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Chapter 9: Problem 3

We have a choice of two investment strategies: stocks and bonds. The returns for each under two possible economic conditions are as follows: $$ \begin{array}{llr} \hline \text { Alternative } & \text { Condition 1 } & \text { Condition 2 } \\\ \hline \text { Stocks } & \$ 10,000 & -\$ 4000 \\ \text { Bonds } & \$ 7000 & \$ 2000 \\ \hline \end{array} $$ a. If we assume the probability of Condition 1 is \(p_{1}=0.75\) and Condition 2 is \(p_{2}=\) \(0.25\), compute the expected values and select the best alternative. b. What probabilities for Conditions 1 and 2 would have to exist to be indifferent toward stocks and bonds? c. What other decision criteria would you consider? Explain your rationale.

Short Answer

Expert verified
a. Choose Stocks ($6,500 > $5,750). b. Indifference occurs at \( p_1 = \frac{2}{3}, p_2 = \frac{1}{3} \). c. Consider minimax regret or risk analysis for comprehensive decision-making.

Step by step solution

01

Expected Value for Stocks

To find the expected value for Stocks, use the formula for expected value: \( EV = p_1 \times \text{outcome 1} + p_2 \times \text{outcome 2} \). Here, the outcomes for Stocks are \(10,000 in Condition 1 and -\)4,000 in Condition 2. Using \( p_1 = 0.75 \) and \( p_2 = 0.25 \), calculate:\[EV_{stocks} = 0.75 \times 10,000 + 0.25 \times (-4,000)\]\[EV_{stocks} = 7,500 - 1,000 = 6,500\]
02

Expected Value for Bonds

Repeat the process for Bonds. The outcomes for Bonds are \(7,000 in Condition 1 and \)2,000 in Condition 2. Using \( p_1 = 0.75 \) and \( p_2 = 0.25 \), compute:\[EV_{bonds} = 0.75 \times 7,000 + 0.25 \times 2,000\]\[EV_{bonds} = 5,250 + 500 = 5,750\]
03

Determine Best Alternative

Compare the expected values calculated in Steps 1 and 2. Stocks have an expected value of $6,500, while Bonds have an expected value of $5,750. The better alternative is the one with the higher expected value, so in this case, Stocks is the better choice.
04

Find Probability for Indifference towards Stocks and Bonds

To be indifferent, set the expected value of Stocks equal to the expected value of Bonds and solve for the unknown probability. Let \( x \) represent the probability of Condition 1:\[x \times 10,000 + (1-x) \times (-4,000) = x \times 7,000 + (1-x) \times 2,000\]\[10,000x - 4,000 + 4,000x = 7,000x + 2,000 - 2,000x\]\[14,000x - 4,000 = 5,000x + 2,000\]\[9,000x = 6,000\]\[x = \frac{6,000}{9,000} = \frac{2}{3}\]Therefore, \( p_1 = \frac{2}{3} \) and \( p_2 = \frac{1}{3} \) would lead to indifference.
05

Consider Other Decision Criteria

When choosing investment strategies, other decision criteria like the **Minimax Regret Criterion** (which minimizes the maximum regret) or **Risk Analysis** (focusing on variability and worst-case scenarios) could be considered. These methods provide additional perspectives, considering not just expected outcomes, but also potential regrets or risks associated with each strategy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value Calculation
The concept of expected value is fundamental in decision making in finance, as it allows investors to make informed choices based on potential future outcomes. The expected value is essentially a weighted average of all possible outcomes, where each outcome is multiplied by its probability of occurring. This helps in estimating the average expected return for an investment over a long period.

For example, when deciding between investing in stocks or bonds, you use the formula:
  • Expected Value (EV) = Probability of Condition 1 ( p_1 ) 脳 Outcome in Condition 1 + Probability of Condition 2 ( p_2 ) 脳 Outcome in Condition 2
This calculation helps determine which investment might yield better returns according to the likelihood of economic conditions.

In our exercise, the expected value for stocks was calculated as EV_{stocks} = 0.75 imes 10,000 + 0.25 imes (-4,000) = 6,500 . For bonds, it was EV_{bonds} = 0.75 imes 7,000 + 0.25 imes 2,000 = 5,750 . This implies that stocks are the better option given the probabilities.
Probability Theory
Understanding probability is essential in finance, especially when predicting outcomes of different investment strategies. Probability theory provides the basis for calculating expected values, making it invaluable for investors attempting to quantify risk and uncertainty.

In the context of our exercise, the probabilities assigned to different economic conditions ( p_1 = 0.75 for condition 1 and p_2 = 0.25 for condition 2) directly influence the expected value calculations. These probabilities reflect the chances of each condition happening, thus impacting the decision-making process significantly.

Moreover, probability helps identify points of indifference between choices by equalizing expected values when varying probabilities. For instance, in step 4 of our solution, setting the expected values of stocks and bonds to be equal revealed the probabilities ( x = rac{2}{3} ) where an investor would be indifferent between the two options. Such calculations assist in understanding how changes in the probability of economic conditions could affect investment decisions.
Investment Strategies
Selecting an investment strategy involves considering various criteria beyond just expected value. Investors need a robust framework that incorporates different perspectives to manage risks effectively and maximize returns.

