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Chapter 5: Problem 2

If you invest \(P\) dollars (the present value of your investment) in a fund that pays an interest rate of \(r\), as a decimal, compounded yearly, then after \(t\) years your investment will have a value \(F\) dollars, which is known as the future value. The discount rate \(D\) for such an investment is given by $$ D=\frac{1}{(1+r)^{t}}, $$ where \(t\) is the life, in years, of the investment. The present value of an investment is the product of the future value and the discount rate. Find a formula that gives the present value in terms of the future value, the interest rate, and the life of the investment.

Short Answer

Expert verified
The present value formula is \( P = \frac{F}{(1+r)^{t}} \).

Step by step solution

01

Understand the Given Information

You are given the formula for the discount rate: \( D = \frac{1}{(1+r)^{t}} \). The present value \( P \) is the product of the future value \( F \) and the discount rate \( D \). We need to find an expression for \( P \) in terms of \( F \), \( r \), and \( t \).
02

Express Present Value in Terms of Future Value and Discount Rate

The relationship for the present value is given by \( P = F \times D \). Substitute the expression for \( D \) into this equation, giving \( P = F \times \frac{1}{(1+r)^{t}} \).
03

Simplify the Expression

The expression can be simplified directly by multiplying: \( P = \frac{F}{(1+r)^{t}} \). This is the formula that gives the present value in terms of the future value \( F \), the interest rate \( r \), and the life of the investment \( t \).

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Key Concepts

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

Future Value
The future value is a core concept in finance. It represents how much an investment made today will grow over a specific period of time at a given interest rate. When discussing future value, you essentially talk about the potential of your money.
Before investing, one must determine the future value to evaluate if the returns will meet expected goals. For example, investing in a fund or savings account helps predict how much that investment will grow by using future value calculations.
  • Future value depends on the interest rate, indicating the growth per period.
  • A higher interest rate results in a higher future value, provided all other variables remain constant.
Understanding the future value helps in planning and making informed financial decisions.
Compound Interest
Compound interest is the key driver behind growing investments over time. Unlike simple interest, which is calculated on the original principal only, compound interest is calculated on the principal and on the accumulated interest of previous periods. This means that interest generates more interest, leading to exponential growth of the investment.
Mathematically, compound interest can be represented by the formula:\[ F = P(1 + r)^{t} \]
This formula shows how an initial principal amount will grow over time at a specified interest rate.
  • The term \((1 + r)^{t}\) accounts for compounding by multiplying the principal by the growth factor for each year.
  • The more frequently the interest is compounded, the more the investment grows.
Understanding compound interest is crucial to maximizing investment returns.
Discount Rate
The discount rate is a critical concept that helps to determine the present value of a future sum of money. In other words, it describes how much less a future value is worth today. The discount rate is inversely related to the future value, as it incorporates the impacts of time and interest rates on the value of money.
Using the formula for the discount rate:
\[ D = \frac{1}{(1+r)^{t}} \]
This formula highlights how the discount rate decreases with increasing interest rate \(r\) and time \(t\), reflecting the diminishing value of future cash flows.
  • A higher discount rate implies a lower present value, assuming the same future value.
  • The discount rate is crucial for evaluating the desirability of investments, as it directly affects profitability.
This understanding allows investors to estimate the present worth of expected returns.
Investment Life
Investment life refers to the duration for which an investment is held, and it plays a significant role in determining both the future and the present values of the investment. The parameter \(t\) in the equations signifies this period in years. The longer the investment life, the more opportunities there are for compound interest to act on the principal, potentially increasing the future value.
In the calculation of both future value and discount rate:
  • As the investment life \(t\) increases, the future value generally increases due to compound interest.
  • Conversely, the same increase in \(t\) results in a lower discount rate, reducing the present value for a given future value.
Understanding how the length of time influences financial calculations is essential for effective investment strategy.

