Maintaining Immunization as Time Passes

Why Immunization Must Be Rechecked

A portfolio that satisfies Redington conditions today does not remove the need to monitor it. As time passes, some cash flows are received or paid, and the remaining payments are closer to the new valuation date. The portfolio’s present value, duration, and convexity must therefore be assessed again using the new date as time zero.

The key is to keep two things distinct: cash flows that have already occurred and cash flows still to come. Past net cash flows are carried forward to the new date under the assumed reinvestment convention. Remaining asset and liability cash flows are discounted from that date, using their new remaining times.

Definition: At a later valuation date, the carried-forward net balance is the accumulated value of asset cash flows received minus liability cash flows paid before or on that date. The remaining asset value and remaining liability value are the present values, at the later date, of their respective future cash flows.

For a fixed effective annual valuation rate $i$, assume interim net cash flows are carried forward at that same rate. If $h$ is the new valuation date and $A_t$ and $L_t$ denote asset and liability cash flows at date $t$, the carried-forward balance is

B_h=\sum_{t\le h}(A_t-L_t)(1+i)^{h-t}.

The future asset and liability present values at $h$ are computed only from dates $t>h$, with remaining time $t-h$. The balance $B_h$ is cash held at the new valuation date; it has zero remaining time and contributes to present value, but not to the first or second time moment.

Redington Conditions at the New Date

For a portfolio of future cash flows $C_t$ valued at rate $i$, define its present value, first time moment, and second time moment by

P=\sum_t C_t v^t,\qquad M_1=\sum_t tC_t v^t,\qquad M_2=\sum_t t(t+1)C_t v^t, \qquad v=\frac{1}{1+i}.

The corresponding duration and convexity measures are $D=M_1/P$ and $K=M_2/P$, when $P>0$. These definitions use times measured from the valuation date. A cash balance held at that date contributes its amount to $P$, and zero to $M_1$ and $M_2$.

At the later date, Redington immunization requires equality of asset and liability values, equality of their first moments, and a strictly larger asset second moment. For positive present values, this is equivalently equality of present values and durations, together with asset convexity strictly exceeding liability convexity.

Theorem: At the valuation rate, local Redington immunization at a new valuation date requires $$ B_h+P_A=P_L,\qquad M_{1,A}=M_{1,L},\qquad M_{2,A}>M_{2,L}, $$ where $P_A,M_{1,A},M_{2,A}$ refer to the remaining asset cash flows, and $P_L,M_{1,L},M_{2,L}$ refer to the remaining liability cash flows. The carried-forward balance $B_h$ is included as an asset at time zero. This test assumes the stated valuation rate and cash-flow model; it is a local rate-sensitivity condition, not a guarantee against every possible rate change or cash-flow disruption.

The reason the second-moment inequality matters follows from the present-value function. Differentiating a dated cash flow $C_t(1+i)^{-t}$ twice with respect to $i$ gives $t(t+1)C_t(1+i)^{-t-2}$. Thus, after present values and first moments match, a larger asset second moment makes the asset-minus-liability surplus curve locally bend upward at the valuation rate. This is the strict convexity condition used in Redington immunization.

A Practical Roll-Forward Method

Use the following sequence whenever the valuation date advances. Keep the interest assumption explicit: changing the rate used to accumulate interim cash flows is a different assumption from simply moving the valuation date.

List cash flows through the new date.
Record asset receipts and liability payments that have already occurred. Include any withdrawals or deposits that affect the portfolio.

Accumulate the net interim balance.
Carry each past net amount to the new date at the specified reinvestment rate. If asset and liability flows occurred at different dates, accumulate each from its own date.

Re-date all remaining cash flows.
For a payment originally due at $t$, use the remaining time $t-h$, not its original time $t$.

Recompute and compare.
Find present values, first moments, and second moments at the new date. Test all three Redington conditions before deciding whether to rebalance.

Worked Examples

Worked Example: Carrying an Initially Immunized Portfolio Forward

At an effective annual rate of 10%, an asset portfolio pays $55 at time 1 and $66.55 at time 3. A liability of $121 is due at time 2. At time 0, verify the Redington conditions, then revalue at time 1. Assume the asset receipt at time 1 is retained as cash and earns 10% until needed.

Method. Calculate present-value-weighted moments at each valuation date. This avoids treating a received cash flow as if it disappeared, or measuring a remaining payment from the old date.

