Redington Checks for Discrete Cash-Flow Portfolios

Why Check All Three Redington Conditions?

A portfolio may have the same present value as a liability stream and still be exposed to interest-rate changes. Its value may change at a different rate, or its curvature may be too low to provide a local surplus around the valuation yield. Redington checks assess these features together for fixed, dated cash flows.

The checks use one common effective annual yield and treat the cash flows as fixed: they do not change when the yield changes. The conclusion is local. Passing the conditions gives a surplus minimum at the valuation yield for sufficiently small parallel yield changes; it does not guarantee a surplus for every possible change in rates.

Definition: For cash flows at times $t=1,2,\ldots,n$, let $A_t$ be the asset cash flow and $L_t$ the liability cash flow at time $t$. At effective annual yield $i>-1$, define $v=1/(1+i)$. The present values are $P_A(i)=\sum_{t=1}^n A_t v^t$ and $P_L(i)=\sum_{t=1}^n L_t v^t$. Cash flows not occurring at a given date are zero. The Redington conditions at valuation yield $i_0$ are: (1) $P_A(i_0)=P_L(i_0)$; (2) $D_A(i_0)=D_L(i_0)$, where $D$ is Macaulay duration; and (3) $C_A(i_0)>C_L(i_0)$, where $C$ is convexity with respect to the effective annual yield $i$.

The model assumes deterministic cash flows, a single common yield applied to every date, and a parallel change in that yield. For fractional payment times, the same formulas apply with the actual times $t$ in years. If payments depend on rates, or the change is nonparallel, these checks alone do not establish immunization.

Duration and Convexity from Dated Cash Flows

Duration is the present-value-weighted average payment time. Convexity measures the normalized second derivative of present value with respect to the effective annual yield. For a positive cash-flow stream, both are calculated from its discounted cash flows.

D_A(i)=\frac{\sum_t tA_t(1+i)^{-t}}{P_A(i)},\qquad C_A(i)=\frac{\sum_t t(t+1)A_t(1+i)^{-t}}{P_A(i)(1+i)^2}.

The corresponding liability measures are calculated by replacing $A_t$ and $P_A$ with $L_t$ and $P_L$. The $t(t+1)$ term follows from differentiating the discount factor twice:

\frac{d}{di}(1+i)^{-t}=-t(1+i)^{-t-1},\qquad \frac{d^2}{di^2}(1+i)^{-t}=t(t+1)(1+i)^{-t-2}.

Therefore $P_A'(i)=-P_A(i)D_A(i)/(1+i)$ and $P_A''(i)=P_A(i)C_A(i)$; the same identities hold for liabilities. These identities explain why duration matching makes the first derivative of surplus zero when present values match, and why the convexity comparison determines the sign of its second derivative.

Theorem: Define surplus as $S(i)=P_A(i)-P_L(i)$. If the first two Redington conditions hold at $i_0$, then $S(i_0)=0$ and $S'(i_0)=0$. If also $C_A(i_0)>C_L(i_0)$, then $S''(i_0)>0$. Since the cash-flow present values are twice differentiable for $i>-1$, Taylor's theorem gives $S(i_0+h)=\tfrac12 S''(i_0)h^2+o(h^2)$, so the surplus is positive for sufficiently small nonzero $h$.

The convexity comparison is made using the same convention and yield for both streams. In particular, the formula above is convexity with respect to $i$, not a convexity measure defined for a different rate variable.

Worked Redington Checks

Worked Example: A Portfolio That Passes All Three Checks

At a 10% effective annual yield, a hypothetical fund has asset payments of $55 at the end of year 1 and $66 at the end of year 3. Its liability is $121 at the end of year 2. Check the Redington conditions.

Method. Discount each payment at $i_0=0.10$, then compare present values, durations, and convexities. The common mistake this avoids is comparing cash amounts at different dates without first discounting them.

Present values. Since $v=1/1.10$, the asset present value is $55/1.10+66/(1.10)^3=50+50=100$. The liability present value is $121/(1.10)^2=100$. The first condition holds.

Durations. The asset duration is $(1(50)+3(50))/100=2$ years. The liability duration is $2(100)/100=2$ years. The second condition holds.

Convexities. The asset's weighted $t(t+1)$ sum is $1(2)(50)+3(4)(50)=100+600=700$. Thus $C_A=700/[100(1.10)^2]=700/121=5.7851$. The liability's weighted sum is $2(3)(100)=600$, so $C_L=600/121=4.9587$. Values are rounded to four decimal places only here.

Since $5.7851>4.9587$, the third condition holds. As a check, $S''(i_0)=P_A C_A-P_L C_L=(700-600)/(1.10)^2=82.6446>0$. The portfolio passes all three conditions and has a local surplus minimum at the 10% yield.

Worked Example: Present-Value Equality Fails First

At a 10% effective annual yield, a hypothetical asset portfolio pays $55 at the end of year 1 and $66 at the end of year 3. The liability is $123.42 at the end of year 2. Determine whether the portfolio passes Redington checks.

Method. Check present values before interpreting the duration or convexity results. This avoids treating later conditions as a substitute for the required value match.

Present values. The assets have present value $55/1.10+66/(1.10)^3=50+50=100$. The liability has present value $123.42/(1.10)^2=123.42/1.21=102$. The first condition fails: the assets are short by $2 at the valuation yield.

