to using standard deviation as the measure of risk. Standard deviation tells us the most when our outcomes are normally distributed (the “bell curve”). If our outcomes have some extreme outliers (for example, a decent chance for a large positive or negative result), then standard deviation tells us very little about the risk associated with these results. If our results are skewed toward positive or negative outcomes, standard deviation tells us nothing of the effect.

11.5 Market History

Over the past few chapters, we have explored different types of financial assets. At this point, it is worth looking at how some of these assets have performed historically.

We see that there is a very strong relationship between the average return and the level of risk (as measured by standard deviation). This corresponds with our understanding that most investors are risk averse; that is, to accept a larger degree of risk (a higher standard deviation) investors demand a larger return. Also, this figure matches our intuition that stocks are riskier than bonds.

While this data is a good starting point for those trying to determine future investor expectations, there are some large caveats. Remember that, during this time period, the United States rose to be one of (if not the only) world “superpowers”. We have no idea what this data would look like if, for instance, the Axis nations won WWII or the Soviet Union still existed. Depending upon these trends to continue might be a mistake, since the world continues to change politically, economically, and culturally.

11.6 The Capital Asset Pricing Model (CAPM)

Riskier assets should demand higher returns. But not all risk is the same, because of diversification. For example:

Figure 11.2 Standard Deviation of a Stock Portfolio vs. Number of Random US Securities

Once we approach about 40 stocks in a portfolio (assuming they are relatively random and not, for example, all technology stocks) we cease to see significant reductions in standard deviation. There are some mutual funds and ETFs that attempt to own virtually the entire market of available securities! Nonetheless, there is a certain minimum amount of risk that must be borne by those investing in stocks due to factors that affect the entire market, which we call market riskMinimum amount of risk that must be borne by those investing in stocks due to factors that affect the entire market. Also called systematic risk or non-diversifiable risk. (also called systematic risk or non-diversifiable risk). Only by adding other asset types (for example, bonds or real estate) can we gain some additional diversification benefits.

The additional risk that any stock has above its market risk is called firm-specific riskThe additional risk that any stock has above its market risk, that encompasses all factors that would affect the company’s stock price, but not the market in general. Also called unsystematic, diversifiable, or unique risk. (also called unsystematic, diversifiable, or unique risk). This encompasses all factors that would affect the company’s stock price, but not the market in general. For example, the CEO being indicted for fraud or the approval of a new popular pharmaceutical would be firm-specific event. Diversification reduces firm-specific risk but leaves market risk.

Since firm-specific risk can be virtually eliminated by diversification, investors shouldn’t demand a higher return to compensate for it. Market risk, however, can’t be diversified away, so investors should demand higher returns for those securities which have a larger sensitivity to broad market events (like the state of the economy). For example, airlines tend to do poorly when the economy tanks (since in a bad economy, people travel less for both business and leisure), so investors should demand a higher return for airlines. Discount retailers, like Walmart, tend to be relatively less affected by a downturn in the economy, so they have less market risk; as a result, investors should demand a lower return for Walmart than an airline. Specifically, we are looking at a stock’s risk premiumThe expected return in excess of the risk free rate. (the expected return in excess of the risk free rate), as this is the portion of the return compensating investors for bearing the extra market risk associated with holding the stock.

One way to attempt to quantify a company’s market risk is to calculate its historical betaThe ratio of a company’s risk premium versus the market’s risk premium.. Beta represents the ratio of a company’s risk premium versus the market’s risk premium. Typically, beta is found by some method of statistical regression of past company returns in excess of the risk-free rate compared to a broad market index’s returns in excess of the risk-free rate on a daily, weekly, or monthly basis. Of course, what investors really want is a company’s beta going forward, so the historical beta will be a proxy for this value. Most financial sites provide their projection of public companies’ betas and will usually describe any adjustments they make from the historical calculation.

So how is beta used? A stock that tends to move in the same direction as the market (most stocks) will have a positive beta. Stocks with a beta of 1.0 will move in the same proportion as the market: if the market is up 1% over the risk-free rate, these stocks will also tend to be up 1% (statistically speaking). Stocks with betas higher than 1.0 are more sensitive to market moves; betas lower than 1.0 indicate less sensitivity. The beta of the broad market index must be, by definition 1.0 (since it must move in perfect tandem with itself).

Since the betas of all stock are perfectly correlated with market risk (and, hence, each other), the beta of a portfolio of stocks is the average (weighted mean) of the betas of the components.

For example, if we have $20 thousand invested in ABC stock (which has a beta of 1.5) and $80 thousand invested in DEF stock (which has a beta of 0.7), then our portfolio’s beta is: $(\frac{20}{20+80}\times 1.5)+(\frac{80}{20+80}\times 0.7)=(0.2\times 1.5)+(0.8\times 0.7)=0.86$

A portfolio holding every stock in the same weights as the market overall should have a risk premium equal to the market’s risk premium (also called the equity risk premium). It should also have a beta of exactly 1.0. Since the relationship of each component to both the portfolio’s return and beta is the same (weighted mean), the two measures must be proportional to each other. Specifically, each component’s contribution to the risk premium of the overall market portfolio must be proportional to the component’s market risk (the beta):

Continuing with our example stocks: if the market risk premium was 4.0%, then ABC (beta=1.5) should have a risk premium of 1.5×0.04=0.06 or 6%, and DEF (beta=0.7) should have a risk premium of 0.7×0.04=0.028 or 2.8%.

Of course, any given observation might deviate from the relationship predicted (beta is a statistical number), but the key idea is that our expected returns should hold to this relationship. If we rearrange this equation to solve for the expected return of stock A, we get what is called the Capital Asset Pricing Model (CAPM):

This is a very important result, as it implies that we can predict a stock’s expected return if we know the risk-free rate, beta of the stock, and the market risk premium.

Thus, if stock GHI has a beta of 1.2, and we know that the risk-free rate is 2% and the expected market returns are 7%, the expected returns for GHI should be 0.02 + 1.2 × (0.07 − 0.02) = 0.08 or 8%.

We can also use the relationship to compare two stocks directly:

11.7 The Bigger Picture

At the core of all of finance is the question of appropriate tradeoff between risk and return. We have looked in detail at one measure of risk, standard deviation, and further separated the portion due to the market and the portion that can be “diversified away”. As financial managers, we will need to evaluate projects based upon their returns and their potential contribution to non-diversifiable risk, since our investors will expect appropriate returns for the risk assumed. This will play a key part in our investigation of the cost of captial, which in turn will affect our evaluation of which projects our companies should undertake.

Furthermore, the concepts introduced in this chapter are the foundation of much of modern portfolio theory. Any who plan to study investments further will use these principles in constructing efficient portfolios.

11.8 End-of-Chapter Exercises