Behavioral Finance and Corporate Finance

Abstract

The seminal works by Daniel Kahneman, Amos Tversky and Richard Thaler have inspired many researches in behavioral economics. Behavioral economics and behavioral finance incorporate the concepts and methods in psychology science into traditional economics (finance).

Appendices

Appendix A: Opportunity Cost in Practice

The following is a true story from Chang (2005). One day a senior student (Jade) came to my office, and told me happily that she just got admitted to a prestigious university to pursue her master degree in finance.

Jade: Professor, I am so happy that I got admitted to X university.

Professor: Congratulations! ... But did you also apply for other disciplines, such as MS or Ph.D. programs in computer science, statistics, or economics?

Jade: No. Why should I? If I pursue these degrees, then my past four-year study of finance would be a waste.

Professor: Apparently, you do not understand the meaning of opportunity cost. Your study of Principles of Economics (Econ101) and Financial Management (Fin101) courses is a waste and futile.

Jade: I don’t quite follow you. Could you explain it to me?

Professor: Let me give you an example. A beautiful girl who just enrolls in a university meets a boy. When the boy asks her to go on a date, she agrees. Two weeks later, the boy asks for more dating, and the girl contemplates: “If I stop dating him, then my two weeks of dating (investment) will be a waste.” Hence, they continue to date for another two years. After two years, the girl contemplates again: “If I stop dating him, then my previous two years of dating him will be a waste.” They continue to date for two more years. After four years, upon graduation, the boy asks the girl to marry him. The girl contemplates: “If I do not marry him, then my four years of dating him will go to waste,” and so she marries the boy. You are that girl.

Jade: No, I’m not. I am not that stupid!

Professor: Oh, yes, you are. If you are this discreet with your marriage, then be even more so in choosing your profession.

What matters in Jade’s choosing a particular graduate program is: Does it provide positive net present value (NPV, the difference between revenues and costs), and is its NPV the largest one among all the mutually exclusive projects (graduate programs)? Costs in the NPV analysis are opportunity costs, which means that you still have the opportunity to make the choice to spend or not to spend, i.e., opportunity costs are ex-ante (Buchanan, 1969). When calculating the NPV of joining a finance graduate program, Jade should consider only how much more costs and time she will spend, and compare them with the revenues (cash flows) she will receive if she finishes the study. Jade’s four-year study of finance is already sunk; it is not an opportunity cost, and therefore should not be considered in decision-making.

Appendix B: Expected Utility Theory and Risky Assets

Example A.1

A consumer decides to buy $\alpha$ units of insurance. When there is an event, every unit of insurance bought will pay one dollar. One unit of insurance costs $q$ dollars (where $q < 1$). Suppose the consumer’s wealth is $w$, the probability of event is $\pi$, and total loss will be $D$ dollars. The consumer has a strictly concave utility function: $u^{\prime}\left( w \right) > 0\;,\;u^{\prime\prime}\left( w \right) < 0$, and is using the expected utility function to make decision:

$$ \mathop {Max}\limits_{\alpha \ge 0} \;\;{ }U = \left( {1 - \pi } \right) \cdot u\left( {w - \alpha q} \right) + \pi \cdot u\left( {w - \alpha q - D + \alpha \cdot 1} \right) $$

(A.1)

or

$$ \begin{gathered} \mathop {Max}\limits_{\alpha ,s} \;\;{ }U = \left( {1 - \pi } \right) \cdot u\left( {w - \alpha q} \right) + \pi \cdot u\left( {w - \alpha q - D + \alpha } \right) \hfill \\ s.t.\quad { } - \alpha + s^{2} = 0 \hfill \\ \end{gathered} $$

By the Lagrangian method:

$$ \mathop {Max}\limits_{\alpha ,\lambda ,s} \;\;{ }L = \left( {1 - \pi } \right) \cdot u\left( {w - \alpha q} \right) + \pi \cdot u\left( {w - \alpha q - D + \alpha } \right) - \lambda \left[ { - \alpha + s^{2} } \right] $$

(A.2)

Since $u\left( \cdot \right)$ is strictly concave, and hence, the expected utility function is also strictly concave. The first-order condition of Eq. (A.2) is both necessary and sufficient conditions:

1st-order condition:

$$ \begin{aligned} & \frac{\partial L}{{\partial \alpha }} = \left( {1 - \pi } \right) \cdot u^{\prime}\left( {w - \alpha q} \right)\left( { - q} \right) + \pi \cdot u^{\prime}\left( {w - D + \alpha \left( {1 - q} \right)} \right) \cdot \left( {1 - q} \right) + \lambda \equiv 0 \\ & \frac{\partial L}{{\partial \lambda }} = - \left[ { - \alpha + s^{2} } \right] \equiv 0 \\ & \frac{\partial L}{{\partial s}} = - 2\lambda s \equiv 0 \\ \end{aligned} $$

