LECTURE 2 — PORTFOLIO DURATION, DURATION GAP, CONVEXITY

Portfolio Duration

D_A = Σ Xᵢ·Dᵢ

Change in Asset Value

ΔA = (D_A·A)/(1+R) × ΔR

Change in Liability Value

ΔL = (D_L·L)/(1+R) × ΔR

Equity Change

ΔE = ΔA − ΔL

Leverage Ratio

k = L / A

Duration Gap

DG = DA − k·DL

Duration Gap Model

ΔE = (DA − k·DL)·A·ΔR/(1+R)

Equity Duration

DE = (DA·A − D_L·L) / E

Convexity

CX = [1/(P(1+R/m)²)] × Σ PVₜ·t(t+1/m)

Duration + Convexity Approx.

ΔP/P = MDΔR + ½CX(ΔR)²

Portfolio Convexity

CX_A = Σ Xᵢ·CXᵢ