Why Compare Repricing With an Estimate?
When a bond’s yield changes, its price changes too. Duration and convexity provide useful estimates of that change, but for a bond with known fixed cash flows there is another option: calculate the price directly at the new yield. Comparing these results shows what an approximation does and gives a practical check on the calculation.
The exact calculation is straightforward when every payment is fixed and the bond is valued using one annual effective yield. The main discipline is to distinguish the starting yield from the shocked yield: estimates use sensitivity measures at the starting yield, while exact repricing discounts all cash flows at the shocked yield.
For comparison, modified duration and convexity at the starting yield are $$ D_{\mathrm{mod}}=-\frac{P'(y)}{P(y)}, \qquad C=\frac{P''(y)}{P(y)}. $$ The duration-only estimate is \(P(y)[1-D_{\mathrm{mod}}\Delta y]\). The duration-convexity estimate is \(P(y)[1-D_{\mathrm{mod}}\Delta y+\tfrac12 C(\Delta y)^2]\). In these formulas, \(\Delta y\) is a decimal change in annual yield: a rise of 1 percentage point means \(\Delta y=0.01\), not \(1\).
Why the Estimates Have These Forms
For a fixed set of cash flows, differentiating each discounted payment gives $$ P'(y)=-\sum_{t=1}^{n}\frac{tC_t}{(1+y)^{t+1}}, \qquad P''(y)=\sum_{t=1}^{n}\frac{t(t+1)C_t}{(1+y)^{t+2}}. $$ Thus the first derivative measures the price’s local response to yield, and the second derivative measures its local curvature. Taylor’s formula about the starting yield gives the following justification for the estimates:
Keeping only the first derivative gives the duration-only estimate; keeping the second derivative as well gives the duration-convexity estimate. Neither is the same operation as evaluating the cash-flow formula at the new yield. Repricing directly is exact under the stated fixed-cash-flow, single-yield model. The estimates are local approximations based on sensitivities at the starting yield.
A Consistent Comparison Procedure
Use this sequence to ensure that the three prices are comparable. First calculate the starting price and the sensitivity measures at the same starting yield. Next apply the yield change in decimal form. Finally, discount the original cash flows at the shocked yield and compare the result with both estimates.
Use the same annual effective yield and time units for the starting price, sensitivities, and exact repricing.
Find \(P(y)\), \(D_{\mathrm{mod}}\), and \(C\) at the starting yield.
Use the signed decimal yield change in the duration-only and duration-convexity formulas.
Discount every original cash flow at \(y+\Delta y\), then compare the three resulting prices.
Worked Example: A Zero-Coupon Payment
A hypothetical bond pays $1,000 at the end of year 3 and has no other payments. Its starting annual effective yield is 8%, which rises by 2 percentage points. Find the exact shocked-yield price and both estimates, to the nearest cent. Method: calculate the starting sensitivities, then apply the shocked yield directly to the one cash flow. This avoids using the duration estimate as though it were an exact repricing.
At \(y=0.08\), the starting price is $$ P(0.08)=\frac{1000}{1.08^3} =\frac{1000}{1.259712} \approx \$793.8322. $$ For a single payment at time 3, the derivative formulas give \(D_{\mathrm{mod}}=3/(1+y)\) and \(C=3(4)/(1+y)^2\). Therefore, $$ D_{\mathrm{mod}}=\frac{3}{1.08}=2.777778,\qquad C=\frac{12}{1.08^2}=\frac{12}{1.1664}\approx10.288066. $$
The yield change is \(\Delta y=0.02\). The duration-only estimate is $$ \$793.8322[1-(2.777778)(0.02)] =\$793.8322(0.944444) \approx \$749.73. $$ The duration-convexity estimate is $$ \$793.8322[1-(2.777778)(0.02)+\tfrac12(10.288066)(0.02)^2] \approx \$793.8322(0.946502) \approx \$751.36. $$
For exact repricing, the shocked yield is \(0.08+0.02=0.10\). Discount the payment at 10%: $$ P(0.10)=\frac{1000}{1.10^3} =\frac{1000}{1.331} \approx \$751.31. $$ The exact price is about $751.31, compared with $749.73 from duration alone and $751.36 from duration-convexity. As a check, a higher yield produces a lower price than the starting price of $793.83. The duration-convexity estimate is closer in this example, but it is still an estimate, not the exact price.
Comparing Multiple Fixed Payments
With several payments, exact repricing means applying the shocked yield to each payment separately and adding the present values. The sensitivities are calculated from the starting-yield price and should not be recalculated at the shocked yield when using the stated estimates.
Worked Example: Two Payments With a Zero Starting Yield
A hypothetical fixed-cash-flow note pays $80 at the end of year 1 and $160 at the end of year 2. The starting annual effective yield is 0%, and it rises to 5%. Find the exact price and both estimates, to the nearest cent. Method: use the cash-flow derivative formulas at the starting yield, then discount both payments at 5% for the exact price. This avoids confusing a yield change of 5 percentage points with a 5% relative change in price.
