Why the First-Order Condition Matters
An immunization strategy aims to protect a liability’s funding position against changes in yield. At a chosen valuation yield, matching asset and liability present values makes the initial surplus zero. But equal values alone do not ensure that a small yield change leaves the surplus unchanged: the two present values may respond at different rates. The first-order Redington condition identifies the additional rate-sensitivity match needed for a stationary surplus.
The focus here is the derivative of surplus with respect to a parallel change in yield. Assume that assets and liabilities have fixed cash flows at fixed times, and that all cash flows are valued using the same annual effective yield. The condition derived below is local: it concerns the slope at the valuation yield. It does not, by itself, establish that surplus has a local minimum.
For a portfolio of assets, write its present value as \(A(y)\) and its Macaulay duration as \(D_A(y)\). Write \(L(y)\) and \(D_L(y)\) for the corresponding liability quantities. In the standard immunization setting, the asset and liability cash flows are positive promised payments, so their present values and durations are positive. The algebraic derivative formulas also apply to finite streams with signed cash flows, provided the relevant present values and durations are defined; the usual positive-cash-flow interpretation of duration may then fail.
Deriving the Present-Value Derivative
Consider a finite stream with payments \(C_t\). Its present value as a function of yield is
Each payment is discounted separately. Differentiating each term with respect to \(y\), while holding both the payment and its time fixed, gives
The sum on the right is the present-value-weighted sum of payment times. By the definition of Macaulay duration, that sum is \(P(y)D(y)\). Therefore,
This identity explains why a present value’s first-order response depends on both its size and its duration. A larger present value scales the dollar response; a longer duration increases the magnitude of the response. The negative sign means that, for positive cash flows, a yield increase reduces present value.
From Present-Value Derivatives to the Redington Condition
Define the asset-minus-liability surplus at yield \(y\) by \(S(y)=A(y)-L(y)\). Apply the derivative formula separately to assets and liabilities:
A stationary surplus at the valuation yield \(y_0\) has zero first derivative there. Consequently, the general first-order equation is \(A(y_0)D_A(y_0)=L(y_0)D_L(y_0)\). If present values also match and are positive, the common value can be cancelled, leaving duration equality.
The phrase “if and only if” matters. Under present-value matching, duration equality makes the derivative zero, and a zero derivative requires duration equality. If present values do not match, a zero surplus derivative instead requires equality of the present-value-weighted durations, \(A D_A=L D_L\). It does not generally require the durations themselves to be equal.
Use the same valuation yield and compounding convention to calculate \(A(y_0)\) and \(L(y_0)\).
Confirm that \(A(y_0)=L(y_0)\). Do not infer duration equality from a zero derivative unless this value condition holds.
Divide each stream’s present-value-weighted sum of payment times by its present value.
With matched positive present values, check whether \(D_A(y_0)=D_L(y_0)\). This makes the surplus slope zero at \(y_0\).
Worked Example: Matching Values and First-Order Sensitivity
A liability pays $100 at the end of year 2. At an annual effective valuation yield of 5%, an asset portfolio pays $47.619047619 at the end of year 1 and $52.50 at the end of year 3. Determine whether the portfolio satisfies the first-order condition. Show the present values, durations, and surplus derivative. Use unrounded amounts in the calculations and round dollar results to the nearest cent.
Method. Calculate present-value contributions first. They provide a direct check of both the total values and the duration weights.
The discount factors are \(v=1/1.05\) for year 1, \(v^2=1/1.1025\) for year 2, and \(v^3=1/1.157625\) for year 3. The liability’s present value is
The asset payment at year 1 has present value \(47.619047619/1.05=45.3514739229\). The year-3 payment has present value \(52.50/1.157625=45.3514739229\). Therefore,
The asset duration is the weighted average of years 1 and 3, with equal present-value weights:
The liability has one payment at year 2, so \(D_L(0.05)=2\). Values and durations both match. The common first derivative is
The portfolio satisfies the first-order condition. As a check, the asset and liability present values agree, and their present-value-weighted payment times both equal \(181.4058956916\) dollar-years. Common mistake avoided: the two asset payments are not averaged equally by cash-flow amount; their duration weights are their present values.
