How should an investor choose his trading strategy? Portfolio selection is – besides asset pricing – one of the central problems that we want to solve in financial economics.8
The CAPM in Chap.3was the first asset pricing model that we have encountered and one of its consequences was the Two-Fund Separation Theorem (Sect.3.1.5) that stated that every mean-variance investor in a CAPM market (i.e., in a market where everybody is a mean-variance investor) should hold the same portfolio of risky assets. Only the combination with the riskless asset is used to account for his risk attitudes.
We have criticized the mean-variance approach as a model for rational or behavioral preferences (compare Sect.2.3.2), and so it seems that the consequences of the CAPM can only be seen as a result of using a mathematically appealing, but unfortunately overly simple decision model. This is, however, not entirely true: in fact we will show that in continuous-time trading under certain assumptions on the underlying asset process and the risk attitudes of the investor the so-calledMutual Fund Theoremholds, which is in a certain sense a generalization of the Two-Fund Theorem to the continuous-time setting. This theorem holds in particular not only for a mean-variance investor, but in fact for a large class of rational investors (in the sense of Expected Utility Theory).
The Mutual Fund Theorem has been proved by Merton [Mer72]. We state here a simplified variant of a version stated in the book by Karatzas and Shreve [KS98].
The (not so easy) proof of this theorem can be found there. As preparation for the theorem we need to make a couple of definitions. In particular, we need to define what we call anoptimal trading strategyfor an expected utility maximizing investor.
We assume that there areN1underlying assets driven by aD1dimensional geometric Brownian motionB.t/(see Definition8.2). LetS.t/ 2 RN be the price vector of the assets and2RD Nthe volatility matrix.
Let us now construct the utility that we aim to maximize. First, we notice that in a time-continous framework with finite investment horizon we need to distinguish two utility functions: one that describes the utility derived from consumption during the investment time and one that describes the utility derived from final wealth. We
8Compare also [CV02].
8.6 Time-Continuity and the Mutual Fund Theorem 307 denote these two utility functions byu1andu2. We allowu1to be time-dependent, thusu1WRŒ0;1/!Œ1;C1/andu2WŒ0;1/!Œ1;C1/.9
Our gaol is now to formulate the optimization problem. The investment horizon is denoted byT, i.e., we invest at timetD0and sell at timetDT. In between, we are allowed (and need) to take money out of the investment for consumption. This consumption plus the amount of money we have at timeTdetermines our utility.
The consumption over time is described by the functionc.t/. Moreover, we have a trading strategy.t/such that we never have to face a utility of1:10 Then we want to maximize the expected utility
UWDE Z T
0 u1.t;c.t//dtCu2.X.T//
in consumption plancand self-financing trading strategy.
Let us now assume that the utility functionsu1 andu2 are “reasonable” in the sense that they are strictly concave (i.e., the investor is strictly risk-averse), increas- ing and satisfy the following technical conditions (a more precise formulation of these statements can be found in [KS98]):
Assumption 8.8 We assume that u1and u2satisfy the following conditions11: (i) I1WD.u01/1and I1WD.u02/1have polynomial growth.
(ii) u1.I1/and u2.I2/have polynomial growth.
(iii) I1is Hửlder continuous.
(iv) Either@I1.t;y/=@y is strictly negative for a.e. y or@I20.y/=@y is strictly negative a.e. (or both).
These assumptions still leave ample room for the choice of the utility functions, but nevertheless Robert Merton proved the following result [Mer72]:
Theorem 8.9 (Mutual Fund Theorem) Assume that the volatility of the under- lying assets is given by and the dividend process is given byı. Assume that the investor can invest risk-free for a return of r and borrow money for a return of b.
9It is of course possible thatu1is time-independent and that both utilities are in effect the same, however, in reality this is unlikely to be the case: consider an investment problem over one year without earnings, then the consumption in that year induces probably a smaller utility than the saved money at the end of that year does, simply because the latter has to be used in all the subsequent years still to come.
10More precisely: letX.t/denote the process of the value of the portfolio defined by the investment strategy (i.e.,Xdepends in particular onS,cand), then the expected value of the intertemporal consumption,RT
0 minf0;u1.t;c.t//gdt, has to be larger than1and the expected value of the final wealth, minf0;u2.X.T//ghas to be larger than1, as well.
11For mathematical terminology compare AppendixA.
Assume that, ı, r and b are smooth functions of the time t.12 Then any agent with preferences satisfying Assumption8.8should hold a mutual fund containing the assets in the proportion
.0.t//1.t/D..t/0.t//1.b.t/Cı.t/r.t/1/ plus a risk-free asset in order to maximize the expected utility U.
This generalizes in a certain way the Two-Fund Separation Theorem to a large class of preferences. It shows in particular that the Two-Fund Separation Theorem was not just a weird artifact of the mean-variance assumption, but that there is some deeper insight behind it. Even more so, the Mutual Fund Theorem does not always apply to mean-variance preferences, since we know from Sect.2.3that the mean-variance approach corresponds to an expected utility maximization with a quadratic utility function; a quadratic utility function, however, cannot be strictly concave and increasing, thus the Mutual Fund Theorem would not be applicable. If we assume, e.g., returns with normal distribution, then mean-variance and expected utility preferences coincide, so in this case the Mutual Fund Theorem applies.
What is more important: the Mutual Fund Theorem holds for a large class of strictly concave expected utility functions. But what about its validity in reality?
First, observations about investment decisions show a strong heterogeneity in asset allocations. One possible explanation might be that expectations of investors often are heterogeneous. In fact, many observations on financial markets can be explained by trading expectations, and the Mutual Fund Theorem ignores this aspect completely.
But still, there are situations where expectations should be homogeneous, take e.g., the case of a client advisor at a bank: his (or the bank’s) expectations on the market are likely to be more correct than the ones of his clients. In fact, the clients might believe his expectations completely and just expect him to invest their money in a way that is providing them with an optimal risk profile, i.e., to optimize the client’s utility function, given the advisor’s expectations. Therefore, the expectations are homogeneous and the Mutual Fund Theorem should hold, i.e., the advisor should suggest the same portfolio of risky assets to all of his clients, regardless of their risk attitudes. This is certainly puzzling and quite contrary to our experience: banks structure the risky part of portfolios differently, depending on the client’s risk profile.
Does the Mutual Fund Theorem prove that this is suboptimal and a policy change would lead to a mutual benefit for client and bank?
12Up to now we have only considered the case where the risk-free rate and the volatility were constant and borrowing and investing in the risk-free asset had the same fixed interest rate.
Moreover, we have not considered dividends. All of these extensions are not essential to understand this theorem, but are stated for completeness. These extensions include as special case particular the setting we have previously used whererDb,ıD0andandrbeing constant in time. For details see [KS98].