Yield Level and Convexity in Bond Price Responses

Why Equal Yield Changes Do Not Have Equal Price Effects

A bond’s price and yield move in opposite directions: when the yield used to discount its promised cash flows rises, their present value falls. But the relationship is not a straight line. For a conventional bond with fixed positive cash flows, the price-yield curve bends upward. This curvature means that a yield decrease of a given size generally raises the bond’s price by more than an equal yield increase lowers it, when both changes are measured from the same starting yield.

This feature is called convexity. It matters when comparing rate scenarios: a calculation based only on the direction of the yield change misses the fact that the price response is asymmetric. The effect depends on the bond’s cash flows and starting yield, so exact repricing is the direct way to compare scenarios.

Definition: For a bond with fixed cash flows $CF_t$ paid at times $t=1,\ldots,n$, its price at annual effective yield $y>-1$ is $P(y)=\sum_{t=1}^{n}CF_t/(1+y)^t$. The price-yield relationship is convex over a yield range when its second derivative with respect to yield is positive there. In this fixed-cash-flow model, each cash flow is nonnegative and at least one is positive, so the price is strictly convex.

The model here assumes valuation on a coupon date, annual cash-flow periods, a single annual effective yield applied to every payment, and fixed promised cash flows. It does not incorporate default, changing coupon payments, or embedded options that can alter cash flows. The yield is treated as a discounting input; a yield change does not itself change the promised payments.

The Curvature Formula and Its Meaning

Differentiate the price formula with respect to $y$. The first derivative is negative when the bond has positive cash flows after the valuation date, confirming that price falls as yield rises. Differentiating once more gives the dollar convexity, the second derivative of price:

\frac{dP}{dy} = -\sum_{t=1}^{n}\frac{t\,CF_t}{(1+y)^{t+1}}, \qquad \frac{d^2P}{dy^2} = \sum_{t=1}^{n}\frac{t(t+1)\,CF_t}{(1+y)^{t+2}}.

Each term in the second derivative is nonnegative when $y>-1$ and $CF_t\geq 0$. If at least one positive payment occurs at a time $t>0$, at least one term is positive, and thus $d^2P/dy^2>0$. This is the mathematical reason the conventional fixed-cash-flow bond price curve bends upward.

The dollar convexity $d^2P/dy^2$ describes curvature in dollars per squared yield unit. Analysts also sometimes refer to relative convexity, defined as $(d^2P/dy^2)/P$, which expresses curvature relative to the bond’s price. Since yield changes in calculations are usually written as decimal changes, a one-percentage-point change is $0.01$, not $1$. The formula’s yield units must be kept consistent with the change being considered.

Why the price gain exceeds the equal price loss: For a strictly convex price function and $h>0$, the midpoint lies below the average of the two endpoint prices. Thus $P(y-h)+P(y+h)>2P(y)$, or, equivalently, $P(y-h)-P(y)>P(y)-P(y+h)$. The first difference is the gain from a yield decrease; the second is the loss from an equal yield increase.

This inequality follows from the definition of strict convexity: at the midpoint $y$, the function’s value is less than the average of its values at $y-h$ and $y+h$. Rearranging that inequality gives the comparison above. It establishes the direction of the asymmetry; exact repricing is still needed to find the size of each change.

Worked Example: Comparing Equal Yield Moves for a Coupon Bond

A hypothetical bond has face value $100$, pays an annual coupon of $10$, and repays $100$ at the end of year 2. Its annual effective yield is initially $10\%$. Find its price at the initial yield and at yields one percentage point above and below it. Compare the two price changes.

Method. Discount the two promised payments at each yield, using $10$ at year 1 and $110$ at year 2. The cash flows are unchanged across scenarios.

At the initial yield, the price is

P(0.10)=\frac{10}{1.10}+\frac{110}{(1.10)^2} =9.090909\ldots+90.909090\ldots=100.

At a yield of $9\%$:

P(0.09)=\frac{10}{1.09}+\frac{110}{(1.09)^2} =9.174312\ldots+92.584125\ldots =101.758437\ldots .

