Finance-Theory · Unit 7 · Video 6 · Interactive Practice

Convexity: The Bend That Duration Misses

IKey Formulas

Formula	Name	What it says
$V_m = \dfrac{1}{P}\dfrac{\partial^2 P}{\partial y^2} = \dfrac{1}{(1+y)^2}\displaystyle\sum_{k=1}^{T} k(k+1)\dfrac{C_k}{P\,(1+y)^k}$	Convexity	The second derivative per unit of price, with $y$ and $k$ per payment period. For a semiannual bond quoted in years, discount at $y/2$ per half-year and divide each $k(k+1)$ by $4 = 2^2$
$P(y') \approx P(y)\left[1 - D_m^*(y'-y) + \tfrac12 V_m (y'-y)^2\right]$	Second-order Taylor approximation	Duration supplies the term linear in the yield change, convexity the quadratic term
$D_m^(P) = \displaystyle\sum_j \frac{P_j}{P} D_{m,j}^, \quad V_m(P) = \sum_j \frac{P_j}{P} V_{m,j}$	Portfolio	Value-weighted averages of the bonds' measures, assuming every yield moves by the same amount
$E\!\left[\dfrac{\Delta P}{P}\right] \approx \tfrac12 V_m\,\sigma^2$ when $E[y'-y] = 0$	Volatility	The linear term averages to zero; the quadratic term adds half the convexity times the variance of the yield change

Key Insight: the video's bond (4 years, 6% coupon paid twice a year, priced at par, every yield an annual rate compounded semiannually, so 3% per half-year) has $D_m^* = 3.509846$ and $V_m = 14.805972$ . For a rise from 6% to 8%, $y'-y = 0.02$ and the two-term estimate is $100(1 - 0.0701969 + 0.0029612) \approx 93.276427$ against the exact $93.267255$ (4% per half-year): a miss under $0.01$ . Duration alone gives $92.980308$ , almost $0.29$ too low.

IITwo Terms Track the Curve

Duration's straight line drifts away from the exact price; the convexity term shrinks the miss to almost nothing.

New yield y′ 8.00%

IIIEqual Swings, Unequal Price Changes

For equal moves in yield, the gain from a fall beats the loss from a rise, so volatility alone raises the expected price.

Swing ±Δ 2.00%

IVPortfolio Convexity Is a Weighted Average

Mixing two zeros averages their durations and convexities, so the mix sits above the single zero of equal duration.

Short zero (years) 1 Long zero (years) 7 Value weight in short 0.50

💡 Both averages assume every bond's yield moves by the same amount. Investors know convexity is valuable, so a position with more convexity than another of the same duration trades at a lower yield, meaning a higher price: you pay for convexity when you buy it.

VQuiz Questions

Problem 1 · Apply the Two-Term Formula

Given: a bond priced at $100$ has $D_m^* = 5$ and $V_m = 40$ . Its yield rises by one percentage point, $y' - y = 0.01$ . Find the second-order estimate of the new price.

✅ Correct!

100(1 - 0.05 + 0.002) = 95.20

: duration takes off 5 points and convexity gives back 0.2.

❌ That is duration alone.

100(1 - 5 \times 0.01) = 95

drops the convexity term

\tfrac12 \times 40 \times 0.01^2 = 0.002

❌ Check the sign of the quadratic term.

\tfrac12 V_m (y'-y)^2

is added: convexity is positive and the change is squared.

❌ Missing the one half. The Taylor term is

\tfrac12 V_m (y'-y)^2 = 0.002

, not

V_m(y'-y)^2 = 0.004

❌ Not quite. Use

P(y') \approx P(y)\left[1 - D_m^*(y'-y) + \tfrac12 V_m (y'-y)^2\right]

Show solution

Linear term: $D_m^*(y'-y) = 5 \times 0.01 = 0.05$ . Quadratic term: $\tfrac12 V_m (y'-y)^2 = \tfrac12 \times 40 \times 0.0001 = 0.002$ .

P(y') \approx 100\,(1 - 0.05 + 0.002) = 100 \times 0.952 = 95.20

Duration alone would predict $95.00$ ; the curve bends back above the tangent line, so the convexity term raises the estimate.

Problem 2 · When the Yield Falls

Given: the video's bond at 6% (price $100$ , $D_m^* = 3.509846$ , $V_m = 14.805972$ , yields compounded semiannually). The yield falls to 4%. Find the second-order estimate of the new price.

✅ Correct! With

y'-y = -0.02

both terms raise the price:

100(1 + 0.0701969 + 0.0029612) \approx 107.3158

, within

0.01

of the exact

107.3255

❌ The quadratic term does not change sign.

(y'-y)^2 = (-0.02)^2 = 0.0004

is positive whichever way the yield moves, so convexity adds

0.0029612

here too.

❌ That is duration alone. The duration line understates the price on both sides; add

\tfrac12 \times 14.805972 \times 0.0004 = 0.0029612

❌ That is the estimate for a rise to 8%. Here

y'-y = -0.02

, so

-D_m^*(y'-y) = +0.0701969

and the price goes up.

