Problem 74 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

PRESENT VALUE OF AN INCOME STREAM What is the present value of an investment that will generate income continuously at a constant rate of \(\$ 1,000\) per year for 10 years if the prevailing annual interest rate remains fixed at \(7 \%\) compounded continuously?

Step-by-Step Solution

Answer

The present value is approximately \(\text{\textdollar} 7,189.64\).

Step 1: Identify Given Values

We are given an income stream generating \(\text{\textdollar} 1,000\) per year for 10 years. The interest rate is \(7\text{\textpercent}\) compounded continuously.

Step 2: Set Up the Formula for Present Value

The present value (PV) of a continuous income stream can be calculated using the formula: \[ PV = \frac{R}{r}\big(1 - e^{-rt}\big) \] where \(R = 1,000\), \(r = 0.07\), and \(t = 10\).

Step 3: Insert Values into the Formula

Substitute \(R = 1,000\), \(r = 0.07\), and \(t = 10\) into the present value formula: \[ PV = \frac{1,000}{0.07}\big(1 - e^{-0.07 \times 10}\big) \]

Step 4: Calculate Intermediate Steps

First, calculate \(e^{-0.7}\): \[ e^{-0.7} \approx 0.496585 \] Then, calculate \(1 - e^{-0.7}\): \[ 1 - 0.496585 = 0.503415 \] Finally, calculate \(\frac{1,000}{0.07} \times 0.503415\): \[ \frac{1,000}{0.07} = 14,285.71 \] So, \[ 14,285.71 \times 0.503415 \approx 7,189.64 \]

Step 5: Conclusion

Thus, the present value of the continuous income stream is approximately \(\text{\textdollar} 7,189.64\).

Key Concepts

continuous compounding

Continuous compounding is a concept in finance where interest is calculated and added to the principal balance continuously, rather than at discrete intervals like annually or monthly. This means that the investment grows at every moment in time. The formula used for continuous compounding is:
\[ A = P e^{rt} \]
where:

• P is the principal amount
• r is the annual interest rate (expressed as a decimal)
• t is the time the money is invested or borrowed for, in years
• e is the base of the natural logarithm (approximately equal to 2.71828)

This formula helps to understand how investments grow when interest is compounded continuously. It’s an important concept because it represents the theoretical maximum of compounding.

present value formula

The present value (PV) formula is crucial in determining what a future sum of money is worth in today’s terms. This is essential for comparing the value of money received at different times. Specifically for a continuous income stream, the formula is:
\[ PV = \frac{R}{r} \left(1 - e^{-rt}\right) \]
where:

• PV is the present value
• R is the income rate per period
• r is the annual discount rate
• t is the length of the period in years
• e is Euler's number (approximately 2.71828)

This formula allows us to account for continuous compounding while assessing how much a continuous stream of income is worth in the present day. It’s commonly used when evaluating investments and financial products.

investment income calculation

Investment income calculation is a critical process to understand the returns from an investment over time. In the context of continuously compounding interest, the income stream is ongoing and the calculation involves:
1. Identifying the income flow, annual interest rate, and investment duration.
2. Using the generalized formula:
\[ PV = \frac{R}{r} \left(1 - e^{-rt}\right) \]
For the given problem, it involves plugging in:

• R = $1,000 per year
• r = 0.07 (7% interest rate)
• t = 10 years

Following through the calculations, we determine the present value of the income stream to understand its current worth, taking into account the continuous growth due to compounding.

exponential functions

Exponential functions play a vital role in financial mathematics, particularly when dealing with growth processes such as compounding interest. An exponential function has the form:
\[ f(x) = a e^{bx} \]
where:

• a is a constant
• e is the base of the natural logarithm (approximately 2.71828)
• b is the rate of growth or decay
• x is the exponent

In the context of continuous compounding, exponential functions describe how investments grow over time. For instance, the term \( e^{-rt} \) in the present value formula represents the decay factor, showing how the value of future income decreases when considered in today’s terms. Understanding exponential functions allows us to model and predict financial scenarios accurately.