Convexity Matching and Surplus Stability

Why Convexity Matters After Duration Matching

Duration matching is designed to offset the first-order effect of an interest-rate change on an asset-liability position. It does not, by itself, make the position insensitive to larger rate changes. Asset and liability values generally bend at different rates as yields change; this difference in curvature is measured by convexity. When the first-order effects are matched, convexity determines the leading approximate change in surplus.

This analysis assumes that the relevant cash flows are fixed and that one flat effective annual yield is used to value every cash flow. A rate movement is a parallel shift: the same change is applied to the yield at every maturity. The methods below are aligned with the SOA Exam FM (Financial Mathematics) syllabus. They do not, on their own, address nonparallel shifts, changing cash flows, or securities whose cash flows change with rates.

Definition: Let $A$ be the present value of assets and $L$ the present value of liabilities at the current yield. The surplus is $S=A-L$. For a value function $V(y)$ at flat effective annual yield $y$, define modified duration and convexity by $$ D=-\frac{V'(y)}{V(y)}, \qquad C=\frac{V''(y)}{V(y)}. $$ For assets and liabilities, write these measures as $D_A,C_A$ and $D_L,C_L$, respectively. The dollar duration is $VD$, and the dollar convexity is $VC$.

From Value Sensitivity to Surplus Sensitivity

For a small yield change $\Delta y$, a second-order Taylor approximation to a value function is

V(y+\Delta y)\approx V(y)+V'(y)\Delta y+\frac{1}{2}V''(y)(\Delta y)^2 =V\left(1-D\Delta y+\frac{1}{2}C(\Delta y)^2\right).

The approximation has a linear duration term and a quadratic convexity term. Subtract the liability change from the asset change to obtain the change in surplus:

\Delta S\approx -(A D_A-L D_L)\Delta y +\frac{1}{2}(A C_A-L C_L)(\Delta y)^2.

This expression uses dollar sensitivities. Matching $D_A$ and $D_L$ is enough to cancel the linear term only when $A=L$. More generally, first-order matching requires $A D_A=L D_L$. If both present values and durations match, then $A=L$ and $D_A=D_L$, so the remaining second-order change is

\Delta S\approx \frac{1}{2}A(C_A-C_L)(\Delta y)^2.

Consequently, when present values and durations are matched, equal convexities make the second-order surplus change zero. If asset convexity is greater than liability convexity, the approximation gives a positive surplus change for either an upward or downward parallel shift. If asset convexity is lower, it gives a negative change for either direction. This statement concerns the local second-order approximation, not a guarantee about the exact change for a large movement.

Why the Convexity Formula Holds

Consider a portfolio of fixed cash flows $C_k$, paid at times $t_k$ years, valued at a flat effective annual yield $y>-1$. Its present value is

V(y)=\sum_k C_k(1+y)^{-t_k}.

Each term is discounted from its payment time. Differentiating one term once gives $-t_k C_k(1+y)^{-t_k-1}$; differentiating twice gives $t_k(t_k+1)C_k(1+y)^{-t_k-2}$. Therefore,

V'(y)=-\sum_k \frac{t_k}{1+y}C_k(1+y)^{-t_k}, \qquad V''(y)=\sum_k \frac{t_k(t_k+1)}{(1+y)^2}C_k(1+y)^{-t_k}.

Dividing the negative first derivative and the second derivative by $V(y)$ yields the duration and convexity definitions above. Applying the second-order Taylor expansion to assets and liabilities separately, then subtracting, gives the surplus formula. This derivation also shows why the yield shift must be entered as a decimal: a shift of 0.50 percentage points is $\Delta y=0.005$, not $0.50$.

For fixed nonnegative cash flows, the contribution of a payment at time $t_k$ to convexity is proportional to $t_k(t_k+1)$. Later payments therefore tend to contribute more convexity, although a portfolio's total convexity depends on the present-value weights of all its cash flows. If cash flows are rate-dependent, or if a shift is not parallel, the simple flat-yield calculation may not describe the relevant risk.

