Question

Suppose you are considering buying some United States Treasury Bonds, say a \(\$ 1,000,000,6\) year bond, which has a coupon rate of \(3.5 \%\), and is offered after half a year for \(\$ 775,000 ;\) your opportunity cost of capital is \(7 \%\). Would you invest in such a bond?

Short answer

The annual coupon payment for a $\$1,000,000$ Treasury Bond with a coupon rate of $3.5\%$ is $\$35,000$. The present value of the coupon payments is approximately $\$179,776.76$, and the present value of the face value is approximately $\$666,287.07$. The total present value of the bond's cash inflows is approximately $\$846,063.83$, which is greater than the bond's offered price of $\$775,000$. Therefore, it would be a good decision to invest in this Treasury Bond, as its present value is greater than the price (opportunity cost) being offered.

Step-by-step solution

Calculate the annual coupon payment

First, we'll find the annual coupon payment, which is the interest paid by the Treasury Bond to the bondholder. We can use the following formula: Annual Coupon Payment = (Coupon Rate × Bond Face Value) Using the given coupon rate (3.5%) and the bond face value ($1,000,000), we can calculate the annual coupon payment as follows: Annual Coupon Payment = \(0.035 * 1,000,000 = $35,000\)

Calculate the present value of the coupon payments

To calculate the present value of the coupon payments, we'll use the formula: Present Value = \(\sum_{t=1}^{N} \frac{C}{(1 + r)^t}\) Where: - t refers to the period (years) - N is the number of years to maturity (6 years) - C is the annual coupon payment - r is the opportunity cost of capital Therefore, the present value of the coupon payments is: PV(Coupon Payments) = \(\sum_{t=1}^{6} \frac{35,000}{(1 + 0.07)^t}\) Calculating each component and summing them gives us: PV(Coupon Payments) = \(35,000*(1 + 0.07)^{-1} + 35,000*(1 + 0.07)^{-2} +...+ 35,000*(1 + 0.07)^{-6} ≈ \$179,776.76\)

Calculate the present value of the face value

The face value of the investment will be paid out at the end of the 6 years. To calculate its present value, we'll use the formula : Present Value = \(\frac{F}{(1+r)^N}\) Where F is the face value of the bond and N is the number of years to maturity. PV(Face Value) = \(\frac{1,000,000}{(1 + 0.07)^6} ≈ \$666,287.07\)

Sum up the present values and compare to the bond's price:

Now, we'll sum up the present values of the coupon payments and the face value to determine the total present value of the bond's cash inflows. Total Present Value = PV(Coupon Payments) + PV(Face Value) = \(179,776.76 + \)666,287.07 ≈ $846,063.83 Let's now compare the total present value to the bond's price to determine if it's a good investment.

Decide if the bond is a good investment

The bond is offered at a price of \(775,000. If the total present value of the bond's cash inflows (\)846,063.83) is greater than the bond's price ($775,000), then the investment is worth considering. \(846,063.83 > \)775,000 Therefore, it would be a good decision to invest in this Treasury Bond as its present value is greater than the price (opportunity cost) being offered.

Key concepts

Understanding Coupon Rate

Among the key concepts to understand when considering an investment in United States Treasury Bonds is the coupon rate. The coupon rate signifies the annual interest rate the bond issuer will pay to the bondholders. It is usually expressed as a percentage of the bond's face value, which is the amount that will be returned to the bondholder at maturity. For example, a Treasury Bond with a face value of \(1,000,000 and a coupon rate of 3.5% will result in an annual interest payment of \)35,000 to the bondholder, regardless of the bond's price on the market.

This fixed payment is crucial for investors looking for regular income, as it determines the yearly cash flow they can expect from holding the bond. When calculating whether or not to purchase a bond, factoring in the coupon rate allows an investor to assess the bond's annual earnings against other potential investments. It's essential to bear in mind that while the coupon rate remains constant, the actual yield an investor receives can vary depending on the bond's market price.

Calculating Present Value

The present value calculation is a foundational pillar of finance, helpful in comparing the value of cash flows received at different times. It is based on the principle that money available today is worth more than the same amount in the future due to its potential earning capacity, often articulated as 'a dollar today is worth more than a dollar tomorrow. To calculate the present value of future cash flows from a Treasury Bond, an investor must discount each payment by the opportunity cost of capital, which is typically the rate of return expected from the best alternative investment adjusted for risk.

Mathematical Formula for Present Value:

The formula for present value of a series of cash flows (\(C\)) paid at regular intervals is given by:

PV = \sum_{t=1}^{N} \frac{C}{(1 + r)^t}

where:

• \(t\) represents the time period,
• \(N\) is the total number of periods,
• \(C\) is the cash flow per period, and
• \(r\) is the discount rate, equivalent to the opportunity cost of capital.

By applying this formula to each coupon payment and the face value payment at the bond's maturity, an investor can sum them to obtain the total present value of the bond's future cash flows. This present value can then be compared to the bond's current price to determine if the investment is attractive.

Opportunity Cost of Capital

When contemplating any investment, understanding opportunity cost of capital is critical. This concept refers to the returns an investor forgoes from the next best investment alternative when they choose a specific investment. It is the benchmark rate that investors use to compare different investments with varying degrees of risk.

Why It's Important:

Considering the opportunity cost helps ensure that capital is allocated most efficiently. For instance, investing in a bond offering a return of 3.5% when there are alternatives yielding 7% wouldn't make financial sense unless the bond's risk profile significantly differs from that of the alternative investments.

In our bond example, the opportunity cost of capital is set at 7%. It means that any investment should yield at least this rate to be considered over keeping the capital invested in the next best alternative. It's used in the present value calculation to discount future cash flows from the bond. Essentially, by computing the present value of the bond's cash flows at the opportunity cost of capital, investors can objectively assess whether the Treasury Bond is an attractive investment compared to other opportunities given their risk and returns profile.