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Chapter 7: Problem 29

Your client is 40 years old and wants to begin saving for retirement. You advise the client to put \(\$ 5,000\) a year into the stock market. You estimate that the market's return will be, on average, 12 percent a year. Assume the investment will be made at the end of the year. a. If the client follows your advice, how much money will she have by age \(65 ?\) b. How much will she have by age 70 ?

Short Answer

Expert verified
By age 65, the client will have approximately $1,149,200. By age 70, she will have approximately $2,247,100.

Step by step solution

01

Determine Number of Investment Years

First, we need to determine how many years the client will be investing. Since the client is 40 years old, we calculate the period for both retirement age 65 and 70. For age 65, the investment period is \(65 - 40 = 25\) years. For age 70, the investment period is \(70 - 40 = 30\) years.
02

Identify the Future Value of an Annuity Formula

The client will be investing a fixed amount every year, which means we use the Future Value of an Annuity formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where \(FV\) is the future value, \(P\) is the annual payment, \(r\) is the annual interest rate, and \(n\) is the number of years.
03

Calculate Future Value at Age 65

Substitute the values into the formula for age 65. \(P = 5000\), \(r = 0.12\), \(n = 25\). Hence,\[ FV = 5000 \times \frac{(1 + 0.12)^{25} - 1}{0.12} \]Calculate \((1+0.12)^{25} - 1\) and then divide by 0.12, and finally multiply by 5000 to find \(FV\).
04

Calculate Future Value at Age 70

Repeat the calculations for age 70, substituting \(n = 30\) into the formula.\[ FV = 5000 \times \frac{(1+0.12)^{30} - 1}{0.12} \]Evaluate \((1+0.12)^{30} - 1\), divide by 0.12, and multiply by 5000 to determine \(FV\).
05

Perform Calculations

For age 65: \((1 + 0.12)^{25} - 1 \approx 27.58\); divide by 0.12 to get 229.84, then multiply by 5000 to get \(FV \approx 1149200\).For age 70: \((1 + 0.12)^{30} - 1 \approx 53.93\); divide by 0.12 to get 449.42, then multiply by 5000 to get \(FV \approx 2247100\).

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Key Concepts

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

Future Value of Annuity
When planning for retirement savings, understanding the concept of the future value of an annuity is crucial. An annuity is a series of equal payments made at regular intervals. In this scenario, your client is planning to invest a fixed sum annually, making it an ordinary annuity. The future value of an annuity tells us how much these regular investments will be worth at some point in the future, considering a constant rate of return over time.

The formula used to calculate this involves several key aspects:
  • Annual Payment (P): This is the fixed amount of money invested every year. For the client, this is $5,000.
  • Interest Rate (r): The expected rate of return, in this case, 12% or 0.12 as a decimal.
  • Number of Periods (n): The total number of years the investment will be made. Here, n could be 25 years (age 65) or 30 years (age 70).
The future value is calculated with the formula:\[ FV = P \times \frac{{(1 + r)^n - 1}}{r} \]By substituting the appropriate values, we can predict how much the client's investment will grow by their desired retirement age.
Investment Strategy
Investing for retirement involves crafting a strategy that aligns with one's financial goals and risk tolerance. Here, the suggestion is to invest in the stock market, which historically offers substantial returns. However, understanding the nuances of this strategy can make a significant difference.

The key elements of a robust investment strategy include:
  • Diversification: Spreading investments across different assets can mitigate risk.
  • Time Horizon: Longer investment periods allow for compounding, enhancing the growth potential of investments.
  • Risk Assessment: Understanding and balancing risk levels to match the investor's comfort can prevent unnecessary losses.
In this exercise, our client is advised to invest consistently over a long period (25-30 years) in the stock market. This long-term strategy is advantageous because it capitalizes on the power of compound interest and the general upward trend of the market over decades.
Stock Market Returns
Stock market returns refer to the profit generated from investing in stocks, usually expressed as a percentage of the investment. They are inherently volatile but can offer higher returns compared to other investment vehicles like bonds or savings accounts.

Key factors influencing stock market returns include:
  • Market Trends: General economic conditions, political stability, and global events can affect stock prices.
  • Company Performance: Individual company success can directly impact its stock value.
  • Inflation: While inflation erodes the dollar's purchasing power, it often leads to higher nominal returns.
In this problem, a 12% average annual return is assumed. This is based on historical stock market performance, though it's essential to recognize that past performance doesn't guarantee future results. Investors should regularly review and adjust strategies to align with their personal risk levels and the changing market environment.

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Most popular questions from this chapter

What is the future value of a 5 -year annuity due that promises to pay you \(\$ 300\) each year? Assume that all payments are reinvested at 7 percent a year, until Year 5

You are serving on a jury. A plaintiff is suing the city for injuries sustained after falling down an uncovered manhole. In the trial, doctors testified that it will be 5 years before the plaintiff is able to return to work. The jury has already decided in favor of the plaintiff, and has decided to grant the plaintiff an award to cover the following items: (1) Recovery of 2 years of back-pay \((\$ 34,000 \text { in } 2000, \text { and } \$ 36,000 \text { in } 2001) .\) Assume that it is December 31,2001 , and that all salary is received at year end. This recovery should include the time value of money. (2) The present value of 5 years of future salary \((2002-2006)\). Assume that the plaintiffs salary would increase at a rate of 3 percent a year. (3) \(\$ 100,000\) for pain and suffering. (4) \(\$ 20,000\) for court costs. Assume an interest rate of 7 percent. What should be the size of the settlement?

Your parents are planning to retire in 18 years. They currently have \(\$ 250,000\), and they would like to have \(\$ 1,000,000\) when they retire. What annual rate of interest would they have to earn on their \(\$ 250,000\) in order to reach their goal, assuming they save no more money?

You need to accumulate \(\$ 10,000\). Io do so, you plan to make deposits of \(\$ 1,250\) per year, with the first payment being made a year from today, in a bank account that pays 12 percent interest, compounded annually. Your last deposit will be less than \(\$ 1,250\) if less is needed to round out to \(\$ 10,000 .\) How many years will it take you to reach your \(\$ 10,000\) goal, and how large will the last deposit be?

To the closest year, how long will it take \(\$ 200\) to double if it is deposited and earns the following rates? [Notes: (1) See the hint for Problem 7-34. (2) This problem cannot be solved exactly with some financial calculators. For example, if you enter \(\mathrm{PV}=-200, \mathrm{FV}=\) \(400,\) and \(I=7\) in an \(\mathrm{HP}-12 \mathrm{C}\), and then press the \(\mathrm{N}\) key, you will get 11 years for part a. The correct answer is 10.2448 years, which rounds to \(10,\) but the calculator rounds up. However, the HP-10B and HP-17B give the correct answer. You should look up \(\mathrm{FVIF}=\$ 400 / \$ 200=2\) in the tables for parts \(a, b,\) and \(c,\) but figure out part d.] Assume that compounding occurs once a year. a. 7 percent. b. 10 percent. c. 18 percent. d. 100 percent.
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