91Ó°ÊÓ

Present value: The amount of money originally put into an investment is known as the present value \(P\) of the investment. For example, if you buy a \(\$ 50\) U.S. Savings Bond that matures in 10 years, the present value of the investment is the amount of money you have to pay for the bond today. The value of the investment at some future time is known as the future value \(F\). Thus, if you buy the savings bond mentioned above, its future value is \(\$ 50 .\) If the investment pays an interest rate of \(r\) (as a decimal) compounded yearly, and if we know the future value \(F\) for \(t\) years in the future, then the present value \(P=P(F, r, t)\), the amount we have to pay today, can be calculated using $$ P=F \times \frac{1}{(1+r)^{t}} $$ if we measure \(F\) and \(P\) in dollars. The term \(1 /(1+r)^{t}\) is known as the present value factor, or the discount rate, so the formula above can also be written as $$ P=F \times \text { discount rate } . $$ a. Explain in your own words what information the function \(P(F, r, t)\) gives you. For the remainder of this problem, we will deal with an interest rate of \(9 \%\) compounded yearly and a time \(t\) of 18 years in the future. b. Calculate the discount rate. c. Suppose you wish to put money into an account that will provide \(\$ 100,000\) to help your child attend college 18 years from now. How much money would you have to put into savings today in order to attain that goal?

Step by step solution

01

Understand the function

The function \( P(F, r, t) \) calculates the present value \( P \) of an investment. This is the amount of money you need to invest today to achieve a certain future value \( F \), given an interest rate \( r \) compounded annually over \( t \) years.

02

Identify given values

We are given that the interest rate \( r \) is \( 9\% \) per year compounded annually, or \( r = 0.09 \) when expressed as a decimal. The time period is \( t = 18 \) years, and we want a future value \( F = 100,000 \) dollars.

03

Calculate the discount rate

The discount rate is given by the formula \( \frac{1}{(1+r)^{t}} \). Substitute \( r = 0.09 \) and \( t = 18 \) into this formula:\[\text{discount rate} = \frac{1}{(1+0.09)^{18}}\]Evaluate this term to find the discount rate.

04

Evaluate the discount rate

Calculate the value of \((1 + 0.09)^{18}\) first, which is approximately \(4.98262\). Then find the discount rate:\[\text{discount rate} = \frac{1}{4.98262} \approx 0.20007\]This value is the discount rate.

05

Calculate the present value

Using the discount rate from Step 4, compute the present value \( P \) using the formula \( P = F \times \text{discount rate} \). Substitute the values:\[P = 100,000 \times 0.20007 \approx 20,007\]You need to invest approximately \\(20,007 today to achieve a future value of \\)100,000.

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Key Concepts

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

Future Value

Future value represents the amount of money that an investment will grow to after a specified period of time at a given interest rate. It's essentially the value of your investment at a point in the future. When you invest money, you're not only putting away "present value" but also anticipating its growth over time due to interest. For instance, if you invest in a fixed deposit or a bond today, the amount it grows to is its future value. This concept helps you understand your return on investment over time. It's crucial for planning financial goals, allowing you to see how much an investment started today will be worth in the future.

Interest Rate

The interest rate is the percentage charged on the total amount you put into an investment or the cost of borrowing money. It's a critical factor that influences how quickly investments grow. The rate is often expressed annually as a percentage of the investment's original amount or principal. For example, an interest rate of 9% means that each year, 9% of the original investment amount is added to the investment. Compounding is key here, as interest can be earned not just on the initial principal but also on accumulated interest over previous periods. This compounding effect can significantly increase the future value of an investment.

Discount Rate

The discount rate is a factor used to calculate the present value of a future sum of money. It's essentially the inverse of compounding. While interest rates grow money over time, discount rates allow us to work backward to find out how much a future cash amount is worth in today's terms. The discount rate can be calculated using the formula: \[ \text{Discount Rate} = \frac{1}{(1+r)^{t}} \] Here, the variables stand for:

• \( r \) as the interest rate
• \( t \) as the number of periods

By applying the discount rate to a future value, one can determine how much should be invested today to achieve a specific financial goal in the future. It's also a helpful tool for comparing different investment options or financial strategies.

Investment Formula

The investment formula helps calculate the present value of an investment. It involves understanding the relationship between present value, future value, interest rate, and time. The formula is expressed as: \[ P = F \times \text{Discount Rate} \] Using:

• \( P \) as the present value needed to achieve the future value
• \( F \) as the future value
• "Discount Rate" as \( \frac{1}{(1+r)^t} \)

This formula helps investors understand how much they need to invest today to reach a future financial goal. By adjusting variables like the interest rate or the time span, different investment strategies can be evaluated for their future potential.