Bond Prices vs Yields: Why They Move in Opposite Directions
When rates go up, bond prices go down. When rates go down, bond prices go up. This inverse relationship isn't coincidence—it's fundamental financial principle. Understand it completely: from seesaw analogy to mathematical proof to real-world cases.
⏱ 10 min read📊 3 real examples🧮 Interactive calculator checks
The Seesaw Principle — Intuitive Understanding
The easiest way to understand the inverse bond price-rate relationship is the seesaw analogy. When one side goes up, the other goes down. Similarly: when rates rise, bond prices fall; when rates fall, bond prices rise.
Concrete Example: You bought a bond one year ago paying 3% interest for $10,000. Now the bank interest rate has risen to 5%. New bond buyers receive 5% coupons. Your 3% bond becomes relatively unattractive and must be discounted in the secondary market to compete with new 5% offerings. You'd need to sell it for less than $10,000—perhaps $9,500—to compensate buyers for the lower coupon.
Core Principle
Bond markets feature competition between new offerings (higher rates) and existing bonds (lower rates). Existing bonds must lower prices to maintain competitive yields.
Mathematical Understanding — The PV Formula
The inverse bond price-rate relationship is evident in the Present Value formula, which calculates bond prices. A bond's price equals the PV of all future cash flows (coupons + principal).
P = C/(1+r) + C/(1+r)² + ... + C/(1+r)ⁿ + FV/(1+r)ⁿ P: Bond Price | C: Annual Coupon | r: Market Yield (Discount Rate) | n: Years | FV: Par Value
The critical point: the discount rate (r) is in the denominator. When the denominator grows (rates rise), the overall fraction shrinks (prices fall). When the denominator shrinks (rates fall), the fraction grows (prices rise).
📊 Concrete Calculation Example
Bond Terms: Par $10,000, 5% coupon, 3-year maturity
Annual Coupon: $10,000 × 5% = $500
1Market Yield 3%:
P = $500/(1.03) + $500/(1.03)² + $500/(1.03)³ + $10,000/(1.03)³
P = $485.44 + $471.30 + $457.58 + $9,151.42 = $10,565.74
2Market Yield 7%:
P = $500/(1.07) + $500/(1.07)² + $500/(1.07)³ + $10,000/(1.07)³
P = $467.29 + $436.69 + $408.15 + $8,162.71 = $9,474.84
Maturity and Sensitivity — Why Long Bonds React More Dramatically
The same 1% rate change affects different bonds differently. Longer-maturity bonds are more price-sensitive to rate changes than shorter-maturity bonds. This sensitivity measure is called Duration.
Why Are Long Bonds More Sensitive?
Long bonds have cash flows further in the future. When the discount rate changes, far-future cash flows are impacted much more dramatically than near-term cash flows.
Example: $1,000 receivable in 10 years is affected far more by a 1% rate change than $1,000 receivable in 1 year.
Condition
3-Year Bond
10-Year Bond
30-Year Bond
Coupon
4%
4%
4%
Par Value
$10,000
$10,000
$10,000
Yield 3% Price
$10,281
$10,855
$12,256
Yield 4% Price
$10,000
$10,000
$10,000
Yield 5% Price
$9,723
$9,209
$8,207
1% Rise Loss
$277 (2.7%)
$791 (7.9%)
$1,793 (17.9%)
1% Decline Gain
$281 (2.7%)
$855 (8.6%)
$2,256 (22.6%)
Conclusion: A 30-year bond experiences 17%-23% price changes from a 1% rate move, while a 3-year bond experiences only 2.7% changes. This is why long-term bonds are higher-risk, higher-reward assets.
Par $10,000, 3% coupon (annual $300), 30-year maturity. Market yields drop from 4% to 3%. Calculate the price at each yield level and see the duration effect.
Yield 4%: Price ≈ $7,800 (simplified)
Yield 3%: Price ≈ $10,000
Price Gain: $10,000 - $7,800 = About $2,200 (28% gain!)
Interpretation: A simple 1% rate decline caused the 30-year bond to gain 28%. This is why long-term bonds can deliver spectacular returns during rate-cutting cycles.
Problem 3: Rate Rise Scenario — Calculating Losses
You bought a 10-year, 4% coupon bond for $10,000 at 4% yield. Rates rise to 5%. (1) What's the new bond price? (2) What's your loss percentage? (3) Can you avoid this loss?
New Bond Price at 5% Yield:
Annual coupon $400 discounted at 5% for 10 years
P ≈ $400 × 7.72 + $10,000/1.05¹⁰ ≈ $3,088 + $6,139 ≈ $9,383
Loss: $10,000 - $9,383 = $617 (about 6.2% loss)
(3) Avoiding the Loss:
- If you hold to maturity (10 years), you get $10,000 back, and coupon payments ($4,000 total) offset the loss
- If you must sell before maturity, the loss is realized
→ Strategy: Buy bonds only if you can hold to maturity, or expect rates to fall
Portfolio Strategy Implications
If you expect rising rates: reduce long-term bond holdings. If you expect falling rates: increase long-term bond holdings. Bonds aren't just "safe" assets—they're tactical assets that profit from rate predictions.