Understanding Zero-Coupon Bonds in Excel Formulas

A zero-coupon bond is a type of bond that does not pay any interest to the investor during its term. Instead, the investor buys the bond at a lower price than its face value, which is the amount that the bond will pay at maturity. The difference between the purchase price and the face value is the investor’s return or profit from holding the bond.

For example, suppose an investor buys a zero-coupon bond with a face value of $1,000 and a maturity of 10 years. The investor pays $500 for the bond today, which means the bond is sold at a 50% discount. After 10 years, the investor will receive $1,000 from the bond issuer, which is the face value of the bond. The investor’s return is $500, which is the difference between the purchase price and the face value.

Zero-coupon bonds are also known as pure discount bonds or deep discount bonds. They are usually issued by governments, corporations, or financial institutions that need to raise funds for long-term projects. Zero-coupon bonds have some advantages and disadvantages for both the issuers and the investors.

Some of the advantages of zero-coupon bonds are:

• They are easy to calculate and understand, as they do not involve any periodic payments or complex formulas.
• They are attractive to investors who want to lock in a fixed rate of return and do not need regular income from their investments.
• They are less sensitive to changes in interest rates than coupon bonds, as they do not have any reinvestment risk. Reinvestment risk is the risk that the investor will have to reinvest the coupon payments at a lower interest rate than the original bond.

Some of the disadvantages of zero-coupon bonds are:

• They are more volatile than coupon bonds, as they have a longer duration. Duration is a measure of how much the bond’s price changes when the interest rate changes. The longer the duration, the more the bond’s price fluctuates.
• They are subject to higher taxes than coupon bonds, as the investor has to pay taxes on the imputed interest. Imputed interest is the interest that the bond is assumed to earn, even though it does not pay any actual interest. The IRS requires the investor to report the imputed interest as income every year, even though the investor does not receive any cash until maturity.
• They have a lower liquidity than coupon bonds, as they are less traded in the secondary market. Liquidity is the ability to buy or sell a bond quickly and easily without affecting its price. The lower the liquidity, the higher the risk and the cost of trading the bond.

Basic Theory

The basic theory of zero-coupon bonds involves the following key components:

1. Face Value (FV): The future value of the bond, which is paid at maturity.
2. Purchase Price (PV): The current price at which the bond is purchased, which is less than the face value.
3. Maturity Period (n): The number of periods until the bond matures.

The formula for calculating the present value of a zero-coupon bond is given by:

Where:

• PV is the present value (purchase price) of the bond.
• FV is the face value of the bond.
• r is the periodic interest rate.
• n is the number of periods until maturity.

Procedures for Excel Calculation

1. Set Up Excel Table:
   • Column A: Periods (time to maturity)
   • Column B: Discount Rate (interest rate)
   • Column C: Present Value
2. Input Data:
   • Enter the face value of the bond in a cell (let’s say A1).
   • Enter the number of periods (maturity) in another cell (A2).
   • Choose a discount rate (interest rate) and enter it in a cell (B1).
3. Formula for Present Value:
   • In cell C2, enter the formula .
   • Drag this formula down for all periods.

Comprehensive Explanation – Real Numbers Scenario

Let’s consider a scenario with the following values:

• Face Value (): $1,000
• Maturity Period (): 5 years
• Discount Rate (): 4%

Excel Calculation

1. Set Up Excel Table:
   • Column A (Periods): 1 to 5
   • Column B (Discount Rate): 0.04
   • Column C (Present Value): Use the formula
2. Result:
   • For each period, calculate the present value using the formula.
   • The present values will decrease over time due to the discounting effect.

Result

The present values represent the purchase price of the zero-coupon bond at each period. In this scenario, the investor would pay $821.74 to purchase a $1,000 face value bond with a 4% discount rate over 5 years.

Other Approaches

1. Using Excel Functions:
   • Excel provides functions like PV (Present Value) and RATE for financial calculations. You can use these functions to simplify the calculation process.
2. Graphical Representation:
   • Create a line chart to visually represent the decrease in present value over time. This can enhance understanding and analysis.
3. Sensitivity Analysis:
   • Explore the impact of different discount rates and maturity periods on the present value. Use data tables or Excel’s Goal Seek function for sensitivity analysis.