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Chapter 8: Problem 18

Accrued Interest You purchase a bond with an invoice price of \(\$ 1,090\). The bond has a coupon rate of 8.4 percent, and there are 2 months to the next semiannual coupon date. What is the clean price of the bond?

Short Answer

Expert verified
To find the clean price of the bond, follow these steps: 1. Calculate the annual coupon payment: \(1000 \times (8.4/100)\) 2. Calculate the semiannual coupon payment: Annual Coupon Payment / 2 3. Calculate the accrued interest: Semiannual Coupon Payment 脳 (2 / 6) 4. Calculate the clean price: Invoice Price - Accrued Interest

Step by step solution

01

Calculate the annual coupon payment

To get the annual coupon payment, we multiply the bond's face value by the coupon rate. Since no face value is given, we can assume it to be \(1000\) (which is a standard for financial calculations). As the coupon rate is given in percentage, we will convert it to a decimal by dividing it by 100. Annual Coupon Payment = Face Value 脳 Coupon Rate = \(1000 脳 (8.4/100)\)
02

Calculate the semiannual coupon payment

Now, we need to find the semiannual coupon payment, which is half the annual coupon payment (as it is paid twice a year). Semiannual Coupon Payment = Annual Coupon Payment / 2
03

Calculate the accrued interest

Next, we will calculate the accrued interest by multiplying the semiannual coupon payment by the fraction of the time to the next coupon payment (given as 2 months in this problem) and the total time in a semiannual period (6 months). Accrued Interest = Semiannual Coupon Payment 脳 (Months to next coupon date / Total months in a semiannual period) = Semiannual Coupon Payment 脳 (2 / 6)
04

Calculate the clean price

Finally, we can find the clean price by subtracting the accrued interest from the invoice price. Clean Price = Invoice Price - Accrued Interest

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Key Concepts

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

Clean Price
The clean price of a bond is simply the price of the bond without considering any additional interest that may have accrued since the last coupon payment. Bonds can accumulate additional value, known as accrued interest, for periods in between payments. This additional interest is not included when calculating the clean price.
  • Formula: To calculate this, use the formula: \[ Clean\ Price = Invoice\ Price - Accrued\ Interest \]
  • Example: Imagine you buy a bond for \(\\(1090\), and there鈥檚 accrued interest due to an upcoming coupon payment. The clean price would be \(\\)1090\) minus whatever interest has accrued.
  • Significance: The clean price is important because it allows traders and investors to understand the actual price of the bond, excluding fluctuating interest values.
This concept is crucial when comparing bonds, as accrued interest can vary significantly between bonds, impacting overall costs.
Semiannual Coupon Payment
A semiannual coupon payment refers to the interest paid to bondholders every six months. Bonds typically pay interest more than once per year, and a common schedule is semiannual.
  • Calculation: The semiannual coupon payment is half of what the bond pays in interest annually:\[ Semiannual\ Coupon\ Payment = \frac{Annual\ Coupon\ Payment}{2} \]
  • Example: If a bond has an annual coupon payment of \(\\(84\), then its semiannual payment would be \[ \frac{\\)84}{2} = \$42 \] each half-year.
  • Implication: Investors receive a portion of their return more frequently, allowing for potential reinvestment earlier than an annual pay schedule.
Understanding how often interest payments are made helps investors in managing cash flow and investment strategies.
Bond Pricing
Bond pricing is determining how much a bond is worth at any given time. Prices fluctuate based on multiple factors, including interest rates, the bond鈥檚 term, and issuer creditworthiness.
  • Clean vs Dirty Price: The clean price excludes accrued interest, while the dirty price (invoice price) includes it.
  • Market Impact: Changing interest rates affect bond prices inversely. When rates rise, bond prices fall, and vice versa.
  • Time Value: Bonds closer to maturity generally have smaller price fluctuations due to less interest rate risk.
Being knowledgeable about how bonds are priced aids in purchasing decisions and anticipating future changes in the bond鈥檚 value.
Coupon Rate
The coupon rate is the interest rate a bond issuer agrees to pay bondholders. This is expressed as a percentage of the bond鈥檚 face or par value.
  • Annual Rate: The coupon rate determines the annual payment amount. For example, with a face value of \(\\(1000\), an 8.4% coupon rate means an annual payment of \(\\)84\).
  • Fixed vs Variable: Most bonds have a fixed coupon rate, promising certain payments every period. Some have variable rates that adjust with market indices.
  • Choosing Bonds: A higher coupon rate might mean more attractive returns, but could also imply higher risk if the issuer compensates for risk with such rates.
Understanding the coupon rate aids investors in gauging the bond鈥檚 yield and overall attractiveness.

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Most popular questions from this chapter

Valuing Bonds Even though most corporate bonds in the United States make coupon payments semiannually, bonds issued elsewhere often have annual coupon payments. Suppose a German company issues a bond with a par value of \(鈧� 1,000\), 15 years to maturity, and a coupon rate of 8.4 percent paid annually. If the yield to maturity is 7.6 percent, what is the current price of the bond?

Real Cash Flows Paul Adams owns a health club in downtown Los Angeles. He charges his customers an annual fee of \(\$ \mathbf{5 0 0}\) and has an existing customer base of 500. Paul plans to raise the annual fee by 6 percent every year and expects the club membership to grow at a constant rate of 3 percent for the next five years. The overall expenses of running the health club are \(\$ 75,000 \mathbf{a}\) year and are expected to grow at the inflation rate of 2 percent annually. After five years, Paul plans to buy a luxury boat for \(\$ 500,000\), close the health club, and travel the world in his boat for the rest of his life. What is the annual amount that Paul can spend while on his world tour if he will have no money left in the bank when he dies? Assume Paul has a remaining life of \(\mathbf{2 5}\) years and earns 9 percent on his savings.

Real Cash Flows When Marilyn Monroe died, ex-husband Joe DiMaggio vowed to place fresh flowers on her grave every Sunday as long as he lived. The week after she died in 1962, a bunch of fresh flowers that the former baseball player thought appropriate for the star cost about \(\$ 8\). Based on actuarial tables, "Joltin' Joe" could expect to live for 30 years after the actress died. Assume that the EAR is \(\mathbf{1 0 . 7}\) percent. Also, assume that the price of the flowers will increase at 3.5 percent per year, when expressed as an EAR. Assuming that each year has exactly 52 weeks, what is the present value of this commitment? Joe began purchasing flowers the week after Marilyn died.

Valuing Bonds Microhard has issued a bond with the following characteristics: \- Par: \(\$ 1,000\) \- Time to maturity: 25 years \- Coupon rate: 7 percent \- Semiannual payments Calculate the price of this bond if the YTM is: 1\. 7 percent 2\. 9 percent 3\. 5 percent

Accrued Interest You purchase a bond with a coupon rate of 7.2 percent and a clean price of \$904. If the next semiannual coupon payment is due in four months, what is the invoice price?
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