GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option that it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A media budget of $3,000 is available for this promotional campaign. Based on past experience in promoting its other services, GreenLawns has obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media: S = − 2 R 2 − 10 M 2 − 8 R M + 18 R + 34 M , where S = total sales in thousands of dollars R = thousands of dollars spent on radio advertising M = thousands of dollars spent on direct-mail advertising GreenLawns would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the media budget. a. What is the value of sales if $2,000 is spent on radio advertising and $1,000 is spent on direct-mail advertising? b. Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $3,000 on total advertising. c. Determine the optimal amount to spend on radio and direct-mail advertising. How much in sales will be generated?

Question

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Answer 1

Textbook Question

Chapter 13, Problem 1P

GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option that it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A media budget of $3,000 is available for this promotional campaign. Based on past experience in promoting its other services, GreenLawns has obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media:

S = − 2 R 2 − 10 M 2 − 8 R M + 18 R + 34 M ,

where

S = total sales in thousands of dollars

R = thousands of dollars spent on radio advertising

M = thousands of dollars spent on direct-mail advertising

GreenLawns would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the media budget.

a. What is the value of sales if $2,000 is spent on radio advertising and $1,000 is spent on direct-mail advertising?
b. Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $3,000 on total advertising.
c. Determine the optimal amount to spend on radio and direct-mail advertising. How much in sales will be generated?

a.

Expert Solution

To determine

To find the amount of sales generated when $2,000 and $1,000 is spend on radio and direct-mail advertising.

Explanation of Solution

To find the amount of sales generated, substitute $2,000 and $1,000 is spend on radio and direct-mail advertising in the below relationship equation.

S=−2R2−10M2−8RM+18R+34M =−2(22)−10(12)−8(2)(1)+18(2)+34(1) =36

Therefore, sales of $36,000 will be generated with the budget allocated for promotion.

b.

Expert Solution

To determine

To formulate optimization problem to maximize sales when total advertising cost is not more than $3,000.

Explanation of Solution

The formulation is for the problem is shown below,

Max−2R2−10M2−8RM+18R+34Ms.t.

R+M≤3000R,M≥0

c.

Expert Solution

To determine

To solve and find the optimal amount to be spend on advertising to determine the amount of sales generated.

Explanation of Solution

Formula:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 1

Solver:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 2

Output:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 3

The solution is R = $2,500 and M = $500 with Sales of $37,000

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In this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)

Problem 4. Margrabe formula and the Greeks (20 pts) In the homework, we determined the Margrabe formula for the price of an option allowing you to swap an x-stock for a y-stock at time T. For stocks with initial values xo, yo, common volatility σ and correlation p, the formula was given by Fo=yo (d+)-x0Þ(d_), where In (±² Ꭲ d+ õ√T and σ = σ√√√2(1 - p). дго (a) We want to determine a "Greek" for ỡ on the option: find a formula for θα (b) Is дго θα positive or negative? (c) We consider a situation in which the correlation p between the two stocks increases: what can you say about the price Fo? (d) Assume that yo< xo and p = 1. What is the price of the option?

We consider a 4-dimensional stock price model given (under P) by dẴ₁ = µ· Xt dt + йt · ΣdŴt where (W) is an n-dimensional Brownian motion, π = (0.02, 0.01, -0.02, 0.05), 0.2 0 0 0 0.3 0.4 0 0 Σ= -0.1 -4a За 0 0.2 0.4 -0.1 0.2) and a E R. We assume that ☑0 = (1, 1, 1, 1) and that the interest rate on the market is r = 0.02. (a) Give a condition on a that would make stock #3 be the one with largest volatility. (b) Find the diversification coefficient for this portfolio as a function of a. (c) Determine the maximum diversification coefficient d that you could reach by varying the value of a? 2

Answer 2

Textbook Question

Chapter 13, Problem 1P

GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option that it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A media budget of $3,000 is available for this promotional campaign. Based on past experience in promoting its other services, GreenLawns has obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media:

S = − 2 R 2 − 10 M 2 − 8 R M + 18 R + 34 M ,

where

S = total sales in thousands of dollars

R = thousands of dollars spent on radio advertising

M = thousands of dollars spent on direct-mail advertising

GreenLawns would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the media budget.

a. What is the value of sales if $2,000 is spent on radio advertising and $1,000 is spent on direct-mail advertising?
b. Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $3,000 on total advertising.
c. Determine the optimal amount to spend on radio and direct-mail advertising. How much in sales will be generated?

a.