At the starting yield, each cash flow is undiscounted, so $$ P(0)=80+160=\$240. $$ At \(y=0\), the modified duration is $$ D_{\mathrm{mod}}=\frac{1(80)+2(160)}{240} =\frac{400}{240} =\frac53\approx1.666667. $$ The convexity is $$ C=\frac{1(2)(80)+2(3)(160)}{240} =\frac{1120}{240} =\frac{14}{3}\approx4.666667. $$
Here \(\Delta y=0.05\). The duration-only estimate is $$ \$240\left[1-\left(\frac53\right)(0.05)\right] =\$240\left(\frac{11}{12}\right) =\$220.00. $$ Including convexity gives $$ \$240\left[1-\left(\frac53\right)(0.05) +\frac12\left(\frac{14}{3}\right)(0.05)^2\right] =\$240(0.9225) =\$221.40. $$
For exact repricing, discount each payment at 5%: $$ P(0.05)=\frac{80}{1.05}+\frac{160}{1.05^2} \approx \$76.1905+\$145.1247 =\$221.3152 \approx \$221.32. $$ The exact price is $221.32. The duration-only estimate is $220.00, and the duration-convexity estimate is $221.40. The exact calculation checks by separately accounting for the year-1 and year-2 payments; rounding only the final total gives the stated price.
Worked Example: A More Widely Spaced Cash Flow
A hypothetical payment arrangement promises $300 at the end of year 1 and $700 at the end of year 3. Its starting annual effective yield is 0%, and the yield rises by 10 percentage points. Calculate the exact price and both estimates, to the nearest cent. Method: calculate the time-weighted derivative measures at the starting yield, then re-discount each cash flow at 10%. This avoids treating a payment at year 3 as though it were due at year 2.
The starting price is $$ P(0)=300+700=\$1{,}000. $$ The modified duration and convexity at zero yield are $$ D_{\mathrm{mod}}=\frac{1(300)+3(700)}{1000} =\frac{2400}{1000}=2.4, $$ $$ C=\frac{1(2)(300)+3(4)(700)}{1000} =\frac{9000}{1000}=9. $$
With \(\Delta y=0.10\), the duration-only estimate is $$ \$1{,}000[1-(2.4)(0.10)] =\$1{,}000(0.76) =\$760.00. $$ The duration-convexity estimate is $$ \$1{,}000[1-(2.4)(0.10)+\tfrac12(9)(0.10)^2] =\$1{,}000(0.805) =\$805.00. $$
The exact shocked-yield price uses 10% for both payments: $$ P(0.10)=\frac{300}{1.10}+\frac{700}{1.10^3} \approx \$272.7273+\$525.9204 =\$798.6476 \approx \$798.65. $$ The exact price is $798.65; the duration-only and duration-convexity estimates are $760.00 and $805.00, respectively. The exact price lies between these two estimates in this example. The comparison also confirms that the year-3 cash flow must be discounted by \(1.10^3\), not \(1.10^2\).
What Exact Repricing Can and Cannot Tell You
For the specified fixed cash flows and single annual effective yield, exact repricing is a direct calculation, not an approximation. It is therefore a useful benchmark for assessing the two estimates in a problem where all cash flows are known. It also provides a check that the sign and unit of the yield change have been applied consistently.
This benchmark depends on the model. If a payment amount or timing changes with the yield, the original fixed-cash-flow formula does not describe that change. Likewise, a single-yield calculation does not capture a change in the shape of a term structure when different maturities move by different amounts. In such situations, the assumptions behind the calculation must be reconsidered rather than applying the fixed-cash-flow repricing formula without adjustment.
A common exam trap is to reprice at the wrong rate. Suppose the starting yield is 8% and it rises by 2 percentage points. The shocked yield is 10%. The incorrect calculation discounts at 2%: $$ \text{Wrong: }P(0.02)=\sum_t\frac{C_t}{1.02^t} \qquad\text{instead of}\qquad \text{Correct: }P(0.10)=\sum_t\frac{C_t}{1.10^t}. $$ The 2% is the change in yield, not the new yield. Another frequent error is to enter \(2\) rather than \(0.02\) in the approximation formulas. Write the yield change as a decimal before substituting; a 2-percentage-point increase is \(\Delta y=0.02\).
The topic is aligned with the SOA Exam FM (Financial Mathematics) syllabus. In a calculation, keep the three results clearly labelled: the duration-only estimate, the duration-convexity estimate, and the exact price at the shocked yield. Reporting them separately makes the method transparent and prevents an estimate from being mistaken for a direct cash-flow valuation.
Check Your Understanding
For each problem, use annual effective yields and fixed cash flows. Answers are rounded to the nearest cent where a dollar amount is requested.
- A note pays $100 at the end of year 1 and $200 at the end of year 2. Its starting yield is 0% and rises to 4%. Find the exact shocked-yield price. (Answer: \(100/1.04+200/1.04^2\approx \$281.07\))
- A zero-coupon bond pays $1,000 at the end of year 2. Its starting yield is 5% and falls by 1 percentage point. Find its exact repriced price. (Answer: \(1000/1.04^2\approx \$924.56\))
- A fixed-cash-flow instrument pays $250 at the end of year 1 and $500 at the end of year 2. Starting at a 0% yield, the yield rises by 2 percentage points. Find the duration-only estimate, duration-convexity estimate, and exact price. (Answer: \(D_{\mathrm{mod}}=5/3,\ C=14/3\); estimates \(\$725.00\) and \(\$725.70\); exact price \(\$250/1.02+\$500/1.02^2\approx \$725.68\))
- A bond pays $40 at the end of year 1 and $1,040 at the end of year 2. Its starting yield is 5% and rises by 2 percentage points. State the correct shocked yield and exact repricing expression. (Answer: shocked yield \(7\%\); \(40/1.07+1040/1.07^2\approx \$945.76\))
- A fixed-cash-flow asset has payments of $300 at year 1 and $700 at year 3. Its starting yield is 0%, and its yield rises by 10 percentage points. What is the exact price, and which time exponent applies to the $700 payment? (Answer: \(\$300/1.10+\$700/1.10^3\approx \$798.65\); use exponent 3)