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Chapter 3: Problem 8

A bank offers a return of \(7 \%\) interest compounded annually. Find the future value of a principal of \(\$ 4500\) after 6 years. What is the overall percentage rise over this period?

Short Answer

Expert verified
The future value is approximately \$6753.15. The overall percentage rise is about 50.07%.

Step by step solution

01

- Understand the formula for compound interest

The formula for compound interest is given by \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \(A\) is the future value of the investment/loan, including interest,- \(P\) is the principal investment amount (\$4500\ in this case),- \(r\) is the annual interest rate (decimal) (0.07 here),- \(n\) is the number of times that interest is compounded per year (since it is compounded annually, \(n = 1\)),- \(t\) is the time the money is invested for in years (6 years here).
02

- Plug in the given values

Substitute the values into the compound interest formula: \[ A = 4500 \left(1 + \frac{0.07}{1}\right)^{1\times6} \]
03

- Simplify the equation

Simplify inside the parentheses first:\[ 1 + \frac{0.07}{1} = 1.07 \]Then raise this to the 6th power:\[ A = 4500 \times 1.07^6 \]
04

- Calculate the power

\[ 1.07^6 \approx 1.5007 \].
05

- Calculate the future value

Now multiply by the principal amount:\[ A = 4500 \times 1.5007 \approx 6753.15 \]. The future value after 6 years is approximately \$6753.15.
06

- Find overall percentage rise

To find the overall percentage rise:\[ \text{Percentage Rise} = \left(\frac{\text{Future Value} - \text{Principal}}{\text{Principal}}\right) \times 100 \]Substitute the values:\[ \text{Percentage Rise} = \left(\frac{6753.15 - 4500}{4500}\right) \times 100 \approx 50.07\%. \]

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Key Concepts

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

Future Value
The future value (FV) of an investment is the amount of money you will have after a certain period, considering the initial amount invested and any interest earned. In our exercise, the future value after 6 years is calculated using the compound interest formula. For an investment of \(4500 with a 7% annual interest rate, compounded annually, the FV is approximately \)6753.15. Calculating the future value helps in determining how much your money will grow over time, aiding in smart financial planning.
Principal Amount
The principal amount is the initial sum of money invested or loaned. In this exercise, the principal amount is $4500. When calculating compound interest, the principal amount plays a crucial role as it is the base amount from which interest is calculated. Ensuring you understand the principal amount helps in better interpreting the growth of an investment over time due to interest accumulation.
Annual Interest Rate
The annual interest rate is the percentage of the principal that is paid as interest each year. In this problem, the annual interest rate is 7%. This percentage needs to be converted to a decimal (0.07 in this case) when inserted into the compound interest formula. The interest rate directly affects how much the investment will grow annually. Higher interest rates generally lead to more significant growth, providing a larger future value.
Percentage Rise
The percentage rise is an important indicator of how much an investment has increased over a given period. It is calculated by subtracting the principal amount from the future value, dividing by the principal amount, and then multiplying by 100. In our example, after 6 years, the percentage rise is approximately 50.07%. This percentage gives a clear picture of the growth achieved over the investment period, providing a straightforward way to compare investment opportunities.

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Most popular questions from this chapter

(Excel) A civil engineering company needs to buy a new excavator. Model \(A\) is expected to make a loss of \(\$ 60000\) at the end of the first year, but is expected to produce revenues of \(\$ 24000\) and \(\$ 72000\) for the second and third years of operation. The corresponding figures for model \(\mathrm{B}\) are \(\$ 96000, \$ 12000\) and \(\$ 120000\), respectively. Use a spreadsheet to tabulate the revenue flows (using negative numbers for the losses in the first year), together with the corresponding present values based on a discount rate of \(8 \%\) compounded annually. Find the net present value for each model. Which excavator, if any, would you recommend buying? What difference does it make if the discount rate is \(8 \%\) compounded continuously?

If a principal, \(P\), is invested at \(r \%\) interest compounded annually then its future value, \(S\), after \(n\) years is given by $$ S=P\left(1+\frac{r}{100}\right)^{n} $$ (a) Use this formula to show that if an interest rate of \(r \%\) is compounded \(k\) times a year then after \(t\) years $$ S=P\left(1+\frac{r}{100 k}\right)^{k t} $$ (b) Show that if \(m=100 k / r\) then the formula in part (1) can be written as $$ S=P\left(\left(1+\frac{1}{m}\right)^{m}\right)^{r t / 100} $$ (c) Use the definition $$ \mathrm{e}=\lim _{m \rightarrow \infty}\left(1+\frac{1}{m}\right)^{m} $$ to deduce that if the interest is compounded with ever-increasing frequency (that is, continuously) then $$ S=P \mathrm{e}^{r / 100} $$

A government bond that originally cost \(\$ 500\) with a yield of \(6 \%\) has 5 years left before redemption. Determine its present value if the prevailing rate of interest is \(15 \%\).

A shop sells books at ' \(20 \%\) below the recommended retail price (r.r.p.)'. If it sells a book for \(拢 12.40\) find (a) the r.r.p. (b) the cost of the book after a further reduction of \(15 \%\) in a sale (c) the overall percentage discount obtained by buying the book from the shop in the sale compared with the manufacturer's r.r.p.

Find the present value of \(\$ 100000\) in 10 years' time if the discount rate is \(6 \%\) compounded (a) annually (b) continuously
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