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xavier1283
19.04.2021 •
Mathematics
1. A company has a cash portfolio measured in millions. The drift is 0.1 per month, variance is 0.16per month. The initial cash is 2.0. a) Find the probability distribution after 6 months and after 1 year. b) Find the probability of a negative cash position at the end of 6 months and the end of 1 year. c) At what time in the future is the probability of a negative distribution greatest.
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Ответ:
Step-by-step explanation:
From the information given:
The probability distribution at the end of 6 months is determined as follows:
After 6 months;
Mean of probability distribution = value of Initial cash +
T
=2.0 +(0.1 × 6)
=2.6
After 6 months;
The probability distribution's standard deviation is estimated by using the following formula:
Standard deviation:
= 0.9798
Hence, after 6 months;
The company's cash position is supposed to be allocated monthly, with the following expenses.
Mean 2.6
Standard deviation 0.9798
Variance 0.96
After 12 months, the probability distribution is as follows:
Mean = value of Initial cash +
T
= 2.0 +(0.1 × 12)
= 3.2
The standard deviation is:
The standard deviation of probability distribution =![b \sqrt{T}](/tpl/images/1268/1307/f56cd.png)
= 1.3856
Hence, after 6 months;
The company's cash position is supposed to be allocated monthly, with the following expenses.
Mean 3.2
Sandard deviation 1.3856
Variance 1.92
b)
in 6-month distribution, the probability of the negative value of the cash position is as follows.
Now, for us to find the negative cash distribution;
We need to estimate the z -scores value.
The z-score inform us greatly on the concept of how far a particular data point is from the mean.
For a normal distribution;
Here;
the value of x = zero as a result that if it exceeds zero. the cash position will be negative.
∴
Using the standard distribution tables, it is now possible to calculate that the likelihood N(-2.65) equals 0.004 or 0.4 percent.
As a result, there's a 0.4 percent chance of getting a negative cash balance after six months.
For 12 months distribution:
The Probability of negative cash position is calculated as follows:
Using the standard distribution tables,
N(-2.31) equals 0.0104 or 1.04 percent.
As a result, there's a 1.04 percent chance of getting a negative cash balance after 1 year
c) To determine the time period over which the likelihood of achieving a negative cash condition is highest, it's necessary to examine the z-score more closely. Essentially, the z-score measures the difference between a given value(x) and the mean of all potential values
, expressed in terms of the total set's standard deviation ![(\sigma)](/tpl/images/1268/1307/fab8c.png)
This suggests that the higher the z-score, the greater the difference occurring between x and
, and thus the likelihood of receiving x is minimal. As a result, the best chance of finding a certain value is when the z-score is the lowest.
To do so, calculate the derivative of the z-score in relation to the time interval. The point where the derivative is equivalent to zero is where the z-scores are at their lowest.
The first step is to go over the z-score formula in more detail, as seen below.;
Now, compute the derivative of this equation with respect to T as follows:
Now, figure out the value of T at which this derivative is equal to zero by substituting all values as follows:
As a result, the time period in which achieving a negative cash condition is = 20 months.
Ответ:
175/3 minutes
Step-by-step explanation:
Assume that each man peel potatoes at the same rate
The time required
= 5 x 35 / 3
= 175/3 minutes