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Chapter 2: Problem 11

Suppose you are able to find an investment that pays a monthly interest rate of \(r\) as a decimal. You want to invest \(P\) dollars that will help support your child. If you want your child to be able to withdraw \(M\) dollars per month for \(t\) months, then the amount you must invest is given by $$ P=M \times \frac{1}{r} \times\left(1-\frac{1}{(1+r)^{t}}\right) \text { dollars } $$ A fund such as this is known as an annuity. For the remainder of this problem, we suppose that you have found an investment with a monthly interest rate of \(0.01\) and that you want your child to be able to withdraw \(\$ 200\) from the account each month. a. Find a formula for your initial investment \(P\) as a function of \(t\), the number of monthly withdrawals you want to provide, and make a graph of \(P\) versus \(t\). Be sure your graph shows up through 40 years ( 480 months). b. Use the graph to find out how much you need to invest so that your child can withdraw \(\$ 200\) per month for 4 years. c. How much would you have to invest if you wanted your child to be able to withdraw \(\$ 200\) per month for 10 years? d. A perpetuity is an annuity that allows for withdrawals for an indefinite period. How much money would you need to invest so that your descendants could withdraw \(\$ 200\) per month from the account forever? Be sure to explain how you got your answer.

Short Answer

Expert verified
Invest \(\$8520.72\) for 4 years, \(\$18794.41\) for 10 years, and \(\$20000\) for indefinite withdrawals.

Step by step solution

01

Identify Given Values and Write the Formula

Given interest rate, \(r = 0.01\), monthly withdrawal \(M = 200\), and the formula \(P=M \times \frac{1}{r} \times\left(1-\frac{1}{(1+r)^{t}}\right) \). Let's consider this as \(P(t)\) to represent the initial investment as a function of the number of months, \(t\).
02

Graph the Initial Investment for t Months

Use the formula \(P(t) = 200 \times \frac{1}{0.01} \times\left(1-\frac{1}{(1+0.01)^{t}}\right) \) to compute values for \(t\) ranging from 1 to 480 months. Plot these values on a graph, with \(t\) on the x-axis and \(P(t)\) on the y-axis, to visualize how the initial investment varies over time.
03

Calculate Initial Investment for 4 Years

For \(t = 48\) months, plug the value into the formula: \[ P(48) = 200 \times \frac{1}{0.01} \times\left(1-\frac{1}{(1+0.01)^{48}}\right) \approx 8520.72 \]. Evaluate this to see that you would need to invest approximately \(\$8520.72\) to provide 4 years of monthly withdrawals.
04

Calculate Initial Investment for 10 Years

For \(t = 120\) months, use the formula: \[ P(120) = 200 \times \frac{1}{0.01} \times\left(1-\frac{1}{(1+0.01)^{120}}\right) \approx 18794.41 \]. Evaluate this to find you need to invest approximately \(\$18794.41\) for 10 years of monthly withdrawals.
05

Determine Investment for Perpetuity

A perpetuity requires \(t \to \infty\), so the limiting value is given by the simplified formula: \[ P_{\text{perpetuity}} = \frac{M}{r} = \frac{200}{0.01} = 20000 \]. This means the amount needed is \(\$20000\).

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Key Concepts

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

Investment Formula
When planning an investment that allows for regular withdrawals, it's crucial to understand the investment formula at play. In this context, an annuity is a financial element that lets you support consistent withdrawals over a defined period.
The formula in focus here is:
  • \( P = M \times \frac{1}{r} \times \left(1 - \frac{1}{(1 + r)^{t}}\right) \)
Let's break it down:
  • \( P \) stands for the initial amount you need to invest.
  • \( M \) is the monthly amount withdrawn.
  • \( r \) is the monthly interest rate as a decimal.
  • \( t \) is the total number of months for withdrawals.
By adjusting \( t \), you can determine how your initial investment must change to accommodate different withdrawal durations.
Monthly Withdrawals
Monthly withdrawals represent the steady cash flow your investment generates over time. These withdrawals are crucial for providing financial support predictably and regularly. For this exercise, we aim to let your child withdraw \\(200 monthly.
Monthly withdrawals in an investment involve the following elements:
  • Deciding the fixed amount to withdraw each month, which here is \\)200.
  • Changing the duration of the withdrawals to see how it affects the required initial investment.
  • Ensuring the total withdrawals do not exceed planned investment returns.
This structured approach helps ensure financial sustainability, depending on how long the withdrawals are meant to last.
Perpetuity
A perpetuity differs from a regular annuity by allowing withdrawals to continue indefinitely. This means that the investment is structured so that it can generate endless series of cash flows without exhausting the principal.
The formula for a perpetuity simplifies because \( t \to \infty \):
  • \( P_{\text{perpetuity}} = \frac{M}{r} \)
For our example with a monthly withdrawal of \\(200 and an interest rate of 0.01 (or 1%), the needed investment is simple:
  • \( P_{\text{perpetuity}} = \frac{200}{0.01} = 20000 \)
Thus, \\)20,000 should be invested to ensure your child鈥攐r even their descendants鈥攃an withdraw \$200 forever.
Initial Investment Calculation
Calculating the initial investment is crucial for setting up a reliable annuity. In this context, the amount invested initially determines the feasibility and sustainability of future withdrawals.
To calculate this initial investment:
  • Input all known values into the formula \( P = M \times \frac{1}{r} \times \left(1 - \frac{1}{(1 + r)^{t}}\right) \)
  • Solve for \( P \) with given values (like interest rate and monthly withdrawal defined in the problem).
  • Use the formula iteratively for different durations (\( t \)). For example:
    • To sustain 4 years, you calculate \( P(48) \), which results in approximately \\(8520.72.
    • For 10 years, it amounts to approximately \\)18794.41.
Understanding this calculation ensures that your investment plan meets desired future withdrawals.

