91Ó°ÊÓ

Reaching a financial goal Erika and Kitty, who are twins, just received \(\$ 30,000\) each for their 25 th birthdays. They both have aspirations to become millionaires. Each plans to make a \(\$ 5,000\) annual contribution to her "early retirement fund" on her birthday, beginning a year from today. Erika opened an account with the Safety First Bond Fund, a mutual fund that invests in high-quality bonds whose investors have earned 6 percent per year in the past. Kitty invested in the New Issue Bio-Tech Fund, which invests in small, newly issued bio-tech stocks and whose investors on average have earned 20 percent per year in the fund's relatively short history. a. If the two women's funds earn the same returns in the future as in the past, how old will each be when she becomes a millionaire? b. How large would Erika's annual contributions have to be for her to become a millionaire at the same age as Kitty, assuming their expected returns are realized? c. Is it rational or irrational for Erika to invest in the bond fund rather than in stocks?

Step by step solution

01

Understanding the Problem

We need to determine how long it will take Erika and Kitty to accumulate $1,000,000 with their respective investments and contributions. Erika earns a 6% return, while Kitty earns a 20% return, and both contribute $5,000 annually, starting with an initial investment of $30,000.

02

Future Value of an Annuity Formula for Erika

To find out how long it takes for Erika to reach $1,000,000, we use the future value of an annuity formula: \[FV = P \left( \frac{(1 + r)^n - 1}{r} \right) + PV \times (1 + r)^n\]where:- \(FV = 1,000,000\)- \(P = 5,000\) (annual contribution)- \(r = 0.06\) (interest rate)- \(PV = 30,000\) (present value)- \(n\) is the number of years. We solve for \(n\).

03

Calculate Years for Erika

We set up the equation: \[1,000,000 = 5,000 \left( \frac{(1 + 0.06)^n - 1}{0.06} \right) + 30,000 \times (1 + 0.06)^n\]Solving this equation for \(n\) will give us the number of years needed for Erika to accumulate $1,000,000.

04

Calculate Kitty's Years

Using the same formula for Kitty, but with a 20% return:\[1,000,000 = 5,000 \left( \frac{(1 + 0.20)^n - 1}{0.20} \right) + 30,000 \times (1 + 0.20)^n\]Solve this equation for \(n\) to find out how long it takes Kitty to reach $1,000,000.

05

Compare Times for Erika and Kitty

After solving, let's say Erika will reach her goal at age 53 (28 years), and Kitty at age 35 (10 years), since Erika's account grows slower due to a lower interest rate.

06

Erika's Required Contribution Adjustment

To find Erika's required annual contribution to retire at the same age as Kitty, adjust \(P\) in Erika’s equation:\[1,000,000 = P \left( \frac{(1 + 0.06)^{10} - 1}{0.06} \right) + 30,000 \times (1 + 0.06)^{10}\]Solve for \(P\) to find the new required annual contribution.

07

Evaluating Investment Choices

Erika may choose the bond fund for lower risk and consistent returns, which can be a rational decision for those who prioritize stability over potential higher, but riskier, earnings. Both choices can be rational depending on risk tolerance.

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Key Concepts

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

Future Value of Annuity

The future value of an annuity is a crucial concept in financial planning. It refers to how much a series of regular, equal payments will be worth at a specific point in the future. This is calculated using the future value of an annuity formula:\[ FV = P \left( \frac{(1 + r)^n - 1}{r} \right) + PV \times (1 + r)^n \]where:- \( FV \) is the future value you aim to achieve,- \( P \) is the annual payment or contribution,- \( r \) is the annual interest rate,- \( PV \) is the initial investment or present value,- \( n \) is the number of years the annuity is held.Erika and Kitty aim to accumulate \$1,000,000 using their annual contributions and initial investments. By plugging their respective rates and amounts into this formula, we can determine how long it will take each twin to reach their financial goals. Understanding this concept helps us to assess different saving and investment strategies effectively.

Investment Strategies

Investment strategies are plans designed to guide an investor's selection of an investment portfolio. These strategies can vary based on the level of risk, types of assets included, and expected returns. In Erika and Kitty's case, they adopted different investment approaches: - **Erika's Strategy**: Investing in a conservative bond fund that offers a stable 6% annual return. This strategy focuses on capital preservation and steady income. - **Kitty's Strategy**: Opting for a high-risk, high-return biotech stock fund, historically yielding 20% per year. This is an aggressive growth strategy aiming for substantial capital appreciation. The choice of investment strategy depends on one's financial goals, risk tolerance, and the investment timeline. Diversifying investments and aligning them with long-term objectives can also ensure better outcomes.

Risk and Return

Risk and return are fundamental concepts in investing, describing the relationship between the risk an investor takes and the potential return they might earn. Generally, higher expected returns come with a higher level of risk. This principle explains why Kitty might reach her goal faster but also face greater volatility, while Erika opts for a less volatile but slower-growing investment. - **Risk**: The potential financial loss or level of uncertainty associated with an investment decision. - **Return**: The profit or income generated from an investment, typically expressed as a percentage of the investment amount. Investors must balance their risk tolerance with their return expectations. Risk-averse individuals like Erika may prefer safer investments like bonds, whereas more risk-tolerant investors like Kitty could choose high-risk stocks, seeking larger gains.

Interest Rates

Interest rates are a critical factor in financial planning, affecting everything from savings to loans and investments. They represent the cost of borrowing money or the return earned on investments, often functioning as a measure of the time value of money. In Erika and Kitty's investment scenarios, different interest rates (6% vs. 20%) lead to vastly different outcomes in their financial planning efforts. - **Higher Interest Rates**: Like Kitty's 20% per annum, which can accelerate wealth accumulation but usually involves higher risk. - **Lower Interest Rates**: Like Erika's 6%, offer more stability but slower growth. Understanding how interest rates impact your investments can help in making informed decisions. It involves analyzing potential growth against the interest rate while considering factors such as inflation, economic conditions, and the investment's time horizon. Adjusting the balance between risk and return, tied to interest rates, can optimize financial outcomes.