Its Kuhn-Tucker condition is:

$$ \left\{ \begin{aligned} & \left( {\text{I}} \right)\quad \left( {1 - \pi } \right) \cdot u^{\prime}\left( {w - \alpha^{*} q} \right)\left( { - q} \right) + \pi \cdot u^{\prime}\left( {w - D + \alpha^{*} \left( {1 - q} \right)} \right) \cdot \left( {1 - q} \right) \le 0 \\ & \left( {{\text{II}}} \right)\;\;\left[ {\left( {1 - \pi } \right) \cdot u^{\prime}\left( {w - \alpha^{*} q} \right)\left( { - q} \right) + \pi \cdot u^{\prime}\left( {w - D + \alpha^{*} \left( {1 - q} \right)} \right) \cdot \left( {1 - q} \right)} \right] \cdot \alpha^{*} = 0 \\ & \left( {{\text{III}}} \right)\; \alpha^{*} \ge 0,\lambda^{*} \ge 0 \\ \end{aligned} \right. $$

(A.3)

Assume that the insurance company is risk-neutral (i.e., it treats expected value as certain value) and earns no excess profits:

$$ q - \left[ {\pi \cdot 1 + \left( {1 - \pi } \right) \cdot 0} \right] = 0 \Rightarrow q = \pi $$

Hence, Eq. (A.3)’s (I) and (II) become:

$$ \left\{ {\left( {\text{I}} \right)\quad u^{\prime}\left( {w - D + \alpha^{*} \left( {1 - \pi } \right)} \right) - u^{\prime}\left( {w - \alpha^{*} \pi } \right) \le 0} \right. $$

(A.4a)

$$ \left\{ {\left( {{\text{II}}} \right)\;\;\left[ {u^{\prime}\left( {w - D + \alpha^{*} \left( {1 - \pi } \right)} \right) - u^{\prime}\left( {w - \alpha^{*} \pi } \right)} \right] \cdot \alpha^{*} = 0} \right. $$

(A.4b)

If in Eq. (A.4a), $u^{\prime}\left( {w - D + \alpha^{*} \left( {1 - \pi } \right)} \right) - u^{\prime}\left( {w - \alpha^{*} \pi } \right) < 0$, then from Eq. (A.4b) we get: $\alpha^{*} = 0$. But because of $u^{\prime}\left( \cdot \right) > 0$ and $u^{\prime\prime}\left( \cdot \right) < 0$, there will be a contradiction:

$\begin{aligned}u^{\prime}\left( {w - D + \alpha^{*} \left( {1 - \pi } \right)} \right) - u^{\prime}\left( {w - \alpha^{*} \pi } \right)\langle {0 \Rightarrow w - D + \alpha^{*} \left( {1 - \pi } \right)} \rangle w - \alpha^{*} \pi \\ \Rightarrow \alpha^{*} > D. \\ \end{aligned}$

It is impossible to have $\alpha^{*} > D$. Therefore, from equation (A.4a), we know that $u^{\prime}\left( {w - D + \alpha^{*} \left( {1 - \pi } \right)} \right) - u^{\prime}\left( {w - \alpha^{*} \pi } \right)$ must be equal to zero. Also, because $u\left( w \right)$ is monotonically increasing in $w$, we have:

$$ \begin{aligned} u^{\prime } \left( {w - D + \alpha ^{*} \left( {1 - \pi } \right)} \right) - u^{\prime } \left( {w - \alpha ^{*} \pi } \right) & = 0 \Rightarrow w - D + \alpha ^{*} \left( {1 - \pi } \right) \\ & = w - \alpha ^{*} \pi \Rightarrow \alpha ^{*} = D. \\ \end{aligned} $$

That is, this consumer will buy 100% insurance. However, to prevent the moral hazard problem (i.e., the insured may handle her asset carelessly), the insurance company may not sell 100% insurance. But according to the endowment effect, the insurance company will sell 100% insurance because the insured feels attached to her asset.

Example A.2

Assume a one-period model, and there are two kinds of assets: one gives certain outcome (one dollar for one dollar), another gives uncertain payoff z where $\smallint zdF\left( z \right) > 1$, $F\left( z \right)$ is the cumulative distribution of $z$. At the beginning of the period, the consumer will allocate her wealth $w$ to the two assets: $w = \alpha + \beta$. At the end of the period, she will get wealth: $\alpha \cdot 1 + \beta \cdot z$. Suppose that her utility function is strictly concave: $u^{\prime}\left( w \right) > 0$ and $u^{\prime\prime}\left( w \right) < 0$, and she makes decision based on the expected utility function:

$$ \mathop {Max}\limits_{\alpha ,\beta \ge 0} \;\;{ }U = \smallint u\left( {\alpha + \beta z} \right)dF\left( z \right) $$

$$ s.t.\quad { }w = \alpha + \beta $$

(A.5)

Substitute $\alpha = w - \beta$ into the objective function:

$$ \mathop {Max}\limits_{\beta } \;\;{ }U = \smallint u\left( {w + \beta \left( {z - 1} \right)} \right)dF\left( z \right) $$

$$ s.t.\quad { }0 \le \beta \le w\;,\quad {\text{or}}\quad \left\{ {\begin{array}{*{20}c} {\beta \ge 0 \Rightarrow - \beta + s_{1}^{2} = 0\quad } \\ {\beta \le w \Rightarrow \beta - w + s_{2}^{2} = 0} \\ \end{array} } \right. $$

