Chapter 1
Introduction: Risk-Return Trade-offs

Imagine you are the Chief Risk Officer (CRO) of a major corporation. The Chief Executive Officer (CEO) wants your views on a major new venture. You have been inundated with reports showing that the new venture has a positive net present value and will enhance shareholder value. What sort of analysis and ideas is the CEO looking for from you?

As CRO it is your job to consider how the new venture fits into the company's portfolio. What is the correlation of the performance of the new venture with the rest of the company's business? When the rest of the business is experiencing difficulties, will the new venture also provide poor returns, or will it have the effect of dampening the ups and downs in the rest of the business?

Companies must take risks if they are to survive and prosper. The risk management function's primary responsibility is to understand the portfolio of risks that the company is currently taking and the risks it plans to take in the future. It must decide whether the risks are acceptable and, if they are not acceptable, what action should be taken.

Most of this book is concerned with the ways risks are managed by banks and other financial institutions, but many of the ideas and approaches we will discuss are equally applicable to nonfinancial corporations. Risk management has become progressively more important for all corporations in the last few decades. Financial institutions in particular are finding they have to increase the resources they devote to risk management. In spite of this, financial institutions still make mistakes. In March 2021, Archegos, a hedge fund managing the assets of a rich private individual named Bill Hwang, went bankrupt as a result of highly levered risky investments. Several banks that the hedge fund had been dealing with lost huge sums. It is reported that Credit Suisse lost over $5 billion and Nomura lost almost $3 billion as a result of their dealings with Archegos.

Risk management is not about minimizing risks. It is about ensuring that the risks taken are manageable and that the expected returns are commensurate with the risks being taken. This chapter sets the scene. It starts by reviewing the classical arguments concerning the risk-return trade-offs faced by a well-diversified investor who is choosing a portfolio of stocks and bonds. It then considers whether the same arguments can be used by a company in choosing new projects and managing its risk exposure. The chapter concludes that there are reasons why all companies, including financial institutions, should be concerned with the total risk they face, not just with the risk from the viewpoint of a well-diversified shareholder.

1.1 Risk vs. Return for Investors

We now examine the theoretical trade-off between risk and return when money is invested. The first point to note is that the trade-off is actually between risk and expected return, not between risk and actual return. The term "expected return" sometimes causes confusion. In everyday language an outcome that is "expected" is considered highly likely to occur. However, statisticians define the expected value of a variable as its average (or mean) value. Expected return is therefore a weighted average of the possible returns, where the weight applied to a particular return equals the probability of that return occurring. The possible returns and their probabilities can be either estimated from historical data or assessed subjectively.

Suppose, for example, that you have $100,000 to invest for one year. Suppose further that Treasury bills yield 5%. One alternative is to buy Treasury bills. There is then no risk and the expected return is 5%. Another alternative is to invest the $100,000 in a stock. To simplify things, we suppose that the possible outcomes from this investment are as shown in Table 1.1. There is a 0.05 probability that the return will be +50%; there is a 0.25 probability that the return will be +30%; and so on. Expressing the returns in decimal form, the expected return per year is:

This shows that, in return for taking some risk, you are able to increase your expected return per annum from the 5% offered by Treasury bills to 10%. If things work out well, your return per annum could be as high as 50%. But the worst-case outcome is a -30% return or a loss of $30,000.

One of the first attempts to understand the trade-off between risk and expected return was by Markowitz (1952). Later, Sharpe (1964) and others carried the Markowitz analysis a stage further by developing what is known as the capital asset pricing model. This is a relationship between expected return and what is termed systematic risk, which will be explained later in this chapter. In 1976, Ross developed arbitrage pricing theory, which can be viewed as an extension of the capital asset pricing model to the situation where there are several sources of systematic risk. The key insights of these researchers have had a profound effect on the way portfolio managers think about and analyze the risk-return trade-offs they face. We will now review these insights.

Table 1.1 One-Year Return from Investing $100,000 in a Stock

1.1.1 Quantifying Risk

How do you quantify the risk you are taking when you choose an investment? A convenient measure that is often used is the standard deviation of the return over one year. This is

where is the return per annum. The symbol denotes expected value so that is the expected return per annum. In Table 1.1, as we have shown, . To calculate we must weight the alternative squared returns by their probabilities:

The standard deviation of the annual return is therefore or 18.97%.

1.1.2 Investment Opportunities

Suppose we choose to characterize every investment opportunity by its expected return and standard deviation of return. We can plot available risky investments on a chart such as Figure 1.1, where the horizontal axis is the standard deviation of the return and the vertical axis is the expected return.

Figure 1.1 Alternative Risky Investments

Once we have identified the expected return and the standard deviation of the return for individual investments, it is natural to think about what happens when we combine investments to form a portfolio. Consider two investments with returns and . The return from putting a proportion of our money in the first investment and a proportion in the second investment is

The portfolio expected return is

where is the expected return from the first investment and is the expected return from the second investment. The standard deviation of the portfolio return is given by

where and are the standard deviations of and and is the coefficient of correlation between the two.

Suppose that is 10% per annum and is 16% per annum, while is 15% per annum and is 24% per annum. Suppose also that the coefficient of correlation, , between the returns is 0.2 or 20%. Table 1.2 shows the values of and for a number of different values of and . The calculations show that by putting part of your money in the first investment and part in the second investment a wide range of risk-return combinations can be achieved. These are plotted in Figure 1.2.

Figure 1.2 Alternative Risk-Return Combinations from Two Investments (as Calculated in Table 1.2)

Table 1.2 Expected Return, , and Standard Deviation of Return, , from a Portfolio Consisting of Two Investments

The expected returns from the investments are 10% and 15%; the standard deviations of the returns are 16% and 24%; and the correlation between returns is 0.2.

Most investors are risk-averse. They want to increase expected return while reducing the standard deviation of return. This means that they want to move as far as they can in a "northwest" direction in Figures 1.1 and 1.2. Figure 1.2 shows that forming a portfolio of the two investments we have been considering helps them do this. For example, by putting 60% in the first investment and 40% in the second, a portfolio with an expected return of 12% and a standard deviation of return equal to 14.87% is obtained. This is an improvement over the...