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Chapter 4: Problem 8

How fast do exponential functions grow? At age 25 you start to work for a company and are offered two rather fanciful retirement options. Retirement option 1: When you retire, you will be paid a lump sum of \(\$ 25,000\) for each year of service. Retirement option 2: When you start to work, the company will deposit \(\$ 10,000\) into an account that pays a monthly interest rate of \(1 \%\). When you retire, the account will be closed and the balance given to you. Which retirement option is more favorable to you if you retire at age 65? What if you retire at age 55?

Short Answer

Expert verified
Option 1 is more favorable for both ages, as it provides more funds in both cases.

Step by step solution

01

Understanding Retirement Option 1

Retirement option 1 gives you a lump sum of $25,000 for each year you work for the company. If you retire at age 65, you will have worked for 40 years (from age 25 to 65). The lump sum would then be \( 25,000 \times 40 = 1,000,000 \) dollars. If you retire at age 55, you will have worked for 30 years, so you would receive \( 25,000 \times 30 = 750,000 \) dollars.
02

Calculating Compound Interest for Retirement Option 2

Retirement option 2 involves compound interest. The account starts with \(10,000, and we compound monthly at a rate of 1%. Using the formula for compound interest, \( A = P(1 + \/frac{r}{n})^{nt} \), where \( P = \\)10,000 \), \( r = 0.01 \), \( n = 12 \) (compounded monthly), and \( t \) is the total number of years. For retirement at 65, \( t = 40 \); for 55, \( t = 30 \).
03

Calculating Amount for Age 65 in Option 2

Substitute the values for retirement at age 65 into the compound interest formula: \[ A = 10,000(1 + \/frac{0.01}{12})^{12 \times 40} \]. Compute it as follows: \( A \approx 10,000 \times (1.0008333)^{480} \). The calculation results in approximately \( A \approx 452,593 \) dollars.
04

Calculating Amount for Age 55 in Option 2

Now substitute for retirement at age 55: \[ A = 10,000(1 + \/frac{0.01}{12})^{12 \times 30} \]. Simplifying gives \( A \approx 10,000 \times (1.0008333)^{360} \), which results in approximately \( A \approx 116,095 \) dollars.
05

Comparing the Two Options at Both Retirement Ages

For retirement at age 65, Option 1 provides \( \\(1,000,000 \), and Option 2 provides approximately \( \\)452,593 \). For retirement at age 55, Option 1 provides \( \\(750,000 \), and Option 2 provides approximately \( \\)116,095 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compound Interest
Compound interest is a powerful financial concept where the interest earned on an investment is reinvested to earn even more interest. This is often said to be "interest on interest," and it accelerates the growth of an investment over time.

Consider a simple formula for compound interest: - **A** - the amount of money accumulated after n years, including interest.- **P** - the principal amount (the initial sum of money).- **r** - the annual interest rate (as a decimal).- **n** - the number of times that interest is compounded per year.- **t** - the time the money is invested for, in years.
The formula is given by:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] When interest is compounded monthly, as is the case in retirement Option 2, it causes the investment to grow much faster than with simple interest. This is because the principal amount grows each month, allowing the interest to be calculated on an increasingly larger amount.
Lump Sum
A lump sum refers to a single payment of money, as opposed to smaller payments or installments over time. In the context of retirement planning, a lump sum payment provides you with a large amount of money upfront when you retire.

In Retirement Option 1, you receive a lump sum of $25,000 for each year of service. If you retire at age 65, having worked 40 years, the lump sum totals $1,000,000. For retirement at age 55 after 30 years, the amount is $750,000.

This option provides certainty and immediate access to a large sum of money, which can be beneficial for planning large purchases, paying off debts, or investing further. However, it also means you need to wisely manage the funds to ensure they last throughout your retirement.
Retirement Planning
Retirement planning is the process of determining retirement income goals and the actions and decisions necessary to achieve them. It involves recognizing how much money you will need in retirement and how to accumulate it.

When contemplating the two retirement options provided by your company, it's crucial to consider your long-term financial goals, your expected expenses in retirement, and the amount you have already saved.

