Why Aggregate the Liability Stream?
An institution may owe payments at several future dates rather than one amount on one date. To assess a proposed asset portfolio, first treat those obligations as one dated cash-flow stream. Its present value measures the amount needed at the valuation rate; its duration and convexity summarize how that value responds locally to a change in the rate.
For Redington immunization, the aggregate liability measures must be compared with the corresponding measures for the assets. Matching only the total present value is not enough: the first derivatives must also match, and the assets must have strictly greater second derivative at the valuation rate.
The same definitions apply to an asset portfolio, using its dated asset cash flows and the subscripts \(A\). Payments at the same date can be added together before calculating the measures. Alternatively, each obligation can be calculated separately and the discounted amounts combined. The two methods agree because present values, and the sums in the duration and convexity numerators, are additive.
Why the Three Redington Conditions Work
Let the surplus be \(S(i)=V_A(i)-V_L(i)\), where both present values are computed at the same annual effective rate. Redington matching concerns the behavior of this surplus near a valuation rate \(i_0\). The present-value and duration conditions make the surplus and its first derivative zero at \(i_0\); the strict convexity condition makes the second derivative positive there.
For \(V(i)=\sum_t C_t(1+i)^{-t}\), differentiation term by term gives \(V'(i)=-\sum_t tC_t(1+i)^{-(t+1)}\). Factoring out \(1/(1+i)\) gives \(V'(i)=-V(i)D/(1+i)\).
A second differentiation gives \(V''(i)=\sum_t t(t+1)C_t(1+i)^{-(t+2)}=V(i)K/(1+i)^2\).
At \(i_0\), equal present values and equal durations imply \(S(i_0)=0\) and \(S'(i_0)=0\). If \(K_A>K_L\), then \(S''(i_0)=V_A(i_0)(K_A-K_L)/(1+i_0)^2>0\), given the matched positive present values.
This yields the three Redington checks at \(i_0\):
The convexity measure here is the cash-flow quantity \(K\) defined above, not a differently scaled price-convexity convention. The derivative identities show exactly how the chosen definition relates to the surplus test. These conditions give a local result: they do not guarantee that the assets cover the liabilities after an arbitrarily large rate change, or after a nonparallel change in rates by maturity.
Aggregate by Discounted Cash Flow
A useful calculation table has one row per payment date. For each row, record the payment, its discount factor, and its discounted value. The discounted values are the weights for both aggregate measures. In particular, duration is not an average weighted by the undiscounted payments or by the number of obligations.
Worked Example: Three Liability Payments at a 5% Valuation Rate
A hypothetical fund owes $105,000 at the end of year 1, $220,500 at the end of year 2, and $347,287.50 at the end of year 3. Assume certain payments and a flat annual effective valuation rate \(i=0.05\). Find the aggregate present value, duration, and convexity. Report duration and convexity to four decimals.
Method. Discount each payment at the stated annual rate, then use the discounted values as weights. This avoids treating the larger later payments as though they had the same value today as payments due sooner.
| Time \(t\) | Payment | Discount factor \(1.05^{-t}\) | Present value | \(t\) times present value | \(t(t+1)\) times present value |
|---|---|---|---|---|---|
| 1 | $105,000 | 0.95238095 | $100,000 | $100,000 | $200,000 |
| 2 | $220,500 | 0.90702948 | $200,000 | $400,000 | $1,200,000 |
| 3 | $347,287.50 | 0.86383760 | $300,000 | $900,000 | $3,600,000 |
| Total | $600,000 | $1,400,000 | $5,000,000 |
For example, the year-2 present value is \(220{,}500/1.05^2=220{,}500/1.1025=200{,}000\). The other payments similarly discount to $100,000 and $300,000. Therefore,
The duration is \(7/3\) years and the convexity is \(25/3\) squared years before rounding. As a check, the duration must lie between the earliest and latest payment dates, years 1 and 3; \(2.3333\) does. The convexity numerator also exceeds the duration numerator because every positive payment has \(t(t+1)>t\).
Weighted-Average Interpretation
Suppose the liability stream is divided into separate obligations, indexed by \(j\), with present value \(V_j\), duration \(D_j\), and convexity \(K_j\) at the same valuation rate. Since the total present value is \(\sum_j V_j\), and each duration numerator is \(V_jD_j\), the aggregate measures satisfy:
Thus the aggregate duration and convexity are present-value-weighted averages of the component measures. This is a calculation shortcut, not permission to average durations or convexities equally. It applies when all components are measured at the same valuation rate and under the same timing and compounding conventions.
Worked Example: Combining Obligations with Different Payment Dates
A hypothetical organization has three certain liabilities. At annual effective valuation rate \(i=0.04\), their present values are $80,000 due at year 1, $120,000 due at year 4, and $100,000 due at year 6. Find the aggregate present value, duration, and convexity. Report the measures to four decimals.
Method. The data give the discounted value of each obligation directly, so use those amounts as the weights. No further discounting is needed to calculate the aggregate measures.
