When Yield Shocks Are Too Large For The Approximation

Why the Size of a Yield Shock Matters

The duration-convexity formula is a local estimate: it uses the price’s slope and curvature at the starting yield to estimate the effect of a change in yield. It can be useful for small changes, but it is not an exact pricing formula. As the yield shock grows, terms beyond convexity can become material, and the estimate may differ substantially from the price obtained by repricing the cash flows.

The key exam skill is not to memorize a universal shock size at which the approximation fails. There is no such cutoff that works for every bond. Instead, compare the estimate with exact repricing when the cash flows and new yield are known, and judge the error in relation to the decision or tolerance in the question.

Definition: For a bond with price $P(y)$ at annual effective yield $y$, modified duration is $D=-P'(y)/P(y)$, and convexity is $C=P''(y)/P(y)$. For a yield change $\Delta y$, the duration-convexity estimate is $$ \frac{\Delta P}{P}\approx -D\Delta y+\frac{1}{2}C(\Delta y)^2, \qquad P_{\text{est}}=P(y)\left(1-D\Delta y+\frac{1}{2}C(\Delta y)^2\right). $$ Here, $\Delta y$ is expressed as a decimal, not as a number of percentage points or basis points.

These definitions and the approximation assume that the bond’s cash flows are fixed and that the same annual effective yield is used to discount them before and after the shock. If cash flows are $CF_t$ at times $t$, exact repricing at the new yield is

P(y+\Delta y)=\sum_t \frac{CF_t}{(1+y+\Delta y)^t}.

The formula requires $y+\Delta y>-1$ for these annual-compounding discount factors to be defined. It does not account for cash flows changing in response to yield, or for changes in the yield curve that cannot be represented by the single yield shift in the model.

Why a Larger Shock Can Increase the Error

The approximation comes from a Taylor expansion of the price function about the starting yield. For a sufficiently smooth price function,

P(y+\Delta y) = P(y)+P'(y)\Delta y+\frac{1}{2}P''(y)(\Delta y)^2+R_3,

where $R_3$ contains the third- and higher-order effects. Dividing by $P(y)$ and using $P'(y)/P(y)=-D$ and $P''(y)/P(y)=C$ gives the duration-convexity formula. The omitted terms are generally more important as the change moves farther from the point $y$ where the derivatives were evaluated. This explains why including convexity improves on a duration-only estimate locally without making the result exact for a large shock.

Fix the starting point.
Record the current yield, price, duration and convexity. These describe the price function locally at the current yield.

Convert the shock to a decimal.
For example, a rise of 5 percentage points is $\Delta y=0.05$. Keep the sign: a yield fall has a negative change.

Calculate the estimate.
Use the duration term $-D\Delta y$ and the convexity term $\frac12 C(\Delta y)^2$, then multiply the estimated percentage change by the starting price.

Reprice and compare.
When cash flows are available, discount them at the new yield. Compare exact and estimated prices using unrounded values, then report money to the cent.

Worked Examples: Comparing Approximation and Exact Repricing

Worked Example: A Zero-Coupon Bond Under Increasing Shocks

A hypothetical bond pays $1,000 in 10 years and no interim coupons. Its annual effective yield is 5%. Compare duration-convexity estimates and exact prices for yield increases of 1, 5, 10 and 15 percentage points. Give prices to the nearest cent.

Method and starting values. For a zero-coupon bond with maturity $n$, $P(y)=1000(1+y)^{-n}$. Differentiating gives $D=n/(1+y)$ and $C=n(n+1)/(1+y)^2$. At $y=0.05$, this gives

P_0=\frac{1000}{1.05^{10}}=613.9133\ldots, \quad D=\frac{10}{1.05}=9.5238095\ldots, \quad C=\frac{110}{1.05^2}=99.7732426\ldots.

Thus, for shock $x=\Delta y$, the estimated price is

P_{\text{est}}=613.9132535\ldots \left(1-9.5238095\ldots x+49.8866213\ldots x^2\right).

For a 1-percentage-point increase, $x=0.01$, so the factor is $1-9.5238095(0.01)+49.8866213(0.01)^2=0.9097506\ldots$. The estimate is $613.9132535(0.9097506)=558.5079\ldots$, or $558.51. The exact price is $1000/1.06^{10}=558.394\ldots$, or $558.39.

Repeating those steps for the larger shocks gives the following results. The differences are calculated using unrounded prices before rounding to cents.

Yield increase	New yield	Estimated price	Exact price	Absolute error
1 percentage point	6%	$558.51	$558.39	$0.11
5 percentage points	10%	$398.14	$385.54	$12.60
10 percentage points	15%	$335.49	$247.18	$88.31
15 percentage points	20%	$425.98	$161.51	$264.47

For instance, at a 10-percentage-point increase, the factor is $1-(10/1.05)(0.10)+\frac12(110/1.05^2)(0.10)^2$. The estimate is $335.49. Exact repricing gives $1000/1.15^{10}=247.1847\ldots$, or $247.18; their unrounded difference is $88.31. At the largest shock, the estimate even rises above the estimate for the 10-point shock, although the exact price continues to fall. This is a clear warning that the local approximation is no longer reliable.

Check. Each exact price uses the original $1,000 payment discounted at the new yield. The estimates use the same starting price, duration and convexity throughout; they are not recalculated as though the yield had first moved to an intermediate value. The example avoids the common mistake of treating a large-shock estimate as an exact price.