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Most popular questions from this chapter

Table \(5.3\) gives the length \(L\), in inches, of a flying animal and its maximum speed \(F\), in feet per second, when it flies. \({ }^{27}\) (For comparison, 10 feet per second is about \(6.8\) miles per hour.) $$ \begin{array}{|l|c|c|} \hline \text { Animal } & \text { Length } L & \begin{array}{c} \text { Flying speed } \\ F \end{array} \\ \hline \text { Fruit fly } & 0.08 & 6.2 \\ \hline \text { Horse fly } & 0.51 & 21.7 \\ \hline \begin{array}{l} \text { Ruby-throated } \\ \text { hummingbird } \end{array} & 3.2 & 36.7 \\ \hline \text { Willow warbler } & 4.3 & 39.4 \\ \hline \text { Flying fish } & 13 & 51.2 \\ \hline \text { Bewick's swan } & 47 & 61.7 \\ \hline \text { White pelican } & 62 & 74.8 \\ \hline \end{array} $$ a. Judging on the basis of this table, is it generally true that larger animals fly faster? b. Find a formula that models \(F\) as a power function of \(L\). c. Make the graph of the function in part b. d. Is the graph you found in part c concave up or concave down? Explain in practical terms what your answer means. e. If one bird is 10 times longer than another, how much faster would you expect it to fly? (Use the homogeneity property of power functions.)

The power \(P\) required for level flight by an airplane is a function of the speed \(u\) of flight. Consideration of drag on the plane yields the model $$ P=\frac{u^{3}}{a}+\frac{b}{u}. $$ Here \(a\) and \(b\) are constants that depend on the characteristics of the airplane. This model may also be applied to the flight of a bird such as the budgerigar (a type of parakeet), where we take \(a=7800\) and \(b=600\). Here the flight speed \(u\) is measured in kilometers per hour, and the power \(P\) is the rate of oxygen consumption in cubic centimeters per gram per hour. \({ }^{68}\) a. Make a graph of \(P\) as a function of \(u\) for the budgerigar. Include flight speeds between 25 and 45 kilometers per hour. b. Calculate \(P(39)\) and explain what your answer means in practical terms. c. At what flight speed is the required power minimized?

A rock is thrown downward, and the distance \(D\), in feet, that it falls in \(t\) seconds is given by \(D=16 t^{2}+3 t\). Find how long it takes for the rock to fall 400 feet by using a. the quadratic formula. b. the crossing-graphs method.

Northern Yellowstone elk: The northern Yellowstone elk winter in the northern range of Yellowstone National Park. \({ }^{6}\) A moratorium on elk hunting was imposed in 1969 , and after that the growth of the elk population was approximately logistic for a time. The following table gives data on the growth. $$ \begin{array}{|c|c|} \hline \text { Year } & N \\ \hline 1968 & 3172 \\ \hline 1969 & 4305 \\ \hline 1970 & 5543 \\ \hline 1971 & 7281 \\ \hline 1972 & 8215 \\ \hline 1973 & 9981 \\ \hline 1974 & 10,529 \\ \hline \end{array} $$ a. Use regression to find a logistic model for this elk population. b. According to the model you made in part a, when would the elk population reach half of the carrying capacity? Note: At one time the gray wolf was a leading predator of the elk, but it was not a factor during this study period. The level at which the elk population stabilized suggests that food supply (and not just predators) can effectively regulate population size in this setting.

Many science fiction movies feature animals such as ants, spiders, or apes growing to monstrous sizes and threatening defenseless Earthlings. (Of course, they are in the end defeated by the hero and heroine.) Biologists use power functions as a rough guide to relate body weight and cross-sectional area of limbs to length or height. Generally, weight is thought to be proportional to the cube of length, whereas cross-sectional area of limbs is proportional to the square of length. Suppose an ant, having been exposed to "radiation," is enlarged to 500 times its normal length. (Such an event can occur only in Hollywood fantasy. Radiation is utterly incapable of causing such a reaction.) a. By how much will its weight be increased? b. By how much will the cross-sectional area of its legs be increased? c. Pressure on a limb is weight divided by crosssectional area. By how much has the pressure on a leg of the giant ant increased? What do you think is likely to happen to the unfortunate ant? \({ }^{15}\)
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