The aggregate present value is
The duration numerator is \(1(80{,}000)+4(120{,}000)+6(100{,}000)=80{,}000+480{,}000+600{,}000=1{,}160{,}000\). The convexity numerator is \(1(2)(80{,}000)+4(5)(120{,}000)+6(7)(100{,}000)=160{,}000+2{,}400{,}000+4{,}200{,}000=6{,}760{,}000\). Hence,
As a check, use the present-value weights \(80/300\), \(120/300\), and \(100/300\). Their weighted average payment time is \(3.8667\) years, between years 1 and 6, as required. The convexity calculation includes the \(t+1\) factor; using only \(t^2\) would calculate a different measure.
Testing a Portfolio Against a Liability Stream
A complete Redington test compares all three quantities at one valuation rate. The asset cash flows and liabilities must be expressed on the same time scale, discounted at the same rate, and valued under the same parallel-rate-shift convention. The present-value test is an equality, duration is compared only after present values are matched, and convexity must be strictly greater for assets.
Worked Example: Testing Assets Against Three Liabilities
At annual effective rate \(i_0=0.05\), a fund has the three liabilities from the first worked example: $105,000 at year 1, $220,500 at year 2, and $347,287.50 at year 3. Its proposed assets pay $350,000 at year 1 and $324,135 at year 4. Assume all cash flows are certain and that only a parallel change in the annual effective rate is being considered. Test the three Redington conditions. Report duration and convexity to four decimals.
Method. Discount both streams at \(5\%\), then compare present values, duration, and convexity. The common trap is to stop after finding equal present values; that alone does not establish first-order matching or the strict convexity condition.
The liabilities have present-value weights $100,000, $200,000, and $300,000, as calculated earlier. Therefore \(V_L=600{,}000\), \(D_L=2.3333\), and \(K_L=8.3333\).
For the assets, the year-1 payment has present value \(350{,}000/1.05=333{,}333.3333\). Since \(1.05^4=1.21550625\), the year-4 payment has present value \(324{,}135/1.21550625=266{,}666.6667\). Thus,
The asset duration numerator is \(1(333{,}333.3333)+4(266{,}666.6667)=333{,}333.3333+1{,}066{,}666.6668=1{,}400{,}000\), to the displayed precision. Its convexity numerator is \(1(2)(333{,}333.3333)+4(5)(266{,}666.6667)=666{,}666.6666+5{,}333{,}333.334=6{,}000{,}000\), to the displayed precision. Therefore,
The first condition passes because \(V_A=V_L=600{,}000\). The second passes because \(D_A=D_L=2.3333\). The third passes strictly because \(K_A=10.0000>8.3333=K_L\). Equivalently, the convexity surplus is \(10-25/3=5/3\) squared years. This portfolio satisfies the local Redington conditions at \(5\%\). The result does not by itself promise a surplus for every possible rate change.
Common Mistakes and Scope of the Test
- Using undiscounted payments as weights. Duration and convexity use present-value weights at the valuation rate.
- Averaging component measures equally. Aggregate duration and convexity are weighted by component present values, not by the number of obligations.
- Omitting \(t+1\) from convexity. Under the definition used here, the second-derivative weight is \(t(t+1)\), not \(t^2\).
- Using inconsistent rates or dates. Both sides must use the same annual effective valuation rate and time units.
- Calling equality of value and duration sufficient. Redington’s strict third condition is \(K_A>K_L\), not merely \(K_A\ge K_L\).
The aggregate calculation is useful because many obligations can be reduced to three comparable quantities at the valuation rate. A successful test establishes a local surplus minimum under the stated assumptions; broader rate movements, changing liability amounts, or nonparallel yield changes require additional analysis.
Check Your Understanding
For each problem, use annual effective discounting where a rate is stated. Treat all cash flows as certain. Convexity means the \(K\) measure defined in this tutorial.
- At \(i=0.06\), a liability pays $53 at year 1 and $112.36 at year 2. Find its present value, duration, and convexity. (Answer: Present value $150; duration 1.6667 years; convexity 4.6667.)
- At a valuation rate, three liabilities have present values $40 at year 1, $60 at year 3, and $100 at year 5. Find aggregate duration and convexity. (Answer: Duration 3.6000 years; convexity 19.0000.)
- Liabilities have present values $50 at year 1, $100 at year 2, and $50 at year 3. Assets have present values $100 at year 1 and $100 at year 3. Test all three Redington conditions. (Answer: Both present values are $200; both durations are 2.0000 years; asset convexity 7.0000 exceeds liability convexity 6.5000, so all three conditions pass.)
- Liabilities have present values $100 at year 1 and $100 at year 3. A proposed asset portfolio has present value $200 entirely at year 2. Test all three conditions. (Answer: Present values match at $200 and durations match at 2.0000 years, but asset convexity 6.0000 is less than liability convexity 7.0000; the strict convexity condition fails.)
- At \(i=0.05\), liabilities pay $105 at year 1 and $220.50 at year 2. Find their present value, duration, and convexity, then state the minimum asset convexity required if assets match both present value and duration. (Answer: Liability present value $300; duration 1.6667 years; convexity 4.6667. Asset convexity must be strictly greater than 4.6667.)