4.1 Introduction to Risk and Return

Almost everyone recognizes that risk must be considered in determining value and making investment decision. Some investors are willing to take more risk than others because they believe risk can be read as“opportunity”. To most investors who are risk-neutral, they require higher expected return from risky investment. In fact, valuation and an understanding of the trade-off between risk and return form the foundation for maximizing shareholder wealth.

To make effective financial decision, managers need to understand how risk should be measured and the effect of risk on the rate of return required by investors. In this chapter, we will focus our discussion on the risk and return using the framework of portfolio theory, and CAPM which links expected return to risk.

4.1.1 Defining Return and Risk

The return and risk of an investment are basic concepts in finance.

Return on an investment is the financial outcome for the investor, usually expressed as an annualized percentage, which is also called rate of return.

Risk is present whenever investors are not certain about the outcomes an investment will produce.

Consider the two examples below：

（a）A government bond with exactly one year to maturity，yield 5%.

（b）A share of common stock in any company and hold it for one year.

If you invest on（a）, you will realize a government guaranteed 5% return on your investment. On（b）, the cash dividend that you anticipate receiving may or may not materialize as expected, and what is more, the year-end price of the stock might be even lower than you started with.

Therefore, we can define risk as the variability of returns from those that are expected. This would include both potential better-than-expected and worse-than-expected returns. Government bond is risk-free asset whereas the common share would be risky. The greater variability, the riskier the investment is said to be.

A financial decision typically involves risk. For example, a company that built a new factory faces the risk that product sales may be lower than expected; an investor who purchased common shares may suffer losses due to the decrease in share price.

Generally, investors do not enjoy taking risk, unless they are satisfied with the return on that investment. Whether the return is high enough to compensate the risk, we now need to measure them mathematically before making trade-off decision.

4.1.2 Measuring Return-Expected Return

Illustration 4.1

An investment would have different outcome due to economic uncertainty, suppose that we identify each possible outcome with assigned possibility.

The expect return is：

Where,Ri is the return for the i th possibility,Pi is the probability of that return occurring,n is the total number of possibility.

For the distribution of possible returns illustrated in Table 4.1, the expected return is shown to be 8.7%.

4.1.3 Measuring Risk-Standard Deviation

離散程度越大，風險越大。離散程度用方差來衡量。方差（樣本方差）是每個樣本值與全體樣本值的平均數之差的平均數。

In the above Illustration 4.1, risk is present because any one of four outcomes might result from the investment. We need a measure of the dispersion, or variability, around our expected return. The more dispersed or widespread the distribution, the greater the risk involved. Statisticians have developed a number of measures to represent dispersion. The conventional measure is the variance（or its square root, the standard deviation）which measures how far a set of numbers are spread out from their average value.

The variance, σ2, can be expressed mathematically as：

In the case in Table 4.1, the variance is：

σ2=（-12%-8.7%）2×0.15+（4%-8.7%）2×0.2+（9%-8.7%）2×0.3+（25%-8.7%）2×0.35=0.01891

Thus the standard deviation is：

That is, the greater the standard deviation of returns, the greater the variability of returns, and the greater the risk of the investment.

Example 4.1

In Table 4.2, there are three mutually exclusive investments available to investors.

Question:What do you suggest for investors？

Solution

Firstly, the project A should be ruled out because it offers lower return than project B with the same level of risk（same standard deviation）, there are two options left, project B and C. Now the choice depends on the investor's risk tolerance.