Duration And Convexity As Price Derivatives

Why View Duration and Convexity as Derivatives?

A bond’s price changes when its yield changes. Modified duration and convexity describe two different features of that relationship: duration measures the price’s local slope, while convexity measures its local curvature. Defining them through price derivatives makes the signs and denominators precise, and provides a direct check on calculations based on cash-flow present values.

The formulas here use an annual effective yield and cash flows at integer year-end times. The cash flows are assumed known, fixed, and independent of yield. This is a yield-to-maturity pricing model, not a model for cash flows that change when interest rates change.

Definition: For fixed cash flows $C_t$ payable at the ends of years $t=1,\ldots,n$, and an annual effective yield $y>-1$, the bond price is $$ P(y)=\sum_{t=1}^{n}\frac{C_t}{(1+y)^t}. $$ The modified duration is $D_{\mathrm{mod}}(y)=-P'(y)/P(y)$. The convexity is $C_{\mathrm{vx}}(y)=P''(y)/P(y)$. Here the primes mean derivatives with respect to the decimal annual yield $y$.

These are normalized derivatives: dividing by price expresses sensitivity relative to the bond’s current price. The minus sign in modified duration makes the measure positive for a bond with nonnegative cash flows, since its price normally falls as its yield rises. Convexity is positive for such cash flows as well. Both measures are evaluated at a particular yield, so the yield used to price the bond must also be used to calculate them.

Deriving the Cash-Flow Derivative Formulas

Differentiate the price contribution from one payment, $C_t(1+y)^{-t}$, while holding $C_t$ and $t$ fixed. The power rule gives its first derivative as $-tC_t(1+y)^{-t-1}$, and differentiating again gives $t(t+1)C_t(1+y)^{-t-2}$. Summing the derivatives of the individual payments gives the derivatives of the whole price.

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}}.

Substituting these derivatives into the definitions yields the cash-flow formulas. They apply when the payment amounts and payment times remain fixed as $y$ varies. They do not, without further modeling, describe a callable bond whose expected cash flows change with yield or any other instrument whose payments depend on rates.

D_{\mathrm{mod}}(y) = \frac{\displaystyle\sum_{t=1}^{n}\frac{tC_t}{(1+y)^{t+1}}}{P(y)}, \qquad C_{\mathrm{vx}}(y) = \frac{\displaystyle\sum_{t=1}^{n}\frac{t(t+1)C_t}{(1+y)^{t+2}}}{P(y)}.

An equivalent form uses each payment’s present value at the yield $y$. Let $PV_t=C_t/(1+y)^t$, so that $P=\sum_t PV_t$. Factoring out the additional powers of $1+y$ in the derivative formulas gives

D_{\mathrm{mod}}(y) = \frac{\sum_{t=1}^{n}tPV_t}{(1+y)P}, \qquad C_{\mathrm{vx}}(y) = \frac{\sum_{t=1}^{n}t(t+1)PV_t}{(1+y)^2P}.

This form is useful as a calculation check. The present values sum to the price, and the weighted sums then supply the numerators. The denominators still contain $1+y$ for duration and $(1+y)^2$ for convexity; omitting those factors changes the measures.

Worked Calculations for a Fixed-Cash-Flow Bond

Worked Example: Calculate Both Derivatives at the Bond’s Yield

A hypothetical bond pays $110 at the end of year 1 and $121 at the end of year 2. Its annual effective yield is 10%. Find its price, modified duration, and convexity by differentiating the price function. Report duration and convexity to four decimal places.

Method. Write the fixed-cash-flow price as a function of $y$, differentiate it twice, and evaluate the price and derivatives at $y=0.10$. This avoids treating duration or convexity as an unnormalized time measure.

The price function and its derivatives are

P(y)=\frac{110}{1+y}+\frac{121}{(1+y)^2}, $$ $$ P'(y)=-\frac{110}{(1+y)^2}-\frac{242}{(1+y)^3}, \qquad P''(y)=\frac{220}{(1+y)^3}+\frac{726}{(1+y)^4}.

At $y=0.10$, the two discounted payments are $110/1.10=100$ and $121/1.10^2=100$. Therefore $P=200$. For the first derivative, the two terms are

-\frac{110}{1.21}=-90.9090909, \qquad -\frac{242}{1.331}=-181.8181818, \qquad P'=-272.7272727.

For the second derivative, the terms are

\frac{220}{1.331}=165.2892562, \qquad \frac{726}{1.4641}=495.8677686, \qquad P''=661.1570248.

Thus

D_{\mathrm{mod}}=-\frac{-272.7272727}{200} =1.3636364\approx 1.3636, \qquad C_{\mathrm{vx}}=\frac{661.1570248}{200} =3.3057851\approx 3.3058.

The negative first derivative is consistent with price falling as yield rises, while both reported measures are positive. The common mistake this method avoids is to report $P'$ itself as duration: modified duration is the negative derivative divided by price.

Worked Example: Verify the Measures with Present-Value Weights

Use the same bond and 10% annual effective yield. Calculate modified duration and convexity from present-value weights, then verify that the derivative results agree.