Expert Solution

To determine

To find the amount of sales generated when $2,000 and $1,000 is spend on radio and direct-mail advertising.

Explanation of Solution

To find the amount of sales generated, substitute $2,000 and $1,000 is spend on radio and direct-mail advertising in the below relationship equation.

S=−2R2−10M2−8RM+18R+34M =−2(22)−10(12)−8(2)(1)+18(2)+34(1) =36

Therefore, sales of $36,000 will be generated with the budget allocated for promotion.

b.

Expert Solution

To determine

To formulate optimization problem to maximize sales when total advertising cost is not more than $3,000.

Explanation of Solution

The formulation is for the problem is shown below,

Max−2R2−10M2−8RM+18R+34Ms.t.

R+M≤3000R,M≥0

c.

Expert Solution

To determine

To solve and find the optimal amount to be spend on advertising to determine the amount of sales generated.

Explanation of Solution

Formula:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 1

Solver:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 2

Output:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 3

The solution is R = $2,500 and M = $500 with Sales of $37,000

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Answer 3

Textbook Question

Chapter 13, Problem 1P

GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option that it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A media budget of $3,000 is available for this promotional campaign. Based on past experience in promoting its other services, GreenLawns has obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media:

S = − 2 R 2 − 10 M 2 − 8 R M + 18 R + 34 M ,

where

S = total sales in thousands of dollars

R = thousands of dollars spent on radio advertising

M = thousands of dollars spent on direct-mail advertising

GreenLawns would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the media budget.

a. What is the value of sales if $2,000 is spent on radio advertising and $1,000 is spent on direct-mail advertising?
b. Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $3,000 on total advertising.
c. Determine the optimal amount to spend on radio and direct-mail advertising. How much in sales will be generated?

a.

Expert Solution

To determine

To find the amount of sales generated when $2,000 and $1,000 is spend on radio and direct-mail advertising.

Explanation of Solution

To find the amount of sales generated, substitute $2,000 and $1,000 is spend on radio and direct-mail advertising in the below relationship equation.

S=−2R2−10M2−8RM+18R+34M =−2(22)−10(12)−8(2)(1)+18(2)+34(1) =36

Therefore, sales of $36,000 will be generated with the budget allocated for promotion.

b.

Expert Solution

To determine

To formulate optimization problem to maximize sales when total advertising cost is not more than $3,000.

Explanation of Solution

The formulation is for the problem is shown below,

Max−2R2−10M2−8RM+18R+34Ms.t.

R+M≤3000R,M≥0

c.

Expert Solution

To determine

To solve and find the optimal amount to be spend on advertising to determine the amount of sales generated.

Explanation of Solution

Formula:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 1

Solver:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 2

Output:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 3

The solution is R = $2,500 and M = $500 with Sales of $37,000

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Textbook Question

Chapter 13, Problem 1P

GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option that it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A media budget of $3,000 is available for this promotional campaign. Based on past experience in promoting its other services, GreenLawns has obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media:

S = − 2 R 2 − 10 M 2 − 8 R M + 18 R + 34 M ,

where

S = total sales in thousands of dollars

R = thousands of dollars spent on radio advertising

M = thousands of dollars spent on direct-mail advertising

GreenLawns would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the media budget.

a. What is the value of sales if $2,000 is spent on radio advertising and $1,000 is spent on direct-mail advertising?
b. Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $3,000 on total advertising.
c. Determine the optimal amount to spend on radio and direct-mail advertising. How much in sales will be generated?

a.

Expert Solution

To determine

To find the amount of sales generated when $2,000 and $1,000 is spend on radio and direct-mail advertising.

Explanation of Solution

To find the amount of sales generated, substitute $2,000 and $1,000 is spend on radio and direct-mail advertising in the below relationship equation.

