Question

1. (50 points) Brian is about to take early retirement at the age of 45. His current wealth is $2 million and he plans to use this wealth to fund his consumption over the remaining three periods of his life. 1 Any money that he does not spend on consumption in period 0 can be put in the bank where it will earn interest at the rate of 100% per period. That is, r = 1, so if he chooses not to consume an amount s of his $2 million in period 0 he will have s(1 + r) = 2s available at the beginning of period 1. Similarly, any money he does not spend on consumption in period 1 can be put in the bank where it will also earn interest at the rate of r = 1. (a) (5 points) Explain why his intertemporal budget set B can be described as follows: B = {(c0, c1, c2) ? (0, 0, 0) : c0 + 1/2 c1 + 1/4 c2 ? 2} Initially suppose that Brian's choices over consumption streams are governed by an exponential discounted utility family of preference relations, characterized by an "instantaneous" utility function u(c) = 2c - c²/2 and a discount factor ? = 1/2. 1 You can think of each these periods as corresponding to fifteen years. (b) (10 points) Utilizing the fact that his marginal utility of consumption mu(c) = 2 - c, show that in period 0 his optimal consumption plan (measured in millions of dollars) is (c0, c1, c2) = (8/7, 8/7, 8/7). (c) (5 points) Explain qualitatively how this optimal plan would change if the interest rate was greater than 1. Explain qualitatively how this optimal plan would change if the interest rate was less than 1. (d) (10 points) If Brian chooses to consume c0 = 8/7 in period 0, explain what his intertemporal budget set B¹((c0, c1, c2)) will be in period 1. Show that the continuation of his original consumption plan (c1, c2) = (8/7, 8/7) is indeed the optimal consumption plan for him to choose in period 1 from this budget set. Explain what property of his choice behavior does this reflect. Now suppose that Brian is a naive (quasi-)hyperbolic discounted utility maximizer characterized by an "instantaneous" utility function u(c) = 2c - c²/2, a long-term discount factor ? = 1/2, and a short-term discount factor (or present bias) ? = 0.5. (e) (10 points) Show that his optimal consumption plan (c0, c1, c2) from the perspective of his period 0 self is equal to (1.4, 0.8, 0.8). Given he consumes c0* = 1.4 in period 0 (and hence saves 0.6) work out the amounts ?1 and ?2 that he will actually choose to consume in periods 1 and 2. For the last question, suppose that instead of being able to put money in the bank to earn interest at the rate of 100% per period, Brian can instead purchase a quantity q ? 0 of an annuity at a per-unit price p. That is, each unit of the annuity costs him p million dollars in period 0 and pays out 1 million dollars in period 1 and 1 million dollars in period 2. For example, if he purchases the fraction 0.2 of a unit of the annuity in period 0 at a cost of 0.2 × p then the annuity will pay him 0.2 (of a million dollars) in period 1 and 0.2 in period 2.

          1. (50 points) Brian is about to take early retirement at the age of 45. His current wealth is $2 million and he plans to use this wealth to fund his consumption over the remaining three periods of his life. 1 Any money that he does not spend on consumption in period 0 can be put in the bank where it will earn interest at the rate of 100% per period. That is, r = 1, so if he chooses not to consume an amount s of his $2 million in period 0 he will have s(1 + r) = 2s available at the beginning of period 1. Similarly, any money he does not spend on consumption in period 1 can be put in the bank where it will also earn interest at the rate of r = 1.

(a) (5 points) Explain why his intertemporal budget set B can be described as follows:
B = {(c0, c1, c2) ? (0, 0, 0) : c0 + 1/2 c1 + 1/4 c2 ? 2}

Initially suppose that Brian's choices over consumption streams are governed by an exponential discounted utility family of preference relations, characterized by an "instantaneous" utility function u(c) = 2c - c²/2 and a discount factor ? = 1/2.

1 You can think of each these periods as corresponding to fifteen years.

(b) (10 points) Utilizing the fact that his marginal utility of consumption mu(c) = 2 - c, show that in period 0 his optimal consumption plan (measured in millions of dollars) is
(c0*, c1*, c2*) = (8/7, 8/7, 8/7).

(c) (5 points) Explain qualitatively how this optimal plan would change if the interest rate was greater than 1. Explain qualitatively how this optimal plan would change if the interest rate was less than 1.

(d) (10 points) If Brian chooses to consume c0* = 8/7 in period 0, explain what his intertemporal budget set B¹((c0*, c1*, c2*)) will be in period 1. Show that the continuation of his original consumption plan (c1*, c2*) = (8/7, 8/7) is indeed the optimal consumption plan for him to choose in period 1 from this budget set. Explain what property of his choice behavior does this reflect.

Now suppose that Brian is a naive (quasi-)hyperbolic discounted utility maximizer characterized by an "instantaneous" utility function u(c) = 2c - c²/2, a long-term discount factor ? = 1/2, and a short-term discount factor (or present bias) ? = 0.5.

