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Welcome to our tutorial on risk analysis -- from Frontline Systems, developers of the Excel Solver and Risk Solver software.  Before we talk about methods for risk analysis, let's make sure we understand the basics:

#### What is Risk?

Uncertainty, which is constantly present in our daily lives, frequently impacts our decisions and actions.  When we talk about risk, we normally mean the chance that some undesirable impact will occur.  Hence, we normally seek to avoid or minimize risk.  If there is a chance of rain, and we don't want to get wet, we may choose to stay indoors -- avoiding that risk -- or we may take an umbrella to minimize the impact of rain upon us.

Uncertainty can impact our decisions and actions in desirable as well as undesirable ways.  If we own shares of stock, the future price is uncertain -- it may go higher, which is desirable, or it may go lower, which is undesirable.  When contemplating large payoffs or penalties, most people are risk averse.  For example, suppose we can choose between a coin toss that gains \$50 or breaks even, and a coin toss that gains \$150 or loses \$100.  Most people would choose the first coin toss, even though the average or 'expected' outcome of both tosses is \$25.  Hence, in risk analysis we usually focus on what can go wrong -- the outcomes that represent loss or damage -- although a good analysis will also help us understand what can go right as well.

#### What are Sources of Uncertainty?

Uncertainty can arise in several ways:

• If the quantity we'd like to know is a competing firm's planned product price, uncertainty arises from our lack of knowledge:  The price may be well-known to that firm's employees, but it's unknown to us.
• If the quantity is market demand for products like ours, uncertainty arises from the complexity of the process:  Demand depends on economic factors, fashions and preferences, and our and other firms' actions -- and even if we knew all of these, we couldn't precisely calculate their net impact on final demand.
• If the quantity is a material thickness in nanometers, uncertainty may arise from limits on our ability to measure this physical quantity.  We may also have limits on our ability to control fabrication of the material.
• Many processes that we want to model -- from the failure rate of an electronic component to the behavior of a macromolecule -- have inherent randomness for all intents and purposes.

Uncertainty that is inherent in nature is sometimes called irreducible uncertainty.  You may be able to reduce the effect of the random variation on your model, or reduce your model's sensitivity to this variation, but it will always be there.  In other situations you may be dealing with reducible uncertainty -- through market research, physical tests, better calibration, or other means you may be able to reduce the uncertainty itself.

#### How are Uncertainty and Risk Different?

Uncertainty is normally an intrinsic feature of some part of nature -- it is the same for all observers.  But risk is specific to a person or company -- it is not the same for all observers.  The possibility of rain tomorrow is uncertain for everyone; but the risk of getting wet is specific to me, if (i) I intend to go outdoors and (ii) I view getting wet as undesirable.  The possibility that stock A will decline in price tomorrow is an uncertainty for both you and me; but if you own the stock long and I do not, it is a risk only for you.  If I have sold the stock short, a decline in price is a desirable outcome for me.

Many, but not all, risks involve choices.  By taking some action, we may deliberately expose ourselves to risk -- normally because we expect a gain that more than compensates us for bearing the risk.  If you and I come to a bridge across a canyon that we want to cross, and we notice signs of weakness in its structure, there is uncertainty about whether the bridge can hold our weight, independent of our actions.  If I choose to walk across the bridge to reach the other side, and you choose to stay where you are, I will bear the risk that the bridge will not hold my weight, but you will not.  Most business and investment decisions are choices that involve "taking a calculated risk" -- and risk analysis can give us better ways to make the calculation.

#### How Can We Best Deal with Risk?

If the stakes are high enough, we can and should deal with risk explicitly, with the aid of a quantitative model.  As humans, we have heuristics or "rules of thumb" for dealing with risk, but these don't serve us very well in many business and public policy situations.  In fact, much research shows that we have cognitive biases, such as over-weighting the most recent adverse event and projecting current good or bad outcomes too far into the future, that work against our desire to make the best decisions.  Quantitative risk analysis can help us escape these biases, and make better decisions.

It helps to recognize up front that when uncertainty is a large factor, the best decision does not always lead to the best outcome.  The "luck of the draw" may still go against us.  Risk analysis can help us analyze, document, and communicate to senior decision makers and stakeholders the extent of uncertainty, the limits of our knowledge, and the reasons for taking a course of action.