Stock A’s expected return and standard deviation are E[rA] = 6% and σA= 12%, while stock B’s expected return and standard deviation are E[rB] = 10% and σB= 20%.
(a) Using Excel to compute the expected return and standard deviation of the return on a portfolio with weights wA=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1, for the following alternative values of correlation between A and B: ρAB=0.6 and ρAB= -0.4. Under the two different correlations, plot the expected return—standard deviation pairs on a graph (with the standard deviations on the horizontal axis, and the expected returns on the vertical axis).
(b) How would you construct a portfolio p with expected return of 8% using Stock A and Stock B? What is the standard deviation of the portfolio? (Assume ρAB = 0.4)
(c) How would you construct a portfolio q with standard deviation of 15% using Stock A and Stock B? What is the expected return of the portfolio? (Assume ρAB = 0.4)
(d) If you want to have the minimum variance for your portfolio z, what will be your portfolio weights? In this case, what are the expected return and variance of your portfolio? (Assume ρAB = 0.4)

In R,

Draw 1000 samples from the following distributions and create histograms for each. Be sure to comment about each histogram. Remember all histograms should have four items addressed.

(a) X ∼ N (0, 1) using rnorm function.  

(b) X ∼ Gamma(2, 3) using the rgamma function.

(c) X+Y where X ∼ N(5,2) and Y ∼ χ2(15).

(d) X ∼ Binomial(1, 0.3).

(e) Calculate a mean of a vector X = (X1, X2, ..., Xn ) 1000 times, where X_i ~ Binomial(1, 0.4) for n = 2, 5, 10, 20, 50, 100 and create a histogram of the means. Each n should be dealt with separately. For example, you draw 1000 samples of size n = 2 (you may use for() loop for that). Every time a sample is drawn, you calculate its mean. Once you will end up with n means, you create a histogram of those means. Then repeat for n = 5, 10, etc.  There should be 6 separate histograms with comments for each. Restrict the x-axis to stay at 0 to 1 for all plots. Comment on the differences between plots as n increases.

Developing filters using convolution theorem and Fourier Transform.

You have been hired as an Engineering Mathematician at a consulting firm located in Saint Louis. On your first job, you have been asked to mathematically design a frequency filter that removes from a standard beacon signal a periodic interference generated by a rotatory machine located in the basement of the company. Please see below for more details: Let s(t) be the standard beacon signal that is being communicated. Below you can find its Fourier Series representation ?(?)=2.5+2 sin(?)+3cos(t)+0.5cos(2?)+ 0.3 sin (2?). The periodic interference is given by ?(?)=0.5 cos (120 ?) and the measurement signal with noise is given as follows: ?(?)=?(?)+?(?) Let g( t) be the filter function and let ? Z(?) be the function that results from applying ? (?) to ?(?). Using the Fourier transform of the convolution theorem, propose the design of a filtering function g(t) which removes from ?(?) the effect of the periodic noise ?(?) assuming that we only know that its fundamental period is equal to 2pi/120. Make sure to write the analytical expression of ?(?) and its Fourier transform. Also, please write the mathematical expression that relates ? Z(?) as a function of ?(?)and ?(?)

The human resources department needs to forecast the number of sexual harassement investigations for the entire company. The data for several months is supplied below. Be careful since the data is listed beginning with the most recent. The forecasting method to be used here is the 4 month weighted moving average adjusting for seasonality where the weights, starting with the most recent time period, are 0.4, 0.3, 0.2, 0.1. Again, you must find the seasonality factors for the data. Please round your forecast to the nearest whole number.

Most analytical machines used in science can either measure absorbance of light, like the spectrophotometers used in BioZ 151 and 152 labs, or can measure change in electrical conductance.

In order to convert things like absorbance values into concentration data, a linear regression is used.

In order to determine nitrite concentration in water samples, nitrite is reacted with several chemicals to produce a purple color. The absorbance of the solution is measured from known amounts of nitrite to produce a standard curve to analyze samples.

Use the data in the provided table, or from the linked spreadsheet to make a graph and perform a linear regression using treadline.