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Most popular questions from this chapter

We most often hear of the power of earthquakes given in terms of the Richter scale, but this tells only the power of the earthquake at its epicenter. Of more immediate importance is how an earthquake affects the location where we are. Seismologists measure this in terms of ground movement, and for technical reasons they find the acceleration of ground movement most useful. For the purpose of this problem, a major earthquake is one that produces a ground acceleration of at least \(5 \%\) of \(g\), where \(g\) is the acceleration 4 due to gravity near the surface of the Earth. In California, the probability \(p(n)\) of one's home being affected by exactly \(n\) major earthquakes over a 10 -year period is given approximately \({ }^{5}\) by $$ p(n)=0.379 \times \frac{0.9695^{n}}{n !} $$ See Exercise 7 for an explanation of \(n !\). a. What is the probability of a California home being affected by exactly 3 major earthquakes over a 10 -year period? b. What is the limiting value of \(p(n)\) ? Explain in practical terms what this means. c. What is the probability of a California home being affected by no major earthquakes over a 10 year period? d. What is the probability of a California home being affected by at least one major earthquake over a 10-year period? Hint: It is a certainty that an event either will or will not occur, and the probability assigned to a certainty is 1. Expressed in a formula, this is Probability of an event occurring \+ Probability of an event not occurring \(=1\). This, in conjunction with part c, may be helpful for part d.

Between the ages of 7 and 11 years, the weight \(w\), in pounds, of a certain girl is given by the formula $$ w=8 t $$ Here \(t\) represents her age in years. a. Use a formula to express the age \(t\) of the girl as a function of her weight \(w\). b. At what age does she attain a weight of 68 pounds? c. The height \(h\), in inches, of this girl during the same period is given by the formula $$ h=1.8 t+40 . $$ i. Use your answer to part b to determine how tall she is when she weighs 68 pounds. ii. Use a formula to express the height \(h\) of the girl as a function of her weight \(w\). iii. Answer the question in part \(i\) again, this time using your answer to part ii. c. The height \(h\), in inches, of this girl during the same period is given by the formula $$ h=1.8 t+40 . $$ i. Use your answer to part b to determine how tall she is when she weighs 68 pounds. ii. Use a formula to express the height \(h\) of the girl as a function of her weight \(w\). iii. Answer the question in part \(i\) again, this time using your answer to part ii.

The circulation \(C\) of a certain magazine as a function of time \(t\) is given by the formula $$ C=\frac{5.2}{0.1+0.3^{t}} $$ Here \(C\) is measured in thousands, and \(t\) is measured in years since the beginning of 1992 , when the magazine was started. a. Make a graph of \(C\) versus \(t\) covering the first 6 years of the magazine's existence. b. Express using functional notation the circulation of the magazine 18 months after it was started, and then find that value. c. Over what time interval is the graph of \(C\) concave up? Explain your answer in practical terms. d. At what time was the circulation increasing the fastest? e. Determine the limiting value for \(C\). Explain your answer in practical terms.

In this exercise we develop a model for the growth rate \(G\), in thousands of dollars per year, in sales of a product as a function of the sales level \(s\), in thousands of dollars. \({ }^{30}\) The model assumes that there is a limit to the total amount of sales that can be attained. In this situation we use the term unattained sales for the difference between this limit and the current sales level. For example, if we expect sales to grow to 3 thousand dollars in the long run, then \(3-s\) gives the unattained sales. The model states that the growth rate \(G\) is proportional to the product of the sales level \(s\) and the unattained sales. Assume that the constant of proportionality is \(0.3\) and that the sales grow to 2 thousand dollars in the long run. a. Find a formula for unattained sales. b. Write an equation that shows the proportionality relation for \(G\). c. On the basis of the equation from part b, make a graph of \(G\) as a function of \(s\). d. At what sales level is the growth rate as large as possible? e. What is the largest possible growth rate?

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