Worked Example: Equal Present Values but Unequal Durations
At a 5% annual effective yield, a liability pays $100 at the end of year 2. An asset portfolio pays $47.619047619 at the end of year 1 and $50 at the end of year 2. Find the surplus derivative at the valuation yield and interpret its sign. Round dollar results to the nearest cent.
Method. First check present values, then compare durations. The liability present value is \(100/1.1025=90.7029478458\). The asset present-value contributions are
Thus \(A(0.05)=90.7029478458=L(0.05)\). The liability duration is 2. For the assets, the two equal present-value contributions occur at years 1 and 2:
The asset portfolio is less sensitive to yield than the liability. Substituting into the surplus derivative gives
The derivative is approximately $43.19 per unit change in the annual effective yield. For a small yield increase of 0.01, the first-order estimate of the surplus change is \(43.1918808790(0.01)=\$0.4319188088\), or $0.43. The positive sign means surplus initially increases as yield rises. The zero initial surplus remains intact, but the surplus is not stationary. Common mistake avoided: present-value matching by itself is not enough to conclude that the first derivative is zero.
Worked Example: Zero Slope Without Duration Equality
At a 5% annual effective yield, assets have present value $120 and duration 2. Liabilities have present value $100 and duration 2.4. Confirm whether the surplus derivative is zero, and explain why this does not establish the first-order Redington condition. Round dollar results to the nearest cent.
Method. Use the general derivative equation without cancelling present values; cancellation is permitted only if the values match. The asset derivative is
The liability derivative is
Therefore \(S'(0.05)=A'(0.05)-L'(0.05)=0\), even though \(D_A=2\) and \(D_L=2.4\). The present-value surplus is \(120-100=\$20\), not zero. The equality here is between \(A D_A=240\) and \(L D_L=240\), not between the durations. Common mistake avoided: treating duration equality as the general stationary-surplus equation. It follows from the general equation only after equal positive present values are established.
Common Traps and Scope of the Result
A frequent calculation error is to omit the factor \(1/(1+y)\) when differentiating present value. For example, a single $100 payment at the end of year 2 has present value \(100/1.05^2=90.7029478458\) and Macaulay duration 2 at a 5% yield. Its derivative is
not \(-90.7029478458(2)=-181.4058956916\). The latter incorrectly uses Macaulay duration as though it were modified duration.
The derivation assumes a common parallel change in yield: every cash flow is repriced using the same changed annual effective yield. If yields at different maturities move by different amounts, a single yield variable does not describe the change, and this derivative test cannot simply be applied as written. The cash flows must also be fixed as yield changes; if a payment depends on yield, differentiating its present value requires accounting for that dependence.
Finally, stationarity means \(S'(y_0)=0\), not that the surplus is necessarily nonnegative around \(y_0\), nor that \(y_0\) is a local minimum. Determining the curvature or the behavior of surplus on either side requires additional conditions. Those questions are distinct from deriving the first-order condition.
Check Your Understanding
For each problem, use an annual effective yield and the surplus convention \(S=A-L\). Dollar answers are rounded to the nearest cent.
- At a 4% yield, assets have present value $200 and duration 3; liabilities have present value $200 and duration 2.5. Find the surplus derivative. (Answer: \(-\$96.15\) per unit yield change.)
- At a 6% yield, assets and liabilities each have present value $150 and duration 4. What is the surplus derivative? (Answer: \(0\).)
- At a 5% yield, asset present-value contributions are $40 at year 1 and $60 at year 3. Liabilities have present value $100 and consist of a single payment at year 2. Find the surplus derivative. (Answer: \(-\$19.05\) per unit yield change.)
- At a 5% yield, assets have present value $120 and duration 2, while liabilities have present value $100 and duration 2.4. State the surplus derivative and whether this satisfies the first-order Redington condition. (Answer: Derivative \(0\); no, because present values do not match.)
- At a 5% yield, a liability has present value $100 and duration 2.2. An asset portfolio has present-value contributions at years 1 and 4, totaling $100. What present-value contributions at years 1 and 4 produce duration matching? (Answer: $60 at year 1 and $40 at year 4.)