# Chapter 5: Understanding Risk

In calculating the present value and IRR in the previous chapter, we made important assumptions about the certainty of the payments. In reality, for many borrowers and bonds, there is some positive probability that the promised payments will not be made at all or will be late. This is known as a default. Understanding and quantifying the risk of default, along with many other types of risk are key to the pricing of financial instrument. Furthermore, the desire to manage various risks is the motivation for financial markets in derivative securites and insurance. The role of risk in finance cannot be overemphasized. Recall core principle 2: Risk requires compensation. To understand why and how this happens, we first need to understand risk.

Defining Risk

Risk is fundamentally about uncertainty. In the context of finance it is uncertainty about future payments of a financial instrument. More specifically, risk is not just a concept, but a measure that can be quantified so the risks of different securities can be compared. Risk depends on time horizon and risk is typically a relative measure, meaning that risk is compared to some benchmark, such as a risk-free investment or the risk of similar investments.

Measuring Risk

So how do we quantify risk? There are many measures used in quantitative financial analysis, some of which are quite complicated. However, we will focus on the basic measures involving probabilities, expected value, and standard deviation.

Probability

Probabilities measure the likelihood of certain events. A probability measure is between 0 and 1. The closer the probability is to 1, the more likely an event is to occur. The probabilities of all possible events must add to 1. For example, consider flipping a coin. Here there are 2 possible outcomes: heads or tails. If the coin is fair, the probability of each possible outcome is 1/2 or .5. So there are 2 outcomes, each with .5 probability and this of course adds to 1.

Expected Value

Knowing all the possible outcomes and their respective probabilities we can compute the expected value. The expected value is something like an average outcome and is also known as the mean. Mathematically the expected value is the sum of each possible outcome, weighted by its probability. Let's do a couple of examples. Below are two alternative investment projects involving and initial investment of \$1000. The possible payoffs for each alternative, and the respective probabilities of that payoff are listed below:

 Investment 1 Investment 2 Payoff Probability Payoff Probability \$500 .20 \$800 .25 \$1000 .40 \$1000 .35 \$1500 .40 \$1375 .40

The expected value of investment 1 = (\$500)(.20) + (\$1000)(.40) + (\$1500)(.40) = \$1100

The expected value of investment 2 = (\$800)(.25) + (\$1000)(.35) + (\$1375)(.40) = \$1100

Note that in each case the expected value is not a possible payoff. Instead, expected value measures the center of the probability weights: if we were to make either of these investments over and over, we would average \$1100 on a \$1000 investment, or a 10% return.

So are we indifferent between these two investments? After all, they have the same expected value. But expected value is not the whole story. The options here different in how far the payoffs are from the expected value and in how likely those payoffs are. So we need an additional measure of the spread of possible payoffs.

Variance (s2) and Standard Deviation (s)

The most common measures of spread are variance and standard deviation. To calculate various we subtract the expected value from each possible outcome and square the result, weight each result by its probability, and sum the final products. Standard deviation is simply the square root of the variance. Standard Deviation is a more convenient measure because it is in the same units as expected value (\$) while variance is measured in squared units (\$ squared). Let's calculate the variance for each of the investments above:

Variance of investment 1 =

Standard deviation of investment 1 =

Variance of investment 2 =

Standard deviation of investment 2 =

Lower standard deviation means lower risk. The \$341 standard deviation is a large spread in likely payoffs than the \$237 standard deviation. In other words, investment 1 is more likely to have a payoff that is a larger deviation from its expected value, relative to investment 2.

Risk Aversion and the Risk Premium

In economics and finance, we assume the people are risk averse. This means people do not like risk ALL ELSE BEING EQUAL. Consider the example above, with expected value of \$1100 or a 10% expected return. Would you take investment 1 over the choice of a GUARANTEED 10% return? No, you would not. Furthermore, risk averse investors will prefer investment 2 to investment 1 since it has the same expected return, but a lower risk.

Risk aversion is a simple but powerful concept. It is a key building block to all modern asset pricing and portfolio theory in finance. If goes back to the core principle that risk requires compensation. To entice investors to buy securities with higher risky, the seller must offer is higher expected payoff. This is known as the risk premium. This also means that investments with higher expected payoffs also carry higher risk. This is the risk-return tradeoff. You don't get both low risk and high return.

Sources of Risk

Risk comes from numerous sources but all sources can be placed into two categories:

(1) Idiosyncratic risk (also known as nonsystematic risk) are risks that affect specific firms or a small number of firms without affecting others.

(2) Systematic risk (also known as macro risk) are risks that affect all firms and even the entire economy.

An example of idiosyncratic risk would be the problems faced by Microsoft due to European antitrust litigation. This is specific to Microsoft and the value of their stock. It will not impact most other stocks, or the value of U.S. Treasury bonds or home prices, etc. Another example would be the risk of labor strikes for United Airline employees. This would have a huge impact on UAL, but virtually no impact on General Electric or Google or Wal Mart.

Systematic risk comes from widespread macroeconomic factors: inflation, falling consumer confidence, business cycles. Few financial instruments, if any, would be immune from the impacts of a recession.

Diversification

Risk is unavoidable but this does not mean it cannot be managed and minimized. Ironically, by holding multiple assets with multiple risks in a portfolio, the combination of risks is often less risky than any single asset. This is the principle of diversification. It works either through hedging risk or spreading risk.

Hedging risk means holding assets whose payoffs offset each other--one does well when one does poorly. Derivative instruments are specifically created for hedging, allowing investors to take a position that offsets the risks of their other assets. We will see this in action in later chapters. However, in reality it is not always possible to construct a portfolio where payoffs offset each other in a predictable, consistent way. We know that systematic risk impacts an entire economy and may not be possible to offset.

Spreading risk means holding a portfolio of assets whose payoffs are unrelated. This minimizes idiosyncratic risks since each asset represents a small part of the portfolio payoffs. The assets may also differ in their sensitivity to systematic risks. Risk spreading is at the heart of the insurance industry. In any given year, only a small fraction of investors paying life insurance premiums will die. The predictable mortality rate in this pool of policy holders allow life insurance companies to profitably take on this risk.

As we will see in the coming chapters, risk is an important component of asset pricing and a key motivation for many of the services and products offered by financial institutions.