Suppose that 50,000 from a retirement account is invested...

Suppose that $\$ 50,000$ from a retirement account is invested in a large cap stock fund. After 20 yr, the value is $\$ 194,809.67$. a. Use the model $A=P e^{r t}$ to determine the average rate of return under continuous compounding. b. How long will it take the investment to reach onequarter million dollars? Round to 1 decimal place.

Suppose that $\$ 50,000$ from a retirement account is invested in a large cap stock fund. After 20 yr, the value is $\$ 194,809.67$.
a. Use the model $A=P e^{r t}$ to determine the average rate of return under continuous compounding.
b. How long will it take the investment to reach onequarter million dollars? Round to 1 decimal place.

Key Concepts

Solving Exponential Equations Using Logarithms

When working with exponential equations like A = P e^(rt), logarithms play a key role in isolating variables such as the rate of return or time. Taking the natural logarithm of both sides of the equation allows one to solve for the unknown by transforming the exponential relationship into a linear one, making it easier to compute the desired values.

Time Value of Money

The time value of money is a fundamental financial concept that states the value of money changes over time due to its potential earning capacity. Continuous compounding models exemplify this concept by illustrating how investments grow exponentially over time, emphasizing the importance of early and consistent investment to maximize returns.

Continuous Compounding

Continuous compounding refers to the method of calculating interest where the frequency of compounding is increased to an infinite number of times per period. In such cases, the accumulated amount is modeled using the exponential function, reflecting how the investment grows continuously over time rather than in discrete steps.

Exponential Growth Model

The exponential growth model, typically represented by A = P e^(rt), is a mathematical formulation used to describe growth processes where the rate of increase is proportional to the current amount. This model is commonly used in finance to calculate the future value of investments that compound interest continuously.

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