# 2.3: Measures of Risk - Putting It Together

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- In this section, you will learn how to compute several common measures of risk using various methods and statistical concepts.

Having developed the concept of probability to quantify the relative likelihood of an uncertain event, and having developed a measure of “expected value” for an uncertain event, we are now ready to try to quantify risk itself. The “expected value” (or mean value or fair value) quantifying the potential outcome arising from an uncertain scenario or situation in which probabilities have been assigned is a common input into the decision-making process concerning the advisability of taking certain actions, but it is not the only consideration. The financial return outcomes of various uncertain research and development, might, for example, be almost identical except that the return distributions are sort of shifted in one direction or the other. Such a situation is shown in Figure \(\PageIndex{1}\). This figure describes the (continuous) distributions of anticipated profitability for each of three possible capital expenditures on uncertain research and development projects. These are labeled A, B, and C, respectively.

Intuitively, in economic terms a risk is a “surprise” outcome that is a consequence of uncertainty. It can be a positive surprise or a negative surprise, as we discussed in "1: The Nature of Risk - Losses and Opportunities".

Using the terms explained in the last section, we can regard risk as the deviation from the expected value. The more an observation deviates from what we expected, the more surprised we are likely to become if we should see it, and hence the more risky (in an economic sense) we deem the outcome to be. Intuitively, the more surprise we “expect” from a venture or a scenario, the riskier we judge this venture or scenario to be.

Looking back on Figure \(\PageIndex{1}\), we might say that all three curves actually represent the same level of risk in that they each differ from their expected value (the mean or hump of the distribution) in identical ways. They only differ in their respective expected level of profitability (the hump in the curve). Note that the uncertain scenarios “B” and “C” still describe risky situations, even though virtually all of the possible outcomes of these uncertain scenarios are in the positive profit range. The “risk” resides in the deviations from the expected value that might result (the surprise potential), whether on the average the result is negative or positive. Look at the distribution labeled “A,” which describes a scenario or opportunity/loss description where much more of the possible results are on the negative range (damages or losses). Economists don’t consider “A” to be any more risky (or more dangerous) than “B” or “C,” but simply less profitable. The deviation from any expected risk defines risk here. We can plan for negative as well as positive outcomes if we know what to expect. A certain negative value may be unfortunate, but it is not risky.

Some other uncertain situations or scenarios will have the same expected level of “profitability,” but will differ in the amount of “surprise” they might present. For example, let’s assume that we have three potential corporate project investment opportunities. We expect that, over a decade, the average profitability in each opportunity will amount to $30 million. The projects differ, however, by the level of uncertainty involved in this profitability assessment (see Figure \(\PageIndex{2}\)). In Opportunity A, the possible range of profitability is $5–$60 million, whereas Opportunity B has a larger range of possible profits, between –$20 million and + $90 million. The third opportunity still has an expected return of $30 million, but now the range of values is from –$40 million to +$100. You could make more from Opportunity C, but you could lose more, as well. The deviation of the results around the expected value can measure the level of “surprise” potential the uncertain situation or profit/loss scenario contains. The uncertain situation concerning the profitability in Opportunity B contains a larger potential surprise in it than A, since we might get a larger deviation from the expected value in B than in A. That’s why we consider Opportunity B more risky than A. Opportunity C is the riskiest of all, having the possibility of a giant $100 million return, with the downside potential of creating a $40 million loss.

Our discussion above is based upon intuition rather than mathematics. To make it specific, we need to actually define quantitatively what we mean by the terms “a surprise” and “more surprised.” To this end, we must focus on the objective of the analysis. A sequence of throws of a pair of colored dice in which the red die always lands to the left of the green die may be surprising, but this surprise is irrelevant if the purpose of the dice throw is to play a game in which the number of dots facing up determines the pay off. We thus recognize that we must define risk in a context of the goal of the endeavor or study. If we are most concerned about the risk of insolvency, we may use one risk measure, while if we are interested in susceptibility of portfolio of assets to moderate interest rate changes, we may use another measure of risk. Context is everything. Let’s discuss several risk measures that are appropriate in different situations.

