From Percentage Change To Dollar Price Change

From a Proportional Estimate to a Dollar Amount

The duration-convexity approximation estimates how a bond’s price changes relative to its starting price when its yield changes. A proportional change is useful for comparing sensitivity across bonds, but a buyer or portfolio manager often needs the corresponding dollar amount: an estimated gain or loss and an estimated new price.

The conversion is simple, but it is easy to misuse. The proportional estimate must be multiplied by the initial price, not by the estimated new price. The sign of the yield change must also be retained: a yield increase usually produces a negative estimated price change, while a yield decrease usually produces a positive one.

Definition: Let $P_0$ be the bond’s price before a yield shock, and let $\Delta y$ be the change in yield, expressed as a decimal in the same yield convention used to calculate modified duration and convexity. If $D_{\mathrm{mod}}$ is modified duration and $C$ is convexity under that convention, the duration-convexity approximation to the proportional price change is $$ r \approx -D_{\mathrm{mod}}\Delta y+\frac{1}{2}C(\Delta y)^2. $$ The estimated dollar price change is $\Delta P_{\mathrm{est}}=P_0r$, and the estimated new price is $P_{1,\mathrm{est}}=P_0+\Delta P_{\mathrm{est}}=P_0(1+r)$.

Here $r$ is a rate of change, not a dollar amount. For example, $r=-0.04$ means an estimated price decline of 4% of the initial price. If the initial price is $1,000, the corresponding dollar change is $-\$40$, and the estimated new price is $960. This conversion does not make the estimated price exact; it carries the approximation in $r$ into dollar terms.

Why Multiplying by the Initial Price Works

By definition, the proportional price change is the price change divided by the price before the change. If the exact new price is $P_1$, then the exact proportional change is $(P_1-P_0)/P_0$. Rearranging gives $P_1-P_0=P_0((P_1-P_0)/P_0)$. The duration-convexity expression approximates the quantity in parentheses, so multiplying that approximation by $P_0$ gives an estimate of the dollar change.

\frac{P_1-P_0}{P_0}\approx r \quad\Longrightarrow\quad P_1-P_0\approx P_0r \quad\Longrightarrow\quad P_1\approx P_0(1+r).

This conversion assumes that the price and the duration-convexity inputs refer to the same bond or portfolio position, at the same starting point and under the same yield convention. The approximation is a local estimate for a specified yield shock, not a guarantee of the price that will actually be observed. It is most useful when the cash flows and yield framework underlying the sensitivity measures remain applicable.

Worked Conversions

Worked Example: A Yield Increase and a Dollar Loss

A bond is priced at $2,500. Its modified duration is 5.4 and its convexity is 42. Using an annual effective yield convention, estimate the dollar price change and new price if the yield rises by 0.0125. Retain full precision until the final step, then round dollar amounts to the nearest cent.

Method. First calculate the approximate proportional price change. Then multiply it by the initial price to obtain the estimated dollar change, and add that change to the initial price. The yield change is positive, so the duration term should contribute a decline.

Step 1: Calculate the duration contribution. $$ -D_{\mathrm{mod}}\Delta y=-5.4(0.0125)=-0.0675. $$

Step 2: Calculate the convexity contribution. $$ \frac{1}{2}C(\Delta y)^2 =\frac{1}{2}(42)(0.0125)^2 =21(0.00015625) =0.00328125. $$

Step 3: Combine the contributions. $$ r\approx -0.0675+0.00328125=-0.06421875. $$ The estimated proportional change is $-6.421875\%$.

Step 4: Convert to dollars and find the estimated new price. $$ \Delta P_{\mathrm{est}}=2500(-0.06421875)=-160.546875. $$ Thus the estimated dollar change is $-\$160.55$. The estimated new price is $$ P_{1,\mathrm{est}}=2500-160.546875=2339.453125\approx \$2,339.45. $$

Check. The estimated new price is below $2,500, consistent with the yield increase and negative proportional change. The dollar loss is about 6.42% of $2,500. A common mistake is to treat $-0.06421875$ as a loss of 6.42 dollars; it is a proportion, so it must be multiplied by the initial price.

Worked Example: A Yield Decrease and a Dollar Gain

A hypothetical bond position is worth $980. Its modified duration is 3.75 and convexity is 22, measured under the same annual effective yield convention. Estimate the dollar change and new value if the yield decreases by 0.012. Round dollar amounts to the nearest cent.

Method. Use a negative yield change for the decrease. The squared yield change is positive, so the convexity contribution is positive as well.

Step 1: Substitute the signed yield change. Here $\Delta y=-0.012$, and $$ -D_{\mathrm{mod}}\Delta y=-3.75(-0.012)=0.045. $$

Step 2: Calculate convexity and the total proportion. $$ \frac{1}{2}C(\Delta y)^2 =\frac{1}{2}(22)(-0.012)^2 =11(0.000144)=0.001584, $$ so $$ r\approx 0.045+0.001584=0.046584. $$ The estimated proportional increase is 4.6584%.