The price gain when yield falls by one percentage point is $101.758437\ldots-100=1.758437\ldots$.

At a yield of $11\%$:

P(0.11)=\frac{10}{1.11}+\frac{110}{(1.11)^2} =9.009009\ldots+89.278141\ldots =98.287150\ldots .

The price loss when yield rises by one percentage point is $100-98.287150\ldots=1.712850\ldots$. Rounded to the nearest cent only at the final step, the price gain is $1.76$ and the price loss is $1.71$. The gain is larger, as the convexity result predicts. Checking the changes against the initial price confirms that the yield decrease raises price and the yield increase lowers it.

Common mistake avoided: Treating equal yield changes as equal and opposite price changes would assign both changes the same absolute size. The exact discounted cash flows show why that shortcut is incorrect.

How the Starting Yield Affects Price Responses

The starting yield matters because it determines the discount factors applied to every payment. For a given bond, the contribution of each payment to both price and curvature changes when the yield changes. Consequently, two yield scenarios of the same size can produce different dollar changes, even if the same bond is being considered.

The following example uses a one-year zero-coupon bond to isolate the effect. A zero-coupon bond has one promised payment at maturity and no interim coupons. Its simple cash-flow structure makes the yield-level comparison transparent; it should not be taken as a claim that dollar price changes are identical across different bonds.

Worked Example: Equal Yield Moves at Two Starting Levels

A hypothetical one-year zero-coupon bond pays $100$ at maturity. Compare a one-percentage-point decrease and increase in yield around starting yields of $5\%$ and $15\%$. Calculate prices and dollar price changes.

Method. For this bond, $P(y)=100/(1+y)$. At a $5\%$ starting yield, the initial price is $100/1.05=95.238095\ldots$. The prices at $4\%$ and $6\%$ are

P(0.04)=\frac{100}{1.04}=96.153846\ldots, \qquad P(0.06)=\frac{100}{1.06}=94.339623\ldots .

The decrease from $5\%$ to $4\%$ gives a gain of $96.153846\ldots-95.238095\ldots=0.915751\ldots$. The increase from $5\%$ to $6\%$ gives a loss of $95.238095\ldots-94.339623\ldots=0.898472\ldots$.

At a $15\%$ starting yield, the initial price is $100/1.15=86.956522\ldots$. Prices at $14\%$ and $16\%$ are

P(0.14)=\frac{100}{1.14}=87.719298\ldots, \qquad P(0.16)=\frac{100}{1.16}=86.206897\ldots .

The decrease from $15\%$ to $14\%$ gives a gain of $87.719298\ldots-86.956522\ldots=0.762776\ldots$. The increase from $15\%$ to $16\%$ gives a loss of $86.956522\ldots-86.206897\ldots=0.749625\ldots$.

Rounded to the nearest cent, the price changes are $+\$0.92$ and $-\$0.90$ around $5\%$, and $+\$0.76$ and $-\$0.75$ around $15\%$. Each pair is asymmetric in the direction predicted by convexity, and the dollar changes are smaller around the higher starting yield in this example. The price calculations provide the check: in each pair, the lower-yield price is higher than the starting price and the higher-yield price is lower.

Common mistake avoided: Comparing only yield changes can hide the different price levels and discount factors. A one-percentage-point move is the same change in yield in both comparisons, but it does not create the same dollar change.

Reading Dollar and Relative Convexity

Dollar convexity and relative convexity answer related but distinct questions. Dollar convexity is the curvature of the price itself. Relative convexity divides that curvature by the current price, making it a price-scaled measure. Neither quantity is an exact price change for a finite yield movement: each describes local curvature at the yield where it is calculated.

For example, a zero-coupon bond paying $100$ in two years has $P(y)=100/(1+y)^2$. Its derivatives are

\frac{dP}{dy}=-\frac{200}{(1+y)^3}, \qquad \frac{d^2P}{dy^2}=\frac{600}{(1+y)^4}.