Worked Example: A Distant Cash Flow and a Large Shock

A hypothetical asset has fixed payments of $100 at the end of year 2 and $100 at the end of year 12. The starting annual effective yield is 0%. Compare the duration-convexity estimate with exact repricing for yield increases of 5 and 10 percentage points.

Method. At a zero yield, the price is $P_0=100+100=200$. For annual compounding, a cash flow $CF_t$ contributes $tCF_t$ to the duration numerator and $t(t+1)CF_t$ to the convexity numerator at $y=0$. Therefore,

D=\frac{2(100)+12(100)}{200}=7, \qquad C=\frac{2(3)(100)+12(13)(100)}{200}=81.

For a 5-point shock, $\Delta y=0.05$. The estimated price is

200\left(1-7(0.05)+\frac12(81)(0.05)^2\right) =200(0.75125)=150.25.

The exact price is $100/1.05^2+100/1.05^{12}=90.70295\ldots+55.6837\ldots=146.3866\ldots$, or $146.39. The unrounded estimate exceeds exact price by $3.86.

For a 10-point shock, the estimate is $200(1-7(0.10)+\frac12(81)(0.10)^2)=200(0.705)=141.00$. Exact repricing gives $100/1.10^2+100/1.10^{12}=82.6446\ldots+31.8631\ldots=114.5077\ldots$, or $114.51. The error is $26.49. The distant payment makes the price response particularly sensitive to the shock; the error grows markedly even though the same duration and convexity are used.

Check. The convexity contribution is positive for both shocks because it contains the square of the yield change. It does not cancel the duration contribution: for the 10-point shock, the estimated percentage change is $-0.70+0.405=-0.295$. The method avoids the mistake of assuming that a positive convexity correction guarantees an accurate estimate for any shock size.

Worked Example: Applying a Dollar Tolerance

A hypothetical two-payment instrument pays $100 at the end of year 1 and $100 at the end of year 2. At a 0% annual effective yield, an analyst must estimate its price after the yield rises by 30 percentage points. The estimate is acceptable only if its absolute error is no more than $5. Should the analyst rely on duration and convexity?

Method. The starting price is $P_0=200$. At $y=0$, the duration and convexity are

D=\frac{1(100)+2(100)}{200}=1.5, \qquad C=\frac{1(2)(100)+2(3)(100)}{200}=4.

The yield change is $\Delta y=0.30$, not $30$. The estimated price is

P_{\text{est}} =200\left(1-1.5(0.30)+\frac12(4)(0.30)^2\right) =200(0.73)=146.00.

At the new yield of 30%, exact repricing gives $100/1.30+100/1.30^2=76.9231\ldots+59.1716\ldots=136.0947\ldots$, or $136.09. The unrounded absolute error is $9.91, which exceeds the $5 tolerance. The analyst should not rely on the approximation for this requirement.

Check. Using $0.30$ rather than $30$ keeps the yield change in the units required by the formula. Exact repricing confirms the decision against the stated tolerance.

Common Traps and a Practical Rule

A frequent error is to insert a percentage-point number directly as $\Delta y$. In the first example, using $10$ instead of $0.10$ for a 10-percentage-point increase would produce the nonsensical factor $1-9.5238095(10)+49.8866213(10)^2$, rather than the correct factor $1-9.5238095(0.10)+49.8866213(0.10)^2$. Another mistake is to omit the square on $\Delta y$ in the convexity term. The convexity contribution must be $\frac12 C(\Delta y)^2$.

Do not infer that a particular shock is always “small” or “large” based only on its percentage-point size. The quality of the estimate also depends on the cash-flow timing, the starting yield and the resulting shape of the price function. If the problem supplies cash flows and asks for a price at a specified new yield, exact repricing is the direct method; the approximation is useful for estimation or comparison, not a substitute when exact calculation is required.

Key takeaway: Duration and convexity describe the price response locally at the starting yield. Convert the shock to a decimal, calculate the estimate, and compare with exact repricing when possible. Treat a growing difference as evidence that omitted higher-order effects matter; do not impose a universal shock-size cutoff.

Check Your Understanding

For each problem, use the stated annual effective yield convention. Round prices to the nearest cent.

A $1,000 zero-coupon bond has two years remaining and a starting yield of 0%. Estimate its price after a 10-percentage-point yield increase, then calculate the exact price. (Answer: estimate $830.00; exact price $826.45.)
An instrument pays $200 at the end of year 1 and $300 at the end of year 2. At a 0% starting yield, calculate duration and convexity, then estimate the price after a 10-percentage-point increase. (Answer: $D=1.6$, $C=4.4$, estimated price $431.00.)
A hypothetical asset pays $1,000 at the end of year 1 and $1,000 at the end of year 3. Starting at a 0% annual effective yield, estimate and exactly reprice after a 20-percentage-point increase. (Answer: estimate $1,480.00; exact price $1,412.04.)
A hypothetical instrument pays $100 at the end of each of years 1 and 2. At a 0% yield, estimate and exactly reprice after a 30-percentage-point increase. If the error tolerance is $5, does the estimate meet it? (Answer: estimate $146.00; exact price $136.09; error $9.91, so it does not meet the tolerance.)
An asset pays $100 at the end of year 2 and $100 at the end of year 8. Starting at a 0% yield, find the duration-convexity estimate and exact price after a 5-percentage-point increase. (Answer: $D=5$, $C=39$, estimated price $159.75; exact price $158.39.)

What You'll Learn