At time 0, $v=1/1.10$. The asset present value is

P_A=\frac{55}{1.10}+\frac{66.55}{1.10^3} =50+50=100.

The liability present value is $P_L=121/1.10^2=100$. The first moments are

M_{1,A}=1(50)+3(50)=200,\qquad M_{1,L}=2(100)=200.

The second moments are

M_{2,A}=1(2)(50)+3(4)(50)=100+600=700,\qquad M_{2,L}=2(3)(100)=600.

Thus present values and first moments match, while $700>600$. The portfolio satisfies the Redington conditions at time 0.

At time 1, the $55 asset receipt is cash at the new valuation date. The remaining $66.55 asset payment is due in two years, and the liability payment is due in one year. The asset value at time 1 is

P_A=55+\frac{66.55}{1.10^2} =55+55=110.

The liability value is $P_L=121/1.10=110$. The cash has zero remaining time, so

M_{1,A}=0(55)+2(55)=110,\qquad M_{1,L}=1(110)=110,

and

M_{2,A}=0(1)(55)+2(3)(55)=330,\qquad M_{2,L}=1(2)(110)=220.

The conditions still hold: values and first moments match, and $330>220$. The asset duration and liability duration are both $110/110=1$ year at time 1. The strict convexity margin also remains positive.

Check. Accumulating the $55 from time 1 to time 2 gives $60.50. At time 2, that amount plus the $66.55 asset payment accumulated from time 3 to time 2 is not appropriate—the latter payment has not yet occurred. Instead, at time 2 the $60.50 cash and the future $66.55 payment at time 3 together have value $60.50+66.55/1.10=121$, exactly the liability due then.

Common mistake avoided: omitting the $55 because it has already been received. It remains part of the asset position as cash; removing it without recording a withdrawal would understate assets.

Worked Example: Recomputing Measures with a Different Remaining Liability Term

At an effective annual rate of 10%, value a portfolio at time 2. An asset paid $55 at time 1, and that receipt was retained at 10%. The asset’s remaining payment is $80.5255 at time 5. A liability payment of $133.10 is due 2.2 years after time 2. Find the asset and liability present values, durations, and convexities at time 2, and determine whether the Redington conditions hold.

Method. Accumulate the already received asset cash flow to time 2, then use only remaining times for future payments. Here the liability has 2.2 years remaining, so its duration is 2.2 years.

The past $55 receipt accumulates for one year:

$$ 55(1.10)=60.50. $$

The asset payment at time 5 is three years from time 2. Its present value is

\frac{80.5255}{1.10^3} =\frac{80.5255}{1.331} =60.50.

Therefore the asset value is $P_A=60.50+60.50=121.00$. Its first moment is $0(60.50)+3(60.50)=181.50$, so

D_A=\frac{181.50}{121.00}=1.500\text{ years}.

Its second moment is $3(4)(60.50)=726.00$, giving $K_A=726/121=6.000$.

The liability present value is

P_L=\frac{133.10}{1.10^{2.2}}\approx 107.93.

Its duration is exactly 2.2 years because it is a single payment 2.2 years from the valuation date. Its convexity measure is $2.2(3.2)=7.04$. Thus $D_L=2.200$ years and $K_L=7.040$.

The portfolio does not satisfy Redington conditions: the asset and liability present values differ, $121.00\ne107.93$; durations differ, $1.500\ne2.200$; and asset convexity is smaller, $6.000<7.040$. Rebalancing is required if the aim is to restore those conditions.

Check. The liability’s duration is its remaining time, 2.2 years, not its original calendar date. For example, if a payment is at time 3 and the new valuation date is time 2, its remaining time is one year. A duration of 2.2 years applies only when the payment is 2.2 years beyond the valuation date.

Common mistake avoided: using original dates in the duration numerator after moving the valuation date. Re-date each future payment before calculating its moments.

Worked Example: Restoring the Conditions After an Interim Withdrawal

At time 1, a liability of $121 is due at time 2. The portfolio has $45 in cash and a zero-coupon asset paying $66.55 at time 3. The effective annual valuation rate is 10%. Determine whether the portfolio is immunized at time 1. If the sponsor can add cash and buy more of the same zero-coupon asset, find the amounts needed to restore the Redington conditions.