At $y=10\%$, the price is $100/(1.10)^2=82.644628\ldots$, and dollar convexity is $600/(1.10)^4=409.808\ldots$. Relative convexity is $409.808\ldots/82.644628\ldots=4.958678\ldots$. These measures are positive, consistent with the upward-bending price-yield curve. Their units and scale differ: the dollar measure is in price units per squared yield unit, while the relative measure is scaled by the price.

It is important not to infer from this example that every bond’s dollar convexity must move in the same way as its relative convexity when yield changes. Cash-flow timing, coupon amounts, and the price level all enter the calculation. The formula, rather than a rule based on yield level alone, determines the curvature for a particular bond.

Limits of the Convexity Conclusion

The strict-convexity result applies to fixed, nonnegative cash flows discounted at one yield. It is not a universal statement about every security called a bond. An embedded option can make future cash flows depend on interest rates or on an exercise decision. In such cases, the promised cash flows may not remain fixed across yield scenarios, so the fixed-cash-flow derivation does not by itself establish that the whole security’s price-yield curve is convex.

For a conventional bond, a sound comparison proceeds in a consistent order:

Specify the cash flows and yield basis.
Use the same promised payments, valuation date, compounding convention, and payment timing in every scenario.

Reprice at each yield.
Discount each payment at the scenario yield and sum the present values.

Compare changes from the same starting price.
Keep the sign convention clear: a price gain is positive and a price loss is negative.

Check the convexity prediction.
For equal upward and downward yield changes, the gain from the yield decrease should exceed the loss from the yield increase under the stated fixed-cash-flow assumptions.

A frequent exam trap is to confuse a percentage-point change in yield with a percentage change in price. A yield rising from $5\%$ to $6\%$ changes by one percentage point, while the proportional yield increase is $20\%$. Price effects should be calculated from the discounting formula or stated clearly as dollar or percentage changes. Another trap is to use a yield expressed as a whole number in a formula that expects a decimal: for example, the yield input $0.01$ represents one percentage point.

Key takeaway: With fixed positive cash flows discounted at one yield, bond price decreases as yield rises and has positive curvature. Therefore, equal yield decreases and increases around a starting yield do not have equal and opposite price effects: the decrease produces the larger price change in the opposite direction.

Check Your Understanding

Use the fixed-cash-flow model and keep yield changes in decimal form. Round money to the nearest cent where needed.

A one-year zero-coupon bond pays $1{,}000$. Its yield starts at $9\%$. Find the prices at $8\%$, $9\%$, and $10\%$, then compare the gain and loss for equal one-percentage-point yield moves. (Answer: Prices are $\$925.93$, $\$917.43$, and $\$909.09$; the gain is $\$8.49$, exceeding the $\$8.34$ loss.)
A bond pays $20$ at year 1 and $120$ at year 2. At yield $y=0$, find its price, dollar convexity, and relative convexity. Use $P''(y)=\sum_t t(t+1)CF_t/(1+y)^{t+2}$. (Answer: Price $140$, dollar convexity $760$, and relative convexity $38/7\approx5.4286$.)
A one-year zero-coupon bond pays $110$ and has a starting yield of $10\%$. Find its price at $9\%$ and $11\%$, then state the gain and loss relative to the starting price. (Answer: Starting price $\$100.00$; the $9\%$ price is $\$100.92$, a gain of $\$0.92$, and the $11\%$ price is $\$99.10$, a loss of $\$0.90$.)
For a fixed-cash-flow bond with a strictly convex price-yield function, write the inequality that compares prices at $y-h$, $y$, and $y+h$, where $h>0$. Explain what the inequality says about the gain and loss. (Answer: $P(y-h)+P(y+h)>2P(y)$, so $P(y-h)-P(y)>P(y)-P(y+h)$; the yield decrease’s gain exceeds the equal yield increase’s loss.)
A callable bond’s promised payments may change if the issuer exercises its option as yields move. Can the fixed-cash-flow convexity formula alone guarantee that the callable bond’s price-yield curve is strictly convex? (Answer: No. The formula assumes fixed cash flows; the option can make cash flows depend on yields, so the conclusion is not guaranteed by that result alone.)

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