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Chapter 5: Problem 33

Find the accumulated present value of an investment for which there is a perpetual continuous money flow of \(\$ 3600\) per year at an interest rate of \(5 \%,\) compounded continuously.

Short Answer

Expert verified
The accumulated present value is $72,000.

Step by step solution

01

Understand the Formula

The accumulated present value of a continuous money flow can be calculated using the formula: \( PV = \frac{F}{r} \), where \( F \) is the annual flow of money and \( r \) is the continuous compounding interest rate.
02

Identify the Values

Identify the values given in the problem: \( F = 3600 \) as the annual money flow, and \( r = 0.05 \) as the interest rate in decimal form (since \(5\% = 0.05\)).
03

Calculate the Present Value

Substitute the given values into the formula: \( PV = \frac{3600}{0.05} \). Calculate the result to find the present value.
04

Perform the Division

Calculate the division: \( 3600 \div 0.05 = 72000 \). This is the accumulated present value.

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Key Concepts

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

Understanding Continuous Money Flow
Continuous money flow refers to a stream of cash that occurs continuously over time without interruptions. Unlike discrete cash flows, which happen at specific intervals, continuous cash flows are constant and unending in nature. This concept is crucial for calculating investments in perpetuity, such as pensions, continuous annuities, or interest accruals over time. When dealing with investments that provide continuous money flow, it is vital to understand how these flows impact present value and future investment growth.
  • Continuous flow ensures that money is always active, never paused.
  • Encompasses concepts like perpetual payments, implying infinite duration.
  • Simplifies calculations as the flow rate stays constant.
The Dynamics of Continuous Compounding
Continuous compounding refers to the process where interest is calculated and added to the account balance an infinite number of times in a time period, usually annually. This method allows the investment to grow at a much faster rate compared to regular compounding, say monthly or annually.
The exponential growth formula used here is: \[ A = Pe^{rt} \]where:
  • \( A \) is the amount of money accumulated after time \( t \), including interest.
  • \( P \) is the principal amount (initial investment).
  • \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
  • \( r \) is the annual interest rate (decimal).
  • \( t \) is the time the money is invested for, in years.
Continuous compounding amplifies the effect of earning interest on prior interest, thus accelerating growth.
Decoding Interest Rates
An interest rate in investment terms is the percentage charged on the total amount that you have borrowed or saved. It's expressed annually but can be compounded in many different ways, such as annually, semi-annually, quarterly, monthly, or continuously.
For continuous compounding, realizing the effect of the rate is even more important due to the exponential nature of growth:
  • An interest rate reflects potential profit or loss from borrowing or investing.
  • With continuous compounding, even a small change in the rate can significantly impact outcome values.
  • Understanding and calculating effectively ensures strategic financial planning.
Here, the importance is placed on using the right rate to ensure accurate valuation in continuous investment scenarios.
The Process of Present Value Calculation
Present value (PV) is a financial concept that discounts the value of future cash flows to reflect their value in today's terms. Calculating the present value for a continuous money flow involves adjusting future cash based on the continuous compounding rate.
In our discussed example, the formula used is simple:\[ PV = \frac{F}{r} \]where \( F \) is the annual flow, and \( r \) represents the interest rate used for continuous compounding.
  • This approach helps in comparing and evaluating financial decisions based on their current worth.
  • Continuous money flow reflects a constant stream of future cash, perpetually discounted back.
  • Understanding PV allows businesses and investors to evaluate the real profitability of seemingly appealing future investments.
Accurately computing the present value helps in making sound financial judgments regarding the long-term management of finances.

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Most popular questions from this chapter

The capitalized cost, \(c,\) of an asset over its lifetime is the total of the initial cost and the present value of all maintenance expenses that will occur in the future. It is computed with the formula $$ c=c_{0}+\int_{0}^{L} m(t) e^{-k t} d t $$ where \(c_{0}\) is the initial cost of the asset, \(L\) is the lifetime (in years), \(k\) is the interest rate (compounded continuously), and \(m(t)\) is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. $$ \begin{array}{l} c_{0}=\$ 300,000, k=5 \%, m(t)=\$ 30,000+\$ 500 t, \\ L=20 \end{array} $$

Before \(1859,\) rabbits did not exist in Australia. That year, a settler released 24 rabbits into the wild. Without natural predators, the growth of the Australian rabbit population can be modeled by the uninhibited growth model \(d P / d t=k P,\) where \(P(t)\) is the population of rabbits \(t\) years after \(1859 .\) (Source: www dpi.vic.gov.au/agriculture.) a) When the rabbit population was estimated to be \(8900,\) its rate of growth was about 2630 rabbits per year. Use this information to find \(k,\) and then find the particular solution of the differential equation. b) Find the rabbit population in \(1900(t=41)\) and the rate at which it was increasing in that year. c) Without using a calculator, find \(P^{\prime}(41) / P(41)\).

Let \(x\) be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. $$ P(-2.01 \leq x \leq 0) $$

Let \(x\) be a continuous random variable that is normally distributed with mean \(\mu=22\) and standard deviation \(\sigma=5 .\) Using Table A, find the following. $$ P(24 \leq x \leq 30) $$

For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard deviation. $$ f(x)=\frac{1}{\ln 5} \cdot \frac{1}{x}, \quad[1.5,7.5] $$
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