Additional diagnostics. The asset duration is $(50+150)/100=2$ years, and the liability duration is $2(102)/102=2$ years. The asset convexity is $700/[100(1.21)]=5.7851$. The liability convexity is $600/[102(1.21)]=600/123.42=4.8618$. Both convexities are rounded to four decimal places.

Although duration matches and asset convexity is greater, the portfolio does not satisfy the three Redington conditions because present values differ. The common mistake is to declare it immunized based only on duration and convexity; those comparisons do not remove the initial $2 shortfall.

Worked Example: Present Values Match but Duration Does Not

At a 10% effective annual yield, a hypothetical fund pays $77 at the end of year 1 and $39.60 at the end of year 3. Its liability is $121 at the end of year 2. Test the conditions.

Method. Convert the cash flows to present-value weights, then use those weights for duration and convexity. This avoids mistakenly using the undiscounted payments as the duration weights.

Present values. The asset payments have present values $77/1.10=70$ and $39.60/(1.10)^3=30$, for a total of $100. The liability present value is $121/1.21=100$. The first condition holds.

Duration. Asset duration is $(1(70)+3(30))/100=160/100=1.6$ years. Liability duration is $2(100)/100=2$ years. The second condition fails.

Convexity diagnostic. The asset weighted sum is $1(2)(70)+3(4)(30)=140+360=500$, giving $C_A=500/121=4.1322$. The liability weighted sum is 600, giving $C_L=600/121=4.9587$. Rounded to four decimal places, asset convexity is also lower.

The portfolio fails Redington checks because its duration does not match the liability duration. The convexity comparison cannot repair this first-order exposure.

Worked Example: Duration Matches but Strict Convexity Fails

At a 10% effective annual yield, a hypothetical asset pays $121 at the end of year 2. The liabilities are $55 at the end of year 1 and $66 at the end of year 3. Check the conditions.

Method. Compare the streams' present-value-weighted timing and then their convexities. This example highlights why equality of present value and duration does not by itself complete the test.

Present values. The asset present value is $121/1.21=100$. The liabilities have present values $55/1.10=50$ and $66/(1.10)^3=50$, for a total of $100. The first condition holds.

Duration. Asset duration is $2(100)/100=2$ years. Liability duration is $(1(50)+3(50))/100=2$ years. The second condition holds.

Convexity. The asset weighted sum is $2(3)(100)=600$, so $C_A=600/121=4.9587$. The liability weighted sum is $1(2)(50)+3(4)(50)=700$, so $C_L=700/121=5.7851$. Both values are rounded to four decimal places.

Here $C_A<C_L$, so the strict third condition fails. In fact $S''(i_0)=(600-700)/(1.21)=-82.6446$, and the matched surplus has local curvature in the wrong direction. Matching the first two quantities is not enough.

Common Calculation Traps and Scope

Using different yields. All asset and liability present values, durations, and convexities must be evaluated at the same valuation yield.
Using cash amounts as weights. Duration and convexity weights are discounted cash flows, not undiscounted payments.
Stopping after duration matching. Present-value equality and duration equality do not establish the strict convexity condition.
Confusing convexity conventions. For the effective annual yield $i$, this tutorial uses $t(t+1)$ and divides the weighted sum by $P(i)(1+i)^2$. Use that same convention for assets and liabilities.
Overstating the result. Redington conditions concern a local parallel yield change under fixed cash flows. They do not prove protection against large changes, nonparallel changes, or cash flows that respond to rates.

A reliable exam approach is to state the valuation yield, calculate the two present values, then compare duration and convexity in that order. If an earlier condition fails, identify the failure clearly; later comparisons may still help diagnose the portfolio but do not make it pass. This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, use a common effective annual yield and treat all cash flows as fixed. Check the requested Redington conditions.

At 5%, assets pay $52.50 at year 1 and $115.7625 at year 3; a liability pays $220.50 at year 2. Find the present values and durations, and state whether the first two conditions hold. (Answer: $P_A=P_L=\$200$; $D_A=D_L=2$ years; both conditions hold.)
At 10%, assets pay $33 at year 1 and $79.86 at year 3; a liability pays $108.90 at year 2. Identify the first Redington condition that fails. (Answer: Present-value equality fails; $P_A=\$90$ and $P_L=\$90$? No—$108.90/1.21=\$90$, so PVs match; duration fails, with $D_A=(30+180)/90=7/3$ years and $D_L=2$ years.)
A portfolio has equal asset and liability present values of $250 and equal durations of 4 years at the valuation yield. Its asset convexity is 19.2 and liability convexity is 18.7. Find the second derivative of surplus and state the local conclusion. (Answer: $S''=250(19.2-18.7)=125>0$; all three conditions hold, so surplus has a local minimum at the valuation yield.)
At 10%, asset present value is $150, liability present value is $150, and both durations are 3 years. The convexities are $C_A=8.4$ and $C_L=8.7$. Does the portfolio pass the Redington conditions? (Answer: No; the strict convexity condition fails, since $8.4<8.7$.)
Suppose all three Redington conditions hold for fixed dated cash flows at a 6% effective annual yield. What does the result establish, and what does it not establish? (Answer: It establishes a local surplus minimum for sufficiently small parallel yield changes; it does not guarantee a nonnegative surplus for large or nonparallel changes.)

What You'll Learn