This is “Risk and Return Basics”, section 11.1 from the book Finance for Managers (v. 0.1).

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- Explain risk and risk aversion.
- Calculate the rate of return and expected rate of return for an investment.

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

- You receive $1,000.
- 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.

Equation 11.1 Rate of Return

$$\begin{array}{ccc}RateofReturn& =& \frac{Return}{Investment}\\ & =& \frac{(CashInflows-CashOutflows)}{CashOutflows}\end{array}$$Often, we will want to know what an investment’s expected rate of returnThe weighted mean of the possible investment outcomes; that is, each possible outcome weighted by the probability of that outcome. is. When we talk of expected return, we will use the weighted mean of the possible investment outcomes; that is, each possible outcome weighted by the probability of that outcome.

Equation 11.2 Expected Rate of Return (weighted mean)

Expected Rate of Return = (probability of outcome 1 × rate of return of outcome 1) + (probability of outcome 2 × rate of return of outcome 2) + … + (probability of outcome n × rate of return of outcome n) =For example, consider an investment that has a 25% chance of gaining 10%, a 50% chance of gaining 5%, and a 25% chance of losing 10% (a return of −10%). Our expected rate of return is .25 × .10 + .50 + .25 × (−.10) = 0.25 = 2.5%.

Other potential ways of thinking about the expectations surrounding the rate of return are also valid, and investors should be encouraged to think about more than just the weighted mean. We could consider the most likely outcome (in our above example a gain of 5%). We could consider the range of outcomes (a return between +10% and −10%). We could plot our expected returns on a graph, and try to consider the shape of the graph. Each gives us a different insight about the uncertainty involved in the investment.

Of course, we rarely know with this level of precision what the future will hold, so we have to make our best educated guess. A common way to guess is to look at the historical returns from similar investments and find the yearly average, usually assuming each year has equal weight (the arithmetic mean). There are some downsides to this: just because an investment behaved one way in the past is no guarantee that it will continue to behave that way! For example, as of this writing, your authors have successfully not died each day of our lives (so far). If we extrapolate these results into the future, we should expect to live forever! While an extreme example, it should underscore a key financial maxim: historical results are not a guarantee of future performance.

- Dealing with risk is a key part of finance, especially since most people are risk averse.
- Expected future returns are the weighted average of the possible outcomes. Historical results are often used as a proxy for future expected results.
- Historical results are not a guarantee of future performance!

- Genny invested $500. She received two payments of $100 and then sold her stake in the investment for $400. What was her rate of return?
- An investment has a 10% chance of returning 20%, a 70% chance of returning 10%, and a 20% chance of returning nothing. What is the expected rate of return?
- An investment has returned 5%, 3%, −4%, 6%, and −7% in each of the last five years, respectively. We have decided that we want to use historical returns as a proxy for expected future returns. What is the expected rate of return? Why might the historical results be a poor estimate of future returns?