Electronic copy available at: http://ssrn.com/abstract=1687272YALE LAW SCHOOLJohn M. Olin Center for Studies in Law, Economics, and Public PolicyResearch Paper No. 413Diversification Across Timeby Ian AyresYale University - Yale Law SchoolYale University - Yale School of ManagementBarry J. NalebuffYale University - Yale School of ManagementElectronic copy available at: http://ssrn.com/abstract=1687272Electronic copy available at: http://ssrn.com/abstract=1687272Diversification Across Time Ian Ayres* & Barry Nalebuff** By employing leverage to gain more exposure to stocks when young, individuals can achieve better diversification across time. Using stock data going back to 1871, we show that early leverage combined with reduced equity exposure when older can reduce lifetime portfolio risk. For example, an initially-leveraged portfolio can produce the same mean accumulation as a constant 75% stock allocation with a 21% smaller standard deviation. Since the mean accumulation is the same, the reduction in volatility does not depend on the equity premium. A leveraged lifecycle strategy can also allow investors to come closer to their utility-maximizing allocation. If risk preferences would lead an investor to allocate 50% of his discounted retirement savings to stocks, that would require a young investor to put well more than 50% of his liquid savings into stocks. We employ leverage (limited to 2:1) to help the investor overcome a limited ability to borrow against human capital. Based on historical returns, we find a 37% improvement in the certainty equivalent (for CRRA=4). Monte Carlo simulations show that these gains continue even with equity premia well below the historical average. Keywords: Diversification, Leverage, Retirement, Investment Strategy JEL Classifications: D31, G1, G11, G18, H5 * William K. Townsend Professor, Yale Law School. [email protected] ** Milton Steinbach Professor, Yale School of Management. [email protected] James Poterba and Robert Shiller provided helpful comments. Isra Bhatty, Jonathan Borowsky, Katie Pichotta, Jake Richardson, Alice Shih, and Heidee Stoller provided excellent research assistance. We thank seminar participants at Yale Law School, SOM, Harvard Law School, University of Chicago, University of Missouri, and Stanford. Preliminary investigations into the issues covered in this paper were published by Forbes and Basic Books (see Ayres and Nalebuff (2005, 2010)). The data and simulations underlying our estimates can be downloaded from http://islandia.law.yale.edu/ayers/retirement.zip.Electronic copy available at: http://ssrn.com/abstract=1687272Electronic copy available at: http://ssrn.com/abstract=1687272 1 [The businessman] can look forward to a high salary in the future; and with so high a present discounted value of wealth, it is only prudent for him to put more into common stocks compared to his present tangible wealth, borrowing if necessary for the purpose . . . . [This point] does justify leveraged investment financed by borrowing against future earnings. But it does not really involve any increase in relative risk-taking once we have related what is at risk to the proper larger basis. – Paul Samuelson (1969) This paper shows that a leveraged lifecycle strategy, one which starts with a leveraged allocation in stock and then gradually decreases leverage and ultimately becomes unleveraged closer to retirement, produces a substantial improvement in expected utility. The gain comes from two factors. The first is improved diversification. Investors use mutual funds to diversify across stocks and over geographies. Even if they are diversified across assets they are insufficiently diversified across time. They have too much invested in stock late in their life and not enough early on. An initially-leveraged portfolio can produce the same mean accumulation with a 21% smaller standard deviation. The second source of gain comes from coming closer to the utility-maximizing investment level. The recommendation from Samuelson (1969) and Merton (1969, 1971) is to invest a constant fraction of wealth in stocks (what we call the Merton-Samuelson share). The mistake in translating this theory into practice is that young people invest only a fraction of their current savings instead of their discounted lifetime savings. For someone in their 30's, investing even 100% of current savings is likely to be less than 10% of their lifetime savings and typically far less than the utility-maximizing Merton-Samuelson share. In the Merton-Samuelson framework, all of a person’s wealth for both consumption and saving was assumed to come at the beginning of the person’s life. Of course that isn’t the situation for a typical worker who starts with almost no savings. Thus, the advice to invest even 50% of the present value of future savings in stocks would imply an investment well more than what would be currently available. This leads to our prescription: buy stocks using leverage when young. More specifically, following Merton-Samuelson, we analyze a strategy that targets investing in stock a constant percentage of the present value of lifetime savings. This strategy calls for leveraged investing in the early years of life when the present value of future contributions is large relative to current savings and then reduced leverage and finally unleveraged investing as current savings grow and the present value of future contributions2 decline. For this strategy to be effective, it requires the ability to borrow at low cost. As we will show, the implied interest cost for 2:1 leverage is remarkably low; in August 2010, it was below 1.0%. Our leveraged strategy still falls short of the investment target in initial years because the incremental cost of borrowing to achieve more than 2:1 leverage increases rapidly and outweighs the benefits. Hence, we analyze leveraged lifecycles which initially allocate to stock a maximum 200% of current savings and then ramp down over time toward the Merton-Samuelson share as the investor nears retirement (and has no future expected saving contributions). To motivate what follows, imagine that a worker has to