Exact Repricing and Duration-Convexity Error

Why Exact Repricing Matters

Duration and convexity provide a useful local estimate of how a present value responds to a yield change. The estimate combines a first-order effect, measured by modified duration, and a second-order adjustment, measured by convexity. It is an approximation, however: for a finite change, the new present value is determined by discounting each cash flow at the new yield, not by stopping after two terms of a local expansion.

This distinction matters when the yield change is not small or when cash flows occur far in the future. Exact repricing provides a direct benchmark for assessing the approximation. It also helps identify whether an error is due to an incorrect sensitivity calculation or simply to using a local estimate for a finite change.

Model and assumptions: At the valuation date, a set of known cash flows $C_t$ is paid at times $t>0$, measured in years. A single annual effective yield $y$ discounts every cash flow, so $P(y)=\sum_t C_t(1+y)^{-t}$, with $y>-1$. The cash flows and their payment dates remain fixed when the yield changes. The formulas below do not directly describe a nonparallel spot-rate shift, cash flows that depend on the yield, or a security whose options change its payments when rates change.

Exact Repricing and the Duration-Convexity Estimate

Let the yield move from $y$ to $y+\Delta y$, where both yields exceed $-1$. The exact new price is found by replacing $y$ with $y+\Delta y$ in the present-value formula. The duration-convexity estimate instead uses the price, modified duration, and convexity at the initial yield.

Definition: For fixed cash flows discounted at one annual effective yield, modified duration and convexity at yield $y$ are $$ D(y)=-\frac{P'(y)}{P(y)},\qquad \mathcal{C}(y)=\frac{P''(y)}{P(y)}. $$ The second-order estimate of the relative price change is $$ \frac{P(y+\Delta y)-P(y)}{P(y)} \approx -D(y)\Delta y+\frac{1}{2}\mathcal{C}(y)(\Delta y)^2. $$

For cash flows at times $t$, differentiation gives the standard cash-flow formulas

D(y)=\frac{\sum_t tC_t(1+y)^{-t-1}}{P(y)},\qquad \mathcal{C}(y)=\frac{\sum_t t(t+1)C_t(1+y)^{-t-2}}{P(y)}.

The duration in this formula is modified duration, not Macaulay duration. In particular, the factor $1/(1+y)$ is already incorporated in the derivative with respect to the effective yield. If all future cash flows are positive, their present values can be used as weights to calculate duration and convexity. If a cash-flow stream includes negative payments, the derivative definitions still apply when $P(y)\ne 0$, but weighted-average interpretations and simple sign conclusions need not hold.

Why the Approximation Has This Form

The exact effect of the yield change on each discounted payment can be written as a ratio to its initial present value. For payment $C_t$,

\frac{C_t(1+y+\Delta y)^{-t}}{C_t(1+y)^{-t}} = \left(1+\frac{\Delta y}{1+y}\right)^{-t}.

For a small change, the second-order Taylor expansion of this factor is

\left(1+\frac{\Delta y}{1+y}\right)^{-t} \approx 1-\frac{t\Delta y}{1+y} +\frac{t(t+1)}{2}\left(\frac{\Delta y}{1+y}\right)^2.

Multiplying by each initial present value $C_t(1+y)^{-t}$ and summing gives the duration term and the convexity term in the definition. This explains why convexity adjusts the straight-line duration estimate: it accounts for the curvature of the price-yield relationship. The expansion is local; it does not make the omitted higher-order terms vanish for a finite change.

Calculate the initial price.
Discount every fixed payment at the initial yield $y$, and add the present values.

Calculate initial sensitivities.
Find modified duration and convexity at $y$, not at the new yield.

Estimate the change.
Apply $-D\Delta y+\frac12\mathcal{C}(\Delta y)^2$ to the initial price to obtain an estimated new price.

Reprice and compare.
Discount the original payments at $y+\Delta y$. Compare this exact price with the estimate, using a consistent sign for the error.

Worked Examples

Worked Example: A Small Increase in Yield

A hypothetical bond pays $100,000 at the end of each of years 1 and 2. Its annual effective yield rises from 10% to 11%. Find the exact new price, the duration-convexity estimate, and the approximation error. Use the error convention “estimate minus exact price.”

Method. First calculate the price and sensitivities at $y=0.10$, then calculate the new price directly at $0.11$. This avoids treating the estimated change as an exact repricing.

