Deriving Second-Order Surplus Stability

Why Duration Matching Is Not the Whole Stability Test

Matching asset and liability values removes an initial surplus or deficit. Matching their durations also removes the first-order change in surplus when yields move slightly. But these two matches alone do not determine what happens at the next order: for a small rate change, assets and liabilities can curve differently as their present values change. This tutorial derives the second-order condition that distinguishes a local surplus minimum from a local surplus maximum.

We consider a portfolio with fixed, deterministic cash flows and a single annual effective yield applied to every cash flow. The yield is shifted in parallel by the same amount at every maturity. This is a local analysis: the Taylor approximation describes small changes near the starting yield, not the effect of arbitrary rate movements.

Definition: Let $A(i)$ and $L(i)$ be the present values of assets and liabilities at annual effective yield $i$. The surplus is $S(i)=A(i)-L(i)$. For a small parallel yield change $h$, the surplus change is $S(i+h)-S(i)$. A portfolio has second-order surplus stability at the starting yield if $S(i)=0$, $S'(i)=0$, and $S''(i)>0$. Under these conditions, surplus has a strict local minimum of zero at that yield.

Price Derivatives and Sensitivity Measures

For a set of fixed cash flows $C_t$ paid at times $t$, measured in years, the value at annual effective yield $i>-1$ is

P(i)=\sum_t C_t(1+i)^{-t}.

The sum is over the payment times. For a liability, $C_t$ may denote promised payments; for an asset, it may denote receipts. We assume the cash flows and payment times do not change when the yield changes. In this model, differentiating the price function gives the first and second derivatives with respect to the annual effective yield.

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

Define modified duration $D(i)$ as the negative price derivative divided by price, and convexity $C(i)$ as the second price derivative divided by price. These are yield sensitivities at the stated yield; they are not dollar amounts.

D(i)=-\frac{P'(i)}{P(i)}, \qquad C(i)=\frac{P''(i)}{P(i)}.

Consequently, $P D=-P'$ is dollar duration, and $P C=P''$ is dollar convexity. Multiplying by value matters when comparing portfolios whose values differ. Dollar duration measures the first derivative in dollars per unit yield; dollar convexity measures the second derivative in dollars per squared unit yield.

Deriving the Second-Order Surplus Condition

For a twice-differentiable value function, the Taylor expansion about the current yield is

P(i+h)=P(i)+P'(i)h+\frac{1}{2}P''(i)h^2+o(h^2),

where $o(h^2)$ denotes terms that become negligible relative to $h^2$ as $h$ approaches zero. Apply this expansion separately to assets and liabilities, then subtract the liability expansion from the asset expansion. The result is

S(i+h)=S(i) -\bigl(A D_A-LD_L\bigr)h +\frac{1}{2}\bigl(A C_A-LC_L\bigr)h^2 +o(h^2).

Here $A=A(i)$, $L=L(i)$; $D_A,D_L$ are modified durations; and $C_A,C_L$ are convexities, all evaluated at the starting yield. The linear coefficient is the difference in dollar durations with a negative sign. The quadratic coefficient is one half the difference in dollar convexities.

Match values.
Set $A=L$, so the initial surplus is zero.

Match dollar durations.
Set $A D_A=L D_L$. With equal values, this is equivalent to matching modified durations and makes the first derivative of surplus zero.

Compare dollar convexities.
Require $A C_A>L C_L$. This makes the second derivative of surplus positive, so the matched position is a strict local minimum of surplus.

These are the second-order Redington-style conditions for local stability in this setting. Equal convexity per unit of value is sufficient to make dollar convexities equal only when asset and liability values are also equal. If the dollar convexities are equal, then $S''(i)=0$, and the second-order test is inconclusive: higher-order terms or the actual cash flows must be examined. If asset dollar convexity is lower, the matched surplus has negative curvature instead.

Worked Example: Deriving the Conditions from Cash Flows

A hypothetical portfolio is valued at yield $i=0$. Its assets pay $500 at the end of year 1 and $500 at the end of year 3. Its liability is a payment of $1,000 at the end of year 2. Assume the cash flows are fixed and the yield shift is parallel. Determine whether the portfolio meets the second-order stability conditions, and estimate surplus after a yield increase of 1 percentage point.

Method. Compute values and derivatives from the cash-flow price functions at $i=0$, then use the Taylor expansion. The common mistake avoided here is checking timing or duration alone without comparing the second derivatives.

At $i=0$, each discount factor is 1. Thus $A=500+500=1{,}000$, and $L=1{,}000$. The values match.

For these cash flows, $P'(0)=-\sum tC_t$. The asset derivative is $A'(0)=-(1\cdot500+3\cdot500)=-2{,}000$, and the liability derivative is $L'(0)=-(2\cdot1{,}000)=-2{,}000$. Therefore dollar durations match. Since both values are $1,000, both modified durations equal $2{,}000/1{,}000=2$ years.

The second derivative at zero is $\sum t(t+1)C_t$. For assets it is $A''(0)=1\cdot2\cdot500+3\cdot4\cdot500=1{,}000+6{,}000=7{,}000$. For liabilities it is $L''(0)=2\cdot3\cdot1{,}000=6{,}000$. Thus $A C_A=7{,}000$ and $L C_L=6{,}000$, so asset dollar convexity exceeds liability dollar convexity by $1,000 per squared unit yield. The portfolio satisfies the strict second-order condition.

For $h=0.01$, the first-order term is zero and the second-order estimate of surplus is $\frac12(7{,}000-6{,}000)(0.01)^2=\$0.05$. The estimate is positive, as the stability condition predicts. As a check using the cash flows directly, the exact shifted values are $A(0.01)=500/1.01+500/(1.01)^3\approx\$980.34$ and $L(0.01)=1{,}000/(1.01)^2\approx\$980.30$. Their difference is about $\$0.05$ to the nearest cent. The Taylor calculation is an approximation; the direct cash-flow calculation confirms its sign and rounded result for this small shift.

