Two-fund Separation Research Articles

A note on reward-risk portfolio selection and two-fund separation

This paper presents a general reward-risk portfolio selection model and derives sufficient conditions for two-fund separation. In particular we show that many reward-risk models presented in the literature satisfy these conditions.

A Note on Reward-Risk Portfolio Selection and Two-Fund Separation

This paper presents a general reward-risk portfolio selection model and derives sufficient conditions for two-fund separation. In particular we show that many reward-risk models presented in the literature satisfy these conditions.

PORTFOLIO CHOICE VIA QUANTILES

A portfolio choice model in continuous time is formulated for both complete and incomplete markets, where the quantile function of the terminal cash flow, instead of the cash flow itself, is taken as the decision variable. This formulation covers a wide body of existing and new models with law‐invariant preference measures, including expected utility maximization, mean–variance, goal reaching, Yaari's dual model, Lopes' SP/A model, behavioral model under prospect theory, as well as those explicitly involving VaR and CVaR in objectives and/or constraints. A solution scheme to this quantile model is proposed, and then demonstrated by solving analytically the goal‐reaching model and Yaari's dual model. A general property derived for the quantile model is that the optimal terminal payment is anticomonotonic with the pricing kernel (or with the minimal pricing kernel in the case of an incomplete market if the investment opportunity set is deterministic). As a consequence, the mutual fund theorem still holds in a market where rational and irrational agents co‐exist.

Ambiguity in Asset Pricing and Portfolio Choice: A Review of the Literature

A growing body of empirical evidence suggests that investors’ behavior is not well described by the traditional paradigm of (subjective) expected utility maximization under rational expectations. A literature has arisen that models agents whose choices are consistent with models that are less restrictive than the standard subjective expected utility framework. In this paper we conduct a survey of the existing literature that has explored the implications of decision-making under ambiguity for financial market outcomes, such as portfolio choice and equilibrium asset prices. We conclude that the ambiguity literature has led to a number of significant advances in our ability to rationalize empirical features of asset returns and portfolio decisions, such as the empirical failure of the two-fund separation theorem in portfolio decisions, the modest exposure to risky securities observed for a majority of investors, the home equity preference in international portfolio diversification, the excess volatility of asset returns, the equity premium and the risk-free rate puzzles, and the occurrence of trading break-downs.

Capital Asset Prices: A Theory of Multiple Equilibria with Two-Fund Separation

Contrary to the case in a Modigliani-Miller context, a pure theory of capital asset prices centered on a trading model of financial markets is established in this paper. Grounded on the dissociation of the model of financial markets from the model of a pure exchange economy, this paper restates the methodological structure of two-fund separation for the novel theory of capital asset prices. On the one fund, a diversified array of unrelated causal elements of the model of financial markets is organized as the information space for the endogenous factor pricing. On the other fund, the state of market structure is introduced into the theory of capital asset prices as the cause of exogenous price effects. The price system thus obtained exhibits the property of multiple equilibria or indeterminacy of equilibrium prices.

Equilibrium Asset Pricing and Portfolio Choice Under Asymmetric Information

We analyze theoretically and empirically the implications of information asymmetry for equilibrium asset pricing and portfolio choice. In our partially revealing dynamic rational expectations equilibrium, portfolio separation fails, and indexing is not optimal. We show how uninformed investors should structure their portfolios, using the information contained in prices to cope with winner's curse problems. We implement empirically this price-contingent portfolio strategy. Consistent with our theory, the strategy outperforms economically and statistically the index. While momentum can arise in the model, in the data, the momentum strategy does not outperform the price-contingent strategy, as predicted by the theory. (JEL G11, G12, G14, D53)

The two-fund separation theorem revisited

This paper resolves two issues regarding the traditional capital asset pricing model with one risk-free asset which seem to have been overlooked in the literature. First, it provides an elementary and complete proof of the two-fund separation theorem which accounts for the fact that asset demand may become undefined if the limiting slopes of the investor’s indifference curves are finite. Second, it shows that an additional limiting condition on investors’ risk aversions is generally necessary to guarantee existence of an equilibrium. Moreover, a generalized existence result is formulated which includes investors who in equilibrium may not invest in risky assets and a simple condition ensuring positive equilibrium asset prices is given.