S=−2R2−10M2−8RM+18R+34M =−2(22)−10(12)−8(2)(1)+18(2)+34(1) =36

Therefore, sales of $36,000 will be generated with the budget allocated for promotion.

b.

Expert Solution

To determine

To formulate optimization problem to maximize sales when total advertising cost is not more than $3,000.

Explanation of Solution

The formulation is for the problem is shown below,

Max−2R2−10M2−8RM+18R+34Ms.t.

R+M≤3000R,M≥0

c.

Expert Solution

To determine

To solve and find the optimal amount to be spend on advertising to determine the amount of sales generated.

Explanation of Solution

Formula:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 1

Solver:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 2

Output:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 3

The solution is R = $2,500 and M = $500 with Sales of $37,000

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

a.

Expert Solution

To determine

To find the amount of sales generated when $2,000 and $1,000 is spend on radio and direct-mail advertising.

Explanation of Solution

To find the amount of sales generated, substitute $2,000 and $1,000 is spend on radio and direct-mail advertising in the below relationship equation.

S=−2R2−10M2−8RM+18R+34M =−2(22)−10(12)−8(2)(1)+18(2)+34(1) =36

Therefore, sales of $36,000 will be generated with the budget allocated for promotion.

b.

Expert Solution

To determine

To formulate optimization problem to maximize sales when total advertising cost is not more than $3,000.

Explanation of Solution

The formulation is for the problem is shown below,

Max−2R2−10M2−8RM+18R+34Ms.t.

R+M≤3000R,M≥0

c.

Expert Solution

To determine

To solve and find the optimal amount to be spend on advertising to determine the amount of sales generated.

Explanation of Solution

Formula:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 1

Solver:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 2

Output:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 3

The solution is R = $2,500 and M = $500 with Sales of $37,000

Answer 8

a.

Expert Solution

To determine

To find the amount of sales generated when $2,000 and $1,000 is spend on radio and direct-mail advertising.

Explanation of Solution

To find the amount of sales generated, substitute $2,000 and $1,000 is spend on radio and direct-mail advertising in the below relationship equation.

S=−2R2−10M2−8RM+18R+34M =−2(22)−10(12)−8(2)(1)+18(2)+34(1) =36

Therefore, sales of $36,000 will be generated with the budget allocated for promotion.

Answer 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

To find the amount of sales generated when $2,000 and $1,000 is spend on radio and direct-mail advertising.

Answer 13

Explanation of Solution

To find the amount of sales generated, substitute $2,000 and $1,000 is spend on radio and direct-mail advertising in the below relationship equation.

S=−2R2−10M2−8RM+18R+34M =−2(22)−10(12)−8(2)(1)+18(2)+34(1) =36

Therefore, sales of $36,000 will be generated with the budget allocated for promotion.

Answer 14

b.

Expert Solution

To determine

To formulate optimization problem to maximize sales when total advertising cost is not more than $3,000.

Explanation of Solution

The formulation is for the problem is shown below,

Max−2R2−10M2−8RM+18R+34Ms.t.

R+M≤3000R,M≥0

Answer 15

b.

Expert Solution

Answer 16

b.

Expert Solution

Answer 17

Expert Solution

Answer 18

To determine

To formulate optimization problem to maximize sales when total advertising cost is not more than $3,000.

Answer 19

Explanation of Solution

The formulation is for the problem is shown below,

Max−2R2−10M2−8RM+18R+34Ms.t.

R+M≤3000R,M≥0

Answer 20

c.

Expert Solution

To determine

To solve and find the optimal amount to be spend on advertising to determine the amount of sales generated.

Explanation of Solution

Formula:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 1

Solver:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 2

Output:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 3

The solution is R = $2,500 and M = $500 with Sales of $37,000

Answer 21

c.

Expert Solution

Answer 22

c.

Expert Solution

Answer 23

Expert Solution

Answer 24

To determine

To solve and find the optimal amount to be spend on advertising to determine the amount of sales generated.

Answer 25

Explanation of Solution

Formula:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 1

Solver:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 2

Output:

Essentials of Business Analytics (MindTap Course List), Chapter 13, Problem 1P , additional homework tip 3

The solution is R = $2,500 and M = $500 with Sales of $37,000

Answer 26

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