Certain factors to consider include: - **Inflation**: Over time, the purchasing power of money decreases. - **Longevity Risk**: Ensuring your savings last throughout your retirement years. - **Income Sources**: Other than the retirement funds, consider pensions, social security, etc.

The benefit of comprehensive retirement planning is having confidence in your financial stability during your retirement years, allowing you to enjoy life without stress about finances.
Financial Modeling
Financial modeling is the task of building a mathematical representation of a financial situation. It helps in evaluating the monetary outcomes of financial decisions.

In retirement planning, financial models can be used to forecast the growth of your investments and to compare different retirement options. In our exercise, financial modeling is used to compare the growth of a lump sum payment versus an account with compound interest.

Key aspects include: - **Assumptions**: Estimations of interest rates, life expectancy, and retirement age. - **Projection**: Visualizing how your investments grow over time and how much they will be worth at retirement. - **Comparison**: Evaluating which option leads to a more favorable financial outcome.

Ultimately, the purpose of financial modeling is to provide a clear picture of your financial future and guide better decision-making. It allows for adjusting variables, like retirement age or investment amount, to see potential impacts on your future wealth.

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Most popular questions from this chapter

Exponential growth with given initial value and growth factor: Write the formula for an exponential function with initial value 23 and growth factor 1.4. Plot its graph.

Decibels: Sound exerts a pressure \(P\) on the human ear. This pressure increases as the loudness of the sound increases. It is convenient to measure the loudness \(D\) in decibels and the pressure \(P\) in dynes per square centimeter. It has been found that each increase of 1 decibel in loudness causes a \(12.2 \%\) increase in pressure. Furthermore, a sound of loudness 97 decibels produces a pressure of 15 dynes per square centimeter. a. Explain why \(P\) is an exponential function of \(D\) and find the growth factor. b. Find \(P(0)\) and explain in practical terms what your answer means. c. Find an exponential model for \(P\) as a function of \(D\). d. When pressure on the ear reaches a level of about 200 dynes per square centimeter, physical damage can occur. What decibel level should be considered dangerous?

Sound pressure: Sound exerts pressure on the human ear. Increasing loudness corresponds to greater pressure. The table below shows the pressure P , in dynes per square centimeter, exerted on the ear by sound with loudness D, measured in decibels. $$ \begin{array}{|c|c|} \hline \text { Loudness } D & \text { Pressure } P \\ \hline 65 & 0.36 \\ \hline 85 & 3.6 \\ \hline 90 & 6.4 \\ \hline 105 & 30 \\ \hline 110 & 50 \\ \hline \end{array} $$ a. Plot the natural logarithm of the data. Does it appear reasonable to model pressure as an exponential function of loudness? b. Find an exponential model of P as a function of D using the logarithm as a link. c. How is pressure on the ear affected when loudness is increased by 1 decibel?

Aphid growth on broccoli: This problem and the following one are based on the results of a study by R. Root and A. Olson of population growth for aphids on various host plants.44 The value of r = 0.243 per day is valid for aphids reared on broccoli plants outdoors. The table on the next page describes the growth of an aphid population reared on broccoli plants indoors (in a controlled environment). Time t is measured in days. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Time } t & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Population } & 25 & 30 & 37 & 44 & 54 \\ \hline \end{array} $$ Calculate r for this table. Did the aphids fare better indoors or outdoors?

National health care spending: The following table shows national health care costs, measured in billions of dollars. $$ \begin{array}{|c|c|} \hline \text { Date } & \text { Costs } \\ \text { in billions } \\ \hline 1960 & 27.6 \\ \hline 1970 & 75.1 \\ \hline 1980 & 254.9 \\ \hline 1990 & 717.3 \\ \hline 2000 & 1358.5 \\ \hline \end{array} $$ a. Plot the natural logarithm of the data. Does it appear that the data on health care spending can be appropriately modeled by an exponential function? b. Find the equation of the regression line for the logarithm of the data and add its graph to the plot in part a. c. By what percent per year were national health care costs increasing during the period from 1960 through 2000?
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