Question

A child's grandparents are considering buying an 80,000face-value, zero-coupon bond at her birth so that she will have enough money for her college education 17 years later. If they want a rate of return of 6 % compounded annually, what should they pay for the bond?

Short answer

They should pay approximately 29,661.52 for the bond.

Step-by-step solution

Understand the Problem

A zero-coupon bond does not pay periodic interest. Instead, it is sold at a discount and pays its face value at maturity. The goal is to determine the current price to pay for this bond to achieve a specified rate of return over a given period.

Identify the Given Values

The face value of the bond (F) is 80,000. The number of years (T) until maturity is 17 years. The annual interest rate (R) is 6% compounded annually.

Set Up the Present Value Formula for Zero-Coupon Bonds

The formula for the present value (P) of a zero-coupon bond is: P = F / (1 + R)^T

Substitute the Given Values into the Formula

Substitute F = 80,000, R = 0.06,andT = 17 years into the formula: P = 80,000 / (1 + 0.06)^{17}

Calculate the Present Value

First, calculate the denominator: (1 + 0.06)^{17} = 1.06^{17} . Then, divide the face value by this result: P = 80,000 / 2.697 (approximately).

Simplify the Calculation

Divide 80,000 by 2.697 to find the present value: P approx = 29,661.52.

Key concepts

present value formula

To determine how much to pay for a zero-coupon bond today, we use the present value formula. This formula calculates the current worth of a future amount of money or a series of cash flows, given a specified rate of return.

For zero-coupon bonds, the present value formula is:
\[ P = \frac{F}{(1 + R)^T} \]
Where:

• P is the present value, or the price to pay today
• F is the face value, which is the bond's value at maturity
• R is the annual rate of return (expressed as a decimal)
• T is the number of years until maturity

Using this formula, we can effectively determine how much to invest now to achieve a specific future value, taking into account the time value of money. This essentially means money today is worth more than the same amount in the future because of its potential earning capacity.

compound interest

Compound interest refers to the process of earning interest on both the initial principal and the interest that has already been accumulated. Unlike simple interest, where interest is calculated only on the principal amount, compound interest grows the investment more rapidly.

In our problem, the rate of return is compounded annually, which means the interest is added to the principal once a year. The formula used in the present value equation accounts for this compounding effect:
\[ (1 + R)^T \]
In this term:

• R represents the annual interest rate
• T represents the number of years the money is invested or borrowed

The compounding effect leads to exponential growth in the value of the investment, which is why it's an important concept in financial calculations.

rate of return

The rate of return is a crucial factor in determining how much an investment grows over time. It is usually expressed as a percentage and represents the gain or loss on an investment over a specific period of time relative to the amount invested.

In our exercise, the grandparents want a rate of return of 6% compounded annually. This means they expect the value of their investment to grow by 6% each year. The rate of return significantly impacts the present value calculation because it influences the discount factor in the formula:
\[ (1 + R)^T \]
A higher rate of return would mean investing less money today for the same future value, while a lower rate of return would mean investing more. Thus, understanding and setting a realistic rate of return is crucial for planning finances effectively.

face value

The face value of a bond is its nominal value, which is the amount paid to the bondholder at maturity. For a zero-coupon bond, this is the amount the holder receives when the bond matures, not including any periodic interest payments.

In the given exercise, the face value (F) is 80,000. This is the amount the child will receive after 17 years. Since zero-coupon bonds do not offer periodic interest payments, the face value is essentially the amount that will be received at the end of the investment period.

Understanding the face value is essential, as it directly affects how much one should invest today (the present value). The greater the face value, the more will be paid out at maturity, meaning a larger present investment is required if all other factors remain constant.