Assignment 14: The LC Model with Survival Probabilities
财务作业代写 We know that individual longevity is highly uncertain, and we can model survival probability using the Gompertz Equation.
Learning Objectives 财务作业代写
- Incorporate survival probabilities into the LC model of saving and consumption.
- Incorporating a personal preference factor into the equation for
- Solving a system of equations using constraints.
Background: The LC Model and Survival Probabilities
The LC model provides a framework for planning one’s consumption and savings over time.In each period, savings is the amount of income not consumed: 𝑆𝑛 = 𝑊𝑛 − 𝐶𝑛, and assets are the accumulation of savings from one period to the next: 𝐴𝑛 = 𝐴𝑛−1 + 𝑆𝑛. By incorporating asset income of 𝐴𝐼𝑛 = 𝑟 × 𝐴𝑛−1 , the annual budget constraint becomes 𝑆𝑛 = 𝑊𝑛 + 𝐴𝐼𝑛 − 𝐶𝑛. Initially, we assumed living to age 100 without longevity risk, and found the highest level of smooth consumption.
We know that individual longevity is highly uncertain, and we can model survival probability using the Gompertz Equation. We can adjust the level of consumption each year to take into account the probability of survival. 财务作业代写
- Plan to consume more when we are more likely to be alive to enjoy the consumption; in return, we accept consuming less each year as we age.
- Social security (or annuity income) will provide a minimum level of consumption even after assets are depleted.
- How much of a decrease in consumption we are willing to accept is a personal decision based on one’s own risk aversion.
We can determine how much to change consumption from one year to the next using the equation:
where:
- 𝐶𝑥 is the level of consumption at age 𝑥, 𝐶𝑥+1 is the level of consumption at age 𝑥 + 1
- 𝑟 is the market’s real interest rate and 𝜌 is one’s personal discount rate. For simplicity, let’s assume that 𝑟 = 𝜌, so they cancel each other out.
- 𝑝(𝑠𝑢𝑟𝑣𝑖𝑣𝑎𝑙𝑥,1 ) is the one-year survival probability from age 𝑥, as given by the Gompertz Equation.
- 𝛾 is a personal measure of longevity risk aversion (i.e., low 𝛾 indicates willingness to reduce consumption if still alive at an old-old age, high 𝛾 indicates unwillingness to reduce consumption at old-old age).
Demonstration and Discussion 财务作业代写
We will begin with a demonstration and we will discuss how these elements work together.
Begin with the spreadsheet LC-model-w-risk-aversion.xlsx.
Case Study
Consider Bob, a 66-year-old who just retired. Bob will receive a Social Security benefit income of $25,000 each year and has $500,000 of accumulated financial assets. Assume he owns his own house where he will live rent-free for the rest of his life. Bob knows that he is very likely to be alive at age 66 or 67, and not very likely to be alive at age 96 or 97.How much should Bob consume each year for the rest of his life? It depends on his social security income, the real interest rate, and his longevity risk aversion.
Tasks 财务作业代写
To help guide Bob’s decision-making, you will run the following 4 scenarios:
- Smooth consumption each year
- Low longevity risk aversion, 𝛾 = 1
- Moderate longevity risk aversion, 𝛾 = 3
- High longevity risk aversion, 𝛾 = 8
For each scenario: 财务作业代写
- copy your entire worksheet to a new sheet (tab) in your workbook.
- change the risk aversion factor
- use Goal Seek (or guess and check) to find the level of consumption in the first year.
After running all 4 scenarios, make a line graph comparing the level of consumption in each scenario. Hint: it will be easiest to create a new sheet onto which you summarize the amount of consumption for each of your 4 scenarios, and then make the graph on that sheet.
What to Submit
Submit your Excel spreadsheets with all calculations.
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