Problem 83 - Algebra and Trigonometry

Question

Here are two ways of investing \(\$ 30,000\) for 20 years. $$ \begin{array}{ccc} {\text { Lump-Sum Deposit }} & {\text { Rate }} & {\text { Time }} \\ {\$ 30,000} & {5 \% \text { compounded }} & {20 \text { years }} \\ {} & {\text { annually }} \end{array} $$ $$ \begin{array}{ll} {\text { Periodic Deposits }} & {\text { Rate } \quad \text { Time }} \\ {\$ 1500 \text { at the end }} & {5 \% \text { compounded } 20 \text { years }} \\ {\text { of each year }} & {\text { annually }} \end{array} $$ After 20 years, how much more will you have from the lump-sum investment than from the annuity?

Step-by-Step Solution

Answer

The difference between the lump-sum deposit and the annuity, calculated as above, reveals how much more you will have from the lump-sum investment than the annuity after 20 years.

Step 1: Calculate Future Value from the Lump-Sum Investment

To calculate the future value of a lump-sum investment, use the formula \(FV = PV (1 + r/n) ^ {(nt)}\) where \(PV\) is the present value (his initial investment), \(r\) is the annual interest rate, \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested for in years. Plugging in the values, we get \(FV = $30000 (1 + 0.05/1) ^ {(1*20)}\).

Step 2: Calculate Future Value from the Annuity

Next, calculate the future value of the annuity using the formula \(FV = P * [((1 + r/n) ^ nt - 1) / (r/n)]\), where \(P\) is the equal payments in an annuity. The future value of the annuity is then \(FV = $1500 * [((1 + 0.05/1) ^ (1*20) - 1) / (0.05/1)]\).

Step 3: Compare the Future Values of Both Investments

Next, compare these two future values to see which investment yields more after 20 years. The difference between the future values of the lump sum deposit and the annuity should be calculated: difference = FV(lump sum) - FV(annuity).

Key Concepts

Lump-Sum Investment

A lump-sum investment refers to the act of investing a single, substantial amount of money at once, rather than making smaller, periodic contributions. In this exercise, \$30,000\ was invested this way. This approach is appealing because it's simple: you invest once and let your money grow over time. The key factor is the compound interest, which allows the initial principal to grow exponentially instead of merely linearly.

• The initial investment remains unchanged over the investment period.
• Interest accrues on the principal and, in subsequent years, on the accumulated interest as well.

To find out how much your investment grows, you need to calculate the future value. This involves understanding the interest rate, how often it is compounded, and the duration of the investment.
The formula used is: \( FV = PV(1 + r/n)^{nt} \). Here, \(PV\) stands for the present value (the initial investment), \(r\) for the annual interest rate, \(n\) for how many times the interest is compounded per year, and \(t\) for the number of years the investment is held.

Future Value Calculation

Calculating the future value of an investment is crucial to understanding how much your money will be worth after a certain period of growth. Whether it’s a lump-sum investment or an annuity, the future value helps in assessing the outcome of different investment strategies. This calculation considers:

• The initial amount of money invested (or paid in stages).
• The interest rate offered and how it compounds over time.
• The total duration the money is allowed to grow.

For lump-sum investments, the future value formula is \(FV = PV(1 + r/n)^{nt}\). In contrast, annuities use a slightly different approach because they involve regular payments, calculated as \(FV = P \times \left[\frac{(1 + r/n)^{nt} - 1}{r/n}\right]\).
Both methods seek to predict the value of funds in the future, albeit through different schemes of compounding and contributions.

Annuity

An annuity involves making regular payments over the investment period rather than a one-time deposit. In this specific case, an annual payment of \$1,500\ is made at the end of each year for 20 years. This installment strategy is beneficial for those who may not have a large sum to invest initially, enabling contributions in manageable amounts.

• The issuer promises a specified rate of return, compounding periodically.
• It creates a schedule of systematic savings.

The future value for an annuity is calculated using the formula \(FV = P \times \left[\frac{(1 + r/n)^{nt} - 1}{r/n}\right]\), where \(P\) represents the periodic payment.
Annuities are often contrasted with lump-sum investments due to their gradual accumulation and differing payment structures. Evaluating which option is more beneficial or safe often depends on personal preferences and financial circumstances.