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Chapter 10: Problem 19

Suppose you purchase a four-year bond with an APR of \(5.75 \% .\) The face value of the bond is \(\$ 4920 .\) Find the purchase price of the bond.

Short Answer

Expert verified
The purchase price of the bond is approximately $3940.88.

Step by step solution

01

Convert APR to decimal

The given APR is \(5.75\% .\) This needs to be converted to its decimal form by dividing by 100. So, \(5.75\% \) becomes \(0.0575 .\)
02

Compute the present value of the bond using the face value

The face value is the amount that will be received at the end of the bond's term, so it's one of the future cash flows. The formula for the present value (PV) is \(PV = \frac{FV}{(1+i)^n} \) where:\n - \( FV \) is the face value of the bond\n - \( i \) is the interest rate in decimal\n - \( n \) is the number of years until maturity\nSubstituting these values into the equation will result in \[ PV = \frac{4920}{(1+0.0575)^4} \]
03

Calculate the final result

Using a calculator, the value inside the brackets is calculated as \( (1+0.0575)^4 = 1.248177 .\) So the purchase price (present value) of the bond is \( \frac{4920}{1.248177} .\) Approximating to two decimal digits gives the Purchase Price of the bond.

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Key Concepts

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

Understanding Annual Percentage Rate (APR)
When dealing with bonds like the one in the exercise, the Annual Percentage Rate (APR) is crucial. The APR is the annual rate at which interest is charged on the bond's face value. It includes the compounded interest over annual periods.
- APR provides a simple percentage that represents the yearly cost of holding the bond or loan.- In this context, APR helps compare different financial products.- For the bond, the APR is given as a percentage which is converted to a decimal for calculations.
For example, an APR of 5.75% means the bond accrues 5.75% interest each year. Converting this to decimals is essential to use in formulas, such as calculating present value. To do this, divide by 100: \( 5.75\% = \frac{5.75}{100} = 0.0575 \).
By understanding APR, investors can better gauge the cost of the bond over its lifespan.
Delving into Present Value Calculation
The present value calculation is a method of determining how much a future sum of money is worth today, based on a specific interest rate. This is an important financial mathematics concept, especially in bond valuation.
- For the given bond, the goal is to find out what it should be purchased for today, considering future value will be collected after four years.- The present value formula is \( PV = \frac{FV}{(1+i)^n} \), where: - \( PV \) stands for Present Value. - \( FV \) is the Future Value, or the face value of the bond, \( 4920 \) in this case. - \( i \) is the interest rate as a decimal. - \( n \) is the duration in years.
In this exercise, substituting the values: \( PV = \frac{4920}{(1+0.0575)^4} \). The bond's purchase price reflects its worth today, considering the 5.75% APR. This gives investors a clear view of the bond鈥檚 current value versus its future payout.
The Role of Financial Mathematics in Bond Valuation
Financial mathematics serves as the backbone for understanding bond valuation. It encompasses various calculations and concepts like present value that help investors make informed decisions.
- Bond valuation depends on both the future value of the bond and interest applied over time. - Using financial formulas, one considers the impact of time and interest when evaluating current bond prices.
For instance, interest affects how future cash flows are valued today. This requires converting APR to a decimal and applying it over years. Evaluating bond prices using financial mathematics provides crucial insights into investment worth and potential returns.
By assessing bonds through these calculations, investors understand the financial implications of their investments comprehensively.

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Most popular questions from this chapter

For an investment having an APY of \(6 \%\), estimate the number of years needed to double the principal.

Advance America is a payday loan company that offers quick, short-term loans using the borrower's future paychecks as collateral. Advance America charges \(\$ 17\) for each \(\$ 100\) loaned for a term of 14 days. Find the APR charged by Advance America.

Reid's credit card cycle ends on the twenty-fifth of every month. The interest rate on Reid's Visa card is \(21.99 \%,\) and the billing cycle runs from the twenty-sixth of a month to the twenty-fifth of the following month. At the end of the July 26-Aug. 25 billing cycle, Reid's balance was \(\$ 5000\). During the next billing cycle (Aug. 26-Sept. 25) Reid made three purchases, with the dates and amounts shown in Table \(10-12 .\) On September 22 Reid made an online payment of \(\$ 200\) that was credited towards his balance the same day. (a) Find the average daily balance on the credit card account for the billing cycle Aug. 26-Sept. \(25 .\) (b) Find the interest charged for the billing cycle Aug. 26 Sept. \(25 .\) (c) Find the new balance on the account at the end of the Aug. 26-Sept. 25 billing cycle. $$ \begin{array}{|l|c|} \hline \text { Date of purchase } & \text { Amount of purchase } \\ \hline 8 / 31 & \$ 148.55 \\ \hline 9 / 12 & \$ 30.00 \\ \hline 9 / 19 & \$ 103.99 \\ \hline \end{array} $$

There were four different sections of Financial Mathematics 101 offered last semester. In section \(\mathrm{A}, 31\) out of 38 students passed the class; in section B, 47 out of 56 students passed the class; in section \(\mathrm{C}, 34\) out of 45 students passed the class; and in section \(\mathrm{D}, 45\) out of 52 students passed the class. Compare the passing rates in each section and rank them from best to worst.

How much should a retailer mark up her goods so that when she has a \(25 \%\) off sale, the resulting prices will still reflect a \(50 \%\) markup (on her cost)?
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