Immunization and Cash Flow Matching Construction | Free Exam FM Lesson

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:

$P V_{A} = P V_{L}$
$D_{mac} (A) = D_{mac} (L)$
$C_{mac} (A) > C_{mac} (L)$ (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:

$P_{1} + P_{2} = P V_{L}$ $t_{1} P_{1} + t_{2} P_{2} = D_{mac} (L) \cdot P V_{L}$

Solve: subtract $t_{1}$ times equation 1 from equation 2:

$(t_{2} - t_{1}) P_{2} = (D - t_{1}) \cdot P V_{L}, P_{2} = \frac{D - t _{1}}{t _{2} - t _{1}} \cdot P V_{L}$

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

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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 $P_{1} + P_{2} = P V_{L}$ and $t_{1} P_{1} + t_{2} P_{2} = D \cdot P V_{L}$ , then verify strict $C_{mac} (A) > C_{mac} (L)$ .
Shortcut: $P_{2} = (D - t_{1}) / (t_{2} - t_{1}) \cdot P V_{L}$ 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: $P_{2} = (D - t_{1}) / (t_{2} - t_{1}) \cdot P V_{L}$ . "Full needs a bracket, Redington needs three checks." "Cheapest match: highest yield wins each date."

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Learning objectives

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