Feedback loops, fair value accounting and correlated investments

Abstract

This paper presents and tests a model of the price dynamics that arise when investors fail to recognize the redundancy of unrealized gains and losses (“UGLs”) that are correlated with the firm’s past returns. Consistent with the predictions of our model, our experiment shows that a firm’s prices and earnings become highly volatile when correlated investment is large and correlated UGLs are made salient by comprehensive income reporting. The results suggest that including correlated UGLs in performance numbers could induce violations of weak-form efficiency that exacerbate volatility in share prices and earnings.

Appendices

Appendix A

1.1 Analysis of price dynamics

Appendix A provides a mathematical analysis of the price dynamic when price changes include a component that is proportional to past price changes. The recurrence relation of price is:

$${P}_{t} = {K \hbox{CORE}}_{t - 1} + \alpha \varepsilon {KP}_{t - 1} - \alpha \varepsilon {KP}_{t - 2}.$$

(6)

As this price dynamic unfolds, P _t is defined in terms of successively higher powers of $\alpha \varepsilon K$. To illustrate this, below we present the sequence of price changes and unrecognized gains and losses (UGLs) created by a single shock to income that is sufficient to increase prices by $1.

For notational simplicity, let $\omega =\varepsilon \alpha K$. Therefore,

$${P}_{t} = {K \hbox{CORE}}_{t - 1} + \omega {P}_{t - 1} - \omega {P}_{t - 2}.$$

(7)

Equation (1) is what is known as a linear non-homogeneous recurrence relation with constant coefficients. It is a “recurrence relation” because P _t is expressed in terms of one or more of the previous terms of the sequence P ₀, P ₁, ..., $P_{t-1}$. It is linear because the right-hand side is a sum of multiples of the previous terms of the sequence. It is non-homogeneous because some terms (in this case, CORE $_{t-1}$) are not multiples of any of the P _t . Finally, it has constant coefficients because ω and − ω are constants and are not functions of t.

1.1.1 Single-shock process

To analyze a single-shock process without any growth in core earnings, we assume that core earnings are 0 in every period, and introduce an initial shock of 1 into the price sequence by assuming that P ₀ = 0 and P ₁ = 1. This process, which is called the ‘associated homogeneous recurrence relation,’ can be written as:

$${P}_{t} = \omega ({P}_{t - 1} - {P}_{t - 2}).$$

(8)

This relation can also be described by a vector equation

$$x_{t + 1} = Ax_t$$

(9)

where

$$x_t = \left({\begin{array}{*{20}c} {P_t } \\ {P_{t - 1} } \\ \end{array} } \right) \quad A = \left[ {\begin{array}{*{20}c} \omega & { - \omega } \\ 1 & 0 \\ \end{array} } \right].$$

(10)

To solve this system, we find the eigenvalues and eigenvectors for matrix A. This is done by setting the determinant of the matrix (A−λ I) = 0 and solving for λ. If ω ≠ 0, then there are two eigenvalues, determined by the equation

$$\lambda = \frac{\omega} {2}\left[ {1 \pm \sqrt {1 - \frac{4} {\omega }} } \right].$$

(11)

To determine the long-term behavior of P _t , we examine the absolute values of the eigenvalues: if either eigenvalues has an absolute value greater than 1, the values of P _t will be unbounded. We separate our analysis into 5 cases:

• Case 1: If $0<\omega<1$, both eigenvalues are complex, but have an absolute value less than 1. Because the roots are imaginary, prices oscillate in two dimensions, as shown in Fig. 1, Panel A.

• Case 2: If $1<\omega<4$, both eigenvalues are complex, but have an absolute value greater than 1. (The absolute value of a complex number is the square root of the sum of its real and imaginary elements.) Because the roots are imaginary, prices oscillate in two dimensions, as shown in Fig. 1, Panel B.

• Case 3: If $\omega \ge 4$, both eigenvalues are real and greater than or equal to 1 and P _t increases or decreases without bound (depending on whether the first shock to price is positive or negative), as shown in Fig. 1, Panel C.

• Case 4: If −0.5 < ω < 0, both eigenvalues are real, with absolute values less than 1. P _t converges to a bounded value, as shown in Fig. 1, Panel D.

• Case 5: If ω < −0.5, both eigenvalues are real, with absolute values greater than 1. In this case, P _t oscillates between positive and negative values, as shown in Fig. 1, Panel E.

1.1.2 Adding growth

The non-homogeneous form of the recurrence relation includes a term reflecting the growth in net income (CORE _t ), which varies as time passes. Assuming net income starts at CORE₀ = 1 and grows at a constant rate g, the non-homogeneous form is

$${P}_t = K(1 + g)^{t} + \omega ({P}_{t - 1} - {P}_{t - 2}).$$

(12)

Our interest now is in the price ratio y _t  = P _t /K(1+ g)^t. If this mispricing ratio converges to a constant, then the price path converges to one that would arise if investors simply ignored correlated gains and losses, and applied a multiple of K to net income. If this price ratio diverges, the price changes created by income growth are amplified, just as the initial price shock in the homogeneous recurrence relation was.

The price ratio is described by the relation

$${y}_{t} = k + \omega {y}_{t - 1} /(1 + g) - \omega y_{t - 2} /(1 + g)^2.$$

We can write the associated homogeneous recurrence relation as

$$z_{t + 1} = Mz_t$$

(13)

where

$$z_t = \left({\begin{array}{*{20}c} {y_t } \\ {y_{t - 1} } \\ \end{array} } \right)M = \left[ {\begin{array}{*{20}c} {\omega /(1 + g)} & { - \omega /(1 + g)^2 } \\ 1 & 0 \\ \end{array} } \right].$$

(A9)

Solving for the eigenvalues of M in the same way as for A yields the eigenvalues

$$\lambda = \frac{\omega } {{2(1 + g)}}\left[ {1 \pm \sqrt {1 - \frac{4} {\omega }} } \right]. $$

Again, we can identify five cases, which are closely related to those in the analysis of a single price shock. This analysis assumes that g  $<(\sqrt{2}-1)$, or approximately 42%, thus reflecting all reasonable perpetual growth rates.