## Some Common Measures of Risk

As we mentioned previously, intuitively, a risk measure should reflect the level of “surprise” potential intrinsic in the various outcomes of an uncertain situation or scenario. To this end, the literature proposes a variety of statistical measures for risk levels. All of these measures attempt to express the result variability for each relevant outcome in the uncertain situation. The following are some risk measures.

## The Range

We can use the **range** of the distribution—that
is, the distance between the highest possible outcome value to the
lowest—as a rough risk measure. The range provides an idea about
the “worst-case” dispersion of successive surprises. By taking the
“best-case scenario minus the worst-case scenario” we define the
potential breadth of outcomes that could arise in the uncertain
situation.

As an example, consider the number of claims per year in Location A of Table 2.1. Table 2.1 shows a low of seven claims per year to a high of fourteen claims per year, for a range of seven claims per year. For Location B of Table 2.2, we have a range in the number of claims from a low of five in one year to a high of fifteen claims per year, which gives us a range of ten claims per year. Using the range measure of risk, we would say that Location A is less risky than Location B in this situation, especially since the average claim is the same (10.2) in each case and we have more variability or surprise potential in Location B. As another example, if we go back to the distribution of possible values in Table 2.3, the extremes vary from −$10.2 million to +$8 million, so the range is $18.2 million.

This risk measure leaves the picture incomplete because it cannot distinguish in riskiness between two distributions of situations where the possible outcomes are unbounded, nor does it take into account the frequency or probability of the extreme values. The lower value of –$10.2 million in Table 2.3 only occurs 1 percent of the time, so it’s highly unlikely that you would get a value this small. It could have had an extreme value of –$100 million, which occurred with probability 0.0000000001, in which case the range would have reflected this possibility. Note that it’s extremely unlikely that you would ever experience a one-in-a-trillion event. Usually you would not want your risk management activities or managerial actions to be dictated by a one-in-a-trillion event.

## Deviation from a Central Value

A more sophisticated (and more traditional) way to measure risk would consider not just the most extreme values of the distribution but all values and their respective occurrence probabilities. One way to do this is to average the deviations of the possible values of the distribution from a central value, such as the expected value \(E(V)\) or mean value discussed earlier. We develop this idea further below.

## Variance and Standard Deviation

Continuing the example from Table 2.1 and Table 2.2, we now ask what differentiates the claims distribution of Location A and B, both of which possess the same expected frequency and severity. We have already seen that the range is different. We now examine how the two locations differ in terms of their deviation from the common mean or expected value. Essentially, we want to examine how they differ in terms of the amount of surprise we expect to see in observations form the distributions. One such measure of deviation or surprise is by calculating the expected squared distance of each of the various outcomes from their mean value. This is a weighted average squared distance of each possible value from the mean of all observations, where the weights are the probabilities of occurrence. Computationally, we do this by individually squaring the deviation of each possible outcome from the expected value, multiplying this result by its respective probability or likelihood of occurring, and then summing up the resulting products.Calculating the average signed deviation from the mean or expected value since is a useless exercise since the result will always be zero. Taking the square of each deviation for the mean or expected value gets rid of the algebraic sign and makes the sum positive and meaningful. One might alternatively take the absolute value of the deviations from the mean to obtain another measure called the absolute deviation, but this is usually not done because it results in a mathematically inconvenient formulation. We shall stick to the squared deviation and its variants here. This produces a measure known as the variance. Variance provides a very commonly used measure of risk in financial contexts and is one of the bases of the notion of efficient portfolio selection in finance and the Capital Asset Pricing Model, which is used to explicitly show the trade-off between risk and return of assets in a capital market.

We first illustrate the calculation of the variance by using the probability distribution shown in Table 2.2. We already calculated the expected value to be $1.008 million, so we may calculate the variance to be \((.01) × (–10.2 –1.008)^2 + (.1) × (–.5 –1.008)^2+ (.4) × (.1 – 1.008)^2+ (.4) × (1 – 1.008)^2 + (.09) × (8 – 1.008)^2= 7.445\). Usually, variance is denoted with the Greek symbol sigma squared, \(σ^2\), or simply V.

