Why Bond Prices Do Not Respond Symmetrically to Yield Changes
A bond’s price falls when its yield rises, but the price-yield graph is not generally a straight line. That bend matters: for equal-sized yield changes around a starting yield, a fixed-cash-flow bond typically gains more when the yield falls than it loses when the yield rises. This is the practical meaning of positive price curvature.
Duration describes the local, straight-line part of price sensitivity. Curvature explains why that straight-line estimate does not capture the full price response as the yield moves away from its starting point. Here the focus is on reading that shape and interpreting repriced values, rather than deriving a derivative-based convexity formula.
The model assumes the cash-flow amounts and payment dates do not change when the yield changes. It also assumes a single annual effective yield is used to discount all payments. A bond with an embedded option may have cash flows that change as yields move; the fixed-cash-flow conclusion should not be applied automatically to that bond. Likewise, if a problem uses a different compounding convention, use the pricing model specified there rather than silently treating the yield as an annual effective rate.
What Upward Curvature Looks Like
Imagine moving left or right from a starting yield on the horizontal axis. A yield decrease moves left and raises the bond price; a yield increase moves right and lowers it. Because the graph bends upward, the price gain from an equal-sized decrease is larger than the price loss from an increase, when the two changes are measured around the same yield. Equivalently, the average of the two endpoint prices lies above the price at the central yield.
The graph is still downward sloping. “Bends upward” does not mean that a yield increase raises price. It means that the negative slope becomes less negative as yield rises: at lower yields, an additional small fall in yield tends to have a larger price effect than an equal small rise in yield at the same central point.
Here \(h>0\) is a yield change, and both endpoint yields must satisfy the model’s conditions. The inequality compares the sum of the two endpoint prices with twice the central price. It is a useful numerical check on the upward bend; it does not say that either endpoint price alone must exceed the central price.
Why the Midpoint Comparison Holds
The inequality can be justified directly from the discounting of each positive cash flow. For a payment at year \(t\), its present value at yield \(i\) is \(C_t(1+i)^{-t}\). Set \(x=1+i\), so \(x>0\), and let the yield change be \(h\), with \(0<h<x\). The two endpoint discount bases are \(x-h\) and \(x+h\).
For a positive payment, the product of its two endpoint discount factors, relative to the square of its central discount factor, is greater than one:
The two endpoint discount factors are positive. By the arithmetic-geometric mean inequality, their average is at least the square root of their product. Since that product is greater than the square of the central discount factor, their average is greater than the central discount factor. Multiplying by \(C_t>0\) gives the midpoint comparison for that payment. Adding the comparisons for all positive cash flows gives \( [B(i-h)+B(i+h)]/2>B(i)\). If all cash flows are zero, prices are zero and there is no curvature to compare.
Worked Example: Comparing Equal Yield Moves Around a Coupon Bond
A hypothetical two-year bond pays $40 at the end of year 1 and $1,040 at the end of year 2. Use a constant effective annual yield and compare its prices at 3%, 4%, and 5%. How do the price effects of the equal-sized yield moves from 4% compare?
Method. Discount each fixed payment at the yield being tested, then compare both endpoint prices with the price at 4%. This avoids assuming that a one-percentage-point yield change has the same dollar effect in each direction.
At 4%, the two present values are \(40/1.04=\$38.4615\) and \(1{,}040/1.04^2=\$961.5385\). Thus \(B(4\%)=\$1{,}000.0000\).
At 3%, the present values are \(40/1.03=\$38.8350\) and \(1{,}040/1.03^2=\$980.2997\). Thus \(B(3\%)=\$1{,}019.1347\).
At 5%, the present values are \(40/1.05=\$38.0952\) and \(1{,}040/1.05^2=\$943.3107\). Thus \(B(5\%)=\$981.4059\).
Rounded to cents only at the end, the prices at 3%, 4%, and 5% are $1,019.13, $1,000.00, and $981.41. The fall in yield gains \(\$1{,}019.1347-\$1{,}000.0000=\$19.1347\); the rise in yield loses \(\$1{,}000.0000-\$981.4059=\$18.5941\). The gain exceeds the loss by \(\$19.1347-\$18.5941=\$0.5406\), or about $0.54. As a check, the endpoint sum is \(\$2{,}000.5406\), which exceeds twice the central price, \(\$2{,}000.0000\). The common mistake is to assume the two effects have equal magnitude simply because the yield moves are equal.
Reading Curvature Alongside Duration
A duration-only estimate uses the local slope at a starting yield to approximate the price change. It is useful for a small yield move, but it treats the price-yield graph as a straight line. When the graph bends, that tangent line does not pass through the actual price at the new yield. For a positive fixed-cash-flow bond, the upward bend puts the actual price above the tangent-line estimate on either side of the starting point, for a nonzero move within this model.
This has a practical interpretation. For a yield increase, duration alone generally predicts a price that is too low, so it overstates the price loss. For a yield decrease, duration alone generally predicts a price that is too low as well, so it understates the price gain. The size of the difference depends on the bond, the starting yield, and the size of the yield change; curvature is not a fixed dollar adjustment that can be used for every bond.