• Case 1: If 0 < ω < (1 + g)², both eigenvalues are complex, but have an absolute value less than 1. Because the roots are imaginary, prices oscillate in two dimensions, as shown in Fig. 2, Panel A.

• Case 2: If (1 + g)² ≤ ω < 4, both eigenvalues are complex, but have an absolute value greater than 1. (The absolute value of a complex number is the square root of the sum of its real and imaginary elements.) Because the roots are imaginary, prices oscillate in two dimensions, as shown in Fig. 2, Panel B.

• Case 3: If ω ≥ 4, both eigenvalues are real and greater than or equal to 1 and y _t increases without bound, similar to Fig. 2, Panel C.

• Case 4: If (1 − 2g − g ²)/(2 + g) $<\omega <0$, both eigenvalues are real, with absolute values less than 1. P _t converges to a bounded value, similar to Fig. 2, Panel D.

• Case 5: If $\omega <$ (1–2g-g²)/(2+g), both eigenvalues are real, with absolute values greater than 1. In this case, P _t oscillates between positive and negative values, similar to Fig. 2, Panel E.

Appendix B

1.1 Laboratory market instructions

1.1.1 Overview

During this session, you will trade shares in several firms. The firms are fictitious, but are modeled after real firms. The value of each firm is determined by a panel of experts who are given only the information given to your group (except, of course, that the panel will not know the market prices that develop in your market). You record a gain whenever you buy shares for less than value or sell shares for more than value. You record a loss whenever you buy shares for more than value or sell shares for less than value. All trading gains and losses are denominated in “laboratory dollars.” We will convert your trading gains into U.S. dollars to determine your payment. At the end of trading, we will also ask you a series of questions about your experience during the trading session.

1.1.2 The task

You will trade shares in three firms. You will trade each one for a number of periods we call “years.” Every time you start trading in a new firm, you will get some basic financial information about the firm. You will trade a “practice” firm for 8 years, and then two other firms for 8 years.

1.1.3 Financial information

When you begin trading a new firm, you will be given the following information:

• A fictitious name for the firm

• The firm’s financial statements for the previous year, which is called “year 0”

  • Income Statement for year 0

  • Balance Sheet as of the end of year 0

  • Statement of Changes in Shareholders’ Equity for year 0

After each year is ended, the financial statements will be updated to reflect the actual earnings for the year. You also will see a history of the prices for the firm over all prior year. The financial statements and the history of prices will be available to you at all times, by clicking the appropriate link on the bottom right of your computer screen.

1.1.4 Accounting methods and investment practices

All of the firms use exactly the same operating and accounting methods:

• Ordinary operating income is generated by cash sales to customers.

• Net income is used to purchase property, plant, and equipment.

• Each firm invests a portion of its assets in an index of companies that are identical to itself. Therefore, changes in the stock price of the firm are mirrored in the value of the index. For example, if the stock price increases by 10% from one year to the next, the value of the index investment also increases by 10%. The firms do not buy or sell any shares of the index over the course of the trading session. This is an investment in “available for sale” securities. Therefore, consistent with GAAP, unrealized gains and losses due to changes in the value of the index investment are recorded as part of “other comprehensive income” in shareholders equity.

• The companies have no long-term debt, so all of their liabilities are current.

1.1.5 How to trade

During each trading period, traders enter orders to buy and sell shares. You can enter multiple orders at each price. Here are your options:

• Entering a bid. A bid is an order to buy a share at a stated price.

• Entering an ask. An ask is an order to sell a share at a stated price.

• Removing a bid or ask. You can remove (cancel) a bid or ask by right-clicking on it.

The only restriction on bids and asks is that you cannot enter an ask lower than a bid you have already entered, or enter a bid higher than an ask you have already entered.

1.1.6 Clearing the market

Once the trading period is over, a computer will determine the “market-clearing range” of prices at which the number of shares traders are willing to buy equals the number of shares they are willing to sell. Every bid above the range and every ask below the range is executed at the midpoint of the market-clearing range.

1.1.7 Making money

You start each security with $0 and 0 shares. However, negative cash and share balances are permitted. Thus, you can buy shares even if you don’t have money to pay for them, and you can sell shares you don’t own (“short selling”).

At the end of trading for each security, the shares you own pay a liquidating dividend equal to their true value. If you have a positive balance of shares, the dividend is added to your cash balance for each share you own. If you have a negative balance of shares, the dividend is subtracted from your cash balance for each share you own. The resulting number is your trading gain (if positive) or trading loss (if negative).

You make money every time you buy a share for less than true value or sell a share for more than true value. For example, buying a share worth $540 at a price of $500 creates a gain of $40. Selling that share at that price creates a loss of $40.

1.1.8 Converting laboratory dollars into US dollars

Laboratory winnings, as described above, will be converted into US$ according to the formula

$$\hbox{US}\$ \hbox{Payment}=\hbox{(Net Gain/loss in laboratory} \$ + “\hbox{Adjustment}”) \times \hbox{Exchange rate}.$$

You are guaranteed a minimum payment of US$5 for every hour that the experiment is scheduled to run.

You will not learn the exact adjustment or exchange rates. However, the exchange rate is positive, meaning that the more laboratory dollars you win, or the fewer you lose, the more $US you take home. The parameters are set so that the average winnings will be approximately $10–15 per scheduled hour.