Convexity in Asset-Liability Matching

Why Convexity Matters After Duration Matching

Matching asset and liability present values makes their initial values equal. Matching their durations also makes their first-order responses to a small interest-rate change equal. But these two matches do not generally make the responses equal for larger changes: the values may curve differently as the rate moves. Convexity measures that curvature.

This tutorial considers fixed, deterministic cash flows valued using one flat annual effective yield. The rate change is assumed to be a small parallel shift: every cash flow continues to be discounted at the same new annual effective yield. Cash flows do not change when rates move, and there are no embedded options. Under these assumptions, comparing asset and liability convexities estimates the second-order change in surplus after present value and duration have been matched.

Definition: For a fixed cash-flow value $P(i)$, where $i$ is the annual effective yield, its normalized convexity is $C=P''(i)/P(i)$. Here $P''(i)$ is the second derivative of the value with respect to the yield. Convexity is measured in squared years when cash-flow times are measured in years. The corresponding dollar convexity is $P''(i)=P(i)C$.

For a portfolio of assets, write its value as $A(i)$, and for the liabilities write their value as $L(i)$. Define surplus as $S(i)=A(i)-L(i)$. The asset convexity and liability convexity are $C_A=A''(i)/A(i)$ and $C_L=L''(i)/L(i)$, respectively. These are normalized by each side’s own present value.

Calculating Convexity from Dated Cash Flows

Suppose fixed amounts $C_t$ are payable at times $t$, in years, and the same flat annual effective yield $i$ discounts each payment. The value is the sum of each cash flow’s present value. Differentiating that sum twice gives the convexity formula below.

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

C=\frac{P''(i)}{P(i)} =\frac{\displaystyle\sum_t C_t(1+i)^{-t}\,t(t+1)} {\displaystyle P(i)(1+i)^2}

The numerator of the final ratio uses each present value $C_t(1+i)^{-t}$, multiplied by $t(t+1)$. Thus, if $w_t=C_t(1+i)^{-t}/P(i)$ is the fraction of total present value contributed by the cash flow at time $t$, then

C=\frac{\sum_t w_t\,t(t+1)}{(1+i)^2}, \qquad \sum_t w_t=1

This present-value weighting is useful for portfolios as well as individual cash-flow streams. The weights are not the proportions of undiscounted payments; they are the proportions of present value at the valuation yield. For a portfolio of components with values $P_j$ and convexities $C_j$, total convexity is $\sum_j (P_j/P)C_j$, where $P=\sum_jP_j$, provided the components are valued under the same yield-change assumption.

Why the Formula Holds and How It Supports an Estimate

For one payment $C_t$, its value is $C_t(1+i)^{-t}$. Its first derivative is $-tC_t(1+i)^{-t-1}$, and differentiating once more multiplies by $t+1$ and introduces another factor of $(1+i)^{-1}$. Therefore its second derivative is $t(t+1)C_t(1+i)^{-t-2}$. Summing over the fixed payments gives $P''(i)$. Dividing by $P(i)$ gives normalized convexity.

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

\Delta P \approx P'(i)\Delta i+\frac{1}{2}P''(i)(\Delta i)^2

Modified duration is $D_{\mathrm{mod}}=-P'(i)/P(i)$. Dividing the approximation by $P(i)$ therefore gives a first-order duration term and a second-order convexity term. If initial asset and liability present values are equal and their modified durations are equal, their first-order dollar changes cancel in surplus. The resulting second-order estimate is

Result: Under the fixed-cash-flow, flat-yield, parallel-shift assumptions, if $A(i)=L(i)=P$ and $D_{\mathrm{mod},A}=D_{\mathrm{mod},L}$, then $$ \Delta S \approx \frac{1}{2}P(C_A-C_L)(\Delta i)^2. $$ A positive difference $C_A-C_L$ gives an estimated positive second-order surplus change; a negative difference gives an estimated negative one. This is an approximation for a small rate change, not an exact repricing formula.

