Question

The sum of \(\$ 1,000\) is invested in five-year bonds with face value \(\$ 100\) and \(\$ 8\) coupons paid annually. All coupons are reinvested in bonds of the same kind. Assuming that the bonds are trading at par and the interest rate remains constant throughout the period to maturity, compute the number of bonds held during each consecutive year of the investment.

Short answer

Year 1: 10.8 bonds, Year 2: 11.6 bonds, Year 3: 12.4 bonds, Year 4: 13.2 bonds, Year 5: 14.0 bonds.

Step-by-step solution

Calculate the Number of Initial Bonds

Initially, with an investment of \$ 1,000\, and bonds trading at par \$ 100\ each, calculate: \[ \frac{1000}{100} = 10 \text{ bonds} \]

Determine the Annual Coupon Payment

Each bond pays a coupon of \$ 8\ each year. So, for 10 bonds: \[ 10 \times 8 = 80 \text{ dollars per year} \]

Reinvest the Coupons in New Bonds

Since the coupons are reinvested in new bonds of the same kind, calculate the number of new bonds bought each year with the \$ 80\ coupon payment: \[ \frac{80}{100} = 0.8 \text{ bonds per year} \]

Compute Bonds Held Each Year

Calculate the total number of bonds at the end of each year by adding the coupons reinvested as bonds: Year 1: \[ 10 + 0.8 = 10.8 \] Year 2: \[ 10.8 + 0.8 = 11.6 \] Year 3: \[ 11.6 + 0.8 = 12.4 \] Year 4: \[ 12.4 + 0.8 = 13.2 \] Year 5: \[ 13.2 + 0.8 = 14.0 \]

Key concepts

annual coupon payments

When you invest in bonds, the bond issuer pays you interest. This interest is called a coupon payment. In our example, each bond has a face value of \(100 and offers an annual coupon payment of \)8. This means for every bond you hold, you receive \(8 each year.
It's important to note that coupon payments can be made annually, semi-annually, or quarterly, depending on the bond's terms. In this case, we focus on annual payments. Therefore, if you initially hold 10 bonds, you will receive \)80 in total each year: \[ 10 \times 8 = 80 \text{ dollars} \]
Understanding annual coupon payments is crucial as it affects your bond investment strategy and the potential returns you can expect.

reinvestment of coupons

Reinvestment of coupon payments is a strategy to grow your investment. Instead of taking the interest payments as cash, you use them to buy more bonds.
In our example, you receive \(80 each year from the initial 10 bonds. Instead of using the \)80 for other purposes, you reinvest it into buying more bonds. Since each bond is priced at \(100, the \)80 reinvested each year buys you 0.8 of a bond: \[ \frac{80}{100} = 0.8 \text{ bonds annually} \]
This reinvestment strategy allows your investment to grow at a compound rate, as you are continuously acquiring more bonds which, in turn, will generate more coupons in future years.

• Year 1: 10 + 0.8 = 10.8 bonds
• Year 2: 10.8 + 0.8 = 11.6 bonds
• Year 3: 11.6 + 0.8 = 12.4 bonds
• Year 4: 12.4 + 0.8 = 13.2 bonds
• Year 5: 13.2 + 0.8 = 14 bonds

bonds trading at par

When bonds are trading at par, it means they are being bought and sold at their face value. For instance, our bond with a face value of \(100 is currently trading at \)100.
This is a key concept because it simplifies calculations for investment strategies, especially when reinvesting coupons. The face value provides a baseline for assessing other financial metrics, like yield and expected returns.
Trading at par indicates a stable market condition for the bond, where the interest rate matches the coupon rate offered by similar investments in the market. Therefore, there's no premium or discount on the bond's price. Investors can easily gauge their returns without factoring in additional complexities.

compound interest in bonds

Compound interest allows your investment to grow exponentially over time. By reinvesting coupon payments to buy more bonds, you essentially earn interest on your interest.
Let's break this down. Initially, investing \(1,000 gets you 10 bonds. The annual coupon payment for these bonds is \)80. When you reinvest these payments to buy more bonds, the number of bonds increases gradually each year. These additional bonds will also produce coupon payments, which will again be reinvested. This cycle continues, enhancing your investment's growth.
This growth can be calculated year by year. For example, in Year 1, you hold 10.8 bonds, earning more than just the original $80 coupon in Year 2, and so on. This demonstrates how compound interest can significantly boost your returns. Embracing this concept can make a considerable difference in long-term investments.