Convexity Corrections To Duration-Based Loss Estimates

Why Add Convexity to Duration?

Duration gives a first-order estimate of how a bond’s price responds to a yield change. For small changes, that linear estimate can be useful. For a larger yield movement, however, the bond price generally does not move along a straight line: its relationship with yield is curved. A duration-only estimate can therefore miss a material part of the price change.

Convexity measures this curvature and supplies a second-order correction. For a conventional fixed-cash-flow bond with positive convexity, the correction is positive in the price-change approximation for either an upward or downward yield movement. Thus, compared with duration alone, the estimated price loss from rising yields is smaller, while the estimated price gain from falling yields is larger.

Definition: Let $P(y)$ be a bond’s price as a function of its annual effective yield $y$. Its modified duration is $D=-\frac{1}{P}\frac{dP}{dy}$, and its convexity is $C=\frac{1}{P}\frac{d^2P}{dy^2}$. For a yield change $\Delta y$, the duration-convexity approximation is $\frac{\Delta P}{P}\approx-D\Delta y+\frac{1}{2}C(\Delta y)^2$.

Here, $P$ is the price before the yield change, and $\Delta P$ is the estimated price change. Use yield changes as decimals: 100 basis points is $0.01$, not $1$. Duration is measured in years under the annual-yield convention used here; convexity has units of years squared. The approximation assumes a sufficiently smooth price-yield relationship near the starting yield and a consistent yield convention. For fixed cash flows, it also assumes those cash flows do not change as yields move.

Deriving the Approximation

The price function can be approximated around its current yield by a Taylor expansion. Keeping the first two derivative terms gives the reason duration and convexity appear in the estimate:

P(y+\Delta y)\approx P(y)+P'(y)\Delta y+\frac{1}{2}P''(y)(\Delta y)^2.

Subtract $P(y)$ from both sides and divide by the starting price $P(y)$. By definition, $P'(y)/P(y)=-D$ and $P''(y)/P(y)=C$. Therefore,

\frac{\Delta P}{P}\approx\frac{P'(y)}{P(y)}\Delta y+\frac{1}{2}\frac{P''(y)}{P(y)}(\Delta y)^2 =-D\Delta y+\frac{1}{2}C(\Delta y)^2.

The squared yield change is nonnegative, so a positive convexity contributes a positive amount to the estimated price change. That does not mean every estimated price change is positive: when yields rise, the negative duration term can still dominate. Nor should the sign be confused when reporting a loss, which is the negative of the price change.

Fixed-Cash-Flow Bond Formulas

Suppose a bond pays fixed cash flows $CF_t$ at times $t$, and its price is calculated using annual effective yield $y$ and annual compounding. Then

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

Differentiating each discounted cash flow gives formulas for modified duration and convexity:

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

With positive cash flows and a positive yield, every term in the second derivative is positive, so this fixed-cash-flow bond has positive convexity. These formulas use annual effective yield and time measured in years. If a bond is priced using a different compounding convention, the yield units, cash-flow timing, and derivatives must be made consistent. Do not combine a duration or convexity calculated under one convention with a yield change under another.

Worked Example: A Zero-Coupon Bond and a 100-Basis-Point Increase

A hypothetical bond pays $1,000 in two years and no other cash flow. Its annual effective yield is 5%. Estimate the price change if yield rises by 100 basis points, and compare the estimate with the repriced bond. State the method and identify the error the calculation must avoid.

Method. Calculate the starting price, modified duration, and convexity from the fixed cash flow, then apply the second-order approximation. The yield change is $+100$ basis points $=+0.01$.

The starting price is $P=1{,}000/(1.05)^2=1{,}000/1.1025=\$907.0295$. For a zero-coupon bond with payment time $t=2$, the formulas reduce to $D=t/(1+y)$ and $C=t(t+1)/(1+y)^2$. Thus $D=2/1.05=1.904762$ and $C=6/(1.05)^2=6/1.1025=5.442177$.

The estimated percentage price change is

-D\Delta y+\frac{1}{2}C(\Delta y)^2 =-(1.904762)(0.01)+\frac{1}{2}(5.442177)(0.01)^2 =-0.01877551.

The estimate is a price decrease of 1.877551%, or $907.0295(0.01877551)=\$17.03$ to the nearest cent. The estimated new price is $\$907.0295-\$17.03=\$890.00$, rounded to the nearest cent.

For a check, the exact price at 6% is $1{,}000/(1.06)^2=1{,}000/1.1236=\$889.9964$, or $\$890.00$ to the nearest cent. The exact decrease is $\$17.03$ to the nearest cent. Agreement at the cent level here is due to rounding; the approximation is not generally exact.

Common mistake avoided. Using $100$ instead of $0.01$ for the yield change would make both terms vastly too large. A basis-point change must first be converted to a decimal yield change, and the convexity term uses that decimal change squared.

