scenario response topic · the example scenario response topic provided herein is meant to help...

Scenario Response Topic2018-19 Example 1: D.I.G. Mining

A program of The Actuarial Foundation

Modeling The Future Challenge

For the 2018-19 Modeling the Future Challenge, Scenario Response topics serve as the qualifying phase to allow teams to enter the main research and report phase. All teams passing the Scenario Responses will be invited to participate in the research and report phase. Teams will also be connected with an actuarial mentor to provide guidance on data identification, model building, and analysis.

The example Scenario Response topic provided herein is meant to help teams understand what to expect when the scenarios are released for the qualification phase. This scenario will not be part of the actual competition, but follows the same template that will be used by the competition scenarios.

How Scenario Response Topics are StructuredEach scenario will include 3 parts: (1) a written description, (2) a dataset, and (3) a series of 10 questions (for the example scenarios we also provide an answer key to the questions). The questions will cover a variety of topics related to mathematical modeling, data analysis, and the statistics and probability tools that are used in actuarial science. In general, the 10 questions will be grouped into 4 basic statistics or probability questions, 4 advanced statistics or probability questions, and 2 critical thinking or analytical questions.

How Scenario Response Topics are ScoredQuestions in the scenarios are scored on this rubric:

Background on Scenario Response Topics

0 Points 1 Point 2 Points 3 points

Question 1-4 (2pt max)

Wrong answer

Correct answer, wrong or no methodology

Correct answer, strong methodology

Question 5-10 (3pt max)

Wrong answer

Correct answer, wrong or no methodology

Correct answer, weak or incomplete methodology

Correct answer, strong methodology

1. Perfect scores are not required to qualify. If your team doesn’t answer all 10 questions correctly in a scenario, don’t fret! Perfect scores are not a requirement to qualify for the main research and report phase of the MTF Challenge. The Scenario Response topics ask a wide range of questions with varying levels of difficulty; however, we do not expect that for teams to be successful in the research and report, they need to have all answers to all types of questions. So if you aren’t certain about a few questions on the scenarios, answer what you know as well as you can, and you’ll get feedback on what else you might want to pick up for your report.

2. In the Challenge, only 2 out of 3 Scenarios are required. The MTF Challenge will present teams with three Scenario Response topics; however, teams only need to complete passing responses to two of the scenarios. In other words, your top two scores will be considered. So you may still want to complete all three scenarios to see which ones you score the best on, but if you’re really pressed for time, completing two can still get you through to the main research & report phase.

3. Don’t worry about a specific methodology.In the answer keys we present straight-forward methodologies to determine the answers; however, these may not be the only methodologies to come to a correct response. So long as you show logical reasoning and how you got to the correct response, it will be counted. You do not need to have the exact methodology that we note in the answer keys.

4. Consider each scenario as an opportunity for practice and feedback.More than anything, the MTF Challenge’s Scenario Response topics are meant to help prepare you for how to create strong mathematical models analyzing data to project future trends. They are not attempting to create “gotcha” moments. Try your best, answer what you can, and look forward to the feedback and responses to inform how you create your models and report for the main competition – or to prepare you for next year if you aren’t quite ready for the research this time.

Scenario Response Tips

Example Scenario: D.I.G. Mining

D.I.G. Mining is a gold mining company based out of Portland, Oregon. Currently, the company has five mines operating in Oregon. It has been performing exceptionally well, and is evaluating five possible states for expansion: Alaska, Colorado, Nevada, Utah, and West Virginia.

The company has hired your actuarial consulting team to perform a statistical analysis of all five states, as well as to provide recommendations for the five states D.I.G. is evaluating. The senior actuary on your team has gathered sample information on the gold output in ounces/cubic yard (oz/y3) of 50 mines in each of the five states, as well as other figures such as employee pay.

Data Summary: You have been given 250 observations of different mines, with 50 observations in each state. These observations were collected by mining a cubic yard of dirt at each mine, and then recording how many ounces of gold was in the cubic yard of dirt. Each mine is identified by its state (column A), and its gold output per cubic yard (column B).

Data is provided in text format on the following page, but is also attached as an excel spreadsheet to this scenario. For ease of use you can use the excel spreadsheet or another program to manipulate the data and make calculations.

