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Section 4.2 Bond Yield

The bond yield, which is a measure of the income generated by a bond, is calculated as the interest divided by the price. Therefore, if a \(\$ 10,000\) bond with 10% annual Coupons (probably broken up into two semi-annual payments of \(\$ 500\) each) is selling at \(\$ 10,000 \text{,}\) or par, the coupon payment is equal to 10%, i.e. the Yield. However bond prices can be affected by market conditions and so may fluctuate. As they do so does the yield.

In this example, if the market price fluctuated and valued your bond to be worth \(\$ 9,000 \text{,}\) your yield would now be 11.11% (1000/9000), but the \(\$ 500\) semi-annual coupon payments would not change. Conversely, if the bond price were to shoot up to \(\$ 11000\text{,}\) your yield would decrease to 9.09% (1000/11000), but again, you would still receive the same \(\$ 500 \) semi-annual coupon payments.

Example 4.2.1. Zero Coupon Bond Purchase Price.

A bond with no coupons is valued as a normal compound investment with the Maturity Value the specified future value. To compute the purchase price of a 20 year \(\$ 5000 \) bond with yearly rate of 4.95%

\begin{equation*} P = \$ 5000 (1+0.0495)^{-20} = \$ 1173.54. \end{equation*}

A bond with annual coupons will pay a simple interest coupon each year in the amount of \(P \cdot r_{eff}\text{,}\) where \(r_{eff}\) is the effective annual interest rate. Since the interest is paid out each year then there is no compounding effect and so the total value including interest of the bond follows a simple interest model with Total Amount Accumulated

\begin{equation*} = P + P \cdot r_{eff} \cdot t = P(1 + r_{eff} \cdot t)\text{.} \end{equation*}
Example 4.2.3.

Consider a 20 year 9% bond with annual coupons purchased in 1998 for \(\$ 93000\text{.}\) The amount of each coupon is the same \(\$ 93000 \cdot 0.09 = \$ 8370\text{.}\) So, by 2012 the accumulated interest would have been \(8370 \cdot 14\) years = \(\$ 117180\text{.}\) For the entire length of the bond, the total amount accumulated by 2018 would be \(\$ 93000(1+0.09 \cdot 20) = \$ 260,400 \text{.}\)