Initial price. $$ P(0.10)=\frac{100{,}000}{1.10}+\frac{100{,}000}{1.10^2} =90{,}909.0909+82{,}644.6281 =173{,}553.7190. $$

The first payment has present-value weight $90{,}909.0909/173{,}553.7190=11/21$, and the second has weight $10/21$. Thus $$ D(0.10)=\frac{11}{21}\frac{1}{1.10}+\frac{10}{21}\frac{2}{1.10} =\frac{310}{231} =1.341991. $$ For convexity, the corresponding terms are $t(t+1)/(1+y)^2$: $$ \mathcal{C}(0.10)=\frac{11}{21}\frac{2}{1.10^2}+\frac{10}{21}\frac{6}{1.10^2} =\frac{8200}{2541} =3.227076. $$

Estimated change. The yield change is $0.11-0.10=0.01$, or one percentage point. Therefore, $$ \frac{\Delta P}{P}\approx -(1.341991)(0.01) +\frac12(3.227076)(0.01)^2 =-0.01325856. $$ The estimated new price is $173{,}553.7190(1-0.01325856)=171{,}252.65$, to the nearest cent.

Exact repricing. $$ P(0.11)=\frac{100{,}000}{1.11}+\frac{100{,}000}{1.11^2} =90{,}090.0901+81{,}162.2433 =171{,}252.33. $$ The estimate-minus-exact error is $171{,}252.65-171{,}252.33=\$0.32$, to the nearest cent. The small error is consistent with a relatively small yield change and with using a second-order estimate.

Check. The yield increased, so the exact price fell from $173,553.72 to $171,252.33. The estimated price also fell. The estimate is slightly higher than the exact price, as expected for positive fixed cash flows and an upward yield change.

Common mistake avoided. The change of one percentage point is entered as $0.01$, not $1$. Also, both sensitivities are calculated at the initial yield of 10%.

Worked Example: A Larger Yield Change

A hypothetical project pays $100 at the end of year 1 and $200 at the end of year 3. The annual effective yield rises from 0% to 10%. Compare exact repricing with the duration-convexity estimate.

Method. Use the initial yield to calculate price and sensitivities, then separately discount each payment at the new yield. Because the initial yield is zero, the initial present values equal the payments.

The initial price is $P(0)=100+200=\$300$. Modified duration is $$ D(0)=\frac{1(100)+3(200)}{300} =\frac{700}{300} =\frac{7}{3}. $$ Convexity is $$ \mathcal{C}(0)=\frac{1(2)(100)+3(4)(200)}{300} =\frac{2600}{300} =\frac{26}{3}. $$

For $\Delta y=0.10$, the estimated relative change is $$ -\frac{7}{3}(0.10)+\frac12\frac{26}{3}(0.10)^2 =-0.2333333+0.0433333=-0.19. $$ The estimated new price is $300(1-0.19)=\$243.00$.

The exact new price is $$ P(0.10)=\frac{100}{1.10}+\frac{200}{1.10^3} =90.9091+150.2629 =\$241.17 $$ to the nearest cent. The estimate is higher than the exact price by $\$243.00-\$241.17=\$1.83$, to the nearest cent.

Check. Both calculations show a decrease from the initial price of $300. The estimate captures the first- and second-order effects, but it does not include the higher-order terms that matter more for this larger yield change.

Common mistake avoided. The $t(t+1)$ factor in convexity is applied at each payment time. Year 3 contributes $3(4)$, not $3^2$.

Worked Example: Timing Changes the Approximation Error

At a 0% annual effective yield, compare two hypothetical fixed-payment streams, each with an initial price of $100: Stream A pays $100 at year 2; Stream B pays $50 at year 1 and $50 at year 3. The yield rises to 10%. Find each stream's duration, convexity, exact new price, and duration-convexity estimate.

Stream A. Its initial price is $100. Its duration is $D_A=2$, and its convexity is $\mathcal{C}_A=2(3)=6$. The estimated relative change is $$ -(2)(0.10)+\frac12(6)(0.10)^2=-0.20+0.03=-0.17. $$ Thus the estimated new price is $\$100(0.83)=\$83.00$. Exact repricing gives $$ P_A(0.10)=\frac{100}{1.10^2}=\$82.64 $$ to the nearest cent. The estimate-minus-exact error is $\$0.36$, to the nearest cent.