(e) (10 points) Show that his optimal consumption plan (c0*, c1*, c2*) from the perspective of his period 0 self is equal to (1.4, 0.8, 0.8). Given he consumes c0* = 1.4 in period 0 (and hence saves 0.6) work out the amounts ?1 and ?2 that he will actually choose to consume in periods 1 and 2.

For the last question, suppose that instead of being able to put money in the bank to earn interest at the rate of 100% per period, Brian can instead purchase a quantity q ? 0 of an annuity at a per-unit price p. That is, each unit of the annuity costs him p million dollars in period 0 and pays out 1 million dollars in period 1 and 1 million dollars in period 2. For example, if he purchases the fraction 0.2 of a unit of the annuity in period 0 at a cost of 0.2 × p then the annuity will pay him 0.2 (of a million dollars) in period 1 and 0.2 in period 2.

$1. (50 points) Brian is about to take early retirement at the age of 45. His current wealth is 2 million and he plans to use this wealth to fund his consumption over the remaining three periods of his life. 1 Any money that he does not spend on consumption in period 0 can be put in the bank where it will earn interest at the rate of 100% per period. That is, r = 1, so if he chooses not to consume an amount s of his $2 million in period 0 he will have s(1 + r) = 2s available at the beginning of period 1. Similarly, any money he does not spend on consumption in period 1 can be put in the bank where it will also earn interest at the rate of r = 1. (a) (5 points) Explain why his intertemporal budget set B can be described as follows: B = (c0, c1, c2) ? (0, 0, 0) : c0 + 1/2 c1 + 1/4 c2 ? 2 Initially suppose that Brian's choices over consumption streams are governed by an exponential discounted utility family of preference relations, characterized by an "instantaneous" utility function u(c) = 2c - c²/2 and a discount factor ? = 1/2. 1 You can think of each these periods as corresponding to fifteen years. (b) (10 points) Utilizing the fact that his marginal utility of consumption mu(c) = 2 - c, show that in period 0 his optimal consumption plan (measured in millions of dollars) is (c0*, c1*, c2*) = (8/7, 8/7, 8/7). (c) (5 points) Explain qualitatively how this optimal plan would change if the interest rate was greater than 1. Explain qualitatively how this optimal plan would change if the interest rate was less than 1. (d) (10 points) If Brian chooses to consume c0* = 8/7 in period 0, explain what his intertemporal budget set B¹((c0*, c1*, c2*)) will be in period 1. Show that the continuation of his original consumption plan (c1*, c2*) = (8/7, 8/7) is indeed the optimal consumption plan for him to choose in period 1 from this budget set. Explain what property of his choice behavior does this reflect. Now suppose that Brian is a naive (quasi-)hyperbolic discounted utility maximizer characterized by an "instantaneous" utility function u(c) = 2c - c²/2, a long-term discount factor ? = 1/2, and a short-term discount factor (or present bias) ? = 0.5. (e) (10 points) Show that his optimal consumption plan (c0*, c1*, c2*) from the perspective of his period 0 self is equal to (1.4, 0.8, 0.8). Given he consumes c0* = 1.4 in period 0 (and hence saves 0.6) work out the amounts ?1 and ?2 that he will actually choose to consume in periods 1 and 2. For the last question, suppose that instead of being able to put money in the bank to earn interest at the rate of 100% per period, Brian can instead purchase a quantity q ? 0 of an annuity at a per-unit price p. That is, each unit of the annuity costs him p million dollars in period 0 and pays out 1 million dollars in period 1 and 1 million dollars in period 2. For example, if he purchases the fraction 0.2 of a unit of the annuity in period 0 at a cost of 0.2 × p then the annuity will pay him 0.2 (of a million dollars) in period 1 and 0.2 in period 2.$

Added by Doreen R.