Worked Comparisons and Interpretation

Worked Example: Duration Match with Insufficient Convexity

At a stated yield, a hypothetical asset portfolio and liability portfolio each have value $1,000 and modified duration 3. The asset convexity is 25 and the liability convexity is 27. Under a small parallel shift, assess the local surplus behavior and estimate the surplus change for a yield increase of 0.5 percentage points.

Method. First check value and dollar-duration equality; then compare dollar convexities. The trap is treating equal values and durations as proof of second-order stability.

Values match at $1,000. Dollar duration is $1{,}000\cdot3=\$3{,}000$ for each side, so the first-order surplus change is zero. Asset dollar convexity is $1{,}000\cdot25=\$25{,}000$; liability dollar convexity is $1{,}000\cdot27=\$27{,}000$. The asset has $2,000 less dollar convexity, so $S''<0$: at the matched yield the surplus is locally a maximum, not a strict local minimum.

For $h=0.005$, the second-order estimate is $\frac12(25{,}000-27{,}000)(0.005)^2 =-1{,}000(0.000025)=-\$0.025$, or approximately $-\$0.03$ to the nearest cent. The estimate indicates a small deficit on either side of the starting yield, to second order. It does not by itself establish behavior for large changes.

Worked Example: Applying the Taylor Approximation to a Rate Decrease

At the current yield, a hypothetical asset portfolio and liability portfolio each have value $2,000 and modified duration 4. Their convexities are 30 for assets and 24 for liabilities. Estimate the surplus change for a parallel yield decrease of 1 percentage point.

Method. Use the surplus Taylor expansion with signed $h$. This avoids the mistake of assigning a negative sign to the quadratic term merely because the yield decreases.

The values match, so initial surplus is zero. Dollar durations are both $2{,}000\cdot4=\$8{,}000$, so the linear term vanishes. Asset dollar convexity is $2{,}000\cdot30=\$60{,}000$; liability dollar convexity is $2{,}000\cdot24=\$48{,}000$. Their difference is $12,000.

The yield change is $h=-0.01$. The second-order surplus estimate is $\frac12(12{,}000)(-0.01)^2 =6{,}000(0.0001)=\$0.60$. The sign is positive because $h^2$ is positive. The surplus has positive curvature at the matched yield, so the second-order estimate is positive for either direction of a sufficiently small parallel shift.

Assumptions, Limits, and Exam Traps

The derivation assumes a common parallel shift in a single annual effective yield, fixed known cash flows, and values that are twice differentiable around the starting yield. In particular, it is not a proof of stability under arbitrary nonparallel changes in rates, changes to cash flows, or large yield shocks. A Taylor approximation omits third- and higher-order terms, so its accuracy generally improves as the shift becomes smaller.

Compare dollar convexities, not just the unscaled figures. In general the relevant quantities are $A C_A$ and $L C_L$. For example, if both portfolios are worth $500, with convexities 22 for assets and 24 for liabilities, their dollar convexities are $500\cdot22=\$11{,}000$ and $500\cdot24=\$12{,}000$, respectively.
Use modified duration for a change in annual effective yield. In the price expansion, the first derivative is $-P D$, where $D=-P'/P$. Substituting a different duration convention without converting it can give an incorrect rate sensitivity.
Keep the sign of the rate change in the linear term. The first-order change is $-\bigl(A D_A-LD_L\bigr)h$. The second-order term uses $h^2$, so it does not change sign when the yield shift changes direction.
Do not claim more than the test establishes. Positive second derivative gives a local minimum when value and first derivative are matched. It does not guarantee nonnegative surplus for every possible rate scenario.

Key takeaway: For second-order local surplus stability under a small parallel yield change, match asset and liability values, match their dollar durations, and make asset dollar convexity strictly greater than liability dollar convexity. The comparison is made at the starting yield and relies on the stated fixed-cash-flow model.

Check Your Understanding

For each question, use the local Taylor conditions and state whether the conclusion is exact or a second-order estimate.

A hypothetical asset and liability portfolio each have value $800 and modified duration 3.2. Their convexities are 18 and 16, respectively. Find both dollar convexities and determine whether the strict second-order condition holds. (Answer: Asset dollar convexity is $14,400 and liability dollar convexity is $12,800; the strict second-order condition holds.)
At a stated yield, assets and liabilities each have value $1,200 and modified duration 5. Asset convexity is 28 and liability convexity is 31. Classify the local curvature of surplus. (Answer: Asset dollar convexity is $33,600 and liability dollar convexity is $37,200; surplus has negative second derivative and is locally a maximum at the matched yield.)
Asset and liability values are each $900. Their modified durations are 4 and 3.5, respectively. What is the first-order surplus change for a parallel yield increase of 0.002? (Answer: Dollar durations are $3,600 and $3,150, so the first-order surplus change is $-\$450(0.002)=-\$0.90$.)
A hypothetical matched position has asset and liability values of $500, equal modified durations, and convexities of 22 and 24. Calculate the dollar convexities and say whether strict second-order stability holds. (Answer: Asset dollar convexity is $11,000 and liability dollar convexity is $12,000; strict second-order stability does not hold.)
Assets and liabilities each have value $2,500 and modified duration 6.1. Their convexities are 40 and 36. Estimate the surplus change for a parallel yield decrease of 0.005 using terms through second order. (Answer: Dollar durations match; dollar convexities are $100,000 and $90,000. The second-order estimate is $\frac12(10{,}000)(0.005)^2=\$0.125$, or $0.13 to the nearest cent.)

What You'll Learn