Interpreting the Convexity Adjustment

For positive convexity, the correction $\frac{1}{2}C(\Delta y)^2$ is positive whether $\Delta y$ is positive or negative. Its effect depends on what is being reported. If yields rise, duration estimates a price decline; positive convexity makes that decline less severe. If yields fall, duration estimates a price increase; positive convexity makes that increase larger. The adjustment’s dollar amount is $P\frac{1}{2}C(\Delta y)^2$.

The correction becomes more important as the yield movement grows because it depends on the square of that movement. This does not make the approximation reliable for arbitrarily large shocks. It is still a local approximation, and omitted higher-order terms, changing cash flows, or an unusual price-yield shape can matter. For a large scenario, full repricing is preferable when the model and cash flows permit it.

Worked Example: A Coupon Bond at a Zero Yield

A hypothetical two-year bond pays a $60 coupon at the end of year 1 and $1,060 at the end of year 2. Its annual effective yield is 0%. Estimate the price change for a 100-basis-point increase, then check it by repricing. The zero yield is part of the stated scenario; do not substitute a positive yield.

Method. Use the fixed-cash-flow derivative formulas at the stated yield, then compare the second-order estimate with direct repricing. The calculation avoids applying an assumed yield different from the one in the problem.

At $y=0$, the starting price is $P=60+1{,}060=\$1{,}120$. Modified duration is

D=\frac{1(60)+2(1{,}060)}{1{,}120} =\frac{2{,}180}{1{,}120}=1.946429.

Convexity is

C=\frac{1(2)(60)+2(3)(1{,}060)}{1{,}120} =\frac{120+6{,}360}{1{,}120} =5.785714.

With $\Delta y=+0.01$, the estimated percentage change is

-(1.946429)(0.01)+\frac{1}{2}(5.785714)(0.01)^2 =-0.019175.

The estimated change is $-1.9175\%$, or $1{,}120(-0.019175)=-\$21.476$, which rounds to a decrease of $\$21.48$. The estimated new price is $\$1{,}098.52$.

At 1%, direct repricing gives $60/1.01+1{,}060/(1.01)^2=\$59.4059+\$1{,}039.1138=\$1{,}098.5198$, or $\$1{,}098.52$ to the nearest cent. The check confirms the estimate at the reported precision. Duration alone would estimate a larger decline of $1{,}120(1.946429)(0.01)=\$21.80$; the positive convexity correction reduces that estimated loss.

Combining Convexity Across a Portfolio

For a portfolio of bonds exposed to the same small parallel yield change, dollar price changes add. If bond $j$ has market value $P_j$, modified duration $D_j$, and convexity $C_j$, the portfolio measures are market-value-weighted averages:

D_{\mathrm{port}}=\frac{\sum_j P_jD_j}{\sum_jP_j}, \qquad C_{\mathrm{port}}=\frac{\sum_j P_jC_j}{\sum_jP_j}.

These weights are the bonds’ shares of the starting portfolio value. The result assumes the same yield change applies to each bond and that the individual measures use consistent yield conventions. A nonparallel yield curve move requires a different treatment of exposures at different maturities; a single parallel-shift estimate should not be mistaken for a general curve-risk calculation.

Worked Example: Correcting a Portfolio Loss Estimate

A hypothetical insurer’s bond portfolio contains Bond A with market value $6 million, modified duration 2.8, and convexity 12, and Bond B with market value $4 million, modified duration 6.2, and convexity 55. Estimate the portfolio value change for a parallel yield increase of 150 basis points. Give the duration-only estimate as a comparison.

Method. First calculate market-value-weighted duration and convexity, then apply the approximation to the total starting value. This avoids treating the two bonds as if they contributed equally despite their different market values.

The portfolio value is $\$6\text{ million}+\$4\text{ million}=\$10\text{ million}$. Its duration is $(6(2.8)+4(6.2))/10=(16.8+24.8)/10=4.16$. Its convexity is $(6(12)+4(55))/10=(72+220)/10=29.2$. The yield change is $+150$ basis points $=+0.015$.

The estimated percentage change is

-(4.16)(0.015)+\frac{1}{2}(29.2)(0.015)^2 =-0.0624+0.003285=-0.059115.

The estimated portfolio loss is $0.059115(\$10{,}000{,}000)=\$591{,}150$, and the estimated ending value is $\$9{,}408{,}850$. Duration alone gives a loss of $0.0624(\$10{,}000{,}000)=\$624{,}000$. The convexity adjustment reduces the estimated loss by $\$32{,}850$. The check is that the correction is positive in the price-change formula, so it offsets part of the negative duration term for this upward yield shock.