Method. First measure the current portfolio, then solve for the present-value portions allocated to cash and the two-year asset. Cash has duration zero and second moment zero.

At time 1, the liability value is $121/1.10=110$. The asset value is

P_A=45+\frac{66.55}{1.10^2}=45+55=100.

The first asset moment is $2(55)=110$, while the liability first moment is $1(110)=110$. Their durations are both $110/100=1.10$ years for assets and $110/110=1.00$ year for liabilities. Since present values do not match, the portfolio is not immunized despite matching first moments.

Let $x$ be the present value allocated to cash and $y$ the present value allocated to the zero-coupon asset due in two years. Matching present value and first moment requires

x+y=110,\qquad 0x+2y=1(110)=110.

The second equation gives $y=55$. Then $x=110-55=55$. The portfolio currently has cash value $45 and zero-coupon present value $55, so add $10 in cash. The zero-coupon asset needs no change: a present value of $55 corresponds to a payment of $55(1.10)^2=\$66.55$ at time 3.

The resulting second moments are $M_{2,A}=2(3)(55)=330$ and $M_{2,L}=1(2)(110)=220$. Since $330>220$, the strict convexity condition holds. Present values, first moments, and the required second-moment inequality are all restored.

Check. The revised asset value is $55+66.55/1.21=110$, equal to the liability value. Both durations equal one year, and the asset second moment exceeds the liability second moment.

Common mistake avoided: trying to correct a present-value shortfall by changing only the maturity allocation. The cash addition is needed to restore value; the longer-dated asset allocation is what supplies the required first moment and convexity.

When Rebalancing Is Needed

Recalculate rather than assume that conditions persist. If interim cash flows are accumulated at the same rate used for valuation and no cash is withdrawn, the roll-forward may preserve matching relationships, as the first example demonstrates. But an unexpected withdrawal, an unrecorded payment, a changed liability amount, or a different reinvestment assumption can alter the later-date value or moments. The later-date test, not the earlier test, determines whether the portfolio currently satisfies the conditions.

A useful diagnostic is to compare all three quantities, not duration alone. Equal duration with unequal present values does not meet the value condition. Equal present values and durations with asset convexity no greater than liability convexity do not meet the strict Redington condition either. When the test fails, calculate the shortfall and determine whether available instruments can change value, first moment, and second moment in the required directions.

Use the new valuation date as time zero for every remaining payment.
Carry past net cash flows forward explicitly; do not silently discard receipts or payments.
Keep the reinvestment assumption consistent with the problem statement.
Recheck present value, first moment, and strict convexity after any material cash-flow change.

Key takeaway: Maintaining immunization is a valuation-date process. Accumulate interim net cash flows, re-date remaining payments, recompute the Redington measures, and rebalance only after testing the conditions at the new date.

Check Your Understanding

For each problem, use the stated effective annual rate and measure times from the specified valuation date.

At 10%, a liability payment of $242 is due one year from a valuation date. Find its present value and duration. (Answer: Present value $220.00; duration 1 year.)
At 10%, an asset pays $30 immediately and $44 in two years. A liability pays $72.60 in one year. Find both present values and the asset duration, then state whether the values and durations match. (Answer: Asset present value $66.36 and duration 1.0959 years; liability present value $66.00 and duration 1 year; neither measure matches.)
At time 1, an asset payment of $50 has just been received and a liability payment of $44 has just been paid. The net cash is retained at 10%. The remaining asset pays $60.50 at time 3, and the remaining liability pays $66 at time 2. Find the carried-forward balance, total asset value, liability value, and surplus or deficit at time 1. (Answer: Balance $6.00; total asset value $56.00; liability value $60.00; deficit $4.00.)
At 10%, a liability of $110 is due in one year. Assets may be held as cash or as a zero-coupon payment due in two years. Find the present-value allocation that matches value and first moment, then check strict convexity. (Answer: $50 in cash and $50 present value in the two-year asset, corresponding to a $60.50 payment; asset second moment 300 exceeds liability second moment 200.)
At 10%, a portfolio at a new valuation date has $40 cash and an asset payment of $66.55 due in two years. A liability of $121 is due in one year. Find the value surplus or deficit and compare first moments. (Answer: Asset value $95.00, liability value $110.00, deficit $15.00; asset first moment $110.00 equals liability first moment $110.00.)

What You'll Learn