Increases in Risk Aversion and the Distribution of Portfolio Payoffs

Hart proved the difficulty of deriving general comparative statics in portfolio weights. Instead, we derive new comparative statics for the distribution of payoffs: A is less risk averse than B iff A’s payoff is always distributed as B’s payoff plus a non-negative random variable plus conditional-mean-zero noise. If either agent has nonincreasing absolute risk aversion, the non-negative part can be chosen to be constant. The main result also holds in some incomplete markets with two assets or two-fund separation, and in multiple periods for a mixture of payoff distributionsover time (but not at every point in time).

Risky asset pricing based on safety first fund management

We study a portfolio selection model based on Kataoka's safety-first criterion (KSF model in short). We assume that the market is complete but without risk-free asset, and that the returns are jointly elliptically distributed. With these assumptions, we provide an explicit analytical optimal solution for the KSF model and obtain some geometrical properties of the efficient frontier in the plane of probability risk degree z α and target return r α. We further prove a two-fund separation and tangency portfolio theorem in the spirit of the traditional mean-variance analysis. We also establish a risky asset pricing model based on risky funds that is similar to Black's zero-beta capital asset pricing model (CAPM, for short). Moreover, we simplify our risky asset pricing model using a derivative risky fund as a reference for market evaluation.

OPTIMAL CONSUMPTION AND PORTFOLIO DECISIONS WITH PARTIALLY OBSERVED REAL PRICES

We consider optimal consumption and portfolio investment problems of an investor who is interested in maximizing his utilities from consumption and terminal wealth subject to a random inflation in the consumption basket price over time. We consider two cases: (i) when the investor observes the basket price and (ii) when he receives only noisy observations on the basket price. We derive the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds in both cases. The compositions of the funds in the two cases are the same, but in general the investor's allocations of his wealth into these funds will differ. However, in the particular case when the investor has constant relative risk‐aversion (CRRA) utility, his optimal investment allocations into these funds are also the same in both cases.

In which financial markets do mutual fund theorems hold true?

The mutual fund theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are: (i) Let N be the wealth process of the numéraire portfolio (i.e., the optimal portfolio for the log utility). If any path-independent option with maturity T written on the numéraire portfolio can be replicated by trading only in N and the risk-free asset, then the MFT holds true for general utility functions, and the numéraire portfolio may serve as mutual fund. This generalizes Merton’s classical result on Black–Merton–Scholes markets as well as the work of Chamberlain in the framework of Brownian filtrations (Chamberlain in Econometrica 56:1283–1300, 1988). Conversely, under a supplementary weak completeness assumption, we show that the validity of the MFT for general utility functions implies the replicability property for options on the numéraire portfolio described above. (ii) If for a given class of utility functions (i.e., investors) the MFT holds true in all complete Brownian financial markets S, then all investors use the same utility function U, which must be of HARA type. This is a result in the spirit of the classical work by Cass and Stiglitz.

Markowitz’s mean-variance asset-liability management with regime switching: A continuous-time model

This paper considers an asset-liability management (ALM) problem under a continuous-time Markov regime-switching model. By adopting the techniques of [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model. SIAM J. Control Optim. 42, 1466–1482], we investigate the feasibility, obtain the optimal strategy, delineate the efficient frontier, and establish the associated mutual fund theorem.

Risk-sensitive benchmarked asset management

This paper extends the risk-sensitive asset management theory developed by Bielecki and Pliska and by Kuroda and Nagai to the case where the investor's objective is to outperform an investment benchmark. The main result is a mutual fund theorem. Every investor following the same benchmark will take positions, in proportions dependent on his/her risk sensitivity coefficient, in two funds: the log-optimal portfolio and a second fund which adjusts for the correlation between the traded assets, the benchmark and the underlying valuation factors.

