Two-fund Theorem Research Articles

Rational pricing of leveraged ETF expense ratios

This paper studies the general relationship between the gearing ratio of a Leveraged ETF and its corresponding expense ratio, viz., the investment management fees that are charged for the provision of this levered financial service. It must not be possible for an investor to combine two or more LETFs in such a way that his (continuously-rebalanced) LETF portfolio can match the gearing ratio of a given, professionally managed product and, at the same time, enjoy lower weighted-average expenses than the existing LETF. Given a finite set of LETFs that exist in the marketplace, I give necessary and sufficient conditions for these products to be undominated in the price-gearing plane. In an application of the duality theorem of linear programming, I prove a kind of two-fund theorem for LETFs: given a target gearing ratio for the investor, the cheapest way to achieve it is to combine (uniquely) the two nearest undominated LETF products that bracket it on the leverage axis. This also happens to be the implementation with the lowest annual turnover. For completeness, we supply a second proof of the Main Theorem on LETFs that is based on Carathéodory’s theorem in convex geometry. Thus, say, a triple-leveraged (“UltraPro”) exchange-traded product should never be mixed with cash, if the investor is able to trade in the underlying index. In terms of financial innovation, our two-fund theorem for LETFs implies that the introduction of new, undominated 2.5\(\times \) products would increase the welfare of all investors whose preferred gearing ratios lie between 2\(\times \) (“Ultra”) and 3\(\times \) (“UltraPro”). Similarly for a 1.5x product.

Portfolio optimization under a minimax rule revisited

In this paper, we revisit the bi-criteria portfolio optimization model where the short selling is permitted, and a trade-off is sought between the expected return rate of a portfolio and the maximum of the uncertainty measured by a general deviation measure for all the investments comprising a portfolio. We solve this bi-criteria model by first converting it into a collection of weighted sum piecewise linear convex programs, and then analysing their optimality conditions. We not only provide explicit analytical formulas for all the efficient portfolios, but also explore as a whole the set of all the efficient portfolios and its structure such as dimensionality and distribution. We generalize the classical Two-fund Theorem by providing some collections of finitely many efficient portfolios to generate or estimate the set of all the efficient portfolios. We also notice that our efficient portfolios are almost the risk parity ones in the sense that the risks are allocated equally across the investments. Moreover, we illustrate the reliability of our model by carrying out Monte Carlo simulations to test the performance of some efficient portfolios versus inefficient ones.

Efficiency, Spanning, and the Fiduciary in 401(k) Plans

We use regression-based tests for mean-variance spanning to justify a systematic implementation of efficient and prudent default fund selections in 401(k) plans. Though our approach ultimately rests on recommending two funds, it differs from the classical two-fund theorem, as the funds are not required to be efficient. In essence, our results illustrate the difference between, on one hand, theoretical mean-variance efficiency, and on the other, empirical mean-variance spanning. Furthermore, we address collinearity and the preponderance of numerical instability in regression-based testing for mean-variance spanning. Finally, we show that published statistical hypotheses for mean-variance testing of 401(k) plans are incorrectly specified.

An Asset−Liability Version of the Capital Asset Pricing Model with a Multi-Period Two-Fund Theorem

An Asset−Liability Version of the Capital Asset Pricing Model with a Multi-Period Two-Fund Theorem

An Asset–Liability Version of the Capital Asset Pricing Model with a Multi-Period Two-Fund Theorem

Background: The optimal role of postoperative radiotherapy for patients with prostate cancer remains undefined. Materials and Methods: The medical records of 70 patients (median age: 66 years), who had received radical radiotherapy (RT) between the years 1996 and 2004 after radical prostatectomy (RP), were analyzed. Fifteen patients had received immediate adjuvant RT, while the other 55 patients had received salvage therapy. Hormonal therapy had been performed in 28 patients before RT and continued in two of them concurrently with RT. A median dose of 60 Gy was delivered to the prostate bed. Pelvic node irradiation was performed in all patients. Results: After a median follow-up period of 23 months, 21 patients had experienced biochemical failure. Actuarial 3- and 5-year biochemical relapse-free survival estimates were 67.4%. No patient had local failure, although distant metastases with biochemical failure were found in five patients. On univariate analysis, the following were significant for biochemical failures: seminar vesicle involvement, serum PSA level >1 ng/ml before RT, pathological pelvic node involvement, RT indication (adjuvant vs. salvage) and Gleason score. However, only the serum PSA level before RT was significant on multivariate analysis. Conclusion: Postoperative RT with a pre-RT PSA level <0.1 ng/ml seemed to be effective in patients with prostate cancer.

Revisiting Static Portfolio Theory for Hara Investors

The implications of the two-fund separation theorem have been carefully examined in the literature for the case of mean-variance preferences. However, even though the two-fund theorem applies to the whole class of HARA utility functions, its implications for the e¢ ciency sets spanned by these preferences are much less known. Without dealing with general equilibrium issues, the goal of this paper is to show how most of the wellknown constructions which arise in connection with the former subclass, extend in a relatively natural way to the whole latter set of preferences. Furthermore, graphical illustrations of the HARA portfolio problem that parallel mean-variance geometry are also provided. Along the same lines, It is also shown how the general problem can be seen as a choice between two parameters, one measuring reward and the other one measuring risk. The youth years of …nancial economics are clearly dominated by the early results of portfolio choice theory. Among other things, the contributions of Markowitz (1957) together with the derivation by Tobin (1958) of the two-fund theorem, always in the context of mean-variance preferences, play a crucial role in developing the …rst general equilibrium model of asset pricing, i.e. the CAPM. These two …elds of …nance marched closely together until the seminal work of Lucas (1978) which implies a radical change of perspective. From this point onwards, it becomes clear that it is not necessary to solve a portfolio choice problem in order to model the behavior of asset prices. The consumption-based approach allows asset pricing to (partially) abandon its early companion and walk its way mostly alone. The GMM technology of Hansen (1982) pushes further in this direction and it helps creating a new approach where the stochastic discount factor (SDF) is to be the main focus.