Question

A bond has two years to mature. It makes a coupon payment of \(\$ 100\) after one year and both a coupon payment of \(\$ 100\) and a principal repayment of \(\$ 1000\) after two years. The bond is selling for \(\$ 966 .\) What is its effective yield?

Short answer

The bond's effective yield or yield to maturity is approximately 7%

Step-by-step solution

Understand the problem

A bond is a financial instrument that pays periodic interest payments and returns the principal at the maturity date. The problem gives coupon payments of \$100 at the end of first and second years, and a principal repayment of \$1000 at the end of second year. The price of the bond is \$966, which is the present value of future payments. The yield or discount rate, is the interest rate that makes the present value of the bond’s payments equal to its price.

Formulate Equation for Yield

The present value of the bond's payments is calculated using the formula: \[PV = \sum_{i=1}^{n} \frac{C}{(1+y)^i} + \frac{M}{(1+y)^n}\] Where PV = Present value or price of the bond = \$966, C = annual coupon payment = \$100, M = Maturity value or face value of the bond = \$1000, y = yield to maturity which we aim to find, n = number of years to maturity = 2 years. Given y is the same for the 2 periods, we can write: \[ \$966 = \frac{\$100}{(1+y)} + \frac{\$100 + \$1000}{(1+y)^2}\] Solve it to find the yield 'y'.

Solve the equation

Solving this equation might be difficult algebraically due to nature of the equation. Therefore, a numerical approach such as bisection method or trial and error can be used to find the yield. The solution falls around 0.07.

Key concepts

Financial Instruments

To begin our exploration of financial concepts, let's start with the basics: financial instruments. These instruments are the building blocks of the financial markets and can take many forms, such as stocks, bonds, derivatives, and currencies.

In the context of our exercise, the financial instrument in question is a bond. Bonds are a type of fixed-income investment that represents a loan made by an investor to a borrower, typically a corporation or government. A bond detail includes the end date when the principal of the loan is due to be paid back, which is known as its maturity date, and the fixed interest payments, often referred to as coupon payments.

Understanding bonds is crucial as they're a stable and predictable form of investment compared to others. However, the key aspect that affects a bond's value is the interest rate environment. If interest rates rise, new bonds are issued at these higher rates, making existing bonds with lower coupon rates less attractive, hence their prices decrease.

Yield to Maturity

When we consider the whole lifespan of a bond, we encounter a core concept: yield to maturity (YTM). YTM is the total return anticipated on a bond if the bond is held until it matures. It's a complex calculation that includes the bond's current market price, its nominal interest rate, the amount and frequency of interest payments, and the time remaining until the bond reaches its maturity date.

YTM is an essential metric for investors because it enables them to compare bonds with different prices and coupon rates on an equal footing. It's expressed as an annual rate and reflects all the future coupon payments, which the bond will make from the time of the calculation until maturity, and the difference between the bond's current price and its face value.

Present Value Calculation

Delving deeper, the concept of present value (PV) is a cornerstone of finance. It is predicated on the principle that money available today is worth more than the same amount in the future due to its potential earning capacity. This core principle allows investors to assess the value of a bond or any future cash flows in today's terms.

Applying the present value to our bond example, we calculate the present value of the bond's future payments by discounting them back to today's dollars. This involves using a discount rate, which is the yield to maturity in our case. The present value formula is: \[ PV = \sum_{i=1}^{n} \frac{C}{(1+y)^i} + \frac{M}{(1+y)^n} \] for a series of cash flows (C), over a period (n), discounted by the YTM (y), and adding the bond's face value (M) at the end. Investors use this calculation to determine if the bond's price is fair based on its risk and return. A bond's price fluctuates with changes in the interest rate environment and the issuer's creditworthiness but the YTM allows investors to decide whether these changes are adequately compensated by the bond's payments.