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: