Question

One year ago, you bought a two-year bond for \(\$ 900\). The bond has a face value of \(\$ 1,000\) and has one year left until maturity. It promises one additional interest payment of \(\$ 50\) at the maturity date. If the interest rate is 5 percent per year, what capital gain (or loss) would you get if you sell the bond today?

Short answer

The capital gain from selling the bond today would be \$100.

Step-by-step solution

Calculate present value of the face value

The bond will pay out its face value of $1000 in one year. Hence, using the concept of present value, which informs us about the amount an investment made today will yield in the future factoring in the interest rate, we can calculate this as follows: \[ PV_1 = \frac{FV}{(1+r)^n} = \frac{1000}{(1+0.05)^1} = \$952.38 \] where \( PV_1 \) is the present value one year from now, \( FV \) is the face value, \( r \) is the interest rate, and \( n \) is the number of years.

Calculate present value of interest payment

In addition to the face value, the bond also promises a payment of $50 at maturity. The present value of this payment can be calculated as follows: \[ PV_2 = \frac{FV}{(1+r)^n} = \frac{50}{(1+0.05)^1} = \$47.62 \] where \( PV_2 \) is the present value of the additional interest payment.

Calculate actual price of the bond

The actual price of the bond today is the sum of the present values calculated in step 1 and step 2. \[ BondPrice = PV_1 + PV_2 = \$952.38 + \$47.62 = \$1000 \]

Calculate the capital gain (or loss)

The capital gain (or loss) is the difference between the current bond price and the price it was bought for. \[ GainOrLoss = BondPrice - PurchasePrice = \$1000 - \$900 = \$100 \]

Key concepts

Capital Gain or Loss

In the world of investments, understanding the concept of capital gain or loss is crucial for any investor. It represents the change in value of an investment over a period of time. A capital gain occurs when the value of an investment increases, while a capital loss occurs when it decreases.

For instance, if you purchase a bond or a stock at one price and sell it at a higher price, you realize a capital gain. Inversely, selling the investment for less than you paid results in a capital loss. This concept is vital for making informed decisions in the financial markets as it affects your net investment income and potentially your tax liabilities.

Bond Pricing

The concept of bond pricing is fundamental to understanding how bonds are valued in financial markets. Bond pricing involves calculating the present value of all expected future cash flows from the bond—this typically includes periodic interest payments, known as coupon payments, and the return of the bond's face value at maturity.

The calculation takes into consideration the current market interest rate, and the bond's price will fluctuate inversely with changes in interest rates. Understanding how bond pricing works is essential for both investors looking to profit from bond trading and for issuers determining the cost of borrowing.

Interest Rates

When dealing with financial instruments such as bonds, one cannot overlook the impact of interest rates. Interest rates are essentially the cost of borrowing money, expressed as a percentage of the amount borrowed. They impact almost every aspect of financial economics, from individual loans like mortgages to the valuation of entire companies.

Rising interest rates generally lead to lower bond prices, and vice versa, because existing bonds with lower interest payments become less attractive. Conversely, lower interest rates can increase the prices of existing bonds in the market. Grasping this seesaw relationship between bond pricing and interest rates is crucial for anyone involved in financial investments.

Financial Mathematics

At the heart of many financial decisions is a field known as financial mathematics, which arms us with the tools needed to analyze and interpret monetary transactions. It encompasses a broad range of topics, including the time value of money, probabilistic models, and risk assessment.

Using mathematical formulas and concepts, such as present value and future value, individuals can determine the current worth of an investment, foresee its potential growth, and assess the profitability of different financial instruments. A strong foundation in financial mathematics is essential for anyone looking to navigate the complexities of the financial world successfully.