Why Bond Prices Fall When Interest Rates Rise, With the Math

"When interest rates rise, bond prices fall" is one of the most repeated lines in investing, and one of the least explained. FINRA lists it as the first of three cardinal rules about bonds and rates, followed by the mirror image (when rates fall, prices rise) and a warning that every bond carries interest rate risk. The reason is not mysterious. It is a matter of comparing a fixed payment with a moving market, and you can see it with a few lines of arithmetic.

1. The pieces of a bond

Investor.gov describes a bond as a debt security, like an IOU. You lend money to an issuer, which promises to pay a set rate of interest during the life of the bond and to repay the face value, also called par value, at maturity.

FINRA adds the detail that matters here: the coupon is set when the bond is issued and is quoted as a percentage of par. Its own example is a $1,000 bond with a 4.5% coupon, which pays $45 a year, usually as two $22.50 payments.

The coupon never changes. That is the whole point of a fixed-rate bond. What changes is the rate the market demands from similar bonds today, and that rate decides what a buyer will pay for yours.

2. Why the price has to move

Suppose you own a bond paying 4% a year, and new bonds from similar issuers now pay 5%. Investor.gov explains that rising rates make newly issued bonds more appealing, so to sell an older, lower-rate bond, you might have to sell it at a discount.

Think about it from the buyer's side. Why pay $1,000 for $40 a year when $1,000 buys $50 a year elsewhere? The only way your bond sells is at a lower price. That lower price, plus the full $1,000 repaid at maturity, has to produce a total return that matches the new 5% market rate.

TreasuryDirect shows the same relationship in a short table for notes and bonds: if the yield to maturity is greater than the interest rate, the price is less than par; if equal, the price is par; if less, the price is more than par. Yield to maturity is the annual rate of return a buyer earns at that price.

3. The math on a one-point move

Here is a bond with assumptions stated up front: $1,000 par value, 4% coupon paid twice a year ($20 every six months), and market yields moving from 4% to either 3% or 5%. The price is the present value of every remaining payment, discounted at half the market yield per six-month period.

Three things stand out.

The direction is always opposite. Yields up, price down. Yields down, price up.

Longer bonds move more. A one-point rise costs the 2-year bond about $19, the 10-year about $78, and the 30-year about $155. A longer bond locks you into the old coupon for more years, so the gap between your coupon and the market's compounds over a longer stretch.

The moves are not symmetric. On the 30-year bond, a one-point drop adds about $197, while a one-point rise takes away about $155. Prices rise more for a fall in yields than they drop for an equal rise.

4. Duration: the shortcut for sensitivity

FINRA describes duration, stated in years, as a measure of how much a bond's price is likely to change for a 1 percent change in interest rates. The higher the duration, the more sensitive the bond. The table above is duration in action: the 30-year bond has a much higher duration than the 2-year, so the same rate change hits it harder.

Bond funds report a duration figure too, which gives you a quick way to gauge how much the fund's value could swing if rates move.

5. What this means if you hold to maturity

Investor.gov makes a point that is easy to miss: if you hold a bond to maturity, you receive the face value plus interest. The price swings only turn into real gains or losses if you sell before maturity, or if the issuer fails to pay.

Back to the 10-year example. If yields rise to 5% the day after you buy, your bond's market price falls to about $922. But if you keep it, you still collect $40 a year and get $1,000 back at the end, exactly as promised. What you lose is opportunity: the money is earning 4% while new bonds earn 5%. FINRA calls this opportunity risk.

The picture is different for bond funds, which do not have a single maturity date. A fund's share price reflects the current market value of everything it holds, so rate changes show up in what your shares are worth on any given day.

6. Turning it around

The same logic works in your favor when rates fall. FINRA's example: if you own a bond paying a 6% coupon and rates drop to 4%, your bond becomes attractive compared with new ones, and you could sell it at a premium.

Watch for a catch, though. FINRA notes that some bonds are callable, meaning the issuer can retire them early, and issuers are most likely to do that when rates fall. A call can cut short the very price gain you were counting on.

7. Key takeaways

• A bond's coupon is fixed, so when market rates change, its price adjusts until its yield matches the market.
• Yields up means price down, and yields down means price up.
• On a $1,000 bond with a 4% coupon, a one-point rise in yields cuts the price to about $981 for a 2-year bond, $922 for a 10-year, and $845 for a 30-year.
• Longer maturities and higher duration mean bigger price swings.
• Holding an individual bond to maturity still returns face value if the issuer pays, but bond funds show rate changes in their share price.