Worked Redington Portfolio Construction

Constructing a Redington-Immunized Portfolio

A liability portfolio and an asset portfolio can have the same present value at a chosen valuation rate yet respond differently when that rate changes. Redington immunization adds conditions on how their values change locally: match value, match the first derivative, and give the assets strictly greater second derivative than the liabilities. The practical task is to translate those conditions into equations for asset holdings, solve them, and check the convexity inequality.

The construction below assumes deterministic cash flows at fixed dates and a single effective annual rate applied consistently to all cash flows. This is a local protection against small parallel shifts in that rate; it is not a guarantee for large shocks or nonparallel yield-curve movements.

Definition: At valuation rate $i_0$, let $A(i)$ and $L(i)$ be the present values of asset and liability cash flows. Redington matching requires $A(i_0)=L(i_0)$, $A'(i_0)=L'(i_0)$, and $A''(i_0)>L''(i_0)$. The strict final inequality is the convexity condition.

Turning the Conditions into Equations

For a cash flow $C_t$ paid at time $t$ years, its present value at effective annual rate $i>-1$ is $C_t(1+i)^{-t}$. For portfolios with cash flows at dates $t$, define $PV_t$ to be the present value of the cash flow at date $t$, evaluated at the common valuation rate. Differentiating the present value with respect to the annual effective rate gives

P'(i)=-\sum_t \frac{t\,C_t}{(1+i)^{t+1}}, \qquad P''(i)=\sum_t \frac{t(t+1)C_t}{(1+i)^{t+2}}.

At the fixed rate $i_0$, the first derivative is $-1/(1+i_0)$ times the sum of $tPV_t$; the second derivative is $1/(1+i_0)^2$ times the sum of $t(t+1)PV_t$. Since these factors are identical for assets and liabilities, the first two Redington conditions can be written as present-value equations.

\sum_t PV_t^{A}=\sum_t PV_t^{L}, \qquad \sum_t tPV_t^{A}=\sum_t tPV_t^{L}, \qquad \sum_t t(t+1)PV_t^{A}>\sum_t t(t+1)PV_t^{L}.

These equations are especially convenient when the instruments are zero-coupon securities: each instrument contributes present value at one date only. For coupon instruments, include the present value of every coupon and principal payment at its own date. The equations assume all cash flows use the same valuation rate and time units. Changing compounding conventions or measuring some dates in months and others in years invalidates a direct comparison.

Why Matching Value and Slope Gives Local Protection

Let the surplus be $S(i)=A(i)-L(i)$. If value and first derivative match at $i_0$, then $S(i_0)=0$ and $S'(i_0)=0$. For cash-flow present values that are twice differentiable near $i_0$, a Taylor expansion gives

S(i_0+h)=S(i_0)+hS'(i_0)+\frac{h^2}{2}S''(i_0)+o(h^2) =\frac{h^2}{2}\bigl(A''(i_0)-L''(i_0)\bigr)+o(h^2).

If the convexity inequality is strict, the leading term is positive for either sign of a sufficiently small nonzero $h$. Thus the surplus is locally positive on both sides of the valuation rate. The argument is local: higher-order terms can matter for larger shifts, and a nonparallel change cannot generally be represented by a single $h$.

Worked Example: Solving a Two-Instrument Portfolio

A liability portfolio will pay $121,000 at year 2 and $146,410 at year 4. At an effective annual valuation rate of 10%, construct a portfolio using zero-coupon securities maturing at years 1 and 5. Find each security’s face amount and verify all three Redington conditions.

Method and assumptions. Cash flows are certain, occur at the stated integer-year dates, and are discounted at an effective annual rate of 10%. Let $x$ and $y$ be the present values, in thousands of dollars, of the year-1 and year-5 securities at the valuation rate.

Step 1: Find the liability present values. The year-2 discount factor is $1/(1.10)^2=1/1.21$, so the year-2 liability has present value $121/1.21=100$ thousand dollars. The year-4 discount factor is $1/(1.10)^4=1/1.4641$, so the year-4 liability has present value $146.410/1.4641=100$ thousand dollars. Total liability present value is $200$ thousand dollars.