Effective Convexity from Repricing

For a bond whose cash flows or exercise behavior can change with yields, fixed-cash-flow derivative formulas may not describe its price response. A practical alternative is to reprice the bond under symmetric yield shifts. With starting price $P_0$, yield bump $h$, price $P_-$ at yield $y-h$, and price $P_+$ at yield $y+h$, define effective duration and effective convexity by

D_{\mathrm{eff}}=\frac{P_--P_+}{2P_0h}, \qquad C_{\mathrm{eff}}=\frac{P_-+P_+-2P_0}{P_0h^2}.

These are finite-difference estimates of the first and second price derivatives, scaled by the starting price. They depend on the repricing model, the selected bump, and the assumption that the scenarios shift the relevant yield consistently. They should not be treated as fixed-cash-flow convexity values when embedded options materially alter cash flows.

Worked Example: Effective Convexity from Scenario Prices

A hypothetical option-embedded bond has current price $P_0=\$100.00$. A pricing model gives a price of $\$103.60$ after a 100-basis-point yield decrease and $\$96.80$ after a 100-basis-point increase. Estimate its effective duration and convexity, then estimate the price for a 150-basis-point yield increase from the current yield.

Method. Apply the symmetric finite-difference formulas with $h=0.01$, then use the resulting measures for the requested yield change. The common error to avoid is using $h^2$ incorrectly in the convexity denominator.

Effective duration is $(103.60-96.80)/(2(100)(0.01))=6.80/2=3.4$. Effective convexity is $(103.60+96.80-2(100))/(100(0.01)^2)=0.40/0.01=40$.

For $\Delta y=+0.015$, the estimated percentage price change is

-(3.4)(0.015)+\frac{1}{2}(40)(0.015)^2 =-0.051+0.0045=-0.0465.

The estimated price decrease is $4.65\%$, or $\$100(0.0465)=\$4.65$, giving an estimated price of $\$95.35$. The positive convexity correction is 0.45 percentage points of price, offsetting part of the 5.10% duration-only decline. This is an estimate based on the local scenario measures, not a substitute for direct repricing at the full 150-basis-point shock.

Common Exam Traps

A frequent sign error is to subtract the convexity term from the price-change estimate. For a positive-convexity bond and $\Delta y=+0.01$, duration $D=4$, and convexity $C=30$, the correct estimate is

\frac{\Delta P}{P}\approx -(4)(0.01)+\frac{1}{2}(30)(0.01)^2 =-0.04+0.0015=-0.0385.

The estimated decline is 3.85%. The wrong calculation, $-0.04-0.0015=-0.0415$, would imply a 4.15% decline and would make positive convexity worsen the estimated loss. The correction belongs with a plus sign in the price-change formula because it comes from the second derivative. For a loss amount, negate the entire estimated price change.

Do not square the number of basis points as a whole number. Convert the yield change to decimal form first; for example, 75 basis points is $0.0075$.
Do not omit the one-half. The second-order Taylor term is $\frac{1}{2}C(\Delta y)^2$.
Do not confuse price change with loss. The convexity adjustment is positive for positive convexity in the price-change formula, but its effect on a reported loss is the opposite sign.
Do not assume the approximation is exact. It omits higher-order terms and may be inadequate for a large shock or a price response with changing cash flows.

Key takeaway: For a consistent yield convention, estimate a bond’s percentage price change as $-D\Delta y+\frac{1}{2}C(\Delta y)^2$. Positive convexity softens the estimated loss when yields rise and increases the estimated gain when yields fall; the yield change must be entered as a decimal.

Check Your Understanding

Work each problem using decimal yield changes and the stated assumptions.

A bond portfolio is worth $2,000 and has modified duration 4.5 and convexity 28. Estimate its price change for a 100-basis-point yield decrease. (Answer: estimated gain of $92.80, or 4.64%.)
A bond is worth $5,000, with modified duration 3.2 and convexity 18. Estimate its price and loss for a 200-basis-point yield increase. (Answer: estimated loss of $302; estimated price $4,698, a 6.04% decrease.)
A portfolio has two positions: $3 million with duration 2 and convexity 10, and $7 million with duration 5 and convexity 40. Find the portfolio measures and estimated loss for a parallel 100-basis-point increase. (Answer: duration 4.1, convexity 31, estimated loss $394,500.)
A bond has current price $100, and model prices of $104.00 and $97.20 under yield changes of $-100$ and $+100$ basis points, respectively. Find effective duration and convexity, then estimate its price after a 150-basis-point increase. (Answer: effective duration 3.4, effective convexity 120, estimated price $96.25.)
A portfolio is worth $1 million, with modified duration 5 and convexity 40. For an 80-basis-point yield increase, compare the duration-only loss with the duration-convexity loss. (Answer: duration-only loss $40,000; corrected estimated loss $38,720.)

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