way that depends on how returns move together, leads to the primary insight of portfolio theory-that diversification, the spreading of investments across less correlated assets, tends to reduce overall portfolio risk. This risk reduction benefit of diversification can be a free lunch for investors. Given the limited appetite each investor has for risk, the diversification benefit itself creates the opportunity to generate higher expected returns. An additional diversification benefit accrues over time. Due to the relatively high degree of independence of returns during different intervals of time, risk generally compounds at a rate close to the square root of time, a rate that is much less than the additive rate at which returns accrue.4 This difference between the rate at which return grows over time and the rate at which risk grows over time leads to the second insight of portfolio theory-that patience in investments is rewarded and that total risk should be spread relatively evenly over time. Consider a simple example. Taking one percentage point of risk per day creates only about 16 percent5 of risk per year. If this one percentage point of risk per day is expected to create two basis points6 of return per day, then over the course of 252 business days in a year this amount of risk would generate an approximately 5 percent return. If, in contrast, the same total amount of risk, 16 percent, were concentrated in one day rather than spread over the year, at the same rate of expected return, two basis points per percentage point, it would generate only 2 16 = 32 basis points of expected return, less than one-fifteenth as much. So time diversification-that is, distributing risk evenly over a long time horizon-is another potential free lunch for investors. All of us are familiar with the trade-offs between quality and cost in making purchases. Higher-quality goods generally are more expensive; part of being a consumer is figuring out how much we can afford to spend on a given purchase. Similarly, optimal investing depends on balancing the quality of an investment (the amount of excess return an investment is expected to generate) against its cost (the contribution of an investment to portfolio risk). In an optimal consumption plan, a consumer should generate the same utility per dollar spent on every purchase. Otherwise, dollars can be reallocated to increase utility. Similarly, in an optimal portfo- 4In fact, as noted earlier, due to compounding, returns accrue at a rate greater than additive. To develop an intuition as to why risk does not increase linearly in time, suppose the risk in each of two periods is of equal magnitude, but independent. The additive nature of the variance of independent returns implies that the total volatility, the square root of total variance, sums according to the same Pythagorean formula that determines the hypotenuse of a right triangle. Thus, in the case of equal risk in two periods, the total risk is not two units, but the square root of 2, as per the Pythagorean formula. More generally, if there are the square root of t units of risk (after t periods), and we add one more unit of independent risk in period t + 1, then using the same Pythagorean formula there will be the square root of t + 1 units of risk after the t + 1st period. Thus, the total volatility of independent returns that have a constant volatility per unit of time grows with the square root of time. This will be a reasonable first-order approximation in many cases. 5Note that 16 is just slightly larger than the square root of 252, the number of business days in a year. &A basis point is one-hundredth of a percent.