Worked Convexity Comparisons

Worked Example: A Two-Payment Asset Portfolio Against One Liability

At a flat annual effective yield of 10%, a hypothetical liability is a payment of \$100,000 at the end of year 2. An asset portfolio pays \$45,454.55 at the end of year 1 and \$55,000 at the end of year 3. Verify the initial present value and duration match, calculate both convexities, and estimate the surplus change if the yield rises by 1 percentage point.

Method. Discount each payment at 10%, then use the present-value weights in the convexity formula. The common mistake to avoid is comparing the timing of undiscounted payments instead of comparing present-value contributions.

The liability present value is

L=\frac{100{,}000}{1.1^2} =82{,}644.6281

The asset present values are $45{,}454.55/1.1\approx \$41{,}322.3182$ and $55{,}000/1.1^3\approx \$41{,}322.3140$. The small difference comes from rounding the stated year-1 payment to cents. Using the exact matching payment $ \$45{,}454.54545\ldots$, both asset present values are $ \$41{,}322.3140$, and their total is $ \$82{,}644.6281$, matching the liability.

With those exact present values, each asset payment contributes one-half of the total present value. The asset Macaulay duration is therefore $(1+3)/2=2$ years. The liability pays at year 2, so its Macaulay duration is also 2 years. Dividing each by $1.1$ gives a modified duration of $2/1.1$ for both sides.

For the assets, the present-value-weighted average of $t(t+1)$ is $\{1(2)+3(4)\}/2=7$. For the liability, it is $2(3)=6$. Hence

C_A=\frac{7}{1.1^2}=5.785124, \qquad C_L=\frac{6}{1.1^2}=4.958678

The yield change is $\Delta i=0.01$, and $C_A-C_L=1/1.21=0.8264463$. Thus

\Delta S\approx \frac{1}{2}(82{,}644.6281)(0.8264463)(0.01)^2 =\$3.4151

Rounded to the nearest cent, the estimated surplus increase is \$3.42. The calculation is a second-order estimate, not an exact valuation at 11%.

Worked Example: Equal Value and Duration but Lower Asset Convexity

At a flat annual effective yield of 10%, an asset portfolio has present-value contributions of \$25,000 at year 1, \$50,000 at year 2, and \$25,000 at year 3. Liabilities have present-value contributions of \$50,000 at year 1 and \$50,000 at year 3. Both sides have total present value \$100,000. Find their durations and convexities, then estimate the surplus change for a 1 percentage point yield increase.

Method. Use the given present-value contributions directly as weights. The common mistake is to treat the stated contributions as the actual payments; they are already discounted values.

The asset Macaulay duration is

D_{\mathrm{Mac},A} =\frac{25{,}000(1)+50{,}000(2)+25{,}000(3)}{100{,}000} =2

The liability Macaulay duration is $\{50{,}000(1)+50{,}000(3)\}/100{,}000=2$. Both modified durations are $2/1.1$, so the first-order responses match.

The asset’s weighted average $t(t+1)$ is $0.25(2)+0.50(6)+0.25(12)=6.5$. The liability’s is $0.50(2)+0.50(12)=7$. Therefore

C_A=\frac{6.5}{1.21}=5.371901, \qquad C_L=\frac{7}{1.21}=5.785124, \qquad C_A-C_L=-0.413223

For $\Delta i=0.01$,

\Delta S\approx \frac{1}{2}(100{,}000)(-0.413223)(0.01)^2 =-\$2.0661

The estimated surplus change is a decrease of \$2.07. Equal present values and durations remove the initial gap and the first-order rate effect, but do not eliminate the difference in curvature.

Worked Example: Aggregate Cash Flows and Estimate a Rate-Decline Effect

At a flat annual effective yield of 10%, an asset portfolio has present-value contributions of \$60,000 at year 1 and \$40,000 at year 3. Liabilities have present-value contributions of \$60,000 at year 1, \$20,000 at year 2, and \$20,000 at year 4. Both sides have present value \$100,000. Calculate their duration and convexity differences, then estimate the surplus change when the yield falls by 2 percentage points.

Method. Work from the present-value contributions of the complete asset and liability cash flows. The common mistake is to reverse the sign of $\Delta i$ for a rate decline but forget that the convexity term uses its square.

