Question

When a security has a maturity value of \(\$ 100,000\) and the coupon rate is \(5 \%\), what are you willing to pay for it when it is due next year and the yield is \(9 \%\) ?

Short answer

If the yield is \(9 \% \), you are willing to pay \(\$96,330.28\) for the bond today when it is due next year.

Step-by-step solution

Determine the Coupon Payment

First, we need to find the annual coupon payment, which is based on the maturity value and coupon rate. The coupon payment can be calculated as follows: Coupon Payment = Maturity value × Coupon rate Remember that the percentage should be converted to its decimal equivalent: Coupon rate = \(5\% = 0.05\)

Calculate the Coupon Payment

Now, we can calculate the Coupon Payment using the above formula. Coupon Payment = \(100,000 \times 0.05=\$5,000\) The annual coupon payment is \(\$5,000\).

Determine the Present Value of the Coupon Payment and Maturity Value

Now, we will find the present value of the bond using the yield as the discount rate. Present value (PV) is calculated as follows: PV = Cash Flows / (1 + Discount Rate)^n Where Cash Flows represent the coupon payment and maturity value, Discount Rate is the yield, and n is the number of years until maturity. In our case, n = 1. We will calculate the present value of both the coupon payment and the maturity value.

Calculate the Present Value of the Bond

1. Calculate the present value of the coupon payment: PV_Coupon = \(5,000 / (1 + 0.09)^1 =\$4,587.16\) 2. Calculate the present value of the maturity value: PV_Maturity = \(100,000 / (1 + 0.09)^1 =\$91,743.12\) 3. Add the present values of the coupon payment and maturity value to determine the total present value of the bond: Total Present Value = PV_Coupon + PV_Maturity =\(4,587.16+\$91,743.12=\$96,330.28\) Hence, if the yield is \(9 \%\), you are willing to pay \(\$96,330.28\) for the bond today when it is due next year.

Key concepts

Present Value

The present value (PV) is a critical concept in bond valuation and finance in general. It represents how much a future sum of money is worth today, given a specific discount rate. Present value calculations help investors determine the worth of receiving money in the future in today's terms.
In the context of bonds, the present value allows us to determine how much we should be willing to pay for a bond today, considering the future cash flows from the coupon payments and the final maturity value.
The formula for calculating the present value of a bond's cash flows is:

• PV = Cash Flows / (1 + r)^n

Where:
* Cash Flows include both coupon payments and maturity value.
* "r" is the discount rate, which can be thought of as the yield rate for the bond.
* "n" is the number of periods until the cash flow occurs.
This equation helps put future payment amounts in terms of today’s values so that you can assess whether an investment is sound.

Coupon Payment

Coupon payments are the interest payments that a bondholder receives for holding a bond. They are typically paid annually or semi-annually.
The amount of the coupon payment depends on the coupon rate, which is agreed upon when the bond is issued, and the face value of the bond.
The formula to find the annual coupon payment is:

• Coupon Payment = Maturity Value × Coupon Rate

In our example, the coupon payment for a bond with a maturity value of $100,000 and a coupon rate of 5% is $5,000.
These payments are critical for investors, as they represent a return on investment and affect the overall yield of the bond.

Yield Rate

The yield rate, often called simply "yield," is the expected rate of return on a bond. It represents the income an investor can expect from the bond, usually expressed as a percentage.
There are different kinds of yields, such as current yield and yield to maturity (YTM), but in bond valuation, the yield rate is often used as a discount rate to calculate the present value of future cash flows.
The yield rate impacts how much an investor is willing to pay for a bond.
If the yield is higher than the coupon rate, the bond will typically sell for less than its face value, as was in our example where the yield is 9% compared to a 5% coupon rate. This happens because investors demand a higher return due to changes in market conditions, inflation, or interest rates.
Understanding the yield rate helps determine the appropriate price for bonds.

Discount Rate

The discount rate is used to calculate the present value of future cash flows. When considering bond valuation, the discount rate is often synonymous with the yield rate.
This rate factors in things like risk, inflation, and opportunity costs. In a bond valuation scenario, the discount rate reflects the required rate of return on the bond investment.
To calculate present values, you'll often use a discount rate that represents the yield on similar securities. In our example, the yield of 9% serves as the discount rate, meaning it reflects the minimum rate of return investors expect.

• A higher discount rate results in a lower present value, indicating a lower price you'd be willing to pay for future cash flows.
• A lower discount rate increases the present value, suggesting a willingness to pay more for the bond.

Choosing an accurate discount rate is crucial, as it ties into the broader valuation and pricing strategies for bonds.