Bond Math Basics: Price/Yield Relationship

When interest rates rise, your bond portfolio drops in value—sometimes sharply. In 2022, the Bloomberg U.S. Aggregate Bond Index fell approximately 15.7% as the Federal Reserve hiked rates by 375 basis points, the worst calendar-year bond market performance in modern history. The practical antidote isn't avoiding bonds. It's understanding exactly how much your bonds move per unit of yield change—and sizing your exposure accordingly.

TL;DR: Bond prices and yields move in opposite directions. Duration tells you how much: multiply your bond's duration by the yield change, and you have your approximate price move. Longer maturity and lower coupons mean more sensitivity.

Why Prices and Yields Move in Opposite Directions (The Core Mechanism)

A bond's price is the present value of its future cash flows—coupon payments plus the return of par value at maturity—discounted at the current market yield. When yields rise, that discount rate increases, and the present value of those fixed cash flows falls. When yields drop, the discount rate decreases, and present value rises.

The point is: a bond's coupon payments are locked in at issuance. If new bonds offer higher yields, your existing bond's fixed payments become less attractive. The market adjusts by marking your bond's price down until its yield matches the new environment.

Here's the relationship as a chain: Market yields rise → discount rate increases → present value of fixed cash flows falls → bond price drops. Reverse every arrow when yields fall.

A bond trading at par (price = 100, or $1,000 per $1,000 face value) has its coupon rate exactly equal to the prevailing market yield. When the coupon rate falls below the market yield, the bond trades at a discount (below par). When the coupon rate sits above the market yield, it trades at a premium (above par).

Duration: Your Price-Sensitivity Ruler (How to Measure the Move)

Duration quantifies how much a bond's price changes for a given yield shift. Modified duration gives you the approximate percentage price change per 100 basis points (1 percentage point) of yield movement.

The formula is straightforward:

Approximate % price change ≈ −modified duration × yield change (in decimal)

A bond with a modified duration of 7.5 loses approximately 7.5% in price for a 100 bp yield increase. That's the entire calculation. Duration is your single most important number for gauging interest rate risk.

Why this matters: duration varies enormously across the yield curve. A 2-year Treasury has a duration of approximately 1.9 years—modest sensitivity. A 10-year Treasury runs about 7.5–8.5 years. A 30-year Treasury stretches to 17–20 years, carrying roughly 3–4 times the interest rate risk of the 10-year (and about 10 times that of the 2-year).

Worked Example: A 100 Basis Point Yield Increase (What Actually Happens to Your Bond)

Start with a $1,000 par, 4% coupon, 10-year bond priced at par (yield to maturity = 4.00%).

Phase 1: The setup. You hold this bond in your portfolio. It pays $40 per year ($20 semiannually), and at today's 4.00% yield, it's worth exactly $1,000.

Phase 2: The trigger. The Federal Reserve signals tighter policy. Market yields on comparable 10-year bonds rise from 4.00% to 5.00%—a 100 basis point increase.

Phase 3: The outcome. Your bond's new price is approximately $920.87, a decline of $79.13 or 7.9%. Nobody changed the coupon. Nobody altered the maturity date. The market simply re-priced your fixed cash flows at a higher discount rate.

Now reverse it. If yields had fallen from 4.00% to 3.00% (−100 bp), that same bond would rise to approximately $1,085.30, a gain of $85.30 or 8.5%.

The durable lesson: notice the asymmetry. The gain from a 100 bp decline (8.5%) exceeds the loss from a 100 bp increase (7.9%). This asymmetry is called convexity—the price-yield curve bends, giving you slightly more upside than downside for equal yield moves. Convexity becomes material (adding more than 0.25% to the price estimate) for yield changes exceeding approximately 50 bp on bonds with duration above 10 years.

What Affects Your Bond's Sensitivity (Three Drivers)

1. Maturity. Longer maturity means higher duration and more price sensitivity. A 5-year Treasury (duration ~4.5) loses about 4.5% per 100 bp yield increase. A 30-year Treasury (duration ~17–20) loses 17–20% for the same move. The point is: extending maturity amplifies both your gains and your losses.

2. Coupon rate. Lower coupons push duration higher. A 10-year bond with a 6% coupon has a duration of approximately 7.4 years, while a 10-year bond with a 2% coupon stretches to approximately 8.9 years (both at a 4% yield). The extreme case: a zero-coupon bond's duration equals its maturity exactly—a 10-year zero has duration of 10.0 years, the maximum possible for that maturity.

3. Starting yield level. Higher starting yields compress duration slightly (because near-term cash flows carry more relative weight at higher discount rates). This effect is modest for most investment-grade bonds but worth noting.

The test: before adding any bond position, ask yourself—do I know this bond's duration, and am I comfortable with a price swing of that magnitude if yields move 100 bp?

Real Markets Show This Relationship Clearly (Historical Evidence)

2022: The Fed hiked from 0.00–0.25% to 3.75–4.00%. The 10-year Treasury yield rose from approximately 1.51% to 4.24%. A hypothetical 10-year Treasury purchased at par at the start of the year declined roughly 17–20% in price.

Early 2020 (COVID flight to safety): The 10-year Treasury yield fell from 1.88% to a record low of 0.54% between January and March 9, 2020—a 134 bp decline. A 10-year Treasury purchased at the start of the year gained approximately 10–12% in price over that period.

1994 (the "Great Bond Massacre"): The Fed raised rates from 3.00% to 5.50% in nine months. The 10-year Treasury yield surged from approximately 5.60% to 8.05%. Estimated global bond market losses exceeded $1.5 trillion. Orange County, California, declared bankruptcy after losing $1.7 billion on leveraged bond positions.

The durable lesson: the inverse price-yield relationship isn't theoretical—it's the dominant driver of bond portfolio returns over short and intermediate horizons.

Practical Takeaways (Tiered by Impact)

Essential (high ROI):

• Know your portfolio's duration. The Bloomberg U.S. Aggregate Bond Index runs approximately 6.0–6.2 years—that's your benchmark sensitivity
• Use the quick estimate. % price change ≈ −duration × yield change. A portfolio with duration 6 loses roughly 6% per 100 bp rate increase
• Match duration to your time horizon. If you need the money in 3 years, a bond with duration of 15 creates unnecessary risk
• Remember that rising rates hurt existing bonds—but they also mean higher yields on new purchases (the reinvestment offset)

High-impact (for active management):

• Lower-coupon bonds carry more duration than higher-coupon bonds of the same maturity
• Convexity favors you in volatile markets—price gains from falling yields slightly exceed losses from rising yields
• Dollar duration (DV01) scales sensitivity to position size: a $1 million position with duration 7.5 moves approximately $750 per basis point

Your Next Step

Pull up your bond fund or portfolio and find its effective duration (listed on any fund fact sheet or brokerage account page). Multiply that number by 1.00 (representing a hypothetical 100 bp rate move). That's your approximate percentage gain or loss for a full percentage point shift in yields. If that number makes you uncomfortable, you're holding more interest rate risk than you want—and now you know exactly how to adjust it.