The asset Macaulay duration is $0.60(1)+0.40(3)=1.8$ years. The liability Macaulay duration is $0.60(1)+0.20(2)+0.20(4)=1.8$ years. Thus both modified durations equal $1.8/1.1$.

The asset weighted average of $t(t+1)$ is $0.60(2)+0.40(12)=6$. For liabilities it is $0.60(2)+0.20(6)+0.20(20)=6.4$. Consequently,

C_A=\frac{6}{1.21}=4.958678, \qquad C_L=\frac{6.4}{1.21}=5.289256, \qquad C_A-C_L=-\frac{0.4}{1.21}=-0.330579

Here $\Delta i=-0.02$, so the estimated surplus change is

\Delta S\approx \frac{1}{2}(100{,}000)(-0.330579)(-0.02)^2 =-\$6.6116

The estimated change is a decrease of \$6.61. The negative sign of the rate change does not make the second-order term positive: its sign is determined by the convexity difference.

Interpretation, Limitations, and Exam Traps

Convexity is useful because it distinguishes portfolios that have the same initial value and duration. In the examples, matching duration means the first-order asset and liability changes offset; the convexity difference then indicates the direction and approximate size of the remaining second-order change. A portfolio with greater asset convexity than liability convexity has a positive second-order surplus estimate for either a small increase or a small decrease in the common yield, because $(\Delta i)^2$ is positive.

Keep the assumptions in view. The formula uses fixed cash flows and a single flat annual effective yield. It estimates the result of a small parallel shift, and higher-order terms may matter for larger changes. If the yield curve changes differently at different maturities, a single parallel-shift convexity calculation may not describe the repricing. If payments change with rates, or an embedded option alters cash flows, the fixed-cash-flow formula may not apply.

Using payment amounts as weights. Use discounted present values at the valuation yield, not the undiscounted amounts.
Forgetting the rate denominator. The weighted average of $t(t+1)$ must be divided by $(1+i)^2$.
Confusing convexity with duration. Duration governs the first-order term; convexity governs the second-order term.
Using an inconsistent rate unit. A 1 percentage point increase is $\Delta i=0.01$, not $1$.
Assuming a match guarantees no surplus movement. Matching present value and duration cancels the first two initial and first-order differences, but the convexity difference can still produce a surplus change.

Key takeaway: For fixed cash flows valued at a flat annual effective yield, calculate each side’s normalized convexity using present-value weights. When asset and liability present values and durations match, estimate the second-order surplus change as one-half the common present value times the convexity difference times the squared parallel yield change.

Check Your Understanding

For each problem, use the stated flat annual effective yield and fixed-cash-flow assumptions. Round dollar estimates to the nearest cent.

A liability pays \$24,000 at the end of year 2. At a 5% annual effective yield, find its normalized convexity. (Answer: $5.4422$)
Assets and liabilities each have present value \$250,000 and equal modified duration. Their convexities are $4.8$ and $5.1$, respectively. Estimate the surplus change for a parallel yield decrease of 1.5 percentage points. (Answer: a decrease of \$8.44)
At a 4% annual effective yield, matched assets have present-value contributions of 40% at year 1 and 60% at year 3; a liability has all its present value at year 2. Both sides have present value \$80,000 and equal duration. Estimate the surplus change for a parallel yield increase of 0.5 percentage points. (Answer: an increase of \$1.85)
Matched assets and liabilities each have present value \$100,000 and equal duration at a 10% annual effective yield. Asset present-value contributions are \$20,000 at year 1, \$30,000 at year 2, and \$50,000 at year 3. Liability convexity is 5.6. Find the asset convexity and estimate the surplus change for a parallel yield decrease of 1 percentage point. (Answer: asset convexity $6.7769$; estimated surplus increase of \$5.88)
At a 10% annual effective yield, matched assets and liabilities each have present value \$120,000 and equal duration. Asset present-value contributions are 25% at year 1, 50% at year 2, and 25% at year 3; liability contributions are 50% at year 1 and 50% at year 3. Estimate the surplus change for a parallel yield increase of 2 percentage points. (Answer: a decrease of \$9.92)

What You'll Learn