Method. First calculate each payment’s present value. Then use the weighted formulas. The trap to avoid is omitting the yield factors in the denominators when converting from present-value weights to derivatives.

The present values are $PV_1=100$, $PV_2=100$, and $P=200$. The duration numerator before division by $1+y$ is

$$ 1(100)+2(100)=300. $$

Therefore the weighted calculation gives

D_{\mathrm{mod}} = \frac{300}{1.10(200)} = \frac{300}{220} = 1.3636364 \approx 1.3636.

For convexity, the weighted numerator is

$$ 1(2)(100)+2(3)(100)=200+600=800. $$

The denominator is $(1.10)^2(200)=1.21(200)=242$, so

C_{\mathrm{vx}} = \frac{800}{242} = 3.3057851 \approx 3.3058.

As a direct check, the weighted duration calculation implies $P'=-D_{\mathrm{mod}}P=-1.3636364(200)=-272.7273$, which agrees with the first derivative above. The weighted convexity implies $P''=C_{\mathrm{vx}}P=3.3057851(200)=661.1570$, also in agreement. Matching both derivatives confirms that the present values, weights, and yield denominators were used consistently.

Worked Example: A Bond with Three Payments

A hypothetical bond pays $110, $242, and $399.30 at the ends of years 1, 2, and 3, respectively. Its annual effective yield is 10%. Calculate its price, modified duration, and convexity to four decimal places.

Method. Discount each payment at the stated annual effective yield, then use the present-value-weighted formulas. This example checks that the measures account for all payment dates, not just the final maturity.

The present values are

PV_1=\frac{110}{1.10}=100,\quad PV_2=\frac{242}{1.10^2}=200,\quad PV_3=\frac{399.30}{1.10^3}=300.

The price is $P=100+200+300=600$. The duration numerator is $1(100)+2(200)+3(300)=1,400$, giving

D_{\mathrm{mod}} = \frac{1,400}{1.10(600)} = \frac{1,400}{660} = 2.1212121 \approx 2.1212.

The convexity numerator is $1(2)(100)+2(3)(200)+3(4)(300)=200+1,200+3,600=5,000$. The denominator is $1.21(600)=726$, so

C_{\mathrm{vx}} = \frac{5,000}{726} = 6.8870523 \approx 6.8871.

The price is $600, and both numerators include every payment’s present value with the appropriate time weight. In particular, the year-3 payment contributes $3(4)(300)=3,600$ to the convexity numerator, not merely $3^2(300)$.

Common Differentiation and Interpretation Traps

The factor $t(t+1)$ in the second derivative is a frequent source of errors. Differentiating $(1+y)^{-t}$ once produces the factor $-t$ and power $-t-1$. Differentiating a second time multiplies by $-(t+1)$, so the two negative signs give $t(t+1)$. Replacing this factor with $t^2$ does not differentiate the price function correctly.

For example, in the two-payment bond above, the present-value-weighted convexity numerator is $1(2)(100)+2(3)(100)=800$. Using $t^2$ instead would give $1^2(100)+2^2(100)=500$, producing an incorrect convexity of $500/242=2.0661$, rather than $800/242=3.3058$.

Also keep track of what is being differentiated. The derivative is with respect to the yield written as a decimal, not the yield expressed as a number of basis points. For example, a change of one percentage point is a change of $0.01$ in $y$, not a change of $1$. The derivative definitions themselves use the decimal yield variable.

Key takeaway: Under annual effective compounding with fixed scheduled cash flows, modified duration is $-P'/P$, and convexity is $P''/P$. The first derivative has a negative sign for ordinary positive cash flows; the second derivative contains the factor $t(t+1)$. Calculate both from the same yield used to price the bond.

Check Your Understanding

Use annual effective yield and fixed cash flows in each problem. Give duration and convexity to four decimal places where needed.

A bond pays $108 at year 1 and $116.64 at year 2. At a 8% annual effective yield, calculate its price, modified duration, and convexity. (Answer: Price $200; modified duration $1.3889$; convexity $3.4294$.)
A zero-coupon bond pays $146.41 at the end of year 4. Its annual effective yield is 10%. Calculate its price, modified duration, and convexity. (Answer: Price $100; modified duration $3.6364$; convexity $16.5289$.)
At a 5% annual effective yield, a bond’s year-1, year-2, and year-3 payments have present values of $50, $100, and $150. Find the price, modified duration, and convexity. (Answer: Price $300; modified duration $2.2222$; convexity $7.5586$.)
At a specified annual effective yield, a fixed-cash-flow bond has $P=250$, $P'=-425$, and $P''=1{,}530$, where the derivatives are with respect to the decimal yield. Find its modified duration and convexity. (Answer: Modified duration $1.7000$; convexity $6.1200$.)
At a 4% annual effective yield, a bond has present values $80 at year 1 and $120 at year 3. Calculate convexity and identify the result if someone incorrectly uses $t^2$ instead of $t(t+1)$ in the numerator. (Answer: Correct convexity $1600/[1.04^2(200)]=7.3965$; incorrect $t^2$ result $1160/[1.04^2(200)]=5.3622$.)

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