Bond Portfolio Immunization & Convexity Lab

Model institutional Asset-Liability Management (ALM) under classical Redington immunization. Match portfolio duration to target liability horizons, calculate barbell allocation weights, evaluate convexity surplus, and stress-test net worth across yield curve shifts.

Redington Conditions Duration Matching Convexity Surplus Fong-Vasicek M² Risk CFA / Actuarial ALM
Institutional ALM Presets:
1. Liability & Asset Parameters
Target Liability Profile
$
Guaranteed future contractual payment due at horizon date.
%
Asset 1 (Shorter-Term Component)
%
%
Asset 2 (Longer-Term Component)
%
%
2. Immunization Status & Redington Verification
IMMUNIZED
Asset PV ($PV_A)
$7.34M
100.0% funded
Liability PV ($PV_L)
$7.34M
H = 7.00 Yrs
Portfolio Duration
7.00 Yrs
ΔD = 0.00 Yrs
Convexity Surplus
+12.44
M² = 9.85 yrs²
Optimal Immunization Allocation Weights
Asset 1: 35.4% ($2.60M) Asset 2: 64.6% ($4.74M)
D_mac = 2.82 yrs | Conv = 9.8 D_mac = 8.74 yrs | Conv = 94.6
Redington Immunization Compliance Audit
Condition Requirement Asset Portfolio Liability Status
1. Present Value PV_A ≥ PV_L $7,337,130 $7,337,130 PASS
2. Duration Matching D_A = D_L 7.00 Yrs 7.00 Yrs PASS
3. Convexity Surplus C_A > C_L 64.55 52.11 PASS
Portfolio is fully immunized against small parallel yield shifts. Convexity surplus provides a positive equity buffer if yields shift up or down.
3. Revaluation Curve & Convexity Smile
Chart Explanation: The blue line shows the Asset Portfolio Market Value ($V_A$), while the red dashed line tracks the Liability Value ($V_L$). Because durations match, the curves are tangent at 0 bps shift. Because asset convexity exceeds liability convexity ($C_A > C_L$), the asset curve curves upward faster than the liability curve, generating a positive net surplus (green fill) under parallel yield shifts.
4. Yield Shift Stress Test (-300 to +300 bps)
Exact vs Taylor Repricing
Yield Shift Asset Value ($V_A$) Liability ($V_L$) Net Surplus ($NW$) Surplus Return
Note: Net Surplus $NW = V_A - V_L$. Under strict parallel shifts, a positive convexity surplus guarantees $NW ge 0$ across all rate shifts.
5. Sensitivity: Yield Shock vs. Liability Horizon on Surplus ($)
Evaluates net dollar surplus under various parallel yield shocks across different liability horizons H.
6. Sensitivity: Asset Maturity Spread on Convexity Surplus & M²
Evaluates portfolio convexity surplus $Delta C = C_P - C_L$ across varied maturity pairings (Asset 1 Maturity vs Asset 2 Maturity).
7. Institutional ALM & Redington Immunization Theory
Redington's Classical Immunization Framework (1952)

In 1952, British actuary Frank Mitchell Redington introduced the mathematical foundation for protecting life insurance and pension funds against interest rate volatility. The net worth of an institution is defined as the difference between asset market value and liability present value:

$$NW(y) = PV_A(y) - PV_L(y)$$

Expanding $NW(y + Delta y)$ via a second-order Taylor series around current yield $y$:

$$Delta NW approx left[ rac{dPV_A}{dy} - rac{dPV_L}{dy} ight] Delta y + rac{1}{2} left[ rac{d^2PV_A}{dy^2} - rac{d^2PV_L}{dy^2} ight] (Delta y)^2$$

To guarantee that $Delta NW ge 0$ for any small parallel yield change $Delta y e 0$:

  • First Derivative Condition (Duration Matching): $ rac{dPV_A}{dy} = rac{dPV_L}{dy} implies D^*_A cdot PV_A = D^*_L cdot PV_L$. If fully funded ($PV_A = PV_L$), then $D^*_A = D^*_L$.
  • Second Derivative Condition (Convexity Condition): $ rac{d^2PV_A}{dy^2} > rac{d^2PV_L}{dy^2} implies C_A > C_L$.
Fong-Vasicek Immunization Risk ($M^2$) & Curve Twists

While high convexity provides an attractive positive surplus under purely parallel shifts, H. Gifford Fong and Oldrich Vasicek (1984) proved that high convexity comes from cash flow dispersion, introducing severe vulnerability to non-parallel yield curve shifts:

$$M^2 = sum_{i=1}^n w_i (t_i - H)^2 = C_P - C_{ ext{bullet}}$$

Where $M^2$ measures the variance of asset cash flows around the liability horizon $H$:

  • Bullet Portfolio ($M^2 o 0$): Concentrates cash flows close to horizon $H$. Minimum structural risk against curve twists, but lower convexity surplus.
  • Barbell Portfolio (High $M^2$): Combines short and ultra-long maturities. Maximum convexity for parallel shifts, but highly vulnerable if the yield curve steepens or twists.

Rebalancing Mandate: Because duration decreases as time elapses and yields fluctuate, immunized portfolios require periodic dynamic rebalancing to maintain $D_P = D_L$.

8. Fixed Income ALM Self-Assessment Mastery Quiz
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10. Frequently Asked Questions (FAQ)

Bond portfolio immunization is an asset-liability management (ALM) investment strategy that locks in a guaranteed rate of return over a targeted holding period, regardless of interest rate fluctuations. By matching the duration of the asset portfolio to the maturity horizon of the liability, capital price gains/losses and coupon reinvestment income changes exactly offset each other when interest rates shift.

Formulated by actuary F. M. Redington in 1952, classical immunization requires: (1) Present Value Equality or Surplus: The market value of assets must equal or exceed the present value of liabilities (PV_A ≥ PV_L); (2) Duration Matching: The Macaulay duration of the asset portfolio must equal the duration of the liability (D_A = D_L); (3) Convexity Surplus: The convexity of the asset portfolio must exceed the convexity of the liability (C_A > C_L), ensuring net equity increases under any non-zero parallel yield shift.

Because bond price-yield curves are convex, a higher convexity means that when interest rates decline, asset prices increase faster than liability values. Conversely, when interest rates rise, asset prices decrease more slowly than liability values. With C_A > C_L and D_A = D_L, any parallel shift in the yield curve generates a positive net surplus (the convexity smile).

The Fong-Vasicek M2 measure quantifies the cash flow dispersion of the asset portfolio around the liability horizon date H: M2 = sum[w_i * (t_i - H)^2]. While a barbell portfolio with high dispersion yields high convexity, it also has a high M2, making it vulnerable to non-parallel yield curve shifts (such as curve steepening, flattening, or twists). A bullet portfolio matching H minimizes M2 and curve-shaping risk.

Yes. This lab provides a fully sanitized RFC 4180 compliant CSV export containing asset weights, Macaulay durations, convexities, Redington compliance checks, and revalued portfolio balances across -300 bps to +300 bps yield shocks with formula injection protection.