Generalized hyperbolic functions, circulant matrices and functional equations

There is a contrast between the two sets of functional equations f_0(x+y) = f_0(x)f_0(y) + f_1(x)f_1(y), f_1(x+y) = f_1(x)f_0(y) + f_0(x)f_1(y), and f_0(x-y) = f_0(x)f_0(y) - f_1(x)f_1(y), f_1(x-y) = f_1(x)f_0(y) - f_0(x)f_1(y) satisfied by the even and odd components of a solution of f(x+y) = f(x) f(y). J. Schwaiger and, later, W. F\"org-Rob and J. Schwaiger considered the extension of these ideas to the case where f is sum of n components. Here we shorten and simplify the statements and proofs of some of these results by a more systematic use of matrix notation.Comment: 18 pages; corrected and updated versio

The Rees product of posets

We determine how the flag f-vector of any graded poset changes under the Rees product with the chain, and more generally, any t-ary tree. As a corollary, the M\"obius function of the Rees product of any graded poset with the chain, and more generally, the t-ary tree, is exactly the same as the Rees product of its dual with the chain, respectively, t-ary chain. We then study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give a bijective proof that the M\"obius function of this poset can be expressed as n times a signed derangement number. From this we derive a new bijective proof of Jonsson's result that the M\"obius function of the Rees product of the Boolean algebra with the chain is given by a derangement number. Using poset homology techniques we find an explicit basis for the reduced homology and determine a representation for the reduced homology of the order complex of the Rees product of the cubical lattice with the chain over the symmetric group.Comment: 21 pages, 1 figur

Are Retirement Savings Too Exposed to Market Risk?

The stock market, as measured by the broad-based Wilshire 5000, declined by 42 percent between its peak in October 9, 2007 and October 9, 2008. Over that one-year period, the value of equities in pension plans and household portfolios fell by $7.4 trillion. Of that$7.4 trillion decline, $2.0 trillion occurred in 401(k)s and Individual Retirement Accounts (IRAs),$1.9 trillion in public and private defined benefit plans, and $3.6 trillion in household non-pension assets. This brief documents where the declines occurred. This information is interesting and important in its own right. But the declines also highlight the fragility of our emerging pension arrangements. Today the declines were divided equally between defined benefit and defined contribution plans, but in the future individuals will bear the full brunt of market turmoil as the shift to 401(k)s continues. Much of the reform discussion regarding private sector employer-sponsored pensions has focused on extending coverage. But the current financial tsunami also underlines the need to construct arrangements where the full market risk does not fall on pension participants.

Why Are Stocks So Risky?

With the decline in privately and publicly guaranteed benefits for pensions and health care, people increasingly must finance a greater share of their retirement expenses through their own savings. The relatively high long-term return on equity makes investments in stocks seem both an attractive and suitable means of accumulating the substantial wealth that savers will require. Yet, the 50 percent drop in the Standard & Poor’s 500 Index from May 2008 to March 2009 is only the latest reminder that stocks pose considerable risk for investors. In the past, equity returns over periods as long as 10 or 20 years have diverged substantially from their long-term averages, tarnishing the appeal of stocks even as investments for the long run...