CASE STUDY - JUMBO SHRIMP Jumbo Shrimp operates ten (10) restaurants (all leased restaurants - no owned buildings) in the coastal area of South Carolina. The restaurants specialize in local seafood and organically grown fruits and vegetables purchased from local farmers. The business has been operating for 20 years. Jumbo has 25 employees at its business office in Beaufort, and +/- 50 employees per restaurant (525 total), mostly in food prep and service. The restaurants operate from 8:00 am to 10:00 pm each night. However, the highest grossing restaurant (by 50%), located on the beach in a trendy area with a high-income demographic, stays open until 2:00 am. Jumbo earned a net profit of $12M ($1M for nine restaurants and $3M for the 2 am location) last year on total revenues of $55M ($5M for each of the nine restaurants and $10M for the 2 am location). Each of the 10 restaurants has $1.5M worth of contents (tenants' improvements, equipment, stock, and furnishings). The president of Jumbo Shrimp compiled a list of situations that could conceivably force the company to require financing over the next twelve months: 1. Climate changes to local waters could reduce the availability of seafood due to migration pattern changes. This has occurred twice in the last 20 years (ten years ago and five years ago), and both times the disruption lasted for eight weeks and affected all ten restaurants - total lost revenue for all ten at $5M each time. 2. A major hurricane in the Beaufort area could close the highest grossing restaurant for six to 12 months. The last major hurricane occurred 20 years ago, but there have been two minor hurricanes (ten years ago and five years ago) in the area since that time, and the highest grossing restaurant was closed for six months each time. Revenue lost was $7.5M ten years ago and $3M five years ago. 3. A restaurant fire would close a location for six to 12 months. This occurred to one of the restaurants (not largest) ten years ago and was out of business for 12 months. Lost revenue $5M. Claim paid by insurer for contents (tenants' improvements, freezers, table, and chairs usual to a restaurant) and revenue (profit) lost. Claim paid $1.5M for contents and $1M for business interruption. 4. The restaurant business is highly competitive, especially on the beach, and the public could lose interest in Jumbo Shrimp, causing revenues to drop. Two new fancy restaurants are opening in the next twelve months - on the beach. Sales are projected to reduce by $2M per year (for the beach restaurant) once those restaurants open. 5. Employees injured on the job lead to lost productivity and workers' compensation claims expenses. Each year, five employees in each of the restaurants go off work for at least ten days each, equaling 500 total lost labor days. Each ten-day loss is $2K paid by WCB. 6. A patron injured on premises or sickened from eating poorly prepared food could sue the company. One restaurant was sued last year when someone had bad seafood due to a failed freezer. He was off work for one month. The insurer paid a $300K claim (legal liability loss for injury). 7. Spoilage and shrinkage consume about 5 percent of the produce ($16,500,000 of produce purchased each year) before it can be sold. Last year their freezer failed and contributed to the insurance claim in 6. Freezer replacement cost $50,000 and $150,000 in food wasted - no boiler and machinery insurance claimed. 8. Employee theft from the cash registers and/or pilferage from the inventory can reduce company profitability. One year ago, an employee stole $200K in cash from the office location. No insurance purchased. 9. Jumbo has a $20M bank loan that it must refinance now or six months from now. Interest rates will increase and result in materially higher loan payments. The current five-year loan rate is 2.50% and will increase to between 5.50% (if refinanced today) and 7.50% (six months from now) at the end of the term of the loan. The five-year rate for both cases is required for Jumbo due to contractual obligations in the shareholder agreement. $500K has been spent on interest payments each year for the last 4.5 years. 10. Jumbo operates ten delivery trucks that pick up and deliver produce and seafood to each of the restaurants on a daily basis. Each delivery truck averages 2,000 miles per week. Each truck hits one of their buildings at least once per year causing damage to only the truck box for $5K ($50K in total damage to owned vehicles). Five of their trucks hit other vehicles each year averaging $5K in physical damage to third-party vehicles ($25K total damage each year). They do not have collision coverage on the trucks but they have $1M legal liability coverage for each truck (PLPD). Annual Costs (you can assume other details based on the information below - list your assumptions) Insurance Premiums - $1.55M total and $75K in self-insured claims and $50K in liability deductibles. $500K premium - Broad Form Property Insurance (contents), $1K deductible for all losses except Hurricane. Hurricane deductible is 5% of loss incurred. $15M total limit of insurance for all 10 restaurants ($1.5M each). Replacement Cost included. $250K premium - Business Interruption Coverage (Profits). $1M limit for each of the smaller restaurants and $3M for the best performing restaurant. $12M total limit. $350K premium - Commercial General Liability Insurance $10M limit, $10K deductible. $50K premium - Umbrella Policy $20M Excess of $10M underlying, Self-Insured Retention $10K - Total Liability limit $30M (CGL and Umbrella). $100K Premium - Fleet Auto Insurance. PLPD insured. No physical damage for vehicles. $300K Premium - Workers Compensation Insurance for all 525 employees. $75K Self-Insurance payments per year for the truck operations ($50K for physical damage and $25K for damage to others) per year. $50K paid in any given year for small slip and fall claims not paid by the insurance company. Uninsured and NOT part of the self-insurance program claims last five years: Theft from office $200K last year. Freezer Failure last year - $50K for freezer and $150K lost produce). Interest rate budget $500K for the current year. Using all the situations above, analyze the risks and develop risk financing strategies for all ten situations to the best benefit of Jumbo Shrimp. Explain why each financing strategy is best for Jumbo Shrimp for each of the ten risks that you will be using for the case study. When explaining why each strategy is best, use some of the tools shown in the textbook to explain your rationale with clear analysis. Also, using the Prouty Method, rate the impact (frequency x severity = impact) of each risk from most important (#1) to the least impactful (#10). Which risks are the most important for Jumbo Shrimp to focus on and why? Develop a Risk Matrix (heat map) as an appendix to your case study. The heat map must be a minimum of 5 rows x 5 columns. Each axis must show the frequency and severity (give a scale to the dollar amount). We suggest completing the full heat map and considering it closely to prioritize risk.

Sri K.

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