Worked Example: Seeing the Tangent-Line Error on a Zero-Coupon Bond
A hypothetical one-year zero-coupon bond pays $1,000 at maturity. Its effective annual yield is 5%. Compare the duration-only estimate with the exact repricing for a 1-percentage-point yield increase and decrease.
Method. First calculate the starting price and modified duration. Then use the local linear estimate and compare it with prices obtained directly from the fixed cash flow. The comparison illustrates curvature; it does not replace exact repricing when an exact price is requested.
At 5%, the price is \(B(5\%)=1{,}000/1.05=\$952.3810\). For this one-payment bond, Macaulay duration is 1 year, so modified duration is \(1/1.05=0.952381\).
For a yield increase of \(0.01\), the duration-only percentage estimate is \(-0.952381(0.01)=-0.00952381\), or \(-0.952381\%\). The estimated dollar change is \(-\$952.3810(0.00952381)=-\$9.0703\), giving an estimated price of \(\$943.3107\). Exact repricing at 6% gives \(1{,}000/1.06=\$943.3962\), which is about $0.09 above the estimate.
For a yield decrease of \(0.01\), the duration-only estimate is a \(0.952381\%\) gain, or \(\$952.3810(0.00952381)=\$9.0703\). The estimated price is \(\$961.4512\). Exact repricing at 4% gives \(1{,}000/1.04=\$961.5385\), again about $0.09 above the estimate. Rounded to cents, the exact prices at 4%, 5%, and 6% are $961.54, $952.38, and $943.40. The tangent-line estimate misses the upward bend in both directions. A common mistake is to treat the duration estimate as the exact price change rather than as a local approximation.
Curvature Is Not the Same as “More Sensitive Everywhere”
Price sensitivity depends on where you are on the graph. The bond’s dollar price response to a small yield change is related to the graph’s slope at that yield, while curvature describes how that slope itself changes as the yield moves. A statement such as “this bond has more curvature” is incomplete unless the comparison specifies the bonds, yields, cash flows, and the measure being used.
For the fixed-cash-flow model, positive payments at later dates also respond to yield through more periods of discounting. Their contribution to the price graph can therefore differ from that of earlier payments. This is one reason that bonds with different cash-flow timing can have different price-yield shapes, even when their prices or durations happen to be similar. Detailed numerical convexity measures and comparisons of cash-flow patterns require more than the visual midpoint check used here.
| Observation | What it means |
|---|---|
| Yield rises and price falls | The graph slopes downward under the fixed-cash-flow model. |
| Equal moves are made above and below a central yield | The average of the endpoint prices is above the central price for positive cash flows. |
| A duration-only estimate is used for a finite yield move | It follows the local tangent and omits the graph’s bend. |
| Cash flows change when yields change | The fixed-cash-flow curvature result may not describe the actual bond-price response. |
Common Exam Traps
- Confusing upward curvature with an upward-sloping graph. The price still falls as yield rises; the bend describes how the slope changes.
- Comparing unequal yield moves. To interpret the midpoint comparison, use equal yield changes around the same central yield.
- Comparing percentages with dollars as if they were interchangeable. State whether the price effects are dollar changes or percentage changes, and use a consistent basis.
- Rounding each present value too early. Keep intermediate values unrounded and round the reported prices only at the end.
- Assuming cash flows remain fixed for every bond. If an option changes payments or redemption when yields move, first determine whether the fixed-cash-flow model is appropriate.
Check Your Understanding
For each problem, use fixed cash flows and the stated effective annual yield. Round dollar answers to cents only at the end.
- A hypothetical one-year bond pays $500 at maturity. Find its prices at yields of 3%, 4%, and 5%. Compare the gain from a 1-percentage-point yield decrease with the loss from an equal increase around 4%. (Answer: The prices are $485.44, $480.77, and $476.19. The gain is about $4.67 and the loss about $4.58, so the gain is larger.)
- A hypothetical two-year zero-coupon bond pays $1,000 at maturity. Find its prices at yields of 4%, 5%, and 6%, and identify which equal-sized move from 5% has the greater absolute price effect. (Answer: The prices are $924.56, $907.03, and $890.00. The decrease to 4% gains about $17.53; the increase to 6% loses about $17.03, so the decrease has the greater effect.)
- A bond has fixed payments of $30 at the end of year 1 and $1,030 at the end of year 2. Its effective annual yield is 5%. Calculate its price, then explain whether an equal yield decrease and increase around 5% should produce equal absolute price changes. (Answer: The price is \(30/1.05+1{,}030/1.05^2=\$962.8118\), or $962.81. The absolute changes should not be equal; the yield decrease should produce the larger price gain.)
- A positive fixed-cash-flow bond is priced at a central yield and at two yields equally far above and below it. The endpoint prices are $1,026.40 and $978.10. What does the midpoint comparison imply about the central price? (Answer: The central price is less than \((\$1{,}026.40+\$978.10)/2=\$1{,}002.25\).)
- A bond’s estimated price from a duration-only tangent line is below its exact repriced value after a finite yield move. What does this indicate about the graph under the fixed, positive-cash-flow model, and does it determine whether the yield rose or fell? (Answer: It is consistent with the graph bending upward relative to the tangent line. By itself, it does not determine whether the yield rose or fell.)