31. Suppose the market portfolio is equally likely to increase by 30% or decrease by 10%.
a. Calculate the beta of a firm that goes up on average by 43% when the market goes up and goes down by 17% when the market goes down.
b. Calculate the beta of a firm that goes up on average by 18% when the market goes down and goes down by 22% when the market goes up.
c. Calculate the beta of a firm that is expected to go up by 4% independently of the market.
a. Oh, calculating the beta of a firm, are we? Well, if this firm goes up by 43% when the market goes up and goes down by 17% when the market goes down, we can call it a bit of an optimist and pessimist at the same time. To calculate its beta, we need to find the average return when the market goes up and the average return when the market goes down. So, let's crunch those numbers. The average return when the market goes up is 43%, and the average return when the market goes down is -17%. Since we know that the market is equally likely to increase by 30% or decrease by 10%, we can put those into our formula. The beta is equal to (43% - (-17%)) / (30% - (-10%)). Let me grab my calculator... *beep boop beep* ... and the beta comes out to be 1.67! That's one optimistic firm!
b. Now, let's move on to the next firm. This one goes up by 18% when the market goes down and goes down by 22% when the market goes up. It's like this firm has a love-hate relationship with the market! To calculate its beta, we'll find the average return when the market goes up and the average return when the market goes down. The average return when the market goes up is -22% (ouch!), and the average return when the market goes down is 18%. Now, let's put those numbers in our trusty formula. The beta is equal to (18% - (-22%)) / (10% - (-30%)). Let me do some quick calculations here... *calculates furiously* ... and the beta for this firm turns out to be -0.67! It seems to have a bit of a rebellious nature when it comes to the market.
c. Ah, the last firm is a bit of a maverick, isn't it? Expected to go up by 4% independently of the market. It's like it has a mind of its own, not affected by the market's ups and downs! Since it's not influenced by the market, we can say it has a beta of 0. So, for this firm, the beta is 0, making it the wild card in the pack. You never know what it's going to do!
a. To calculate the beta of the firm, we can use the formula:
Beta = (Return on Firm - Risk-Free Rate) / (Return on Market - Risk-Free Rate)
Given that the market portfolio is equally likely to increase by 30% or decrease by 10%, we can assume that the expected return on the market is the average of these two outcomes:
Return on Market = (30% + (-10%)) / 2 = 20% / 2 = 10%
The return on the firm when the market goes up is 43%, and when the market goes down is -17%.
Beta = (43% - Risk-Free Rate) / (10% - Risk-Free Rate)
We need more information about the risk-free rate to calculate beta accurately.
b. Similarly, we can calculate the beta for the second firm. Given that the market portfolio is equally likely to increase by 30% or decrease by 10%, we can assume that the expected return on the market is 10%.
The return on the firm when the market goes up is -22%, and when the market goes down is 18%.
Beta = (-22% - Risk-Free Rate) / (10% - Risk-Free Rate)
Again, we require information about the risk-free rate to calculate beta accurately.
c. In this case, the firm's expected return is independent of the market. So, the beta would be 0 since there is no relationship between the firm's returns and the market returns.
To calculate the beta of a firm, you need to divide the average percentage change in the firm's returns by the average percentage change in the market returns.
a. Calculate the beta of a firm that goes up on average by 43% when the market goes up and goes down by 17% when the market goes down.
First, calculate the average percentage change in the firm's returns:
(43% + (-17%)) / 2 = 13%
Next, calculate the average percentage change in the market returns:
(30% + (-10%)) / 2 = 10%
The beta of the firm is then calculated as the ratio of the two average changes in percentage:
Beta = Average change in firm's returns / Average change in market returns
Beta = 13% / 10% = 1.3
Therefore, the beta of the firm in this case is 1.3.
b. Calculate the beta of a firm that goes up on average by 18% when the market goes down and goes down by 22% when the market goes up.
Similarly, calculate the average percentage change in the firm's returns:
(18% + (-22%)) / 2 = -2%
The average percentage change in the market returns remains the same, 10%.
The beta of the firm is then calculated as:
Beta = Average change in firm's returns / Average change in market returns
Beta = -2% / 10% = -0.2
Therefore, the beta of the firm in this case is -0.2.
c. Calculate the beta of a firm that is expected to go up by 4% independently of the market.
In this case, the average percentage change in the firm's returns is 4%, and the average percentage change in the market returns is still 10%.
The beta of the firm is then calculated as:
Beta = Average change in firm's returns / Average change in market returns
Beta = 4% / 10% = 0.4
Therefore, the beta of the firm in this case is 0.4.