As another example, Table 2.4 and Table 2.5 show the calculation of the variance for the two samples of claims given in locations A and B of Table 2.1 and Table 2.2 , respectively. In this case, the years are all treated equally so the average squared deviation from the mean is just the simple average of the five years squared deviations from the mean. We calculate the variance of the number of claims only.

Year | Number of Fire Claims | Difference between Observed Number of Claims and Mean Number of Claims | Difference Squared |
---|---|---|---|

1 | 11 | 0.8 | 0.64 |

2 | 9 | −1.2 | 1.44 |

3 | 7 | −3.2 | 10.24 |

4 | 10 | −0.2 | 0.04 |

5 | 14 | 3.8 | 14.44 |

Total | 51 | 0 | 26.8 |

Mean | 10.2 | = \(\frac{26.8}{4}\)= 6.7 | |

Variance 6.70 | |||

Standard Deviation = Square Root (6.7) = 2.59 |

Year | Number of Fire Claims | Difference between Observed Number of Claims and Mean Number of Claims | Difference Squared |
---|---|---|---|

1 | 15 | 4.8 | 23.04 |

2 | 5 | −5.2 | 27.04 |

3 | 12 | 1.8 | 3.24 |

4 | 10 | −0.2 | 0.04 |

5 | 9 | −1.2 | 1.44 |

Total | 51 | 0 | 54.8 |

Mean | 10.2 | = \(\frac{54.8}{4}\) = 13.70 | |

Variance 13.70 | |||

Standard Deviation 3.70 |

A problem with the variance as a measure of risk is that by squaring the individual deviations from the mean, you end up with a measure that is in squared units (e.g., if the original losses are measured in dollars, then the variance is measured in dollars-squared). To get back to the original units of measurement we commonly take the square root and obtain a risk measure known as the standard deviation, denoted by the Greek letter sigma (σ). To provide a more meaningful measure of risk denominated in the same units as the original data, economists and risk professionals often use this square root of the variance—the standard deviation—as a measure of risk. It provides a value comparable with the original expected outcomes. Remember that variance uses squared differences; therefore, taking the square root returns the measure to its initial unit of measurement.

Thus, the standard deviation is the square root of the variance. For the distribution in Table 2.3, we calculated the variance to be 7.445, so the standard deviation is the square root of 7.445 or $2.73 million. Similarly, the standard deviations of locations A and B of Table 2.1 and Table 2.2 appear in Tables 2.4 and 2.5. As you can see, the standard deviation of the sample for Location A is only 2.59, while the standard deviation of the sample of Location B is 2.70. The number of fire claims in Location B is more spread out from year to year than those in Location A. The standard deviation is the numeric representation of that spread.

If we compare one standard deviation with another distribution of equal mean but larger standard deviation—as when we compare the claims distribution from Location A with Location B—we could say that the second distribution with the larger standard deviation is riskier than the first. It is riskier because the observations are, on average, further away from the mean (more spread out and hence providing more “surprise” potential) than the observations in the first distribution. Larger standard deviations, therefore, represent greater risk, everything else being the same.

Of course, distributions seldom have the same mean. What if we
are comparing two distributions with different means? In this case,
one approach would be to consider the **coefficient of
variation**, which is the standard deviation of a
distribution divided by its mean. It essentially trades off risk
(as measured by the standard deviation) with the return (as
measured by the mean or expected value). The coefficient of
variation can be used to give us a relative value of risk when the
means of the distributions are not equal.

