Problem 61 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Juan is contemplating buying a zero coupon bond that matures in \(10 \mathrm{yr}\) and has a face value of $$\$ 10,000 .$$ If the bond yields a return of \(5.25 \% /\) year, how much should Juan pay for the bond?

Step-by-Step Solution

Answer

Juan should pay approximately $5,837.63 for the zero coupon bond to achieve a yield of 5.25% per year.

Step 1: Write down the present value formula

The present value (PV) formula is given by: \[PV = \frac{FV}{(1 + r)^n}\] Where: - PV is the present value (the price Juan should pay) - FV is the face value of the bond - r is the yield rate per year (as a decimal) - n is the number of years until the bond matures

Step 2: Convert the given yield rate to a decimal

The yield rate is given as 5.25% per year. To use this in the formula, we need to convert it to a decimal: \[r = \frac{5.25}{100} = 0.0525\]

Step 3: Plug the given values into the present value formula

We are given: - Face value (FV) = $10,000 - Yield rate as a decimal (r) = 0.0525 - Number of years until maturity (n) = 10 years Plug these values into the present value formula: \[PV = \frac{10,000}{(1 + 0.0525)^{10}}\]

Step 4: Calculate the present value

Now, we will simplify the expression to find the present value (PV). \[PV = \frac{10,000}{(1.0525)^{10}}\] \[PV = \frac{10,000}{1.71336}\] \[PV \approx 5837.63\]

Step 5: Interpret the result

Juan should pay approximately $5,837.63 for the zero coupon bond in order to achieve a yield of 5.25% per year.

Key Concepts

Time Value of Money

The principle of the time value of money is a fundamental concept in finance which holds that a sum of money is worth more now than the same sum will be in the future. This is due to the potential earning capacity of money. In essence, a dollar today has the potential to earn interest, making it more valuable than a dollar tomorrow.

For example, if you have \(100 today and you can invest it at an interest rate of 5% per annum, in a year's time you could have \)105. Therefore, \(100 today is worth more than \)100 in the future because of its potential to grow. This concept is crucial when it comes to investment decisions, such as knowing how much to pay for a financial instrument like a bond, so that the return accounts for this time value.

The formula to calculate the present value, given the future value, the interest rate, and the number of periods, reflects this concept. In context, if Juan wants to know the true worth of a zero coupon bond that will pay out in the future, he needs to discount that future value to its present value.

Financial Mathematics

Financial mathematics is the application of mathematical methods to solve problems related to finance. It encompasses a range of topics from simple interest calculations to more complex topics like derivatives pricing and risk management. The use of algebra, calculus, probability, and statistics helps in understanding how money flows over time.

One application of financial mathematics is bond valuation, which employs formulas to determine the fair price of a bond. The present value formula used by Juan is an example of a financial mathematics tool. It takes into account the yield rate and the time until the bond matures to calculate what the bond is worth today, allowing Juan and other investors to make informed decisions.

Bond Valuation

Bond valuation is the process of determining the fair price of a bond. Unlike stocks, bonds come with predefined terms that include the face value (the principal amount), the maturity date (when the face value is due), and the coupon rate (the interest paid, which may be zero in the case of a zero coupon bond).

Valuation of Zero Coupon Bonds

To evaluate a zero coupon bond, we only need to consider the face value and the discount rate that corresponds to the bond's yield. Since zero coupon bonds do not make periodic interest payments, the investor's return is the difference between what they pay initially (the present value) and the face value received at maturity.

Juan calculated the present value to determine what to pay now for the future payoff. The key here is that he pays less than the face value to make a profit. An understanding of bond valuation helps investors in assessing whether the bond is a good investment, given the time until maturity and the prevailing interest rates.

Yield Rate Conversion

Yield rate conversion is essential when working with financial formulas as it involves converting percentage yield rates into decimal form. This is necessary for correct formula application. For instance, a yield rate of 5.25% per year needs to be converted to 0.0525 when used in a mathematical formula.

For Juan, accurately converting the percentage to a decimal is crucial in determining the present value of the zero coupon bond. If he or any other investor fails to convert appropriately, the valuation will be incorrect, potentially leading to a misinformed financial decision.

Understanding how to make this conversion allows investors to switch between percentage yields, which are often quoted in financial news and reports, and decimal rates, which are required by various financial calculations. This knowledge is part of the basic toolkit for anyone dealing in bond markets or any other form of financial investment.