Checking Convexity Estimates Against Repricing

Why Check an Estimate Against Repricing?

Duration and convexity give a practical estimate of how a fixed set of cash flows responds to a yield change. The estimate is useful because it summarizes a bond’s local price sensitivity in two measures. It is still an approximation: it uses information at the original yield, whereas exact repricing discounts each cash flow at the new yield. Checking both is a way to see how well the approximation works for the shock being considered.

The comparison is meaningful only when the methods use the same cash flows, yield convention, and shock. In this tutorial, yields are annual effective rates, cash flows occur at integer times measured in years, and the cash flows do not change when the yield changes. The yield after the shock must be greater than $-1$, so each discount factor is defined. No mortality or probability model is needed: the stated cash flows are treated as certain.

Definition: Let $C_t$ be a fixed cash flow at time $t$, and let $y$ be the annual effective yield. The exact price is $P(y)=\sum_t C_t(1+y)^{-t}$. For a yield shock $\Delta y$, the exact repriced price is $P(y+\Delta y)$. The duration-and-convexity estimate uses measures calculated at the original yield $y$, not at the shocked yield.

The Estimate and the Exact-Price Check

For fixed cash flows under annual compounding, modified duration $D_{\mathrm{mod}}$ and convexity $C$ are defined by the first and second derivatives of price with respect to yield:

D_{\mathrm{mod}}=-\frac{P'(y)}{P(y)}, \qquad C=\frac{P''(y)}{P(y)}.

A second-order Taylor approximation to the new price is therefore:

\frac{\Delta P}{P} \approx -D_{\mathrm{mod}}\Delta y+\frac{1}{2}C(\Delta y)^2, \qquad P_{\mathrm{est}}=P(y)\left(1-D_{\mathrm{mod}}\Delta y+ \frac{1}{2}C(\Delta y)^2\right).

Here $\Delta y$ is a rate change in decimal form: a rise from 10% to 15% is $+0.05$, not $+5$. The comparison error in this tutorial is defined as $P_{\mathrm{est}}-P(y+\Delta y)$. A positive error means the estimate is above the exact price; a negative error means it is below.

Why the formula holds: Differentiate $P(y)=\sum_t C_t(1+y)^{-t}$ term by term. The first derivative is $-\sum_t tC_t(1+y)^{-t-1}$, and the second is $\sum_t t(t+1)C_t(1+y)^{-t-2}$. Taylor’s formula about $y$, retaining terms through $(\Delta y)^2$, gives $P(y+\Delta y)\approx P(y)+P'(y)\Delta y+\tfrac12P''(y)(\Delta y)^2$. Dividing by $P(y)$ gives the estimate above. The omitted terms explain why the estimate need not equal exact repricing.

For a positive-cash-flow bond, the derivative formulas can also be evaluated directly:

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

These formulas assume fixed cash flows and a single yield applied consistently to every payment. If cash flows depend on the yield, or if the yield convention changes between the original price and the repricing, this calculation does not represent the required price change. For ordinary positive cash flows, convexity is positive. The estimate is local, so a larger shock generally gives the omitted higher-order terms more opportunity to matter; the actual discrepancy should be checked rather than assumed.

Worked Example: Build an Estimate at the Original Yield

A hypothetical two-year bond pays $100 at the end of year 1 and $1,100 at the end of year 2. Its annual effective yield is 10%. Calculate its price, modified duration, and convexity, then estimate its price after a 5-percentage-point yield increase and decrease. Use the same fixed payments in both cases.

Method and trap avoided. First calculate the present value of each payment at the original yield. Then calculate duration and convexity from those present values. The common mistake is to insert 5 instead of 0.05 as the yield shock.

The original price is:

P(0.10)=\frac{100}{1.10}+\frac{1100}{1.10^2} =90.909091+909.090909 =\$1{,}000.00.

The present-value weights are $90.909091/1000=1/11$ at year 1 and $909.090909/1000=10/11$ at year 2. The Macaulay duration is $(1)(1/11)+(2)(10/11)=21/11=1.909091$ years. Dividing by $1.10$ gives:

D_{\mathrm{mod}}=\frac{21/11}{1.10}=\frac{210}{121}=1.735537.