Step 2: Match value and first derivative. The liability time-weighted present value is $2(100)+4(100)=600$, in thousand-dollar-years. For the two assets, the equations are

x+y=200,\qquad x+5y=600.

Subtracting the first equation from the second gives $4y=400$, so $y=100$. Then $x=200-100=100$. Both holdings are positive.

Step 3: Convert present values to face amounts. The year-1 security must pay $100(1.10)^1=110$ thousand dollars. The year-5 security must pay $100(1.10)^5=100(1.61051)=161.051$ thousand dollars. Thus the face amounts are $110,000 and $161,051.

Step 4: Verify convexity. For liabilities, the second-moment sum is $2(3)(100)+4(5)(100)=600+2{,}000=2{,}600$, in thousand-dollar-years squared. For assets, it is $1(2)(100)+5(6)(100)=200+3{,}000=3{,}200$. Since $3{,}200>2{,}600$, asset second derivative exceeds liability second derivative. The portfolio satisfies all three conditions.

Check. Asset and liability present values are each $200,000, and both time-weighted present values are $600,000-years. A common mistake is to compare face amounts instead of discounted cash flows; the Redington equations use present values at the valuation rate.

Using a Third Instrument

With three or more instruments, the value and first-derivative equations do not by themselves determine a unique portfolio. A further allocation choice, such as fixing one holding, can be used to select a solution. The resulting holdings must still be checked for feasibility—particularly whether they are nonnegative when short positions are not allowed—and for the convexity condition.

Worked Example: Selecting Holdings in a Three-Instrument Portfolio

At a 10% effective annual valuation rate, liabilities of $242,000 at year 2 and $146,410 at year 4 are to be immunized using zero-coupon securities maturing at years 1, 3, and 6. Suppose the present value of the year-3 security is fixed at $50,000. Find the other two present values and face amounts, then test convexity.

Method and assumptions. All cash flows are certain, the valuation rate is 10% effective annually, and the stated present values are in thousands of dollars. First discount the liabilities: $242/1.21=200$, and $146.410/1.4641=100$. Total liability present value is $300,000; its time-weighted present value is $2(200)+4(100)=800$ thousand-dollar-years.

Step 1: Solve the matching equations. Let $x$ and $z$ be the present values, in thousands, of the year-1 and year-6 securities. The year-3 present value is 50. Matching value gives $x+50+z=300$, or $x+z=250$. Matching the first derivative gives $x+3(50)+6z=800$, or $x+6z=650$. Subtracting yields $5z=400$, so $z=80$, and then $x=250-80=170$.

Step 2: Convert to face amounts. The year-1 face amount is $170(1.10)=187$ thousand dollars. The year-3 face amount is $50(1.10)^3=50(1.331)=66.55$ thousand dollars. The year-6 face amount is $80(1.10)^6=80(1.771561)=141.72488$ thousand dollars. The portfolio uses positive holdings in all three securities.

Step 3: Test convexity. The liability second-moment sum is $2(3)(200)+4(5)(100)=1{,}200+2{,}000=3{,}200$. The asset sum is $1(2)(170)+3(4)(50)+6(7)(80)=340+600+3{,}360=4{,}300$. Because $4{,}300>3{,}200$, the selected portfolio has greater second derivative at the valuation rate. The matching equations and strict convexity inequality are all satisfied.

Check. The asset present values sum to $170+50+80=300$, and their time-weighted sum is $170+150+480=800$, agreeing with the liability totals. Fixing the third instrument’s holding is an allocation choice, not an additional Redington condition.

Exact Repricing After a Small Parallel Shift

The derivative conditions describe local behavior, but an exam problem may ask for the actual surplus after a specified rate change. For that task, reprice every dated cash flow at the new rate rather than treating a local approximation as exact. The next example uses the first portfolio and considers a one-percentage-point shift in either direction.

Worked Example: Checking Surplus Under Two Rate Shifts

Use the portfolio from the first example. Its year-1 asset has present value $100,000 at 10%, its year-5 asset also has present value $100,000, and the liabilities each have present value $100,000 at years 2 and 4. Calculate the exact surplus after rates move to 11% and 9%.