## The Semivariance

The above measures of risk gave the same attention or importance
to both positive and negative deviations from the mean or expected
value. Some people prefer to measure risk by the surprises in one
direction only. Usually only negative deviations below the expected
value are considered risky and in need of control or management.
For example, a decision maker might be especially troubled by
deviations below the expected level of profit and would welcome
deviations above the expected value. For this purpose a
“semivariance” could serve as a more appropriate measure of risk
than the variance, which treats deviations in both directions the
same. The **semivariance** is the average square
deviation. Now you sum only the deviations below the expected
value. If the profit-loss distribution is symmetric, the use of the
semivariance turns out to result in the exact same ranking of
uncertain outcomes with respect to risk as the use of the variance.
If the distribution is not symmetric, however, then these measures
may differ and the decisions made as to which distribution of
uncertain outcomes is riskier will differ, and the decisions made
as to how to manage risk as measured by these two measures may be
different. As most financial and pure loss distributions are
asymmetric, professionals often prefer the semi-variance in
financial analysis as a measure of risk, even though the variance
(and standard deviation) are also commonly used.

## Value at Risk (VaR) and Maximum Probable Annual Loss (MPAL)

How do banks and other financial institutions manage the
systemic or fundamental market risks they face?
**VaR** modeling has become the standard risk
measurement tool in the banking industry to assess market risk
exposure. After the banking industry adopted VaR, many other
financial firms adopted it as well. This is in part because of the
acceptance of this technique by regulators, such as conditions
written in the Basel II agreements on bank regulation.Basel
Committee on Banking Supervision (BCBS), International Convergence
of Capital Measurement and Capital Standards: A Revised Framework
(Basel, Switzerland, 2004). Further, financial institutions need to
know how much money they need to reserve to be able to withstand a
shock or loss of capital and still remain solvent. To do so, they
need a risk measure with a specified high probability. Intuitively,
VaR is defined as the worst-case scenario dollar value loss (up to
a specified probability level) that could occur for a company
exposed to a specific set of risks (interest rates, equity prices,
exchange rates, and commodity prices). This is the amount needed to
have in reserve in order to stave off insolvency with the specified
level of probability.

In reality, for many risk exposures the absolute “worst-case” loss that could be experienced is conceivably unbounded. It’s conceivable that you could lose a very huge amount but it may be highly unlikely to lose this much. Thus, instead of picking the largest possible loss to prepare against, the firm selects a probability level they can live with (usually, they are interested in having their financial risk exposure covered something like 95 percent or 99 percent of the time), and they ask, “What is the worst case that can happen up to being covered 95 percent or 99 percent of the time?” For a given level of confidence (in this case 95 percent or 99 percent) and over a specified time horizon, VaR can measure risks in any single security (either a specific investment represented in their investment securities or loan from a specific customer) or an entire portfolio as long as we have sufficient historical data. VaR provides an answer to the question “What is the worst loss that could occur and that I should prepare for?”

In practice, professionals examine a historical record of returns for the asset or portfolio under consideration and construct a probability distribution of returns. If you select a 95 percent VaR, then you pick the lowest 5 percent of the distribution, and when multiplied by the asset or portfolio value, you obtain the 95 percent VaR. If a 99 percent VaR is desired, then the lowest 1 percent of the return distribution is determined and this is multiplied by the asset or portfolio value to obtain the 99 percent VaR.

We illustrate this further with the Figure \(\PageIndex{3}\), concerning Hometown Bank.

## Case: Hometown Bank Market Risk

Market risk is the change in market value of bank assets and liabilities resulting from changing market conditions. For example, as interest rates increase, the loans Hometown Bank made at low fixed rates become less valuable to the bank. The total market values of their assets decline as the market value of the loans lose value. If the loans are traded in the secondary market, Hometown would record an actual loss. Other bank assets and liabilities are at risk as well due to changing market prices. Hometown accepts equity positions as collateral (e.g., a mortgage on the house includes the house as collateral) against loans that are subject to changing equity prices. As equity prices fall, the collateral against the loan is less valuable. If the price decline is precipitous, the loan could become undercollateralized where the value of the equity, such as a home, is less than the amount of the loan taken and may not provide enough protection to Hometown Bank in case of customer default.

Another example of risk includes bank activities in foreign exchange services. This subjects them to currency exchange rate risk. Also included is commodity price risk associated with lending in the agricultural industry.

