Problem 57 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In the last 5 yr, Bendix Mutual Fund grew at the rate of \(10.4 \% /\) year compounded quarterly. Over the same period, Acme Mutual Fund grew at the rate of \(10.6 \% /\) year compounded semiannually. Which mutual fund has a better rate of return?

Step-by-Step Solution

Answer

The effective annual rates (EAR) for Bendix and Acme Mutual Funds are calculated as \(EAR_{Bendix} \approx 10.862\%\) and \(EAR_{Acme} \approx 10.876\%\), respectively. Since the EAR for Acme Mutual Fund is higher, it has a better rate of return.

Step 1: Convert the interest rates to decimals

To work the calculation easily, we should first convert the percentage rates to their decimal forms. - Bendix: \(10.4\% = 0.104\) - Acme: \(10.6\% = 0.106\)

Step 2: Calculate the EAR for Bendix Mutual Fund

The formula for EAR is given by: \(EAR = (1 + i/n)^{n} - 1\) Substitute i = 0.104 and n = 4(compounded quarterly) in the formula: \(EAR_{Bendix} = (1 + 0.104/4)^{4} - 1\) Calculate the EAR for Bendix \(EAR_{Bendix} = (1 + 0.026)^{4} - 1\) \(EAR_{Bendix} = (1.026)^{4} - 1\) \(EAR_{Bendix} = 1.1086265664 - 1\) \(EAR_{Bendix} = 0.1086265664\)

Step 3: Calculate the EAR for Acme Mutual Fund

Substitute i = 0.106 and n = 2(compounded semiannually) in the EAR formula: \(EAR_{Acme} = (1 + 0.106/2)^{2} - 1\) Calculate the EAR for Acme \(EAR_{Acme} = (1 + 0.053)^{2} - 1\) \(EAR_{Acme} = (1.053)^{2} - 1\) \(EAR_{Acme} = 1.108759 - 1\) \(EAR_{Acme} = 0.108759\)

Step 4: Compare the EARs of the two mutual funds

Now let's compare the two EARs: - \(EAR_{Bendix} = 0.1086265664 \approx 10.862\%\) - \(EAR_{Acme} = 0.108759 \approx 10.876\%\) As we can see, the EAR for Acme Mutual Fund (10.876%) is higher than the EAR for Bendix Mutual Fund (10.862%). Therefore, Acme Mutual Fund has a better rate of return.

Key Concepts

Mutual Fund Performance

Mutual funds are investment vehicles that pool money from multiple investors to purchase a diversified portfolio of stocks, bonds, or other securities. The mutual fund's performance is crucial for investors because it determines the return on investment (ROI) they might expect over time.

The performance of a mutual fund can be evaluated by looking at its past returns, the fund manager's track record, and comparing it to relevant benchmarks. Past performance often reflects the compound growth rate, taking into account interest that accrues on both the initial investment and accumulated interest.

Key aspects to consider in mutual fund performance include:

• Fund's historical returns compared to similar funds or a market index.
• Volatility of returns, which measures the risk associated with the fund.
• Management fees and expenses, as they reduce the overall returns to the investor.

Understanding the effective annual rate (EAR) is vital in assessing how frequently interest is compounded within a mutual fund. The EAR helps investors to get a clearer picture of the actual return on their investments, adjusting for compounding periods.

Compound Interest

Compound interest is a financial concept where the interest earned on an initial principal amount also earns interest in subsequent periods. Compound interest is crucial in determining the growth of an investment, such as in a mutual fund.

The \(EAR (Effective Annual Rate)\) formula helps in understanding the actual annual interest rate earned, considering the effects of compounding.

Compound interest can significantly enhance investment growth through "interest on interest," which means that each period's interest is calculated on the initial principal plus all accumulated interest from previous periods.

• The frequency of compounding (quarterly, semiannually, annually, etc.) impacts total returns.
• A more frequent compounding results in higher interest accumulation over time.
• Investors can optimize returns by choosing investment products with more favorable compounding frequencies.

For example, comparing two investments, one compounded quarterly and another semiannually, the one with more frequent compounding could potentially offer higher returns, underscoring the power of compound interest.

Interest Rate Comparison

Interest rate comparison is crucial when evaluating investment options like mutual funds. To properly compare them, it's essential to understand the Effective Annual Rate (EAR), as it shows the real return on an investment after considering the compounding effects.

When comparing mutual funds or any other similar investment, simply looking at the nominal interest rates might not provide the full picture. Different investments might have varying compounding periods, which affect the actual returns investors would receive.

• Nominal rates don't account for the frequency of interest compounding.
• Effective Annual Rate (EAR) accounts for compounding and reflects total annual returns more accurately.
• Calculating and comparing EARs can help make more informed investment decisions.

In the given exercise, the two mutual funds—Bendix and Acme—are compared on the basis of their EAR. Even though Acme had a slightly higher nominal rate, the difference in compounding periods allowed for a more accurate comparison, showing Acme as the better investment option.