► C = A + B A = Old Portfolio Expected Return B = New Investment Expected Return C = New Portfolio Expected Return FIGURE 2.1 Expected Return Sums Linearly The interesting insights provided by modern portfolio theory arise from the interplay between the mathematics of return and risk. It is important at this juncture to review the different rules for adding risks or adding returns in a portfolio context. These issues are not particularly complex, but they are at the heart of modern portfolio theory. The mathematics on the return side of the investment equation is straightforward. Monetary returns on different investments at a point in time are additive. If one investment creates a $30,000 return and another creates a $40,000 return, then the total return is $70,000. The additive nature of investment returns at a point in time is illustrated in Figure 2.1. Percentage returns compound over time. A 20 percent return one year followed by a 20 percent return the next year creates a 44 percent1 return on the original investment over the two-year horizon. The risk side of the investment equation, however, is not so straightforward. Even at a point in time, portfolio risk is not additive. If one investment creates a volatility2 of $30,000 per year and another investment creates a volatility of $40,000 per year, then the total annual portfolio volatility could be anywhere between $10,000 and $70,000. How the risks of different investments combine depends on whether the returns they generate tend to move together, to move independently, or to offset. If the returns of the two investments in the preceding example are roughly independent, then the combined volatility is approximately3 $50,000; if they move together, the combined risk is higher; if they offset, lower. This degree to which returns move together is measured by a statistical quantity called correlation, which ranges in value from +1 for returns that move perfectly together to zero for independent returns, to -1 for returns that always move in oppo- !The two-period return is z, where the first period return is x, the second period return is y, and (1 + z) = (1 + x)(l + y). ^Volatility is only one of many statistics that can be used to measure risk. Here "a volatility" refers to one standard deviation, which is a typical outcome in the distribution of returns. 3In this calculation we rely on the fact that the variance (the square of volatility) of independent assets is additive.