Convexity - Fixed-Income Cash Flows Step by Step

Convexity is the second-order term that complements duration in rate sensitivity.

highlighted = computed this step

Exact reprice

The exact price-yield relation discounts the bond cash flows directly. At the base yield 10%, price is $1,000.00. After a yield increase of 5%, the new yield is 15% and exact repricing gives rounded $885.84, exact 10778000/12167 dollars.

P(y)=\sum_{t=1}^{3}\frac{C_t}{(1+y)^t},\quad P(15\%)\approx \$885.84

Duration line

Modified duration is 3310/1331. The tangent-line estimate uses price times one minus duration times the yield change, giving rounded $875.66, exact 1165500/1331 dollars.

P_{\text{lin}}=P_0(1-D_{\text{mod}}\Delta y)=\$875.66

What duration misses

The exact reprice is above the duration line by rounded $10.18, exact 164879500/16194277 dollars. The sketch recomputes the exact curve and tangent from the same cash flows, showing the positive gap duration misses.

P_{\text{exact}}-P_{\text{lin}}=\$10.18

Convexity correction

Convexity from the exact cash flows is 128200/14641. Adding one half times convexity times the square of the yield change gives rounded $886.60, exact 12980750/14641 dollars, leaving about $0.76 from the exact price.

P_{\text{conv}}=P_0(1-D_{\text{mod}}\Delta y+1/2C(\Delta y)^2)\approx \$886.60

Approximation note

The convexity correction is still a second-order approximation. The exact price is the discounting computation above; this is a descriptive risk shape, not a forecast or investment advice.

\text{exact discounting is the source of truth}