Application of numerical method of functional differential equations in fair value of financial accounting

Assume that there are N possible assets. Let the asset to be measured is NTH. For a particular seller (firm), let its asset utility function on the balance sheet date be u = U_S(x₁, x₂, … , x_N, m), where U represents the total utility of maintaining the use or holding of these assets in the current state, represents the number of non-monetary assets and m is the number of monetary assets

Assumption: The quantity of non-monetary assets and monetary assets of the seller before the transaction is s₁, s₂, … , s_N, m_s, respectively. After exchanging the NTH asset of unit Q with the price P, the seller’s total asset utility changed as follows: fS(q,p)=US(s1,s2,⋯,sN−q,ms+qp)−US(s1,s2,⋯,sN,ms) {f_S}(q,p) = {U_S}({s_1},{s_2}, \cdots ,{s_N} - q,{m_s} + qp) - {U_S}({s_1},{s_2}, \cdots ,{s_N},{m_s}) For a particular buyer, let the asset utility function on the balance sheet date be u = U_B(x₁, x₂, … , x_N, m), and assume that the number of non-monetary assets and monetary assets of the buyer before the transaction is b₁, b₂, … , b_N, m_b; after buying the NTH asset of q unit at the price p, the change in the total utility of the asset is fB(q,p)=UB(b1,b2,⋯,bN+q,mb−qp)−UB(b1,b2,⋯,bN,mb). {f_B}(q,p) = {U_B}({b_1},{b_2}, \cdots ,{b_N} + q,{m_b} - qp) - {U_B}({b_1},{b_2}, \cdots ,{b_N},{m_b}). Using the asset utility function, the concept of fair value can be re-expressed as:

For a specific amount of asset ( q¯(q¯>0) \overline q (\overline q > 0) the NTH asset), the exchange price p¯ \overline p can be called the fair value corresponding to the quantity supplied q¯ \overline q if: (I)

For the buyer, when the purchase quantity is q¯, fB(q¯,p¯)≥0 \overline q ,\;{f_B}(\overline q ,\overline p ) \ge 0 , and ∂UB∂xN≥p¯∂UB∂m {{\partial {U_B}} \over {\partial {x_N}}} \ge \overline p {{\partial {U_B}} \over {\partial m}} and at any exchange price p<p¯, q¯ p < \overline p ,\;\overline q assets cannot be bought.

Conditions (I) are fair conditions of the purchaser; conditions (II) are fair conditions of the seller. The constraint condition of the increase of total utility is the basis of both parties’ willingness to exchange. The marginal utility constraint represents the rationality of the exchange parties in terms of the quantity exchanged, that is, the exchange of the q¯ \overline q asset is cost-effective, so the exchange of all q¯ \overline q assets can be completed. The constraint condition of exchange price represents the rationality of both parties in exchange price, that is, the buyer cannot lower the price and the seller cannot raise the price [10].

Model

For a given price p, build the following buyer optimisation model: max0≤qfB(q,p)s.t.fB(q,p)≥0. \matrix{ {\mathop {\max }\limits_{0 \le q} {f_B}(q,p)} \cr {s.t.{f_B}(q,p) \ge 0.} \cr } The corresponding relationship between the price determined by the buyer’s optimisation model and the buyer’s optimal purchase quantity is called the buyer’s optimal purchase quantity function, denoted as q = B(p). When there are multiple buyers in the market, the optimal buying quantity function can be obtained by summing their optimal buying quantity function. The optimal buying volume function decreases as the price rises. In a coordinate system with price as the vertical axis and trading volume as the horizontal axis, the curve of the optimal buying volume function is called the optimal buying volume curve (see Figure 1). The inverse function of the optimal buying quantity function is denoted as p = P_B(q).

Fig. 1

Hypothesis: There is a non-zero optimal solution in the buyer’s optimisation model. According to the necessary condition ∂fB(q,p)∂q=0 {{\partial {f_B}(q,p)} \over {\partial q}} = 0 of the optimal solution, it can be known that the non-zero optimal solution satisfies ∂UB∂xN=p∂UB∂m {{\partial {U_B}} \over {\partial {x_N}}} = p{{\partial {U_B}} \over {\partial m}} .

Similarly, for a given price p, the following seller optimisation model is established as follows: max0≤qfS(q,p)s.t.fS(q,p)≥0. \matrix{ {\mathop {\max }\limits_{0 \le q} {f_S}(q,p)} \cr {s.t.{f_S}(q,p) \ge 0.} \cr } The corresponding relationship between the price determined by the seller’s optimisation model and the seller’s optimal selling quantity is called the seller’s optimal selling quantity function, denoted as q = S(p). When there are many sellers in the market, the optimal selling quantity function of the market can be obtained by summing their optimal selling quantity function. The optimal selling function obviously increases as the price increases. In a coordinate system with price as the vertical axis and volume as the horizontal axis, the curve of the optimal selling volume function is called the optimal selling volume curve (see Figure 1). Consider the inverse function of the optimal selling quantity function as p = P_S(q).

