Question

A zero-coupon bond can be redeemed in 20 years for \(\$ 10,000\). How much should you be willing to pay for it now if you want a return of: (a) 5 % compounded monthly? (b) 5 % compounded continuously?

Short answer

(a) 3678.79 dollars. (b) 3678.79 dollars.

Step-by-step solution

Define the Variables

Let the future value (FV) be \(\text{FV} = 10,000 \) dollars, the time to maturity (t) be 20 years, and the desired annual interest rate (r) be 5%. We will calculate the present value (PV) using different compounding frequencies.

Calculate PV for Monthly Compounding

For monthly compounding, use the formula \(\text{PV} = \text{FV} / \big(1 + \frac{r}{n}\big)^{nt} \), where n is the number of compounding periods per year. Here, n = 12.

Plug in the Numbers for Monthly Compounding

Substitute the values into the formula: \(\text{PV} = 10,000 / \big(1 + \frac{0.05}{12}\big)^{12 \times 20} \). Calculate the result: \(\text{PV} = 10,000 / \big(1 + \frac{0.05}{12}\big)^{240} \).

Present Value for Monthly Compounding

Calculate the exponent and division: \(\text{PV} \approx 10,000 / (1.004167)^{240} \approx 10,000 / 2.71864 \approx 3678.79 \).

Calculate PV for Continuous Compounding

For continuous compounding, use the formula \(\text{PV} = \text{FV} \times e^{-rt} \), where e is the base of the natural logarithm.

Plug in the Numbers for Continuous Compounding

Substitute the values into the formula: \(\text{PV} = 10,000 \times e^{-0.05 \times 20} \). Calculate the exponent: \(\text{PV} = 10,000 \times e^{-1} \).

Present Value for Continuous Compounding

Calculate the result: \(\text{PV} = 10,000 \times 0.367879 \approx 3678.79 \).

Key concepts

Zero-Coupon Bond

A zero-coupon bond is a type of bond that does not pay periodic interest (coupons). Instead, it is sold at a discount to its face value and only pays the face value at maturity. This makes it simple to calculate its return using the present value concept.
Zero-coupon bonds are ideal for long-term investors who do not need immediate income from their investments. They are also useful for financial planning, as the eventual payout is a known quantity.

• A zero-coupon bond is bought at a lower price than its face value.
• No periodic interest payments.
• The bond matures at a specific date when the face value is paid out.

Monthly Compounding

Monthly compounding is when interest is calculated and added to the principal balance of an investment every month. This means interest is earned on interest from previous months, leading to a compounding effect.
The formula for calculating the present value with monthly compounding is:
\[ \text{PV} = \frac{\text{FV}}{\big(1 + \frac{r}{n}\big)^{nt}} \]
Where:

• \text{PV} = Present Value
• \text{FV} = Future Value
• r = annual interest rate
• n = number of compounding periods per year (12 for monthly)
• t = time in years

Using monthly compounding enables investors to achieve a higher return than annual compounding due to the more frequent calculation and addition of interest.

Continuous Compounding

Continuous compounding calculates interest assuming it is being added an infinite number of times per year. It is the theoretical limit of the compounding process.
To calculate present value with continuous compounding, use the formula:
\[ \text{PV} = \text{FV} \times e^{-rt} \]
Where:

• e = approximately 2.71828, the base of the natural logarithm
• r = annual interest rate
• t = time in years

Continuous compounding provides a slightly higher return than any finite compounding frequency due to the constant addition of interest.

Present Value

The present value (PV) is the current worth of a future sum of money, given a specific rate of return. It helps to determine how much money should be invested now to achieve a future goal sum.
The main elements involved in calculating PV are:

• Future Value (FV): The amount of money in the future.
• Interest Rate (r): The rate at which money grows over time.
• Time (t): The time period over which the money is invested or considered.

PV is crucial for making informed financial decisions, enabling investors to compare different investment opportunities on a like-for-like basis.

Interest Rate

The interest rate (r) is a critical component in the calculation of the present and future value of money. It represents the rate at which money accrues interest over time.
Interest rates can be:

• Fixed: Unchanging throughout the investment period.
• Variable: Fluctuating based on market conditions.

The annual interest rate is used in various formulas to determine how much an investment will grow. Higher interest rates lead to greater returns over time, magnifying the effects of compounding. Understanding how interest rates work is essential for optimizing investments and planning future financial goals.