One important approach is using the **Minimax Regret Criterion**, which focuses on minimizing the potential regret from not choosing the best decision in hindsight. This involves analyzing the worst-case scenarios of each investment alternative, which is a useful complement to the expected value approach when certainty is unattainable.

Another crucial criterion is **Risk Analysis**. This analysis goes beyond just financial returns to consider the variability in returns and other risk factors like market volatility. Diversification, asset allocation, and understanding one's own risk tolerance are also part of effective investment strategies.

By combining these decision criteria, investors can better align their choices with their financial goals and risk comfort levels, ensuring a comprehensive assessment of their investment opportunities.

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Most popular questions from this chapter

Consider a firm handling concessions for a sporting event. The firm's manager needs to know whether to stock up with coffee or cola and is formulating policies for specific weather predictions. A local agreement restricts the firm to selling only one type of beverage. The firm estimates a \(\$ 1500\) profit selling cola if the weather is cold and a \(\$ 5000\) profit selling cola if the weather is warm. The firm also estimates a \(\$ 4000\) profit selling coffee if it is cold and a \(\$ 1000\) profit selling coffee if the weather is warm. The weather forecast says that there is a \(30 \%\) of a cold front; otherwise, the weather will be warm. Build a decision tree to assist with the decision. What should the firm handling concessions do?

A new energy company, All Green, has developed a new line of energy products. Top management is attempting to decide on both marketing and production strategies. Three strategies are considered and are referred to as \(A\) (aggressive), \(B\) (basic), and \(C\) (cautious). The conditions under which the study will be conducted are \(S\) (strong) and \(W\) (weak) conditions. Management's best estimates for net profits (in millions of dollars) are given in the following table. Build a decision tree to assist the company determine its best strategy. $$ \begin{array}{lcc} \hline \text { Decision } & \text { Strong (with probability } 45 \%) & \text { Weak (with probability 55\%) } \\ \hline A & 30 & -8 \\ B & 20 & 7 \\ C & 5 & 15 \\ \hline \end{array} $$

Assume the following probability distribution of daily demand for bushels of strawberries: $$ \begin{array}{lllll} \hline \text { Daily demand } & 0 & 1 & 2 & 3 \\ \text { Probability } & 0.2 & 0.3 & 0.3 & 0.2 \\ \hline \end{array} $$ Further assume that unit cost is \(\$ 3\) per bushel, selling price is \(\$ 5\) per bushel, and salvage value on unsold units is \(\$ 2\). We can stock \(0,1,2\), or 3 units. Assume that units from any single day cannot be sold on the next day. Build a decision tree and determine how many units should be stocked each day to maximize net profit over the long haul.

For a new development area, a local investor is considering three alternative real estate investments: a hotel, a restaurant, and a convenience store. The hotel and the convenience store will be adversely or favorably affected depending on their closeness to the location of gasoline stations, which will be built in the near future. The restaurant will be assumed to be relatively stable. The payoffs for the investment are as follows: $$ \begin{array}{lccc} \hline & {\text { Conditions }} & \\ \text { Alternative } & \text { 1: Gas close } & \text { 2: Gas medium distance } & \text { 3: Gas far away } \\ \hline \text { Hotel } & \$ 25,000 & \$ 10,000 & -\$ 8000 \\ \text { Convenience store } & \$ 4000 & \$ 8000 & -\$ 12,000 \\ \text { Restaurant } & \$ 5000 & \$ 6000 & \$ 6000 \\ \hline \end{array} $$ Determine the best plan by each of the following criteria: a. Laplace b. Maximin c. Maximax d. Coefficient of optimism (assume that \(x=0.45\) ) e. Regret (minimax)

Baseball has dramatically changed the penalties associated with failing steroid drug testing. The new penalties that Commissioner Bud Selig has proposed are an approach of "three strikes and you're out," which goes as follows: The first positive test would result in a 50-game suspension. The second positive test would result in a 100 -game suspension. Finally, the third positive test would result in a lifetime suspension from Major League Baseball. Let's examine the issue and the choices using decision trees. First, let's consider testing a baseball player for steroids; the test result will be either positive or negative. However, these tests are not flawless. Some baseball players who are steroid free test positive, and some baseball players who use steroids test negative. The former are called false positives; the latter are false negatives. We will extract data from the baseball steroid data table for our analysis. Build a conditional probability tree and use it to help compute all the probabilities associated with the following key issues: If some players test positive, what are the chances that they were a steroid user? If some players test positive, what are the chances that they were not a steroid user? If a test was negative, what are the chances that the player was a steroid user? If a test was negative, what are the chances that the player was not a steroid user?What do these results suggest about this drug testing? $$ \begin{aligned} &\text { Baseball steroid data }\\\ &\begin{array}{lcrr} \hline & \text { Positive results } & \text { Negative results } & \text { Totals } \\ \hline \text { Users } & 28 & 3 & 31 \\ \text { Nonusers } & 19 & 600 & 619 \\ \text { Totals } & 47 & 603 & 650 \\ \hline \end{array} \end{aligned} $$
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