Problem 81 - Algebra and Trigonometry

Question

Present Value The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. (a) Find the present value of \(\$ 10,000\) if interest is paid at a rate of 9\(\%\) per year, compounded semiannully, for 3 years. (b) Find the present value of \(\$ 100,000\) if interest is paid at a rate of 8\(\%\) per year, compounded monthly, for 5 years.

Step-by-Step Solution

Answer

(a) $7,049.82; (b) $67,100.57

Step 1: Identify the Present Value Formula

The present value (PV) formula for compound interest is given by the equation: \[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \] where \(FV\) is the future value, \(r\) is the annual interest rate, \(n\) is the number of times the interest is compounded per year, and \(t\) is the time in years.

Step 2: Calculate Present Value for Part (a)

For part (a), we are given \(FV = 10,000\), \(r = 0.09\) (since 9\% is equal to 0.09), \(n = 2\) (compounded semiannually), and \(t = 3\). Substitute these values into the present value formula: \[ PV = \frac{10,000}{(1 + \frac{0.09}{2})^{2 \times 3}} \] Simplify: \(1 + \frac{0.09}{2} = 1.045\), so \[ PV = \frac{10,000}{1.045^6} \] Calculating \(1.045^6\), we get \(1.419\), and then \[ PV = \frac{10,000}{1.419} \approx 7049.82 \]

Step 3: Calculate Present Value for Part (b)

For part (b), we have \(FV = 100,000\), \(r = 0.08\) (since 8\% is 0.08), \(n = 12\) (monthly compound), and \(t = 5\). Substitute these into the formula: \[ PV = \frac{100,000}{(1 + \frac{0.08}{12})^{12 \times 5}} \] Simplify: \(1 + \frac{0.08}{12} = 1.00667\), so \[ PV = \frac{100,000}{1.00667^{60}} \] Find \(1.00667^{60}\), which is approximately \(1.489\), and then \[ PV = \frac{100,000}{1.489} \approx 67100.57 \]

Step 4: Conclusion and Interpretation

The present value tells us how much needs to be invested today to achieve a future sum at a given interest rate compounded at certain intervals over time. For (a) investing approximately \(\\(7049.82\) today at 9\% per annum, compounded semiannually for 3 years, will result in \(\\)10,000\). For (b), investing \(\\(67,100.57\) today at 8\% per annum, compounded monthly for 5 years, will result in \(\\)100,000\).

Key Concepts

Compound Interest

Compound interest is the foundation of many investment calculations. It refers to earning interest not just on your initial investment, but also on the accumulated interest over previous periods. This can lead to significant growth over time, compared to simple interest.

• Every time interest is compounded, it's calculated on the current total balance, which includes both the initial principal and the previously accumulated interest.
• Compound interest can be compounded at different frequencies: annually, semiannually, quarterly, monthly, daily, etc.

The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (initial investment).
• \( r \) is the annual interest rate (decimal).
• \( n \) is the number of times interest is compounded per year.
• \( t \) is the number of years the money is invested for.

This concept helps grow an investment much faster than earning interest solely on the principal.

Future Value

Future value is how much an investment today will be worth at a specific time in the future. It considers the interest you will earn over time under compound interest. Knowing the future value helps investors understand what kind of returns they can expect from their current outlays.
To calculate future value, you use the compound interest formula, reformulated as:
\[ FV = PV \times \left(1 + \frac{r}{n}\right)^{nt} \]
This equation gives a glimpse into the future of your invested money, allowing you to plan your finances more effectively.

• The future value is critical for comparing the potential outcomes of different investment options.
• It helps in setting financial goals and making informed decisions about where and how much to invest.

Interest Rate

The interest rate is a key player in investment calculations. It refers to the percentage at which your money grows over time when invested. The choice of interest rate directly affects how fast your money will grow.

• It's expressed as a percentage, such as 8% per annum.
• Interest can be simple or compound, with compound usually yielding higher returns over time.
• Interest rates vary according to the type of investment and economic conditions.

To convert a percentage to a decimal for use in formulas, simply divide the percentage value by 100. For example, 9% becomes 0.09.
A higher rate is usually favorable for growth in investments, but it generally comes with higher risk. Therefore, it's essential to evaluate the risk-return profile of investments thoroughly.

Investment Calculation

Investment calculation is the process of determining how much you need to invest presently to meet your future financial goals. It involves understanding and applying formulas like the present value of an investment, which is especially vital when dealing with compound interest.

• The present value formula is: \[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]
• Present value reveals how much a future sum of money is worth today, considering a specific interest rate and compounding interval.
• It's instrumental in financial planning, helping individuals and businesses make sound investment decisions.

By mastering the present value calculation, you can better determine the most efficient way to invest your funds to reach your desired financial outcomes efficiently. Understanding these investment principles also aids in comparing and analyzing different investment opportunities.