Immunization and Cash Flow Matching Construction

Mix a 3-year and a 12-year zero to match a 7-year liability. The combined position brackets the liability in time and is fully immunized against any parallel rate shift.

Three conditions at the current yield:

• $PV_A = PV_L$
• $D_{\text{mac}}(A) = D_{\text{mac}}(L)$
• $C_{\text{mac}}(A) > C_{\text{mac}}(L)$ (strict)

Why it works: conditions 1 and 2 make the surplus zero with zero first derivative. Condition 3 makes the current rate a local minimum, any small shift increases the surplus.

With two zero-coupon bonds at times $t_1, t_2$, let $P_1, P_2$ be the PVs invested:

$$P_1 + P_2 = PV_L$$ $$t_1 P_1 + t_2 P_2 = D_{\text{mac}}(L) \cdot PV_L$$

Solve: subtract $t_1$ times equation 1 from equation 2:

$$(t_2 - t_1)P_2 = (D - t_1) \cdot PV_L, \quad P_2 = \frac{D - t_1}{t_2 - t_1} \cdot PV_L$$

HIGH-FREQUENCY: Two zeros + liability, find amounts. Standard 2x2 system on nearly every FM exam.

Common mistakes

• Confusing PV and face amount. $P_1 = 4270.05$ is the PV invested. Face = $4270.05/v^2 = 4797.82$. Report the quantity asked for.
• Convexity equality does not satisfy Redington. Must be strict: $C_A > C_L$. Equality means zero second derivative, a saddle point. Not immunized.
• Full immunization for multiple liabilities. Does not work. Use Redington (small shifts) or cash flow matching (exact).

Bottom line

• Redington: solve $P_1 + P_2 = PV_L$ and $t_1 P_1 + t_2 P_2 = D \cdot PV_L$, then verify strict $C_{\text{mac}}(A) > C_{\text{mac}}(L)$.
• Shortcut: $P_2 = (D - t_1)/(t_2 - t_1) \cdot PV_L$ eliminates most linear-system algebra errors.
• Full immunization: same system but assets must bracket the single liability ($t_1 < n < t_2$), and it holds for any rate change.
• Cash flow matching: work backwards from the last liability; later bonds' coupons reduce earlier funding needs.

Exam shortcut

Write the 2x2 system immediately. Solve, then verify convexity. Three instruments and two liabilities: add convexity as the third equation. "PV-D-C": Redington: PV match, Duration match, Convexity dominance. Cash flow matching: "Last to First." Shortcut: $P_2 = (D - t_1)/(t_2 - t_1) \cdot PV_L$. "Full needs a bracket, Redington needs three checks." "Cheapest match: highest yield wins each date."

The full lesson (about 3,485 words, 23 min read) adds 8 worked examples, all 8 common mistakes, a self-check, free in the app.

Learning objectives

• 5c

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