Stream B. Its initial price is also $100. Its duration and convexity are $$ D_B=\frac{1(50)+3(50)}{100}=2,\qquad \mathcal{C}_B=\frac{1(2)(50)+3(4)(50)}{100}=7. $$ The estimated relative change is $-(2)(0.10)+\frac12(7)(0.10)^2=-0.165$, so the estimated new price is $\$83.50$. Exact repricing gives $$ P_B(0.10)=\frac{50}{1.10}+\frac{50}{1.10^3} =45.4545+37.5657 =\$83.02 $$ to the nearest cent. The estimate-minus-exact error is $\$0.48$, to the nearest cent.

Check. Both streams have the same initial price and modified duration, but their convexities differ because their payment times differ. For this yield change, the second-order estimate is less accurate for Stream B. This comparison illustrates that duration alone does not determine the approximation error.

Common mistake avoided. Stream B's year-3 payment must be discounted for three years in the exact repricing. Treating both of its payments as if they occurred at the average payment time would not give the exact price.

How to Read the Error

For positive fixed cash flows, each discounted payment has a negative third derivative with respect to $y$, since $$ \frac{d^3}{dy^3}\left[C_t(1+y)^{-t}\right] =-t(t+1)(t+2)C_t(1+y)^{-t-3}<0. $$ Consequently, when both the initial and new yields exceed $-1$, the remainder after the second-order Taylor approximation is negative for an upward change and positive for a downward change. In other words, for an upward yield change, the duration-convexity estimate is above the exact price; for a downward yield change, it is below the exact price. This sign statement assumes nonnegative future payments. It need not hold for cash flows with mixed signs.

The size of the error depends on more than the direction of the yield change. The second-order approximation omits terms of third and higher order. When the change is small, those terms are generally small relative to the retained terms; as the change grows, they can become material. Later cash flows also have larger time factors in the higher derivatives. This is why a long-dated payment can make a finite-change estimate more sensitive to omitted terms, even when duration and convexity have both been calculated correctly.

Common Exam Traps

Using percentage points as whole numbers. A rise from 6% to 7% means $\Delta y=0.01$, not $1$.
Confusing modified and Macaulay duration. The derivative-based formula uses modified duration. For an annual effective yield, the conversion factor $1/(1+y)$ is included.
Forgetting the one-half in the convexity term. The relative-price estimate uses $\frac12\mathcal{C}(\Delta y)^2$.
Repricing at the initial yield. The exact new price uses $y+\Delta y$; the duration and convexity used in the estimate are calculated at $y$.
Calling the estimate exact. Exact repricing discounts every original payment at the new yield. Duration and convexity provide a second-order approximation only.
Applying a single-yield formula to a nonparallel curve change. If different maturities have different yield changes, a single $\Delta y$ does not represent the movement. The calculation must reflect the maturity-specific rates.

Key takeaway: For fixed cash flows and one annual effective yield, exact repricing is $P(y+\Delta y)=\sum_t C_t(1+y+\Delta y)^{-t}$. Duration and convexity estimate the change from the initial price, but the estimate omits higher-order effects. Always compare it with exact repricing when the yield change is finite.

Check Your Understanding

For each problem, use the annual effective yield stated and round dollar answers to the nearest cent.

A hypothetical project pays $100 at the end of years 1 and 2. The yield rises from 0% to 5%. Find the exact new price and the duration-convexity estimate. State whether the estimate is above or below the exact price. (Answer: Exact price $185.94; estimate $186.00, above exact by $0.06.)
A hypothetical obligation pays $200 at the end of year 2. Its yield falls from 0% to $-10\%$. Find the exact new price and the duration-convexity estimate. (Answer: Exact price $246.91; estimate $246.00, below exact by $0.91.)
A hypothetical zero-coupon payment of $100 is due at the end of year 3. The yield rises from 0% to 10%. Find the exact price and the duration-convexity estimate. (Answer: Exact price $75.13; estimate $76.00, above exact by $0.87.)
At a 0% yield, Stream A pays $100 at year 2. Stream B pays $50 at year 1 and $50 at year 3. For each stream, find the exact price and duration-convexity estimate after the yield rises to 10%. Which has the larger estimate-minus-exact error? (Answer: Stream A exact $82.64, estimate $83.00, error $0.36; Stream B exact $83.02, estimate $83.50, error $0.48; Stream B has the larger error.)

What You'll Learn