For convexity, the weighted sum of $t(t+1)$ is $(2)(1/11)+(6)(10/11)=62/11$. The annual-compounding convexity is this quantity divided by $1.10^2$:

C=\frac{62/11}{1.21}=\frac{6200}{1331}=4.658903.

For either 5-percentage-point shock, the convexity contribution to the estimated percentage change is $\tfrac12(4.658903)(0.05)^2=0.00582363$. For an increase, the duration contribution is $-(1.735537)(0.05)=-0.08677686$. Thus the estimated price is:

P_{\mathrm{est}}(0.15)=1000(1-0.08677686+0.00582363) =\$919.05.

For a decrease, the duration contribution changes sign while the convexity contribution remains positive:

P_{\mathrm{est}}(0.05)=1000(1+0.08677686+0.00582363) =\$1{,}092.60.

The estimate predicts a larger price gain for the yield decrease than the price loss for the equal yield increase. These are estimates only; the next example checks them against exact prices.

Compare Several Shocks by Repricing

Exact repricing uses the original cash-flow schedule and the shocked yield directly. For the bond above, for example, the exact price after a 5-percentage-point increase is $100/(1.15)+1100/(1.15)^2$. It is not obtained by changing the original duration or convexity estimate. A useful comparison records the shocked yield, estimated price, exact price, and estimate-minus-exact error.

Worked Example: Check Four Yield Shocks Against Exact Prices

Use the same hypothetical bond, with payments of $100 at year 1 and $1,100 at year 2, and original annual effective yield 10%. Compare the duration-and-convexity estimates for shocks of $-10$, $-5$, $+5$, and $+10$ percentage points with exact repricing.

Method and trap avoided. Keep the original measures $D_{\mathrm{mod}}=1.735537$ and $C=4.658903$ for all four estimates; reprice separately at each new yield. Do not interpret a negative shock as a negative convexity term: $(\Delta y)^2$ is positive for either direction.

For a 5-point shock, the estimate is the one calculated above. For a 10-point shock, the duration contribution has magnitude $1.735537(0.10)=0.1735537$, and the convexity contribution is $\tfrac12(4.658903)(0.10)^2=0.0232945$. Therefore, the estimated prices for the 10-point shocks are:

P_{\mathrm{est}}(-10\text{ points})=1000(1+0.1735537+0.0232945)=\$1{,}196.85, $$ $$ P_{\mathrm{est}}(+10\text{ points})=1000(1-0.1735537+0.0232945)=\$849.74.

Now reprice exactly. At a 5% yield:

P(0.05)=\frac{100}{1.05}+\frac{1100}{1.05^2} =95.238095+997.732426 =\$1{,}092.9705.

At a 15% yield:

P(0.15)=\frac{100}{1.15}+\frac{1100}{1.15^2} =86.956522+831.758034 =\$918.7146.

At 0%, the exact price is $100+1100=\$1{,}200.00$. At 20%:

P(0.20)=\frac{100}{1.20}+\frac{1100}{1.20^2} =83.333333+763.888889 =\$847.2222.

The comparison below rounds prices and errors to the nearest cent. Each error is estimated price minus exact price.

Yield shock	New yield	Estimated price	Exact price	Estimate error
$-10$ points	0%	$1,196.85	$1,200.00	$-$3.15$
$-5$ points	5%	$1,092.60	$1,092.97	$-$0.37$
$+5$ points	15%	$919.05	$918.71	$+$0.33$
$+10$ points	20%	$849.74	$847.22	$+$2.52$

For the 5-point shocks, the estimate is close to exact repricing: it is about 37 cents low for the decrease and 33 cents high for the increase. At 10 points, the errors are larger in dollars: about $3.15 low and $2.52 high. The equal positive and negative shocks also do not produce equal price changes. Exact repricing shows a $92.97 gain when yield falls to 5%, compared with an $81.29 loss when yield rises to 15%.

The pattern is consistent with a local second-order estimate becoming less precise as the shock grows, but it does not make the estimate a substitute for repricing. The sign of the error also depends on the cash flows and the direction and size of the shock. Use the calculated difference, not a rule that the approximation must always overstate or understate price.

Does the Cash-Flow Pattern Change the Check?

Duration and convexity summarize the timing of the particular cash flows being valued. Another bond with the same original price and yield can have different measures and a different exact-price discrepancy. The comparison procedure stays the same: calculate the original measures, apply the estimate, reprice at the new yield, and subtract exact price from estimated price.

