What Is a Combined Single Limit?

Investors in the single-account portfolio theory and the mean-variance portfolio theory consider the asset portfolio as a whole, that is, a single account. They consider the covariance between assets as proposed in [[| Modern Portfolio Theory | Markowitz Theory]] (Markowitz). To some extent, the single-account portfolio theory of portfolio selection is similar to portfolio selection in the mean-variance model. The core of the mean variance theory is the effective boundary of the mean variance in the (, ) plane. Corresponding to the single account portfolio theory is the effective boundary in the (Eh (W), Pr {WA}) plane. In both cases, the investor will choose or Eh (W) with a higher value and or Pr {WA} with a lower value. Therefore, the effective boundary of the mean variance is obtained by fixing the maximum value at , while the effective boundary of the single account behavior combination theory is obtained by fixing the maximum value Eh (W) at Pr {WA}. Therefore, the effective combination in the single-account asset portfolio theory is not exactly the same as the effective combination in the mean variance model.

Single account portfolio theory

Right!

Single account

Assume that the investor's wealth in period 0 is W0, and his purpose is to maximize the expected wealth Eh (W) in period 1.

Theorem 1 , let the single-account portfolio selection model be:

Target: max: Eh (W) = ri Wi

Condition: Pr {WA}

vi Wi W0

Among them, vi Wi W0 is a budget constraint. The model assumes that the states are ordered so that vi / pi decreases by i accordingly. Under this assumption, the optimal solution can be obtained as:

Wi = 0, when i does not belong to T

Wi = A, when i belongs to T \ {sn}

Wn = (W0-vi Wi) / vn. When W0> vn A, it exceeds A. The sum in the formula is from 1 to n-1. T is a subset of states, including the nth state sn, and Pr {T} , but there is no true subset T 'in T such that Pr {T'} .

Theorem 1 gives an effective BPT-SA solution. If A or reaches a sufficiently high value, it is impossible to be limited by probability. Therefore, the optimal solution does not exist.

Theorem 2. In the example of the discrete state, the effective combination of mean and variance has the following form:

Where b is a positive constant.

Theorem 3. If at least three states in the BPT-SA effective combination have positive consumption characteristics and a clear value of vi / pi, then this combination is not an effective combination of mean and variance.

This can determine the effective boundary of the single account behavior combination theory. It is an ordered number pair consisting of many Pr {WA} values and the corresponding maximum value Eh (W) under (Eh (W), Pr {W A}) curve drawn on the plane. Investors will choose the optimal portfolio through the effective boundary maximization function U (Eh (W), D (A)).

From the form of the model solution, we can see the distribution form of effective portfolio returns in the single account behavior combination theory. There are three possible outcomes of its gain: 0, A, a value Wn higher than A. This distribution is similar to the distribution of a combination of a risk-free bond with a return of A or 0 and a lottery with a return of Wn. This is consistent with the phenomenon observed by Friedman and Savage that people buy insurance and lottery tickets at the same time. This simultaneity is exactly the characterization of effective portfolios in the single account behavior portfolio theory.

In the mean variance model, investor preferences can be expressed by the function -2 / d, where d represents risk tolerance. Here, attitudes to risk are measured by a single variable, d. In SP / A theory, risk is multi-dimensional, and its effective boundary is affected by five risk measurement parameters. They are:

qs to measure the degree of fear (the need for safety);

qp to measure the degree of hope (the need for potential);

A, expected level;

, used to determine the relative strength of fear and hope;

, used to determine the degree of desire to obtain the desired level of fear and hope.

Changes in the values of these five parameters will change the investor's choice of portfolio.