Mutual fund theorems when minimizing the probability of lifetime ruin

We show that the mutual fund theorems of Merton [1971. Journal of Economic Theory 3, 373–413] extend to the problem of optimal investment to minimize the probability of lifetime ruin. We obtain two such theorems by considering a financial market both with and without a riskless asset for random consumption. The striking result is that we obtain two-fund theorems despite the additional source of randomness from consumption.

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.

Kimball'sPrudence and Two-Fund Separation as Determinants of Mutual Fund Performance Evaluation

Summary/Zusammenfassung Kimball's Prudence and Two-Fund Separation as Determinants of Mutual Fund Performance Evaluation We consider investors with mean-variance-skewness preferences who aim at selecting one out of F different funds and combining it optimally with the riskless asset and direct stock holdings. Direct stock holdings are either exogenously or endogenously determined. In our theoretical section, we derive and discuss several performance measures for the investor's decision problems with a central role of Kimball's (1990) prudence and of several variants of Sharpe and Treynor measures. In our empirical section, we show that the distinction between exogenous and endogenous stock holding is less important than the issue of skewness preferences. The latter are most relevant for fund rankings, when an investor's skewness preferences are not derived from cubic HARA utility so that the two-fund separation theorem is not valid. (JEL G11) Kimballs Prudence und Zwei-Fonds-Separation als Determinanten d...

Separability and Pension Optimization

The Separation or Mutual Fund Theorem of finance theory is extended to the case surplus optimization, which is relevant for defined-benefit pension plans and others. While Separation theorems in the standard case have limited empirical applications, separation results in the surplus-optimization case are directly relevant to real-world pension management, because one of the “bases” of the surplus-efficient frontier is the full hedge of the pension plan9s liabilities. Surplus-optimal or –efficient portfolios can be expressed as combinations of the full liability hedge and exposure in the overlay with maximum Sharpe ratio. Optimal pension allocation is thus analogous to management of a portable alpha strategy with a “beta” equal to plan liabilities. Our results also indicate that the difference in portfolio weights between asset-optimal (standard) and surplus-optimal efficient frontiers is constant across those respective frontiers, so that the impact of moving from an asset-optimal to a surplus-optimal framework is independent of target returns or risk tolerance. Finally, we explore the details of the full liability-hedge portfolio. In a number of relevant cases, there is no presumption that a dollar of liabilities can be fully-hedged with a dollar of assets, which gives rise to the concept of an effective funding ratio for a pension plan.

Empirical Evaluation of Investor Rationality in the Asset Allocation Puzzle

We examine the portfolio-choice puzzle posed by Canner, Mankiw, and Weil (1997). The idea is to test a conclusion reached by Elton and Gruber (2000), stating that a bonds/stocks ratio which decreases in relation to risk tolerance does not necessarily mean a contradiction of modern portfolio-choice theory and does not cast doubt on the rationality of investors. From data on the portfolio composition of 470 clients of a Canadian brokerage firm, we obtain that the bonds/stocks ratio does decrease in relation to risk tolerance. We also verify the existence of the two-fund separation theorem in the assets data available to the investors in our sample.

Asymptotic behaviour of mean-quantile efficient portfolios

In this paper we investigate portfolio optimization in the Black–Scholes continuous-time setting under quantile based risk measures: value at risk, capital at risk and relative value at risk. We show that the optimization results are consistent with Merton’s two-fund separation theorem, i.e., that every optimal strategy is a weighted average of the bond and Merton’s portfolio. We present optimization results for constrained portfolios with respect to these risk measures, showing for instance that under value at risk, in better markets and during longer time horizons, it is optimal to invest less into the risky assets.

The mean-variance investment problem in a constrained financial market

This paper extends the mean-variance analysis and the two-fund separation theorem to a market with some constraints, such as, the incompleteness, prohibition of short-selling, and partial information, with stochastic interest rate, and with stochastic volatility for risky assets. By maximizing a quadratic utility of terminal wealth, we show that the efficient frontier for the problem is a straight line in the mean-standard-deviation diagram. The quadratic utility function exhibits mean-variance efficiency. Our results apply to portfolios of claims in a single period, multiperiod, and continuous time.