Why Do US Treasury Bond Yields and Prices Have an Inverse Relationship?

US Treasury Bonds are among the most important financial instruments globally, and understanding the relationship between their yields and prices is crucial for investors. This article will explain in detail why US Treasury bond yields and prices exhibit an inverse relationship.

Basic Concepts

• Bond Price: The actual amount investors pay when purchasing a bond
• Face Value: The amount the government promises to pay when the bond matures (usually $100)
• Coupon Rate: The fixed annual interest payment rate of the bond
• Market Yield: The current yield level of similar bonds in the market

The Principle of Inverse Relationship

Suppose there’s a 10-year US Treasury bond with a face value of $100 and a coupon rate of 3$3 in interest annually. When market interest rates change:

1. When market rates rise:

   • Newly issued bonds offer higher interest
   • To make old bonds competitive, their prices must fall
   • This allows buyers to obtain yields comparable to new bonds through lower purchase prices

2. When market rates fall:

   • Newly issued bonds offer lower interest
   • Existing bonds with higher coupon rates become more attractive
   • Investors are willing to pay higher prices to purchase these higher-yielding old bonds

Numerical Example

Let’s illustrate the relationship between bond prices and yields through a detailed example:

Suppose you hold a 10-year US Treasury bond:

• Face Value: $100
• Coupon Rate: 3%
• Annual Interest Payment: $3
• Maturity: 10 years

When market rates rise to 4%, we can calculate the new bond price using the bond pricing formula:

$P = C \times \frac{1 - (1 + r)^{-n}}{r} + F \times (1 + r)^{-n}$

Where:

• P = Bond price
• C = Annual coupon interest ($3)
• r = Market rate (4% = 0.04)
• n = Remaining years (10 years)
• F = Face value ($100)

Substituting the values: $P = 3 \times \frac{1 - (1 + 0.04)^{-10}}{0.04} + 100 \times (1 + 0.04)^{-10}$ $= 3 \times \frac{1 - 0.67556}{0.04} + 100 \times 0.67556$ $= 3 \times 8.1109 + 67.556$ $= 24.3327 + 67.556$ = $91.89

This shows:

1. Bond price drops from $100 to$91.89, a decline of about 8.11%
2. New buyers purchasing this bond at $91.89 receive$3 annual interest
3. Actual yield = 3/91.89 + (100-91.89)/(91.89×10) = 4%
4. This way new buyers can obtain a 4% yield comparable to market rates

This example demonstrates how bond prices adjust to balance different market rates, reflecting the inverse relationship between yields and prices. It also explains why holding long-term fixed-rate bonds may face significant price risk in a rising interest rate environment.

Impact on Investors

1. Long-term Holders:

   • If planning to hold until maturity, price fluctuations have minimal impact
   • Will receive fixed coupon rate returns

2. Traders:

   • Need to closely monitor interest rate movements
   • Price fluctuations create trading opportunities
   • Also face greater risks

Conclusion

Understanding the inverse relationship between US Treasury yields and prices is crucial for investment decisions:

• When expecting interest rates to rise, be cautious about purchasing long-term bonds
• When expecting interest rates to fall, long-term bonds may offer capital appreciation opportunities
• Investment strategies should be formulated based on individual investment goals and market expectations

This inverse relationship is a fundamental principle of fixed-income markets, applicable not only to US Treasuries but also to other types of bond investments.