Question

Dmitri bought a 1,000 dollars bond at par value with a coupon rate of 5 percent. He determines the yield by dividing the amount of interest he earns by the price. a. How much interest would he earn in the first year and what would be the yield? b. How much interest would he earn in the first year and what would be the yield if he had paid 950 dollars for the bond? What would be the interest and yield if he paid 1,050 dollars?

Short answer

Interest is $50 per year; yield at par is 5%. At $950, yield is 5.26%; at $1,050, yield is 4.76%.

Step-by-step solution

Calculate First-Year Interest

To find the interest Dmitri earns in the first year, multiply the bond's par value by the coupon rate. The par value is $1,000 and the coupon rate is 5%. Therefore, the first-year interest is calculated as \( 1,000 \times 0.05 = 50 \) dollars.

Compute Yield at Par Value

The yield is determined by dividing the interest earned by the price paid for the bond. When the bond is bought at par value, which is $1,000, the yield is \( \frac{50}{1,000} = 0.05 \), or 5%.

Calculate Interest and Yield at $950 Purchase Price

First, the interest remains $50 since it is based on the par value and coupon rate. The yield is calculated by dividing the interest earned by the purchase price: \( \frac{50}{950} \approx 0.0526 \), or 5.26%.

Calculate Interest and Yield at $1,050 Purchase Price

Again, the interest earned is \(50. To find the yield when he paid \)1,050, divide the interest by the purchase price: \( \frac{50}{1,050} \approx 0.0476 \), or 4.76%.

Key concepts

Coupon Rate

The coupon rate is a key concept when it comes to understanding bonds. It represents the annual interest rate paid by the bond issuer to the bondholder, based on the bond's par value. For example, if Dmitri owns a bond with a par value of \(1,000\) dollars and a coupon rate of 5%, he will receive interest payments amounting to 5% of \(1,000\), which is \(50\) dollars annually. The coupon rate is typically fixed, meaning it doesn’t change over the life of the bond.

The coupon rate is expressed as a percentage, which helps investors compare bonds with different face values. It's also crucial in determining the bond's interest income, but it does not change with market interest rates. This constancy allows bondholders to foresee the cash they will receive over the bond's life.

Par Value

Par value, also referred to as "face value," is the amount that the bondholder will receive back from the issuer at maturity. It is also the reference amount used to calculate the coupon payments. Dmitri's bond has a par value of \(1,000\) dollars, so he expects to receive this amount at the bond's maturity date unless he decides to sell the bond beforehand.

The par value does not change throughout the bond's life. This makes it vital when calculating the coupon payment—it is the figure to which the coupon rate is applied.

In bond terminology, purchasing a bond at par means buying it for its original or face value. In Dmitri's situation, buying at par implies he paid \(1,000\) dollars for his bond. The market price of a bond could be below or above the par value, affecting the bond's yield more than its interest income.

Interest Income

Interest income from a bond is the return a bondholder earns from the bond's coupon payments. Dmitri, for instance, earns \(50\) dollars each year from his bond, calculated using the formula: \[\text{Interest Income} = \text{Par Value} \times \text{Coupon Rate}\]Plugging in the numbers, for Dmitri's bond: \[1,000 \times 0.05 = 50\]

This \(50\) of interest income is guaranteed annually, as long as the bond issuer does not default. The interest earned is a form of passive income for bondholders and serves as a stable earnings stream over the bond's life.

Unlike the yield, the interest income does not change based on the bond's purchase price. It remains consistent since it's calculated on the bond's par value. This is why, even when the bond is bought for \(950\) or \(1,050\) dollars instead of its par value, Dmitri’s annual interest income remains at \(50\) dollars.

Purchase Price

The purchase price is the amount Dmitri pays to acquire the bond. While the par value is important for calculating the coupon payment, the purchase price is crucial for determining the bond's yield. Dmitri's bond can be bought at different prices—\(950\), \(1,000\) (at par), or \(1,050\) dollars.

When the purchase price of a bond differs from its par value, it influences the yield, which is computed by dividing the interest income by the purchase price. For example:

• If Dmitri buys the bond at \(950\) dollars, the yield becomes approximately 5.26%: \[\frac{50}{950} \approx 0.0526\]
• Buying the bond at \(1,050\) dollars leads to a yield of approximately 4.76%:\[\frac{50}{1,050} \approx 0.0476\]

The varying purchase prices do not affect the actual interest he receives, but they do change the yield, reflecting the bond's relative attractiveness as an investment at the given price.