Method and assumptions. Cash flows remain fixed, and each shifted rate is an effective annual rate applied in parallel to all dates. Let $q=1.10/(1+j)$, where $j$ is the new rate. Each baseline present-value contribution at time $t$ is multiplied by $q^t$. Therefore, in dollars,

S(j)=100{,}000\bigl(q+q^5-q^2-q^4\bigr).

Upward shift to 11%. Here $q=1.10/1.11=110/111$. Factor the expression as $q+q^5-q^2-q^4=(q-1)^2q(q^2+q+1)$. Substituting gives

S(0.11)=100{,}000\left(\frac{1}{111^2}\right) \left(\frac{110}{111}\right) \left[\left(\frac{110}{111}\right)^2+\frac{110}{111}+1\right] \approx \$23.91.

Downward shift to 9%. Here $q=1.10/1.09=110/109$, so

S(0.09)=100{,}000\left(\frac{1}{109^2}\right) \left(\frac{110}{109}\right) \left[\left(\frac{110}{109}\right)^2+\frac{110}{109}+1\right] \approx \$25.72.

The portfolio has a positive surplus under both specified shifts. Dollar amounts are rounded to the nearest cent only at the final step. This check is consistent with the convexity condition, but two favorable results do not establish that every possible rate change will be favorable.

Common mistake avoided. For a downward shift, use the new rate in the denominator: $q=1.10/1.09$, which is greater than 1. Using $1.10/1.11$ for both directions would reprice the downward shock incorrectly. Exact repricing also avoids treating a second-order local approximation as the exact surplus.

Limits and Exam Checks

Redington construction is useful when the objective is to protect a surplus locally around one valuation rate. It does not replace exact cash-flow matching, and it does not ensure protection against every change in rates. In particular, the conditions here concern one parallel shift applied to all cash flows. A yield curve can change nonparallelly, and a sufficiently large shift can make higher-order terms important.

Before accepting a constructed portfolio, verify each of the following:

All asset and liability cash flows are discounted at the same effective annual valuation rate and use the same time unit.
The present-value equation balances at that rate.
The time-weighted present-value equation balances, which is equivalent to matching first derivatives.
The asset second-moment sum is strictly larger than the liability sum, not merely equal.
Any restrictions on holdings, such as nonnegative amounts or no short sales, are met.

A second-moment inequality should be checked at the valuation rate using discounted contributions. Comparing only maturity dates, face amounts, or an informal statement that one portfolio “looks more convex” is not a substitute for calculating the two sums. If asset cash flows can be negative, signs also matter: the usual intuition that later cash flows increase convexity depends on positive contributions.

Key takeaway: Solve the present-value and time-weighted present-value equations to match value and slope, then independently verify that the asset second-moment sum is strictly greater. Reprice cash flows directly when a specific rate shift is given.

Check Your Understanding

For each problem, assume deterministic cash flows and an effective annual valuation rate unless stated otherwise.

At 5%, liabilities have present-value contributions of $50,000 at year 2 and $80,000 at year 4. A portfolio uses zero-coupon assets at years 1 and 5. Find the present-value contribution of each asset required to match value and first derivative, and test convexity. (Answer: $57,500 at year 1 and $72,500 at year 5; asset second-moment sum $2,290,000-years squared versus liability $1,900,000-years squared, so convexity is strictly greater.)
At a fixed valuation rate, a liability portfolio has time-weighted present value $420,000-years. Two zero-coupon assets mature at years 1 and 5; the year-5 asset has present value $72,500. What year-1 present value is required for first-derivative matching? (Answer: $57,500.)
At 10%, liability present-value contributions are $200,000 at year 2 and $100,000 at year 4. Assets mature at years 1, 3, and 6; fix the year-3 asset present value at $50,000. Find the year-1 and year-6 present values that match value and first derivative. (Answer: $170,000 at year 1 and $80,000 at year 6.)
For the portfolio in the first worked example, a rate rises from 10% to 11%. State the correct repricing ratio $q$ and the resulting surplus rounded to cents. (Answer: $q=110/111$; surplus approximately $23.91.)
A portfolio matches value and first derivative, but its asset and liability second derivatives are equal at the valuation rate. Does it satisfy strict Redington immunization as defined here? (Answer: No; the required condition is $A''(i_0)>L''(i_0)$, not equality.)

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