Hypothesis: The seller optimisation model has a non-zero optimal solution Q, and the non-zero optimal solution satisfies ∂US∂xN=p∂US∂m {{\partial {U_S}} \over {\partial {x_N}}} = p{{\partial {U_S}} \over {\partial m}} :

It can be seen that no matter for the seller or the buyer, the quantity willing to exchange is different at different exchange prices. If there is an intersection point between the optimal selling volume curve and the optimal buying volume curve, the volume corresponding to this point is called the market equilibrium volume, and the exchange price corresponding to this point is the market equilibrium price (see Figure 1). If the market is in a state of equilibrium volume and price, then the market is called a market of equilibrium supply and demand or in a state of equilibrium supply and demand [11].

Fair value determination model: it is assumed that on the balance sheet date, the total supply of assets is Q (0>0). In the market environment of free competition, the market optimal selling volume curve of the asset is q = B(p) and the anti-rain number is p = P_S(q). The optimal buying volume curve of the market is q = B(p) and the inverse function is p = P_B(q); there is an intersection point between the optimal selling volume curve and the optimal buying volume curve; the equilibrium trading volume is q¯ \overline q and the equilibrium price is p¯ \overline p . So (q

If Q≤q¯ Q \le \overline q , then the fair value; = P_B(Q).

If Q>q¯ Q > \overline q , then there is no fair value for selling all of Q.

The fair value determination model is proved as follows:

Assumption Q≤q¯ Q \le \overline q .

According to the definition of the optimal buying volume curve, there are one or more buyers, and the technical price P_B(Q) is willing to buy an asset quantity of Q, after buying, meet: f_B(q, p) ≥ 0 and ∂UB∂xN=PB(Q)∂UB∂m {{\partial {U_B}} \over {\partial {x_N}}} = {P_B}(Q){{\partial {U_B}} \over {\partial m}} .

According to the definition of the optimal selling curve, there are one or more sellers, so P_S(Q) is willing to sell a total of Q assets. After selling, meet f_S(q, p) ≥ 0 and ∂US∂xN=PS(Q)∂US∂m {{\partial {U_S}} \over {\partial {x_N}}} = {P_S}(Q){{\partial {U_S}} \over {\partial m}} .

Because of the monotonically selective and decreasing properties of Q≤q¯ Q \le \overline q , the monotonically increasing properties of the optimal buying function and the optimal selling function, then P_S(Q) ≤ P_B(Q) (see Figure 1). So ∂US∂xN≤PB(Q)∂US∂m {{\partial {U_S}} \over {\partial {x_N}}} \le {P_B}(Q){{\partial {U_S}} \over {\partial m}} .

In addition, the marginal utility of the asset to the buyer has reached ∂UB∂xN=PB(Q)∂UB∂m {{\partial {U_B}} \over {\partial {x_N}}} = {P_B}(Q){{\partial {U_B}} \over {\partial m}} , obviously, if the price is higher than P_B(Q), the buyer will be reluctant to buy the whole Q. This means that if the seller raises the price and makes it higher than P_B(Q), then they cannot sell all Q. The sellers have the information about the optimal buying volume curve and take advantage of the buyer’s competition; if the price P_B(Q) sells all the Q, the price will not be lowered. So, the buyer cannot buy Q for less than P_B(Q).

Therefore, if P_B(Q) meets all the conditions for fair value, it must be fair value.

Assume Q>q¯ Q > \overline q .

According to the definition of the optimal buying volume curve, if there are one or more buyers, at a price P_B(Q), willing to buy a total of Q assets, after purchase, the marginal utility of the asset meets ∂UB∂xN=PB(Q)∂UB∂m {{\partial {U_B}} \over {\partial {x_N}}} = {P_B}(Q){{\partial {U_B}} \over {\partial m}} . So, if the price is higher than P_B(Q), the buyer does not want to buy the Qth asset, and so they do not want to buy all of the Qs.

According to the definition of the optimal selling curve, there are one or more sellers; if P_S(Q) is willing to sell a total of Q assets, after the sale, the marginal utility of the asset satisfies ∂US∂xN=PS(Q)∂US∂m {{\partial {U_S}} \over {\partial {x_N}}} = {P_S}(Q){{\partial {U_S}} \over {\partial m}} , and if the price is lower than P_S(Q), the seller will be reluctant to sell the Qth asset, that is, to sell all of Q.

Due to the monotonically decreasing nature of Q>q¯ Q > \overline q , the optimal buying quantity function and the monotonically increasing nature of the optimal selling quantity function, P_S(Q) > P_B(Q) (see Figure 1). Therefore, the seller’s lowest selling price P_S(Q) will not be accepted by the buyer (price higher than P_B(Q) will not be accepted).

So, there is no fair value for exchanging all of Q. After the [12].