Question

A risk-neutral principal (P) must hire a risk-neutral a manager (the agent) in order to undertake some action. If the agent is successful, the principal receives a pay-off of S = 100 and if the agent fails the principal receives F = 0. The agent must exert costly effort e in order to make this task successful and the probability of success p is given by p(e) = √ e. The agent’s cost of effort is c(e) = 100e. Finally, assume that the agent receives a wage of wS if successful and wF if the project fails and that the agent’s opportunity cost of not working for this principal is 0. 1. Assume that effort is fully observable by the principal so that there is no moral hazard. Calculate the optimum amount of effort and wages that the agent should receive. 2. Now assume that there is moral hazard (for the rest of the questions). Show, using calculus, that if wS = wF the agent will not exert any effort. 1 3. Calculate the second best level of effort under moral hazard and show that it is smaller than the amount determined in (1). 4. Suppose that instead of receiving a wage, the agent receives a fraction r ∈ (0, 1] of the principal’s payoff. The principal receives the remaining (1 − r)S or (1 − r)F. How much effort will the agent exert (as a function of r)? 5. How large must r (in the previous question) be in order for the agent to exert the optimum amount of effort determined in (1)? How much will be left-over for the owner. 6. Extra credit Suppose that the manager can fire the worker if she fails. How would that change the behavior of the agent?

Answer 1

Risk Neutral:-
Risk neutral is a concept used in both game theory studies and in finance. It refers to a mindset where an individual is indifferent to risk when making an investment decision. This mindset is not derived from calculation or rational deduction, but rather from an emotional preference. A person with a risk-neutral approach simply doesn't focus on the risk--regardless of whether or not that is an ill-advised thing to do. This mindset is often situational and can be dependent on price or other external factors.

KEY TAKEAWAYS
1-Risk neutral describes a mindset where investors focus on potential gains when making investment decisions.
2-Risk neutral investors may understand that risk is involved, but they aren't considering it for the moment.
3-An investor can change their mindset from risk averse to risk neutral.
4-Risk-neutral measures play an important part in derivatives pricing.

A risk-neutral principal (P) must hire a risk-neutral a manager (the agent) in order to undertake some action. If the agent is successful, the principal receives a pay-off of S = 100 and if the agent fails the principal receives F = 0. The agent must exert costly effort e in order to make this task successful and the probability of success p is given by p(e) = √ e. The agent’s cost of effort is c(e) = 100e. Finally, assume that the agent receives a wage of wS if successful and wF if the project fails and that the agent’s opportunity cost of not working for this principal is 0. 1. Assume that effort is fully observable by the principal so that there is no moral hazard.

Understanding the Concept of Risk Neutral
Risk neutral is a term used to describe the attitude of an individual who may be evaluating investment alternatives. If the individual focuses solely on potential gains regardless of the risk, they are said to be risk neutral. Such behavior, to evaluate reward without thought to risk, may seem to be inherently risky. A risk averse investor would not consider the choice to risk $1000 loss with the possibility of making $50 gain to be the same risk as a choice to risk only $100 to make the same $50 gain. However someone who is risk neutral would. Given two investment opportunities the risk-neutral investor only looks at the potential gains of each investment and ignores the potential downside risk.

Risk Neutral Pricing and Measures
There could be any number of reasons why an individual would reach a risk-neutral mindset, but the idea that an individual could actually change from a risk averse mindset to a risk-neutral mindset based on pricing changes then leads to another important concept: that of risk-neutral measures. Risk-neutral measures have extensive application in the pricing of derivatives because the price where investors would be expected to exhibit a risk-neutral attitude should be a price of equilibrium between buyers and sellers.

Individual investors are almost always risk averse, meaning that they have a mindset where they exhibit more fear over losing money than the amount of eagerness they exhibit over making money. This tendency often results in the price of an asset finding a point of equilibrium somewhat below what might be accounted for by the expected future returns on this asset. When trying to model and adjust for this effect in marketplace pricing, analysts and academics attempt to adjust for this risk aversion by using these theoretical risk-neutral measures are.

For example, consider a scenario where 100 investors are presented and accept the opportunity to gain $100 on if they deposit $10,000 in a bank for 6 months. There is virtually no risk of losing money (unless the bank itself were in danger of going out of business). Then suppose those same 100 investors are subsequently presented with an alternative investment. This investment gives them the opportunity of gaining $10,000, while accepting the possibility of losing all $10,000. Finally suppose we poll the investors over which investment they would choose and give them three responses: (A) I'd never consider that alternative, (B) I need more information about the alternative investment, (C) I'll invest in the alternative right now.

In this scenario, those who responded A, would be considered risk-averse investors, and those who responded C would be considered risk seeking investors, since the investment value is not accurately determinable with only that much information. However those who responded with B recognize that they need more information to determine whether they would be interested in the alternative. They are neither adverse to risk nor seeking it for its own sake. Instead, they are interested in the value of expected returns to know whether or not they prefer to take the risk. So at the moment they seek more information, they are considered risk neutral.

Such investors would probably want to know what the probability of doubling their money might be (in comparison to possibly losing it all). If the probability of doubling were only 50%, then they could recognize that the expected value of that investment is 0 since it has a equal possibility of losing everything or doubling. If the probability of doubling were to shift to 60%, then those who were willing to consider the alternative at that point, would have adopted a risk-neutral mindset, because they were focused on the probability of gain and no longer focused on the risk.

The price at which risk-neutral investors manifest their behavior of considering alternatives, despite the risk, is an important point of price equilibrium. This is a point where the greatest numbers of buyers and sellers may be present in the market.