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Data on the 30 largest stock and balanced funds provided one-year and five- year percentage returns for the period ending March 31,2000 (The Wall Street Journal, April 10,2000 ). Suppose we consider a one-year return in excess of \(50 \%\) to be high and a five-year return in excess of \(300 \%\) to be high. Nine of the funds had one-year returns in excess of \(50 \%\) seven of the funds had five-year returns in excess of \(300 \%\), and five of the funds had both one-year returns in excess of \(50 \%\) and five-year returns in excess of \(300 \%\) a. What is the probability of a high one-year return, and what is the probability of a high five-year return? b. What is the probability of both a high one-year return and a high five-year return? c. What is the probability of neither a high one-year return nor a high five- year return?

Step by step solution

01

Calculate Probability of High One-Year Return

To find the probability of a high one-year return, divide the number of funds with high one-year returns by the total number of funds. Given: 9 funds have a high one-year return, and there are 30 funds in total. \[ P(\text{High One-Year Return}) = \frac{9}{30} = 0.3 \]

02

Calculate Probability of High Five-Year Return

Next, calculate the probability of a high five-year return by dividing the number of funds with high five-year returns by the total number of funds.Given: 7 funds have a high five-year return out of 30 funds.\[ P(\text{High Five-Year Return}) = \frac{7}{30} = 0.2333 \]

03

Calculate Probability of Both High Returns

Now, determine the probability that a fund has both a high one-year return and a high five-year return. Divide the number of funds with both high returns by the total number of funds.Given: 5 funds have both high one-year and five-year returns out of 30 funds.\[ P(\text{Both High Returns}) = \frac{5}{30} = 0.1667 \]

04

Calculate Probability of Neither Return Being High

Calculate the probability of neither return being high using the formula that includes complementary probabilities.Calculate the probability of either return being high:\[ P(\text{One or Five-Year or Both}) = P(\text{One-Year}) + P(\text{Five-Year}) - P(\text{Both}) \]\[ P(\text{One or Five-Year or Both}) = 0.3 + 0.2333 - 0.1667 = 0.3666 \]Now, find the probability of neither by subtracting the above result from 1:\[ P(\text{Neither High}) = 1 - 0.3666 = 0.6334 \]

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Key Concepts

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

One-Year Return

When we talk about a "One-Year Return" in the context of stocks or investment funds, we refer to the total percentage change in the value of an investment over a single year. It's a way to measure how well an investment has performed in a relatively short term. In our case, a one-year return is considered "high" if it exceeds 50%.
To find this probability, you determine the number of funds that achieved a one-year return over 50% out of all the observed funds. Here, we had 9 out of 30 funds that met this condition. To calculate:

• Numerator: Number of funds with a high one-year return (9).
• Denominator: Total number of funds (30).

The probability formula is \[ P(\text{High One-Year Return}) = \frac{9}{30} = 0.3 \].
This means there is a 30% chance, or 0.3 probability, that a randomly chosen fund from our dataset had a high one-year return.

Five-Year Return

A "Five-Year Return" measures the total percentage increase in an investment's value over five years. It gives a longer-term view of performance and is considered "high" in this instance if it exceeds 300%. This metric helps investors see the consistency and growth of their investments over multiple years, offering a more comprehensive picture than just a one-year snapshot.
To find the probability of a high five-year return, identify the number of funds with returns over 300% and divide by the total number of funds:

• Numerator: Number of funds having a high five-year return (7).
• Denominator: Total number of funds (30).

Using the probability formula:\[ P(\text{High Five-Year Return}) = \frac{7}{30} = 0.2333 \].
So, there is about a 23.33% chance, or 0.2333 probability, that any given fund in our dataset experienced a high five-year return.

Complementary Probability

Complementary Probability is a useful concept in probability theory that helps us find the likelihood of an event not occurring. In our scenario, it helps us calculate the chance that neither the one-year return nor the five-year return is high. This can be key for a decent risk assessment or understanding downside probabilities.
First, find the probability of an event happening through direct calculations:

• Both one-year and five-year returns considered high was found to be \( 0.1667 \) for both conditions occurring (5 funds out of 30).
• Probability of having either high one-year or five-year returns is computed as:
  \[ P(\text{One or Five-Year or Both}) = P(\text{One-Year}) + P(\text{Five-Year}) - P(\text{Both}) \]
• Plugging in the values, we get:
  \[ 0.3 + 0.2333 - 0.1667 = 0.3666 \]

Thus, the probability for neither return being high is simply the complement:\[ P(\text{Neither High}) = 1 - 0.3666 = 0.6334 \].
This implies there is a 63.34% probability that a chosen fund from this group did not have a high return in either timeframe.