D.I.G. Mining Data

Colorado Mine Gold per cubic yard

0.4578

0.4922

0.5408

0.5145

0.6776

0.3650

0.6528

0.6922

0.3942

0.5337

0.4683

0.5613

0.6229

0.6490

0.5772

0.4822

0.6244

0.4225

0.4629

0.5331

0.5693

0.5105

0.3282

0.5941

0.5969

0.4447

0.3558

0.5311

0.6985

0.6112

0.3079

0.5719

0.4711

0.6193

0.3604

0.5695

0.5694

0.4872

0.5914

0.5043

0.5419

0.5371

0.4684

0.3898

0.5566

0.5211

0.4902

0.4560

0.2626

0.4750

West Virginia Mine Gold per cubic yard

0.7098

0.3997

0.5387

0.2130

0.5459

0.5313

0.7529

0.6552

0.7482

0.4327

0.6917

0.7624

0.5980

0.7106

0.5762

0.5454

0.4786

0.4501

0.8908

0.8558

0.1617

0.2539

0.7645

0.5255

0.4596

0.3662

0.6755

0.6717

0.4523

0.6129

0.6948

0.7906

0.6137

0.7234

0.9235

0.4530

0.8488

0.3632

0.4164

0.8255

0.3835

1.0635

0.6141

0.4856

0.9163

0.2879

0.5413

0.5753

0.4753

0.5507

Nevada Mine Gold per cubic yard

0.4244

0.4226

0.5072

0.4441

0.4239

0.4733

0.4865

0.5193

0.4918

0.4333

0.4970

0.4874

0.4734

0.4226

0.4210

0.4805

0.5445

0.4829

0.5451

0.3949

0.3456

0.4570

0.3972

0.3812

0.4685

0.4368

0.3762

0.4784

0.4562

0.4438

0.5127

0.4852

0.4265

0.4369

0.4472

0.4823

0.4863

0.4622

0.3883

0.4281

0.4922

0.4558

0.5224

0.4472

0.4045

0.4347

0.4717

0.4275

0.4944

0.3965

Alaska Mine Gold per cubic yard

0.4689

0.6289

0.7132

0.5418

0.4337

0.6923

0.7076

0.6758

0.6896

0.7584

0.5447

0.1110

0.5372

0.7335

0.7218

0.5686

0.6583

0.7018

0.6066

0.6307

0.5564

0.5578

0.6149

0.7179

0.5997

0.7093

0.6120

0.3642

0.5804

0.6850

0.8503

0.5540

0.4952

0.7252

0.8237

0.8475

0.5252

0.5214

0.6919

0.3807

0.3359

0.5118

0.5886

0.5044

0.4075

0.5334

0.6830

0.3311

0.5405

0.5713

Utah Mine Gold per cubic yard

0.4524

0.5039

0.4736

0.4965

0.4357

0.5596

0.5195

0.4657

0.4199

0.4957

0.4450

0.4702

0.4862

0.5188

0.5115

0.6009

0.4237

0.4287

0.3378

0.6245

0.5090

0.4290

0.4014

0.6090

0.4960

0.3982

0.4343

0.5722

0.5009

0.4083

0.5224

0.4257

0.5658

0.5448

0.4785

0.5878

0.4690

0.5073

0.4997

0.4542

0.4288

0.3310

0.5651

0.4817

0.4794

0.5260

0.4787

0.4544

0.4572

0.4422

Questions

As part of the D.I.G. actuarial consulting team, you are tasked with answering the following questions and providing recommendations to D.I.G. Mining:

1. What is the sample mean of gold (oz/y3) produced by a mine in each state?

2. What is the sample variance of gold (oz/y3) produced by a mine in each state?

3. Based upon the expected value of gold produced in a given mine, rank the states from highest to lowest average output.

4. Based upon the variance of gold produced in a given mine, rank the states from lowest to highest variance.

5. Let ut be the population mean of gold produced from mines in Utah. Find a 95% confidence interval for ut.

6. If the company acquires one mine, and promises investors that 97.5% of the time, they will produce at least .36 oz/y3 of gold, in which state should they acquire the mine?

7. D.I.G. is considering buying five mines in one state and two mines in another. What combination of states should D.I.G. select to produce the lowest combined variance? Find the combined variance assuming independence between states.

8. Determine if the expected value of the amount of gold mined in Colorado is greater than the expected value of the amount of gold mined in Utah. Please be 95% confident in your response. Explain the result.

9. D.I.G. has narrowed its potential destinations down to Alaska and West Virginia. The data collection expense for the sample information in each state has already been incurred. After negotiating with local gold miners in each state, D.I.G. agreed to pay them a percentage of total revenue listed below as “Employee Costs” with no legally required hourly wage. Each mine produces .59 ounces of gold per cubic yard. Assume the current price of gold is $1200/oz and the cost of purchasing each mine is equal.

Given the above information, to which state should D.I.G. expand and why?

10. D.I.G. is considering purchasing a warranty on their equipment. Your team has made the decision tree on the following page in order to analyze this option. The warranty is a one-time $35,000 fee upfront that would cover equipment failures for two years. Based on your team’s analysis of the probability of the mining equipment needing maintenance in either year, and the expected cost of that maintenance, will the warranty save the D.I.G. Mining money (ignore any change in the time value of money over the years)?

State Data Collection

Expense

Tax Rate

Employee Costs

Minimum Wage (per hr)

Gold Output

(oz/week)

Operating Weeks

Operating Cost (per

Year)

AK $25,000 24% 55% $9.84 115 28 $400,000WV $10,000 28% 45% $8.75 80 40 $600,000

Answer Key

1. What is the sample mean of gold (oz/y3) produced by a mine in each state?

2. What is the sample variance of gold (oz/y3) produced by a mine in each state?

3. Based upon the expected value of gold produced in a given mine, rank the states from highest to lowest average output.

