Bond Pricing Basics

The Present Value Principle of Bond Pricing

At its core, a bond is a type of loan made by an investor to a borrower (typically a corporation or government). In return for the loan, the borrower promises to make periodic interest payments, known as coupon payments, and to repay the original amount of the loan, called the face value or par value, at a specified maturity date.

The price an investor is willing to pay for a bond today is simply the sum of the present values of all the future cash flows they expect to receive. This includes all the future coupon payments and the final repayment of the face value. The rate used to discount these future cash flows back to their present value is the current market interest rate, also known as the yield to maturity (YTM). This rate reflects the return available on other, similar investments in the market.

Before diving into the full formula, let's review these foundational concepts.

Deriving the Bond Pricing Formula

We can express the price of a bond ($V$) as the sum of the present values of its cash flows:

Where:

• $C$ is the periodic coupon payment.
• $F$ is the face value of the bond.
• $r$ is the market interest rate per period.
• $T$ is the number of periods until maturity.

The stream of coupon payments forms an annuity. As you know from your work with probability and series, we can simplify the sum of the coupon payments using the formula for a finite geometric series. The following reading walks through this derivation, which should appeal to your preference for mathematical formalism.

The derivation shows that the expanded sum elegantly simplifies to the following standard bond valuation formula:

Notice that this formula neatly separates the price into two distinct components:

1. Present Value of the Coupon Annuity: $C\times \left[\frac{1}{r}\left(1-\frac{1}{(1+r{)}^T}\right)\right]$
2. Present Value of the Face Value: $F\times \left[\frac{1}{(1+r{)}^T}\right]$

This structure is not just a mathematical convenience; it reflects the two sources of value a bond offers its holder.

A Step-by-Step Calculation: Annual Coupons

Now let's apply this formula to a concrete example. The following video provides a clear, methodical walkthrough of a bond pricing problem from start to finish, assuming annual coupon payments.

Let's recap the example from the video:

• Face Value (F): $1,000
• Coupon Rate: 5% (annual payments)
• Maturity (T): 20 years
• Market Interest Rate / YTM (r): 7.5% or 0.075

First, the annual coupon payment is calculated: $C=0.05\times \$1000=\$50$.

Plugging these values into our formula:

As the video meticulously demonstrates, calculating this yields a price of approximately $745.14. Notice that because the market interest rate (7.5%) is higher than the bond's coupon rate (5%), the bond sells for less than its face value. This is known as selling at a discount. Investors are only willing to buy a bond with below-market coupons if they can get it at a cheaper price.

Adjusting for Semi-Annual Coupons

In practice, most government and corporate bonds pay coupons twice a year (semi-annually). This requires three simple adjustments to our formula's inputs:

1. Number of periods (T): Multiply the years to maturity by 2.
2. Market interest rate (r): Divide the annual market rate by 2.
3. Coupon payment (C): Divide the annual coupon payment by 2.

The following video demonstrates how to make these adjustments.

Practice Problems

Now it's your turn to apply what you've learned. The best way to solidify this is through practice. Below are two problems. In both cases, assume a face value of $1,000, a coupon rate of 6%, and semi-annual coupon payments.

Problem 1:

• Years to maturity = 5
• Market interest rate = 5%

Problem 2:

• Years to maturity = 5
• Market interest rate = 8%

Please take a few minutes to solve for the price of each bond. Remember to adjust your inputs for semi-annual compounding.

Solution Walkthrough (Problem 2):

Let's work through the solution for the second problem together.

• Face Value (F): $1,000
• Annual Coupon Rate: 6%
• Annual Market Rate: 8%
• Years to Maturity: 5
• Payments: Semi-annual

First, we adjust our inputs:

• $T=5\text{ years}\times 2=10\text{ periods}$
• $r=8\%/2=4\%=0.04$
• Annual coupon = $0.06\times \$1000=\$60$. So, the semi-annual coupon payment is $C=\$60/2=\$30$.

Now, we plug these into the formula:

Let's calculate the two parts:

1. PV of Face Value: $\frac{1000}{(1.04{)}^{10}}=\frac{1000}{1.480244}\approx \$675.56$
2. PV of Coupon Annuity: $\frac{30}{0.04}\left(1-\frac{1}{1.480244}\right)=750\times (1-0.67556)=750\times 0.32444=\$243.33$

Finally, we sum them up:

The price of the bond is $918.89. For further practice, the document below contains a large set of problems with fully worked-out solutions.

Conclusion

In this lesson, we established the fundamental process for valuing a bond. You learned that a bond's price is the net present value of its expected future cash flows, discounted by the current market interest rate.

Key Takeaways:

• A bond's price is determined by its face value ($F$), coupon payment ($C$), time to maturity ($T$), and the market interest rate ($r$).
• The valuation formula, $V=\frac{C}{r}\left(1-\frac{1}{(1+r{)}^T}\right)+\frac{F}{(1+r{)}^T}$, represents the sum of the present values of the coupon annuity and the face value.
• For bonds with semi-annual payments, you must adjust $T$, $r$, and $C$ by dividing or multiplying by two before using the formula.
• There is a crucial inverse relationship between a bond's price and the market interest rate. When the market rate rises, the price of existing bonds falls, and vice-versa.

This final point is the bridge to our next topic. Now that you can calculate bond prices, you are equipped to understand how interest rates are determined in the broader economy. In our next lesson, we will develop a model of the money market, where the supply of and demand for money interact to set the economy's equilibrium interest rate.