Allen Young has always been proud of his personal investment strategies and has done very well over the past several years. He invests primarily in the stock market. Over the past several months, however, Allen has become very concerned about the stock market as a good investment. In some cases, it would have been better for Allen to have his money in a bank than in the market. During the next year, Allen must decide whether to invest $10,000 in the stock market or in a certificate of deposit (CD) at an interest rate of 5%. If the market is good, Allen believes that he could get a 15% return by investing in stocks. With a fair market, he expects to get a 6% return. If the market is bad, he will most likely to get a 5% loss.
Allen estimates that the probability of a good market is p and the probability of a fair market is q. He thinks that CD is the best option given the probabilities. Now, Allen is thinking about paying for a stock market newsletter, which could predict very accurately how the market would be. He determines that he would at most pay $500 for the newsletter. Please offer an upper bound and a lower bound for Allen’s estimate of p.
To determine the upper and lower bounds for Allen's estimate of p, we need to consider the maximum amount he is willing to pay for the stock market newsletter.
Allen is willing to pay at most $500 for the newsletter, which suggests that the potential gain from investing in the stock market must exceed this amount to make it worthwhile for him.
Let's calculate the potential gain from investing in the stock market:
If the market is good, Allen expects a 15% return, which would be 15% of $10,000 = $1,500.
If the market is fair, he expects a 6% return, which would be 6% of $10,000 = $600.
If the market is bad, he expects a 5% loss, which would be a loss of 5% of $10,000 = -$500.
To make investing in the stock market worthwhile, the potential gain should be at least $500 (the amount he is willing to pay for the newsletter).
Therefore, we can set up the following inequality:
Potential gain from stock market - Loss from a bad market ≥ Amount Allen is willing to pay for the newsletter
$1,500 - $500 ≥ $500
$1,000 ≥ $500
This implies that the potential gain from the stock market should be at least $1,000 to make it worthwhile for Allen.
Now let's consider the lower bound. Since Allen believes that the CD is the best option given the probabilities, the potential gain from the stock market should not exceed the gain from the CD, which is 5% of $10,000 = $500.
Therefore, we can set up the following inequality:
Potential gain from stock market ≤ Gain from CD
$1,500 ≤ $500
This inequality shows that Allen's estimate of p should not be greater than the probability that would result in a potential gain from the stock market greater than $1,500.
So, the upper bound for Allen's estimate of p is a probability that would result in a potential gain ≥ $1,000.
The lower bound for Allen's estimate of p is a probability that would result in a potential gain ≤ $500.
Unfortunately, without specific values for the probabilities p and q, we cannot determine the exact upper and lower bounds for Allen's estimate of p.
To determine the upper bound and lower bound for Allen's estimate of p, we need to consider the potential benefits of paying for the stock market newsletter.
First, let's analyze the expected return from investing in the stock market without the newsletter. Based on the given information, Allen expects a 15% return in a good market, a 6% return in a fair market, and a 5% loss in a bad market. We can express this in terms of expected values:
Expected return without newsletter = p * 15% + q * 6% + (1 - p - q) * (-5%)
Now, let's consider the potential benefits of the stock market newsletter. Allen believes that the newsletter can predict the market very accurately, which implies that the probability of a good market would increase, increasing his expected return from stock market investments.
If Allen values the potential increase in expected return from using the newsletter at $500, then we can set up an equation:
Expected return without newsletter + $500 = Expected return with newsletter
We know:
Expected return without newsletter = p * 15% + q * 6% + (1 - p - q) * (-5%)
Expected return with newsletter = p' * 15% + q' * 6% + (1 - p' - q') * (-5%)
Additionally, we want the increase in expected return to be at most $500, so we can write:
Expected return with newsletter - Expected return without newsletter ≤ $500
Substituting the expressions for the expected returns, we have:
(p' * 15% + q' * 6% + (1 - p' - q') * (-5%)) - (p * 15% + q * 6% + (1 - p - q) * (-5%)) ≤ $500
Simplifying the equation gives:
p' * 20% + q' * 11% - p * 20% - q * 11% ≤ $500
Since we are interested in the upper bound for p, we can assume that p' = 1 (the highest possible value). We can also assume q' = 0 (the lowest possible value), as we are interested in finding the maximum value for p. Therefore, the equation becomes:
20% - p * 20% - q * 11% ≤ $500
Simplifying further, we get:
- p * 20% - q * 11% ≤ $500 - 20%
Since we are finding the upper bound for p, we want to maximize p while still satisfying this inequality. To do that, we can assume the worst-case scenario for the right-hand side, where $500 - 20% is the maximum amount we can subtract from $500. Therefore, the upper bound for p is:
- p * 20% - q * 11% ≤ $400
Now, let's consider the lower bound for p. In this case, we assume that Allen would not be willing to pay anything for the newsletter. If he considers the newsletter to have no value, there would be no increase in his expected return. This means there would be no change in the probabilities p and q. Therefore, the lower bound for p is the same as the original estimate.
In summary, the upper bound for Allen's estimate of p is determined by the equation:
- p * 20% - q * 11% ≤ $400
And the lower bound for p remains unchanged.