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Chapter 5: Problem 71

If two equal investments have the same effective interest rate and you graph the future value as a function of time for each of them, are the graphs necessarily the same? Explain your answer.

Short Answer

Expert verified
In conclusion, if two equal investments have the same effective interest rate, their future values over time will be the same, and their graphs will be identical. This is because both investments follow the same future value function, \(FV = PV * (1 + EIR)^{t}\), demonstrating that both investments grow at the same rate over time.

Step by step solution

01

Define the effective interest rate

The effective interest rate is the interest rate on an investment when compounding occurs more frequently than once a year. It takes into account the effects of compounding and is usually higher than the nominal interest rate. The formula to calculate the effective interest rate is: \(EIR = (1 + \frac {r}{n})^{n} - 1\) Where: - EIR: Effective interest rate - r: Nominal interest rate - n: Number of compounding periods per year
02

Define future value

Future value (FV) is the value of an investment at a specific point in the future, taking into account the interest earned over time. The formula to calculate the future value of an investment is: \(FV = PV * (1 + EIR)^{t}\) Where: - FV: Future value of the investment - PV: Present value of the investment (initial amount) - EIR: Effective interest rate - t: Time (in years)
03

Analyze the future value function

Based on the future value formula, we can see that the future value of an investment depends on four factors: present value (initial investment), effective interest rate, time, and the compounding frequency. In the given problem, we have two equal investments with the same effective interest rate. Now, let's consider two equal investments A and B with the same effective interest rate. The future values of the investments according to the formula are: \(FV_A = PV_A * (1 + EIR)^{t}\) \(FV_B = PV_B * (1 + EIR)^{t}\) Since the investments are equal, we can say that \(PV_A = PV_B\). Thus, the formulas become: \(FV_A = PV * (1 + EIR)^{t}\) \(FV_B = PV * (1 + EIR)^{t}\)
04

Compare the graphs of future value

As we have derived, both investments A and B have the same function for future value: \(FV_A = PV * (1 + EIR)^{t}\) \(FV_B = PV * (1 + EIR)^{t}\) These functions indicate that the future value of both investments over time is identical. Therefore, when plotted on a graph, the future values of both investments as a function of time will be the same. In conclusion, if two equal investments have the same effective interest rate, their future values over time will be the same, and their graphs will be identical.

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