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As all fund managers know, there is a trade-off between risk and return when money is invested. The greater the risks taken, the higher the return that can be realized. 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 %.1 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.

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

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.” In 1976, Ross developed the 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 that they face. In this section we review these insights.

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 R is the return per annum. The symbol E denotes expected value so that E(R) is the expected return per annum. In Table 1.1, as we have shown, E(R) = 0.10. To calculate E(R²) we must weight the alternative squared returns by their probabilities:

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

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 R₁ and R₂. The return from putting a proportion w₁ of our money in the first investment and a proportion w₂ = 1 − w₁ in the second investment is

The portfolio expected return is

(1.1)

where μ₁ is the expected return from the first investment and μ₂ is the expected return from the second investment. The standard deviation of the portfolio return is given by

(1.2)

where σ₁ and σ₂ are the standard deviations of R₁ and R₂ and ρ is the coefficient of correlation between the two.

Suppose that μ₁ is 10 % per annum and σ₁ is 16 % per annum, while μ₂ is 15 % per annum and σ₂ is 24 % per annum. Suppose also that the coefficient of correlation, ρ, between the returns is 0.2 or 20 %. Table 1.2 shows the values of μP and σP for a number of different values of w₁ and w₂. 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.

TABLE 1.2 Expected Return, μP, and Standard Deviation of Return, σP, from a Portfolio Consisting of Two Investments

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

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

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 risk-return trade-off for the first investment. (The expected return is 2 % higher and the standard deviation of the return is 1.13 % lower.)