Step 3: Convert the proportion to a dollar change. $$ \Delta P_{\mathrm{est}}=980(0.046584)=45.65232. $$ The estimated gain is $\$45.65$.

Step 4: Add the gain to the starting value. $$ P_{1,\mathrm{est}}=980+45.65232=1025.65232\approx \$1,025.65. $$

Check. The estimated new value exceeds $980, as expected when the yield falls. Do not insert $+0.012$ merely because 0.012 is the magnitude of the change: doing so would reverse the duration contribution and give the wrong direction.

Worked Example: Combining Dollar Changes Across Two Positions

A hypothetical portfolio holds two bonds. Position A is worth $120,000, with modified duration 4 and convexity 30. Position B is worth $80,000, with modified duration 2.5 and convexity 12. Both use the same yield convention and experience a yield decrease of 0.015. Estimate each position’s dollar change and the portfolio’s new value. Round dollar amounts to the nearest cent.

Method. Calculate a proportional change for each position using its own duration and convexity. Convert each proportion using that position’s starting value, then add the dollar changes. The two proportions need not be the same.

Position A. The duration contribution is $$ -4(-0.015)=0.060. $$ The convexity contribution is $$ \frac{1}{2}(30)(-0.015)^2=15(0.000225)=0.003375. $$ Therefore, $$ r_A\approx 0.060+0.003375=0.063375, $$ and the estimated dollar gain is $$ \Delta P_A=120000(0.063375)=7605. $$ Position A’s estimated new value is $\$127,605.00$.

Position B. The duration contribution is $$ -2.5(-0.015)=0.0375. $$ The convexity contribution is $$ \frac{1}{2}(12)(-0.015)^2=6(0.000225)=0.00135. $$ Therefore, $$ r_B\approx 0.0375+0.00135=0.03885, $$ and the estimated dollar gain is $$ \Delta P_B=80000(0.03885)=3108. $$ Position B’s estimated new value is $\$83,108.00$.

Portfolio result. The initial portfolio value is $$ 120000+80000=200000. $$ The estimated dollar gain is $$ 7605+3108=10713, $$ so the estimated new portfolio value is $$ 200000+10713=\$210,713.00. $$

Check. The position values sum to the portfolio value both before the shock, $\$200,000$, and after the estimated changes, $\$210,713$. A common mistake is to apply one bond’s proportional change to the entire portfolio. That is only justified if the relevant portfolio sensitivity has first been established; here the direct position-by-position dollar calculation avoids that assumption.

Common Traps in the Conversion

The arithmetic is short, but several interpretation errors can reverse the result or change its scale. Keep the proportional estimate and dollar estimate separate until the final conversion.

Using the wrong starting value. Multiply the estimated proportion by $P_0$, the price before the shock. Do not multiply by the estimated new price.
Dropping the minus sign on the duration term. The duration contribution is $-D_{\mathrm{mod}}\Delta y$, not $D_{\mathrm{mod}}\Delta y$.
Giving a proportion as a dollar amount. A result such as $-0.02$ means an estimated 2% decline. The dollar change depends on the initial price.
Adding the proportional change directly to the price. The new price is $P_0(1+r)$, not $P_0+r$, unless $r$ has already been converted to dollars.
Rounding too early. Keep intermediate values at full precision, especially when the duration and convexity contributions partly offset one another. Round the requested money amount only at the end.

Key takeaway: Calculate the duration-convexity estimate $r$ first. Then use $P_0r$ for the estimated dollar change and $P_0(1+r)$ for the estimated new price. Always preserve the sign of the yield change and use the initial price as the conversion base.

Check Your Understanding

For each problem, use the stated duration-convexity approximation and round dollar answers to the nearest cent.

A bond is priced at $800, with modified duration 4 and convexity 20. Estimate its dollar price change and new price for a yield increase of 0.005. (Answer: estimated change $-\$15.80$; estimated new price $\$784.20$)
A bond position is worth $1,500, with modified duration 6 and convexity 50. Estimate its dollar gain and new value for a yield decrease of 0.01. (Answer: estimated gain $\$93.75$; estimated new value $\$1,593.75$)
A bond is priced at $2,400, with modified duration 3.2 and convexity 18. Estimate its dollar price change and new price for a yield increase of 0.015. (Answer: estimated change $-\$110.34$; estimated new price $\$2,289.66$)
A portfolio has Position A worth $50,000 with modified duration 4.5 and convexity 28, and Position B worth $75,000 with modified duration 2 and convexity 10. Both yields increase by 0.01. Estimate the portfolio’s dollar change and new value. (Answer: estimated change $-\$3,642.50$; estimated new value $\$121,357.50$)
Position A is worth $40,000, with modified duration 5 and convexity 40; its yield increases by 0.008. Position B is worth $60,000, with modified duration 3 and convexity 16; its yield decreases by 0.005. Estimate the combined dollar change and new value. (Answer: estimated change $-\$636.80$; estimated new value $\$99,363.20$)

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