CFA L1 / FI / M12. Convexity
Module 12 · Fixed Income

Yield-Based Bond Convexity and Portfolio Properties

EN: Second-order term in the price–yield Taylor expansion.
VN: Số hạng bậc 2 trong khai triển giá–yield.

1. Convexity Adjustment Core

About: Second-order term in price–yield Taylor expansion. Always positive for option-free bonds. Bond gains more than ModD predicts on rate falls; loses less on rate rises.Tóm tắt: Hệ số bậc 2 trong khai triển Taylor. Luôn dương cho TP option-free. Tăng giá nhiều hơn duration dự báo khi rate giảm.
\[ \%\Delta P \approx -\text{ModD} \cdot \Delta y + \tfrac{1}{2} \cdot \text{Convexity} \cdot (\Delta y)^{2} \]

Interpretation

  • Duration term Linear, symmetric.
  • Convexity term Always positive — bonds rise more than duration predicts when yields fall, and fall less than duration predicts when yields rise.
Practice problem

ModD = 6, Convexity = 70, Δy = +50bps. % price change?

Show solution
= −6(0.005) + 0.5(70)(0.005)²
= −0.030 + 0.000875
≈ −2.91%

2. Properties of Convexity Core

About: Maturity ↑ → convexity ↑. Coupon ↑ → convexity ↓. YTM ↑ → convexity ↓. Callable: NEGATIVE convexity at low yields. Putable: higher convexity than option-free.Tóm tắt: Maturity ↑ convexity ↑; coupon ↑ convexity ↓; YTM ↑ convexity ↓. Callable: convexity âm tại yield thấp.
Yield Price Option-free Callable Price compressed (call)
Callable bond: negative convexity at low yields — issuer call caps price upside.
  • Maturity ↑ Convexity ↑.
  • Coupon ↑ Convexity ↓.
  • Yield ↑ Convexity ↓.
  • Callable Negative convexity at low yields (price compressed near call).
  • Putable Higher convexity than option-free.
Practice problem

ModD = 7.0, Convexity = 80, ΔYTM = +1% (100 bps). Estimate % price change.

Show solution
% change ≈ −7.0(0.01) + ½(80)(0.01)² = −0.07 + 0.004
≈ −6.6% (vs. −7.0% from duration alone)