Hometown Bank has a total of $65.5 million in investment securities. Typically, banks hold these securities until the money is needed by bank customers as loans, but the Federal Reserve requires that some money be kept in reserve to pay depositors who request their money back. Hometown has an investment policy that lists its approved securities for investment. Because the portfolio consists of interest rate sensitive securities, as interest rates rise, the value of the securities declines.Valuation of bonds is covered in general finance text. Bond value = present value of coupons + present value of face value of bond. Hometown Bank’s CEO, Mr. Allen, is interested in estimating his risk over a five-day period as measured by the worst case he is likely to face in terms of losses in portfolio value. He can then hold that amount of money in reserve so that he can keep from facing liquidity problems. This problem plagued numerous banks during the financial crisis of late 2008. Allen could conceivably lose the entire $65.5 million, but this is incredibly unlikely. He chooses a level of risk coverage of 99 percent and chooses to measure this five-day potential risk of loss by using the 99 percent—the VaR or value at risk. That is, he wants to find the amount of money he needs to keep available so that he has a supply of money sufficient to meet demand with probability of at least 0.99. To illustrate the computation of VaR, we use a historical database to track the value of the different bonds held by Hometown Bank as investment securities. How many times over a given time period—one year, in our example—did Hometown experience negative price movement on their investments and by how much? To simplify the example, we will assume the entire portfolio is invested in two-year U.S. Treasury notes. A year of historical data would create approximately 250 price movement data points for the portfolio.The number 250 comes from a rough estimate of the number of days securities can be traded in the open market during any given year. Fifty-two weeks at five days per week yields 260 weekdays, and there are roughly ten holidays throughout the year for which the market is closed. Of those 250 results, how frequently did the portfolio value decrease 5 percent or more from the beginning value? What was the frequency of times the portfolio of U.S. Treasury notes increased in value more than 5 percent? Hometown Bank can now construct a probability distribution of returns by recording observations of portfolio performance. This probability distribution appears in Figure \(\PageIndex{4}\).

The frequency distribution curve of price movement for the portfolio appears in Figure \(\PageIndex{1}\). From that data, Hometown can measure a portfolio’s 99 percent VaR for a five-day period by finding the lower one percentile for the probability distribution. VaR describes the probability of potential loss in value of the U.S. Treasury notes that relates to market price risk. From the chart, we observe that the bottom 1 percent of the 250 observations is about a 5 percent loss, that is, 99 percent of the time the return is greater than –5 percent. Thus, the 99 percent VaR on the returns is –5 percent. The VaR for the portfolio is the VaR on the return times $65.5 million, or –.05×($65.5 million)= −$3,275,000. This answers the question of how much risk capital the bank needs to hold against contingencies that should only occur once in one hundred five-day periods, namely, they should hold $3,275,000 in reserve. With this amount of money, the likelihood that the movements in market values will cause a loss of more than $3,275,000 is 1 percent.

The risk can now be communicated with the statement: *Under
normal market conditions*, *the most the investment security
portfolio will lose in value over a five-day period is about
$3,275,000 with a confidence level of 99 percent*.Philippe
Jorion, *Value at Risk: The New Benchmark for Managing Financial
Risk*, 2nd ed. (McGraw Hill, 2001), ch. 1.
"1:The Nature of Risk - Losses and Opportunities".

In the context of pure risk exposures, the equivalent notion to VaR is the Maximal Probable Annual Loss (MPAL). As with the VaR measure, it looks at a probability distribution, in this case of losses over a year period and then picks the selected lower percentile value as the MPAL. For example, if the loss distribution is given by Figure 2.1.2, and the 95 percent level of confidence is selected, then the MPAL is the same as the 95 percent VaR value. In insurance contexts one often encounters the term MPAL, whereas in finance one often encounters the term VaR. Their calculation is the same and their interpretation as a measure of risk is the same.