4. Based upon the variance of gold produced in a given mine, rank the states from lowest to highest variance.

State x̄

CO 0.514

WV 0.592

NV 0.454

AK 0.591

UT 0.483

State s²

CO 0.010

WV 0.038

NV 0.002

AK 0.020

UT 0.004

State x̄

WV 0.592Highest

AK 0.591

CO 0.514

UT 0.483

NV 0.454Lowest

State s²NV 0.002Lowest

UT 0.004

CO 0.010

AK 0.020

WV 0.038Highest

Answer Key

5. Let ut be the population mean of gold produced from mines in Utah. Find a 95% confidence interval for ut.

6. If the company acquires one mine, and promises investors that 97.5% of the time, they will produce at least .36 oz/y3 of gold, in which state should they acquire the mine?

Because Nevada is the only state with a lower bound greater than .36 oz/cubic yard, D.I.G. should acquire a mine in Nevada.

7. D.I.G. is considering buying five mines in one state and two mines in another. What combination of states should D.I.G. select to produce the lowest combined variance? Find the combined variance assuming independence between states.

Logically, the least variable state should have the five mines, and the second least variable state should have the two mines. Therefore: V(5NV+2UT) = 0.0646 making purchasing 5 mines in Nevada and 2 mines in Utah the lowest combined variance.

From Questions 1 and 2: x:̄ 0.48255138

s²: 0.00401297

s: 0.06334798

z-score: 1.96

N 50

Lower-Bound: 0.465

Upper-Bound: 0.500

From Questions 1 and 2: State x̄ s² N Lower Bound Upper Bound

CO 0.514 0.010 50 0.318 0.711

WV 0.592 0.038 50 0.210 0.973

NV 0.454 0.002 50 0.368 0.541

AK 0.591 0.020 50 0.313 0.869

UT 0.483 0.004 50 0.358 0.607

z-score: 1.96

From Question 2: State s²

CO 0.010

WV 0.038

NV 0.002

AK 0.020

UT 0.004

Answer Key

8. Determine if the expected value of the amount of gold mined in Colorado is greater than the expected value of the amount of gold mined in Utah. Please be 95% confident in your response. Explain the result.

Because the p-value is less than alpha (.05), we reject the null hypothesis and conclude that Colorado's true mean output is greater than Utah's.

9. D.I.G. has narrowed its potential destinations down to Alaska and West Virginia. The data collection expense for the sample information in each state has already been incurred. After negotiating with local gold miners in each state, D.I.G. agreed to pay them a percentage of total revenue listed below as “Employee Costs” with no legally required hourly wage. Each mine produces .59 ounces of gold per cubic yard. Assume the current price of gold is $1200/oz and the cost of purchasing each mine is equal.

Given the above information, to which state should D.I.G. expand and why?

D.I.G. should expand to Alaska because Alaska has a higher expected net income than West Virginia. *Because the data collection expense is a one-time expense, and it has already occurred, it would be considered a sunk cost. Therefore, it would not be included in any financial decision going forward. *Because the miners are paid on a percentage of revenue basis, they waive the right to minimum wage in the respective state. Therefore, the minimum wage has no effect on net income.

T-Test p value: 0.0307

State Data Collection

Expense

Tax Rate

Employee Costs

Minimum Wage (per hr)

Gold Output

(oz/week)

Operating Weeks

Operating Cost (per

Year)

AK $25,000 24% 55% $9.84 115 28 $400,000WV $10,000 28% 45% $8.75 80 40 $600,000

Inputs

State AK WV

Employee Cost (% of Revenue) 55% 45%

Effective Tax Rate 24% 28%

Ounces of Gold per Cubic Yard 0.59 0.59

Cubic Yards Mined per Week 115.00 80.00

Operating Weeks per Year 28.00 40.00

Price of Gold $ 1,200.00 $ 1,200.00

Data Collection Expense* $ 25,000.00 $ 10,000.00

Minimum Wage** $ 9.84 $ 8.75

Estimated Income Statement

State AK WV

Revenue $ 2,283,624.00 $ 2,273,280.00

Employee Cost $ (1,255,993.20) $ (1,022,976.00)

Operating Costs $ (400,000.00) $ (600,000.00)

Earnings Before Taxes $ 627,630.80 $ 650,304.00

Taxes $ (150,631.39) $ (182,085.12)

Net Income $ 476,999.41 $ 468,218.88

Answer Key

10. D.I.G. is considering purchasing a warranty on their equipment. Your team has made the decision tree on the following page in order to analyze this option. The warranty is a one-time $35,000 fee upfront that would cover equipment failures for two years. Based on your team’s analysis of the probability of the mining equipment needing maintenance in either year, and the expected cost of that maintenance, will the warranty save the D.I.G. Mining money (ignore any change in the time value of money over the years)?

Since the expected value of repairs is greater than the cost of the warrantee, D.I.G. Mining should purchase the warrantee on their equipment.

(((-70000*0.2)+(-50000))*0.3)+((-60000*0.5)*0.7)

Expected value of repairs = -$40,200

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