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 obtained from selling the bond today would be $100.

Step-by-step solution

Calculate future value of bond

To calculate the future value (FV) of the bond, it's needed to take into consideration the bond’s face value plus the additional interest payment. Therefore, the FV of the bond is \(1000 + 50 = 1050\)

Calculate present value of the bond

To calculate the price of the bond today (which is the present value), the future value should be discounted using the interest rate. Using the formula for present value (PV = FV / (1 + r)^[n]), where 'r' is the interest rate and 'n' is the number of periods (or years), the PV works out as \(PV = 1050 / (1 + 0.05)^1 = 1000\)

Calculate capital gain or loss

To calculate the capital gain or loss, the PV of the bond is compared with the initial investment. The capital gain or loss is the difference between the present value (selling price) and the price paid for the bond one year ago. Therefore, capital gain (or loss) = selling price - purchase price = \(1000 - 900 = 100\)

Key concepts

Future Value

The future value (FV) of a bond is the amount it will be worth at a given future date, including the bond's face value and any additional interest accrued. In most cases, an investor is interested in knowing how much money they will get at maturity. Maturity is when the bond's terms are fulfilled, and all payments are made. For the bond in this scenario, the future value is calculated by adding the face value of the bond with its promised interest payment. In this case:

• The bond has a face value of \(1000\) dollars.
• There is also a promised additional interest payment of \(50\) dollars.

When combined, the future value amounts to \(1050\) dollars. Knowing the future value is crucial, as it provides the investor with a clear picture of what to expect when the bond matures.

Present Value

The present value (PV) helps determine how much a future sum of money is worth today, given a specific interest rate. This involves discounting, which adjusts the future value down to a present value by considering the interest rate over time. The formula typically used is:\[PV = \frac{FV}{(1 + r)^n}\]where:

• \(FV\) is the future value of the bond (\(1050\) dollars in this example).
• \(r\) is the interest rate (\(0.05\) or 5% per year).
• \(n\) is the number of periods or years (\(1\) year in this scenario).

After plugging in the values, we calculate the present value as \(1000\) dollars. Understanding present value is essential when assessing whether a bond or other investment is worth its current cost, based on future payout.

Capital Gain

Evaluating a capital gain involves comparing the selling price of an investment with its purchase price. If the item is sold for more than it was paid for, there is a gain. If sold for less, there is a loss. For bonds, this calculation determines the profit or loss made from the investment over time. In this particular scenario:

• The present value (or selling price) of the bond today is \(1000\) dollars.
• The purchase price of the bond one year ago was \(900\) dollars.

To find the capital gain or loss, subtract the initial investment from the present value:\[Capital\ Gain\ (or\ Loss) = 1000 - 900 = 100\]Thus, the capital gain in this instance is \(100\) dollars. Understanding capital gain is important, as it provides insight into the profitability of holding an investment over time. For students and investors alike, capital gain calculations indicate how well an investment has performed.