We also note that debate rages about perceived weaknesses in
using VaR as a risk measure in finance. “In short, VaR models do
not provide an accurate measure of the losses that occur in
*extreme* events. You simply cannot depict the full texture
and range of your market risks with VaR alone.”Gleason, chapter 12.
In addition, the VaR examines the size of loss that would occur
only 1 percent of the time, but it does not specify the size of the
shortfall that the company would be expected to have to make up by
a distress liquidation of assets should such a large loss occur.
Another measure called the expected shortfall is used for this. The
interested reader is referred to Brockett and AiPatrick L. Brockett
and Jing Ai, “Enterprise Risk Management (ERM),” in
*Encyclopedia of Quantitative Risk Assessment and Analysis*,
ed. E. Melnick and B. Everitt (Chichester, UK: John Wiley &
Sons Ltd., 2008), 559–66. for this calculation.

## CAPM’s Beta Measure of Nondiversifiable Portfolio Risk

Some risk exposures affect many assets of a firm at the same time. In finance, for example, movements in the market as a whole or in the entire economy can affect the value of many individual stocks (and firms) simultaneously. We saw this very dramatically illustrated in the financial crisis in 2008–2009 where the entire stock market went down and dragged many stocks (and firms) down with it, some more than others. In "1: The Nature of Risk - Losses and Opportunities" we referred to this type of risk as systematic, fundamental, or nondiversifiable risk. For a firm (or individual) having a large, well-diversified portfolio of assets, the total negative financial impact of any single idiosyncratic risk on the value of the portfolio is minimal since it constitutes only a small fraction of their wealth.

Therefore, the asset-specific idiosyncratic risk is generally ignored when making decisions concerning the additional amount of risk involved when acquiring an additional asset to be added to an already well-diversified portfolio of assets. The question is how to disentangle the systematic from the nonsystematic risk embedded in any asset. Finance professors Jack Treynor, William Sharpe, John Lintner, and Jan Mossin worked independently and developed a model called the Capital Asset Pricing Model (CAPM). From this model we can get a measure of how the return on an asset systematically varies with the variations in the market, and consequently we can get a measure of systematic risk. The idea is similar to the old adage that a rising tide lifts all ships. In this case a rising (or falling) market or economy rises (or lowers) all assets to a greater or lesser degree depending on their covariation with the market. This covariation with the market is fundamental to obtaining a measure of systematic risk. We develop it now.

Essentially, the CAPM model assumes that investors in assets
expect to be compensated for both the time value of money and the
systematic or nondiversifiable risk they bear. In this regard, the
return on an asset A, \(R_A\), is assumed to be equal to the return
on an absolutely safe or risk-free investment, \(r_f\) (the time
value of money part) and a **risk premium**, which
measures the compensation for the systematic risk they are bearing.
To measure the amount of this systematic risk, we first look at the
correlation between the returns on the asset and the returns on a
market portfolio of all assets. The assumption is that the market
portfolio changes with changes in the economy as a whole, and so
systematic changes in the economy are reflected by changes in the
level of the market portfolio. The variation of the asset returns
with respect to the market returns is assumed to be linear and so
the general framework is expressed as

\[R_A= r_f+β_A× (R_m − r_f )+ ε,\]

where ε denotes a random term that is unrelated to the market return. Thus the term \(β_A× (R_m − r_f )\) represents a systematic return and ε represents a firm-specific or idiosyncratic nonsystematic component of return.

Notice that upon taking variances, we have \(σ_A^{2}= .β_A^2×β_m^2,+ σ_ε^2\), so the first term is called the systematic variance and the second term is the idiosyncratic or firm-specific variance.

The idea behind the CAPM is that investors would be compensated for the systematic risk and not the idiosyncratic risk, since the idiosyncratic risk should be diversifiable by the investors who hold a large diversified portfolio of assets, while the systematic or market risk affects them all. In terms of expected values, we often write the equation as

\[E[R_A]= r_f+ β_A×(E[R_m]− r_f),\]

which is the so-called CAPM model. In this regard the expected rate of return on an asset \(E[R_A]\), is the risk-free investment, \(r_f\), plus a market risk premium equal to \(β_A×(E[R_m] − r_f)\). The coefficient \(β_A\) is called the market risk or systematic risk of asset A.