By the Lagrangian method:

$$ \mathop {Max}\limits_{{\begin{array}{*{20}c} {\beta ,\lambda_{1} ,s_{1} ,} \\ {\lambda_{2} ,s_{2} } \\ \; \\ \end{array} }} \quad L = \smallint u\left( {w + \beta \left( {z - 1} \right)} \right)dF\left( z \right) - \lambda_{1} \left[ { - \beta + s_{1}^{2} } \right] - \lambda_{2} \left[ {\beta - w + s_{2}^{2} } \right] $$

1st-order condition:

$$ \begin{aligned} & \frac{\partial L}{{\partial \beta }} = \smallint u^{\prime}\left( {w + \beta \left( {z - 1} \right)} \right)\left( {z - 1} \right)dF\left( z \right) + \lambda_{1} - \lambda_{2} \equiv 0 \\ & \frac{\partial L}{{\partial \lambda_{1} }} = - \left[ { - \beta + s_{1}^{2} } \right] \equiv 0 \\ & \frac{\partial L}{{\partial s_{1} }} = - 2\lambda_{1} s_{1} \equiv 0 \\ & \frac{\partial L}{{\partial \lambda_{2} }} = - \left[ {\beta - w + s_{2}^{2} } \right] \equiv 0 \\ & \frac{\partial L}{{\partial s_{2} }} = - 2\lambda_{2} s_{2} \equiv 0 \\ \end{aligned} $$

Its Kuhn-Tucker condition is:

$$ \left\{ \begin{aligned} & \left( {\text{I}} \right)\quad \smallint u^{\prime}\left( {w + \beta^{*} \left( {z - 1} \right)} \right)\left( {z - 1} \right)dF\left( z \right) = \lambda_{2}^{*} - \lambda_{1}^{*} \\ & \left( {{\text{II}}} \right)\;\; \lambda_{1}^{*} \cdot \beta^{*} = 0 \\ & \left( {{\text{III}}} \right)\; \lambda_{2}^{*} \cdot \left( {\beta^{*} - w} \right) = 0 \\ & \left( {{\text{IV}}} \right)\;{ }w \ge \beta^{*} \ge 0; \lambda_{1}^{*} ,\lambda_{2}^{*} \ge 0 \\ \end{aligned} \right. $$

(A.6)

From the first-order condition and Eq. (A.6):

$$ \begin{gathered} {\text{If}}\quad s_{2}^{*} \ne 0\quad ({\text{i}}.{\text{e}}.,\quad 0 \le \beta ^{*} < w),\;{\text{then}}\;\lambda _{2}^{*} = 0\;{\text{and}}\; \hfill \\ \int {u^{\prime } \left( {w + \beta ^{*} \left( {z - 1} \right)} \right)\left( {z - 1} \right)dF\left( z \right) = - \lambda _{1}^{*} \le 0} \hfill \\ \end{gathered} $$

(A.6a)

$$ \begin{gathered} {\text{If}}\quad s_{1}^{*} \ne 0\quad ({\text{i}}.{\text{e}}.,\quad \beta ^{*} > 0),\;{\text{then}}\;\lambda _{1}^{*} = 0\;{\text{and}}\; \hfill \\ \int {u^{\prime } \left( {w + \beta ^{*} \left( {z - 1} \right)} \right)\left( {z - 1} \right)dF\left( z \right) = \lambda _{2}^{*} \ge 0} \hfill \\ \end{gathered} $$

(A.6b)

If in Eq. (A.6a), $\beta^{*} = 0$, then because $u^{\prime}\left( w \right) > 0$ and $\smallint \left( {z - 1} \right)dF\left( z \right) > 0$,

$$ [\smallint u^{\prime}\left( {w + \beta^{*} \left( {z - 1} \right)} \right) \cdot \left( {z - 1} \right)dF\left( z \right)]_{{\beta^{*} = 0}} = u^{\prime}\left( w \right)\smallint \left( {z - 1} \right)dF\left( z \right) > 0 $$

which contradicts Eq. (A.6a). Thus, we must have $0 < \beta^{*} < w$, i.e., the consumer’s investment portfolio must contain some risky assets.

Problems

• 1. Suppose you're on a game show, and you're given the choice of three doors: Behind one door is $1,000,000; behind the others, $0. You choose a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has $0. He then says to you, “Do you want to choose door No. 2?”

Questions

• (1) How many choices do you have in this game?

• (2) If the host opens two other doors, how many choices do you have in this game?

• (3) If the host opens no door and asks you to choose again, how many choices do you have in this game?

• 2. Suppose that an individual is offered a bet: ($100,000, 0.50; $50,000, 0.50). Mankiw and Zeldes (1991) argue that a person with a coefficient of relative risk aversion of 30 would be indifferent between this gamble and a certain consumption of $51,209. Give your comments on this claim.²⁷

• 3. Some researchers claim that most common errors among individuals are: under-diversification, holding onto losers, chasing winners, buying stocks that catch their attention, systematically ignoring important information, paying too little attention to fees and trading too much. What do you think?