Worked Applications

Worked Example: Duration Matched, but Convexity Is Not

A liability has one fixed payment of $110,250 at time 2 years. An asset portfolio has fixed payments of $52,500 at time 1 year and $57,881.25 at time 3 years. The flat effective annual yield is 5%. Assume no cash-flow changes when yields move. Find the approximate surplus change for a parallel increase of 1 percentage point, and check it by repricing the cash flows exactly.

Method. First find present values and sensitivities at the initial yield. Then use the second-order surplus approximation. The common mistake to avoid is concluding that matching present values and durations also matches convexity.

At 5%, the discount factors for years 1, 2, and 3 are $1/1.05$, $1/1.05^2$, and $1/1.05^3$. The asset present value is

A=\frac{52{,}500}{1.05}+\frac{57{,}881.25}{1.05^3} =50{,}000+50{,}000=100{,}000.

The liability present value is $L=110{,}250/1.05^2=100{,}000$, so initial surplus is zero. For fixed cash flows, modified duration is the present-value-weighted average of payment times, divided by $1+y$. The asset's weighted average payment time is $(1+3)/2=2$ years; the liability is paid at year 2. Thus,

D_A=D_L=\frac{2}{1.05}=1.904762.

For convexity, use the present-value-weighted average of $t(t+1)/(1+y)^2$. Each asset payment has half of the asset's present value, so

C_A=\frac{\tfrac12[1(2)]+\tfrac12[3(4)]}{1.05^2} =\frac{7}{1.1025}=6.349206, \qquad C_L=\frac{2(3)}{1.05^2} =\frac{6}{1.1025}=5.442177.

The first-order surplus effect is zero because the present values and durations match. With $\Delta y=0.01$, the second-order estimate is

\Delta S\approx \frac12(100{,}000)(6.349206-5.442177)(0.01)^2 =\$4.54.

To check, reprice at 6%. Define $q=1.05/1.06=105/106$, the ratio of the old to the new accumulation factor for one year. The new asset and liability values are $100{,}000(q+q^3)/2$ and $100{,}000q^2$, respectively. Thus the exact new surplus is

100{,}000\left(\frac{q+q^3}{2}-q^2\right) =50{,}000q(1-q)^2 =\$4.41 \text{ (approximately)}.

The exact check is close to, but not identical to, the second-order estimate because the Taylor approximation omits higher-order terms. Both calculations show a small positive surplus after the yield increase.

Worked Example: Lower Asset Convexity Reduces Surplus

At a flat effective annual yield of 4%, a fixed-cash-flow asset portfolio and fixed-cash-flow liability each have present value $250,000. Their modified durations are both 3.5 years. Their convexities are $C_A=5.547$ and $C_L=6.469$, respectively. Estimate the surplus change after a parallel yield increase of 2 percentage points. Assume the given convexities are measured at the initial yield and the cash flows do not change.

Method. With equal present values and durations, use the convexity-only surplus expression. The common trap is to give the convexity effect a negative sign merely because yields rose; the quadratic term contains $(\Delta y)^2$, while the sign comes from the convexity difference.

The rate change is $\Delta y=0.02$. The duration term cancels because

A D_A-L D_L=250{,}000(3.5)-250{,}000(3.5)=0.

The convexity difference is $5.547-6.469=-0.922$. Therefore,

\Delta S\approx \frac12(250{,}000)(-0.922)(0.02)^2 =\frac12(250{,}000)(-0.922)(0.0004) =-\$46.10.

The estimated surplus change is a decrease of $46.10. The result is consistent with the asset having less convexity than the liability: after first-order effects cancel, the liability's value has the larger second-order response. The estimate is local and should not be treated as an exact repricing for a large yield movement.

Worked Example: Match Dollar Duration When Present Values Differ

At a flat effective annual yield of 4%, assets have present value $1,200,000, modified duration 4 years, and convexity 22. Liabilities have present value $1,000,000, modified duration 4.8 years, and convexity 30. Treat these as fixed-cash-flow portfolio summaries at the initial yield. Estimate the surplus change after a parallel yield decrease of 0.50 percentage points.