By running a linear regression of the returns experienced on
asset A with those returns experienced on a market portfolio (such
as the Dow Jones Industrial stock portfolio) and the risk-free
asset return (such as the U.S. T-Bill rate of return), one can find
the risk measure \(β_A\). A regression is a statistical technique
that creates a trend based on the data. An actual linear regression
to compute future frequency and severity based on a trend is used
in
"4: Evolving Risk Management - Fundamental Tools" for risk
management analysis. Statistical books showSee Patrick Brockett and
Arnold Levine Brockett, *Statistics, Probability and Their
Applications* (W. B. Saunders Publishing Co., 1984). that
\(β_A. = \tfrac{COV(R_A, R_m)}{β_m^2}\) where \(COV(R_A,R_m)\) is
the covariance of the return on the asset with the return on the
market and is defined by

\[COV(R_A, R_m) = E[{R_A,−E(R_A)} × {R_m ,-E(R_m )}],\]

that is, the average value of the product of the deviation of the asset return from its expected value and the market returns from its expected value. In terms of the correlation coefficient \(ρ_{Am}\) between the return on the asset and the market, we have \(β_A= ρ_{Am}×(\frac{β_A}{β_m})\), so we can also think of beta as scaling the asset volatility by the market volatility and the correlation of the asset with the market.

The \(β\) (beta) term in the above equations attempts to quantify the risk associated with market fluctuations or swings in the market. A beta of 1 means that the asset return is expected to move in conjunction with the market, that is, a 5 percent move (measured in terms of standard deviation units of the market) in the market will result in a 5 percent move in the asset (measured in terms of standard deviation units of the asset). A beta less than one indicates that the asset is less volatile than the market in that when the market goes up (or down) by 5 percent the asset will go up (or down) by less than 5 percent. A beta greater than one means that the asset price is expected to move more rapidly than the market so if the market goes up (or down) by 5 percent then the asset will go up (or down) by more than 5 percent. A beta of zero indicates that the return on the asset does not correlate with the returns on the market.

## Key Takeaways

- Risk measures quantify the amount of surprise potential contained in a probability distribution.
- Measures such as the range and Value at Risk (VaR) and Maximal Probable Annual Loss (MPAL) focus on the extremes of the distributions and are appropriate measures of risk when interest is focused on solvency or making sure that enough capital is set aside to handle any realized extreme losses.
- Measures such as the variance, standard deviation, and semivariance are useful when looking at average deviations from what is expected for the purpose of planning for expected deviations from expected results.
- The market risk measure from the Capital Asset Pricing Model is useful when assessing systematic financial risk or the additional risk involved in adding an asset to an already existing diversified portfolio.

## Discussion Questions

- Compare the relative risk of Insurer A to Insurer B in the
following questions.
- Which insurer carries more risk in losses and which carries more claims risk? Explain.
- Compare the severity and frequency of the insurers as well.

- The experience of Insurer A for the last three years as given in
Problem 2 was the following:
Year Number of Exposures Number of Collision Claims Collision Losses ($) 1 10,000 375 350,000 2 10,000 330 250,000 3 10,000 420 400,000 - What is the range of collision losses per year?
- What is the standard deviation of the losses per year?
- Calculate the coefficient of variation of the losses per year.
- Calculate the variance of the number of claims per year.

- The experience of Insurer B for the last three years as given in
Problem 3 was the following:
Year Number of Exposures Number of Collision Claims Collision Losses 1 20,000 975 650,000 2 20,000 730 850,000 3 20,000 820 900,000 - What is the range of collision losses?
- Calculate the variance in the number of collision claims per year.
- What is the standard deviation of the collision losses?
- Calculate the coefficient of collision variation.
- Comparing the results of Insurer A and Insurer B, which insurer has a riskier book of business in terms of the range of possible losses they might experience?
- Comparing the results of Insurer A and Insurer B, which insurer has a riskier book of business in terms of the standard deviation in the collision losses they might experience?