Worked Example: A Different Two-Year Cash-Flow Pattern

A hypothetical bond pays $110 at year 1 and $1,089 at year 2. Its annual effective yield is 10%. Estimate and exactly reprice its value after a 10-percentage-point yield increase. Report the estimate error to the nearest cent.

Method and trap avoided. Recalculate the measures for this bond rather than reusing the previous bond’s duration and convexity. Equal original prices do not imply equal cash-flow timing or equal price sensitivity.

At 10%, the present values are $110/1.10=\$100$ and $1089/1.10^2=\$900$, so the original price is $1,000. The Macaulay duration is $(1)(0.1)+(2)(0.9)=1.9$ years, giving:

D_{\mathrm{mod}}=\frac{1.9}{1.10}=1.727273.

The weighted sum of $t(t+1)$ is $(2)(0.1)+(6)(0.9)=5.6$, so:

C=\frac{5.6}{1.10^2}=4.628099.

For $\Delta y=0.10$, the estimated percentage change is:

-(1.727273)(0.10)+\frac12(4.628099)(0.10)^2 =-0.1727273+0.0231405=-0.1495868.

Thus the estimated price is $1000(1-0.1495868)=\$850.41$. Exact repricing at 20% gives:

P(0.20)=\frac{110}{1.20}+\frac{1089}{1.20^2} =91.666667+756.25 =\$847.916667.

The estimate error is $850.4132-847.9167=\$2.4965$, or +$2.50 to the nearest cent. The estimate is above the exact price. This check is complete: the new-yield price was calculated from the original payments, and the error sign follows the stated convention.

Reading the Discrepancy Correctly

A comparison is most informative when the calculation conventions are controlled. Before interpreting the error, verify that:

The yield shock is entered in decimals and is measured from the original yield.
The original price, duration, and convexity use the same annual-compounding convention.
Exact repricing discounts every stated cash flow at the shocked yield.
The error is calculated in the stated order: estimate minus exact price.

If the estimate and exact price differ, that difference is not automatically an arithmetic error. The estimate retains only the first- and second-order yield effects; exact repricing includes the full price function at the new yield. As the shock grows, higher-order effects can become more material, which is why a sensitivity estimate should be checked at the shock sizes relevant to the calculation. When the exact cash flows and shocked yield are available, exact repricing is the direct benchmark.

Key takeaway: Duration and convexity give a local, second-order estimate. To assess it, calculate the exact price from the same fixed cash flows at each shocked yield, then report the estimate-minus-exact difference. Larger shocks warrant particular care because the approximation omits higher-order terms.

Check Your Understanding

For each problem, use the stated annual effective yield and fixed cash flows. Round money to the nearest cent.

A certain payment of $105 is due in one year. At an original annual effective yield of 5%, estimate the price after a one-percentage-point yield increase using duration and convexity, then check it by exact repricing at 6%. (Answer: Estimate $99.06; exact price $99.06.)
A bond pays $100 at year 1 and $1,100 at year 2, with original annual effective yield 10%. Use the duration-and-convexity measures calculated for this cash-flow pattern to estimate its price after a 5-percentage-point yield decrease, then find the exact repriced price at 5%. (Answer: Estimate $1,092.60; exact price $1,092.97; estimate error $-$0.37$.)
A bond pays $110 at year 1 and $1,089 at year 2, with original annual effective yield 10%. For a 10-percentage-point yield increase, compare the duration-and-convexity estimate with exact repricing at 20%. (Answer: Estimate $850.41; exact price $847.92; estimate error $+$2.50$.)
For the bond in Question 2, compare the exact price changes for yield shocks of $+5$ and $-5$ percentage points. Are the price changes equal in magnitude? (Answer: No. The price falls by about $81.29 for the increase and rises by about $92.97 for the decrease.)

Yield shock	New yield	Estimated price	Exact price	Estimate error
\(-10\) points	0%	$1,196.85	$1,200.00	\(-$3.15\)
\(-5\) points	5%	$1,092.60	$1,092.97	\(-$0.37\)
\(+5\) points	15%	$919.05	$918.71	\(+$0.33\)
\(+10\) points	20%	$849.74	$847.22	\(+$2.52\)

What You'll Learn