Why Bond Prices Fall When Yields Rise

Bond prices fall when yields rise because the fixed interest payments of existing bonds become less attractive compared to newly issued bonds paying higher rates, forcing sellers to discount the older bonds to compete. In short, price and yield move in opposite directions by mathematical necessity. This article explains the mechanics, the math, and what it means for investors.

Formula

The price of a bond is the present value of its future cash flows: periodic coupon payments plus the face value at maturity. The standard formula is:

P = C × [1 − (1 + r)^−n] / r + F / (1 + r)^n

• P = bond price (currency)
• C = annual coupon payment = (coupon rate × face value) / number of payments per year
• r = market yield per period (annual yield divided by payments per year)
• n = total number of periods (years × payments per year)
• F = face value (currency)

For a bond paying annual coupons, the formula simplifies to the one used in the calculator above.

Worked Example

Using the default values: a 10-year bond with a 3% coupon, face value $1,000, and current market yield of 4%.

1. Annual coupon C = 3% × $1,000 = $30
2. Yield per period r = 4% = 0.04
3. Number of periods n = 10
4. Present value of coupons: $30 × [1 − (1.04)^−10] / 0.04 = $30 × 8.1109 = $243.33
5. Present value of face value: $1,000 / (1.04)^10 = $675.56
6. Bond price = $243.33 + $675.56 = $918.89

The calculator shows $918.89, which is $81.11 below par. The price fell because the market yield (4%) is higher than the coupon (3%).

Why Price and Yield Move Opposite

Imagine you bought a bond that pays 3% interest. If new bonds now pay 4%, your bond’s 3% coupon is less attractive. To sell it, you must lower the price so that a buyer earns a 4% yield. The discount compensates for the lower coupon. Conversely, if yields fall to 2%, your 3% bond becomes more valuable, so its price rises above par.

This relationship is not a market opinion; it is a mathematical identity. The bond’s yield is defined as the return that equates its price to the present value of its cash flows. When the denominator (yield) rises, the present value falls.

The Role of Duration

Duration measures how sensitive a bond’s price is to a change in yield. A bond with a longer maturity has higher duration because its cash flows are further in the future, making them more sensitive to discount rate changes. For example, a 1% rise in yield might cause a 2-year bond to drop 2% but a 30-year bond to drop 15%. Duration is a useful risk metric: the higher the duration, the more your bond’s price will move when yields change.

Real-World Example: The 2020–2023 Rate Hikes

When central banks raise interest rates, bond yields rise. For instance, if the Federal Reserve hikes its benchmark rate, newly issued Treasury bonds offer higher coupons. Existing bonds with lower coupons must be sold at a discount. A 10-year Treasury issued at 2% in 2020 could trade at a 20% discount by 2022 when yields reached 4%, because the price needed to adjust so that a buyer would earn the current market yield.

Common Misconceptions

Myth: “Bonds are safe, so they never lose money.” While bonds are less volatile than stocks, they still lose value when yields rise. If you hold to maturity, you get your principal back, but if you sell before maturity, you can suffer a capital loss.

Myth: “A high coupon protects you from price drops.” A high coupon reduces duration (because more cash flow comes earlier), but it does not eliminate price risk. Even a 5% coupon bond will fall in price if yields rise to 6%.

How to Use the Calculator

Enter the bond’s time to maturity, coupon rate, current market yield, and face value. The calculator returns the fair price you should pay for the bond today. If the market price is lower than the calculated price, the bond is undervalued (assuming your yield estimate is correct). This is useful for comparing bonds with different coupons and maturities.

Limitations of the Formula

The formula assumes a flat yield curve (same yield for all maturities) and that coupon payments are reinvested at the same yield. In reality, yields vary by maturity and reinvestment rates may differ. Also, the formula does not account for call provisions (where the issuer can redeem the bond early) or credit risk (the chance of default). For high-yield or callable bonds, the price-yield relationship is more complex.

Frequently Asked Questions

Q: If yields rise, should I sell my bonds?
It depends on your time horizon. If you can hold to maturity, you will receive the face value regardless of interim price swings. If you need to sell early, you may face a loss. Consider whether the higher yields on new bonds compensate for the capital loss.

Q: Why do bond prices fall when the Fed raises rates?
The Fed’s rate hike pushes up short-term yields, and longer-term yields often follow. As market yields rise, existing bonds with lower coupons become less attractive, so their prices fall to bring their effective yield in line with the new market rate.

Q: What is the difference between coupon rate and yield?
The coupon rate is the fixed annual interest payment as a percentage of face value. The yield (yield to maturity) is the total return you can expect if you buy the bond at its current price and hold it to maturity, including both coupon payments and any capital gain or loss.

Q: Can bond prices fall to zero?
No, because the issuer must repay the face value at maturity (assuming no default). The price will approach the face value as maturity approaches, regardless of yield changes. It may trade below par, but it will not fall to zero unless the issuer defaults.

Q: How does inflation affect bond prices?
Inflation erodes the purchasing power of fixed coupon payments. If inflation rises, investors demand higher yields to compensate, which pushes bond prices down. This is why bonds with longer maturities are more sensitive to inflation expectations.

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in t | Time to maturity | years | 10 | 0
in c | Coupon rate | % per year | 3 | 0
in y | Current market yield | % per year | 4 | 0
in f | Face value | currency | 1000 | 0
out price | Bond price | currency | f * (c/100/y) * (1 - 1/(1+y/100)^t) + f/(1+y/100)^t | 2
out change | Price change from par | % | (price - f) / f * 100 | 2