Problem 5 Consider an asset that costs \(1... [FREE SOLUTION]

Chapter 34: Problem 5

Consider an asset that costs $120$dollars today. You are going to hold it for 1 year and then sell it. Suppose that there is a 25 percent chance that it will be worth $100$dollars in a year, a 25 percent chance that it will be worth $115$dollars in a year, and a 50 percent chance that it will be worth $140$ dollars in a year. What is its average expected rate of return? Next, figure out what the investment's average expected rate of return would be if its current price were $130$ dollars today. Does the increase in the current price increase or decrease the asset's average expected rate of return? At what price would the asset have a zero rate of return?

Short Answer

Expert verified

The average expected rate of return is 4.17% at $120 and decreases to -3.85% at $130. A zero rate of return occurs at a price of $125.

Step by step solution

Define Initial Conditions

Start by defining the probabilities and future values of the asset. Today, the asset costs $120$ dollars. In a year, there is a 25% chance of the asset being worth $100$, 25% chance of it being $115$, and 50% chance of it being $140$.

Calculate Expected Future Value (at $120$)

The expected future value is the weighted average of all possible outcomes. Calculate it as follows: $ \text{Expected Future Value} = (0.25 \times 100) + (0.25 \times 115) + (0.5 \times 140) = 125 $.

Calculate Rate of Return (at $120$)

The rate of return is calculated by \[ \text{Rate of Return} = \frac{\text{Expected Future Value} - \text{Current Price}}{\text{Current Price}} \]. Substituting the values yields: $ \frac{125 - 120}{120} = 0.0417 $ or 4.17%.

Repeat Calculation with New Initial Price ($130$)

Now assume the asset costs $130$ dollars today. The expected future value calculated earlier is still $125$. Use the formula for rate of return: $ \frac{125 - 130}{130} = -0.0385 $ or -3.85%.

Analyze Impact of Price Change on Rate of Return

The rate of return decreased when the asset's price increased from $120$ to $130$ as it went from 4.17% to -3.85%. A higher initial price, with the same future value, decreases the rate of return.

Determine Zero Rate of Return Price

The price where the expected rate of return is zero is where the current price is equal to the expected future value: $125$. This implies $ \frac{125 - 125}{125} = 0 $, confirming a zero rate of return.

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Key Concepts

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

Investment Analysis

Investment analysis is a critical process when deciding whether to invest in a financial asset. This involves evaluating the potential for future returns and understanding the associated risks. In essence, it helps determine whether an investment is suitable given its potential payoff versus the investment cost.
When performing investment analysis, you examine various scenarios and their probabilities. In our exercise, a financial asset costs $120 today. It has different probabilities to be worth $100, $115, or $140 in a year. Calculating these potential outcomes helps us understand the expected future value.
Expected future value is crucial as it represents the average outcome you can anticipate based on the weighted probabilities and specific values. This value helps gauge whether the potential return justifies the current investment price. Investment decision-making isn't just about potential earnings; analyzing the asset's full risk and reward profile is vital.

Asset Valuation

Asset valuation determines the present value of an expected future return of an asset. This concept is at the heart of determining if investments are worth their price. By predicting future value and considering the asset's cost today, investors can derive its value.
In our exercise, the future value of the asset has been calculated based on different scenarios. It is essential to assess this alongside the current price. At an initial cost of $120, the future value calculation shows an expectation of $125. This increment from the present cost indicates profitability.
However, if the current price increases, as illustrated when it rises to $130, asset valuation suggests a poorer investment. The asset would then expect a value of $125, lower than the price paid, resulting in a negative rate of return. Such analysis, using changes in price and returns, is fundamental in asset valuation.

Probability and Statistics

Probability and statistics are the backbone of evaluating potential future values in investments. They give a mathematical framework to make informed predictions about uncertainty.
By assigning probabilities to each possible outcome, we use a statistical approach to estimate the expected value. For example, in this exercise, the future value of $100 has a 25% chance, $115 another 25%, and $140 a 50% chance. Combining these probabilities and values helps compute the expected outcome, which aids in gauging potential returns on investment.
Understanding these statistics helps assess risk levels. In our scenario, the price with a zero return was calculated statistically. When current price equals expected future value, there is no profit or loss. Thus, statistics allow investors to make data-driven decisions based on potential risks and returns, which is vital for strategic investment planning.

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