Method. Compare dollar durations and dollar convexities. The mistake to avoid is comparing the durations alone: they are not equal, but the relevant first-order quantities can still match when present values differ.

The dollar durations are

A D_A=1{,}200{,}000(4)=4{,}800{,}000, \qquad L D_L=1{,}000{,}000(4.8)=4{,}800{,}000.

So the first-order term is zero. The rate change is $\Delta y=-0.005$. The dollar convexities are

A C_A=1{,}200{,}000(22)=26{,}400{,}000, \qquad L C_L=1{,}000{,}000(30)=30{,}000{,}000.

The approximate surplus change is therefore

\Delta S\approx \frac12(26{,}400{,}000-30{,}000{,}000)(-0.005)^2 =\frac12(-3{,}600{,}000)(0.000025) =-\$45.00.

The estimated surplus decreases by $45.00. The negative sign comes from the lower asset dollar convexity; the fact that rates fell rather than rose does not change the sign of this second-order effect.

Interpreting a Convexity Match

A useful matching sequence is to compare present values first, then dollar durations, then dollar convexities. If present values are equal, matching ordinary modified durations also matches dollar durations. If present values are unequal, matching ordinary durations alone is insufficient; check $A D_A=L D_L$ directly. After first-order matching, compare $A C_A$ with $L C_L$. Matching these dollar convexities removes the second-order term in the local surplus approximation.

Establish the valuation model.
Specify the flat effective annual yield, fixed cash flows, and parallel shift assumption.

Check first-order matching.
Calculate $A D_A-L D_L$. If it is not zero, the duration term must be included.

Check second-order matching.
Calculate $A C_A-L C_L$, then multiply by $\tfrac12(\Delta y)^2$.

Interpret the approximation cautiously.
It estimates local surplus change; exact repricing is needed when the shift is large or the assumptions do not hold.

Convexity matching improves the local approximation; it does not make assets and liabilities immune to every interest-rate movement. With a nonparallel change in spot rates, different maturities may move by different amounts, and a single flat-yield duration and convexity pair may not capture the full effect. Likewise, embedded options or other rate-sensitive cash flows can change the payment amounts or dates, invalidating the fixed-cash-flow model used here.

A reliable exam calculation states the sign convention, uses the rate change in decimal form, and keeps the liability subtraction explicit. In particular, the second-order contribution is positive when asset dollar convexity exceeds liability dollar convexity and negative when it is lower, regardless of whether the parallel shift is upward or downward. These checks help separate a genuine risk conclusion from a sign or percentage-point error.

Check Your Understanding

For each problem, use the second-order surplus approximation and assume fixed cash flows under a small parallel shift in a flat effective annual yield.

Assets and liabilities each have present value $200,000 and equal modified duration. Their convexities are 8 and 6, respectively. Estimate the surplus change when the yield falls by 1 percentage point. (Answer: increase of $20.00.)
Assets and liabilities each have present value $600,000 and equal modified duration. Asset convexity is 5.5 and liability convexity is 6.5. Estimate the surplus change when the yield rises by 2 percentage points. (Answer: decrease of $120.00.)
Assets have present value $800,000, modified duration 3.5, and convexity 12. Liabilities have present value $700,000, modified duration 4, and convexity 15. Estimate the surplus change for a 1 percentage-point yield increase. (Answer: decrease of $45.00.)
Assets and liabilities each have present value $500,000. Their modified durations are 4.2 and 4.0; their convexities are 22 and 20. Estimate the surplus change for a 0.50 percentage-point yield increase, including both duration and convexity terms. (Answer: decrease of $487.50.)
Assets and liabilities each have present value $2,000,000 and equal modified duration. Their convexities are 9 and 7. Estimate the surplus change for either a 1 percentage-point increase or a 1 percentage-point decrease in yield. (Answer: increase of $200.00 for either shift.)

What You'll Learn