In order for this to work, samples must be diluted prior to measurement so that the concentration falls within the range of the standard curve. If a sample underwent a 0.01 dilution, and produce an absorbance value of 0.236, what is the resultant concentration of NO₂^- in mg per L? Report your answer to two decimal places.

Tip: The regression equation will give you the final concentration. The dilution is equal to V₁/V₂. The question is asking for the initial concentration.

Direct Labor Variances

Ada Clothes Company produced 14,000 units during April. The Cutting Department used 2,700 direct labor hours at an actual rate of $12.60 per hour. The Sewing Department used 4,500 direct labor hours at an actual rate of $12.30 per hour. Assume there were no work in process inventories in either department at the beginning or end of the month. The standard labor rate is $12.50. The standard labor time for the Cutting and Sewing departments is 0.2 hour and 0.3 hour per unit, respectively.

a. Determine the direct labor rate, direct labor time, and total direct labor cost variance for the (1) Cutting Department and (2) Sewing Department. Enter a favorable variance as a negative number using a minus sign and an unfavorable variance as a positive number.

b. The two departments have opposite results. The Cutting Department has a(n)   rate variance and a(n)   time variance, resulting in a total   cost variance. In contrast, the Sewing Department has a(n)   rate variance but has a(n)   time variance, resulting in a total   cost variance.

To celebrate their Queen’s 90th birthday, the country of Wonderland has decided to bake the world’s largest cake. The main ingredient(input) of the cake will be flour. It has been estimated that the cake will use 50,000 tons of flour, which represents 20% of the current supply of flour. The current price of flour is $4000 per ton. Previous study has shown the elasticity of supply, Es, for flour to be 0.3 and the elasticity of demand, Ed to be -1.2.

a. Assuming linear demand curves, calculate the percentage change in the price of flour that would result due to this project? Also, calculate the change in quantity supplied as well as the change in quantity demanded by private consumers. What is the opportunity cost of flour that should be used in a cost-benefit analysis of this project?

[Hint: Es = (ΔQs/ΔP)(P/Qs) and Ed =(ΔQd/ΔP)(P/Qd) where Qs is current supply and P is current price. Note that the amount of flour demanded by the project Qp = ΔQs – ΔQd

Using the above equations, you should be able to find the change in price, ΔP. Once, you have the value for ΔP, you can plug it into the elasticity formula to find ΔQs & ΔQd]

b. Sketch the supply and demand for flour and show the opportunity cost on your sketch.

Caro Manufacturing has two production departments, Machining and Assembly, and two service departments, Maintenance and Cafeteria. Direct costs for each department and the proportion of service costs used by the various departments for the month of August follow:

Assume that both Machining and Assembly work on just two jobs during the month of August: CM-22 and CM-23. Costs are allocated to jobs based on machine-hours in Machining and labor-hours in Assembly. The number of labor- and machine-hours worked in each department are as follows:

Required:
How much of the service department costs allocated to Machining and Assembly in the direct method should be allocated to Job CM-22? How much should be allocated to Job CM-23? (Round "Department rate" to 2 decimal places.)

Given below is a bivariate distribution for the random variables x and y.

(a)

Compute the expected value and the variance for x and y.

E(x)

=

E(y)

=

Var(x)

=

Var(y)

=

(b)

Develop a probability distribution for

x + y.

(c)

Using the result of part (b), compute

E(x + y)

and

Var(x + y).

E(x + y)

=

Var(x + y)

=

(d)

Compute the covariance and correlation for x and y. (Round your answer for correlation to two decimal places.)

covariancecorrelation

Are x and y positively related, negatively related, or unrelated?

The random variables x and y are  ---Select--- positively related negatively related unrelated .

(e)

Is the variance of the sum of x and y bigger, smaller, or the same as the sum of the individual variances? Why?

The variance of the sum of x and y is  ---Select--- greater than  less than unrelated the sum of the variances by two times the covariance, which occurs whenever two random variables are  ---Select--- positively related negatively related unrelated .