❌ Not quite. Substitute

y'-y = -0.02

into the two-term formula.

Show solution

Linear term: $-D_m^*(y'-y) = -3.509846 \times (-0.02) = +0.0701969$ . Quadratic term: $\tfrac12 \times 14.805972 \times (-0.02)^2 = +0.0029612$ .

P(0.04) \approx 100\,(1 + 0.0701969 + 0.0029612) = 107.31581

The exact price, discounting every cash flow at 2% per half-year, is $107.325481$ : the estimate is $0.0097$ low. Duration alone gives $107.0197$ , off by $0.306$ . Against a rise to 8% the gain of $7.33$ beats the loss of $6.73$ , while the duration line predicts $7.02$ each way.

Problem 3 · A Two-Bond Portfolio

Given: a portfolio holds $60 million of bond A ( $D_m^* = 2.0$ , $V_m = 6.0$ ) and $40 million of bond B ( $D_m^* = 9.0$ , $V_m = 95.0$ ); both yields move by the same amount. Find the portfolio's modified duration, its convexity, and the estimated change in its value if yields rise by 50 basis points.

Portfolio modified duration $D_m^*(P)$ ?

Portfolio convexity $V_m(P)$ ?

Estimated change in value for $y' - y = 0.005$ ?

✅ Correct! Weights

0.6

and

0.4

give

D_m^*(P) = 4.8

and

V_m(P) = 41.6

, so the value falls by about

2.348\%

of $100 million.

❌ The weights are value shares, not one half each. Bond A is

60/100 = 0.6

of the portfolio's value and bond B is

0.4

❌ The weights are swapped. The larger holding, bond A, carries weight

0.6

❌ That is duration alone.

-4.8 \times 0.005 = -0.024

leaves out the convexity term

\tfrac12 \times 41.6 \times 0.005^2 = 0.00052

❌ Check the sign of the quadratic term. The convexity term is added, which makes the loss smaller than duration predicts, not larger.

❌ Missing the one half. The quadratic term is

\tfrac12 V_m (y'-y)^2 = 0.00052

, not

0.00104

❌ Check the duration.

D_m^*(P) = \sum_j \frac{P_j}{P} D_{m,j}^*

is a value-weighted average, never a sum.

❌ Check the convexity.

V_m(P) = \sum_j \frac{P_j}{P} V_{m,j}

is a value-weighted average, never a sum.

❌ Check the change. Use

\frac{\Delta P}{P} \approx -D_m^*(P)(y'-y) + \tfrac12 V_m(P)(y'-y)^2

with the portfolio's own measures.

Show solution

The portfolio is worth $100 million, so the value weights are $\frac{60}{100} = 0.6$ for A and $\frac{40}{100} = 0.4$ for B.

D_m^*(P) = 0.6 \times 2.0 + 0.4 \times 9.0 = 1.2 + 3.6 = 4.8

V_m(P) = 0.6 \times 6.0 + 0.4 \times 95.0 = 3.6 + 38.0 = 41.6

\frac{\Delta P}{P} \approx -4.8 \times 0.005 + \tfrac12 \times 41.6 \times 0.005^2 = -0.024 + 0.00052 = -0.02348

The value falls by about $0.02348 \times 100 = 2.348$ million dollars: −$2.348 million. Duration alone would say −$2.400 million.

Problem 4 · What Volatility Is Worth

Given: the video's bond at 6% (price $100$ , $D_m^* = 3.509846$ , $V_m = 14.805972$ ). Rates grow more volatile: tomorrow the yield will be either 4% or 8%, each with probability $\tfrac12$ . Find the expected price according to the second-order approximation.

✅ Correct!

E[y'-y] = 0

and

E[(y'-y)^2] = 0.0004

, so

E[P] \approx 100(1 + \tfrac12 \times 14.805972 \times 0.0004) \approx 100.30

. The exact average,

(107.325481 + 93.267255)/2 = 100.296368

, agrees.

❌ Only the linear terms cancel.

-D_m^*(y'-y)

averages to zero, but

\tfrac12 V_m (y'-y)^2

is positive in both outcomes.

❌ Missing the one half. The expected gain is

\tfrac12 V_m \sigma^2

, not

V_m \sigma^2

❌ Volatility helps the bondholder here. The quadratic term is added, so the expected price rises above

100

❌ Not quite. Average the two-term formula over the two outcomes.

Show solution

The yield change is $+0.02$ or $-0.02$ with equal probability, so $E[y'-y] = 0$ and $\sigma^2 = E[(y'-y)^2] = 0.02^2 = 0.0004$ .

E\!\left[\frac{\Delta P}{P}\right] \approx -D_m^* \cdot 0 + \tfrac12 V_m \sigma^2 = \tfrac12 \times 14.805972 \times 0.0004 = 0.0029612

So $E[P] \approx 100 \times 1.0029612 \approx 100.30$ . Checking against exact prices: $(107.325481 + 93.267255)/2 = 100.296368$ . Other things equal, the wider swing raises the bond's expected price; in the market, investors pay for that convexity up front.

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