Testing Convexity After Two-Asset Matching

Why Convexity Is the Final Matching Test

Present-value matching makes asset and liability values equal at the valuation rate. Duration matching makes their first-order responses to a small parallel rate change equal. Neither condition alone guarantees that the surplus has a local minimum at the valuation rate. The remaining test compares the assets’ and liabilities’ second derivatives, or equivalently their convexities, at that rate.

This tutorial assumes a fixed set of discrete cash flows and a single effective annual rate applied consistently to every cash flow. The rate changes considered are parallel changes to that rate. There are no defaults, embedded options, or cash flows that change in response to rates. Under these assumptions, the strict Redington condition is a local, second-order condition: it says the asset value curve bends upward more strongly than the liability value curve at the matching rate.

Definition: Let $A(i)$ and $L(i)$ be the present values of asset and liability cash flows at effective annual rate $i$, and define the surplus as $S(i)=A(i)-L(i)$. At a valuation rate $i_0$, present-value matching requires $S(i_0)=0$, and duration matching requires $S'(i_0)=0$. The strict Redington convexity condition is $S''(i_0)=A''(i_0)-L''(i_0)>0$.

For positive cash flows at times $t$, it is convenient to compare convexities normalized by present value. Because matching makes $A(i_0)=L(i_0)$, the normalized comparison gives the same strict inequality as comparing the unnormalized second derivatives. The normalization does not mean the portfolio is valued using a different rate: all present values and derivatives in the test are evaluated at the same $i_0$.

Formula: For cash flow $C_t$ at time $t$, present value at rate $i$ is $C_t(1+i)^{-t}$. For a collection of positive cash flows with present value $P(i)$, define duration $D(i)=-\frac{1+i}{P(i)}P'(i)$ and normalized convexity $C(i)=\frac{1}{P(i)}P''(i)$. For discrete cash flows, $$ D(i)=\frac{\sum_t tC_t(1+i)^{-t}}{P(i)},\qquad C(i)=\frac{\sum_t t(t+1)C_t(1+i)^{-(t+2)}}{P(i)}. $$

Why the Convexity Formula Works

The formulas follow by differentiating each discounted cash flow. For one payment $C_t$ at time $t$, the present-value function is $C_t(1+i)^{-t}$. Its first derivative is negative because a higher rate reduces present value; its second derivative is positive when the payment is positive and $t>0$.

\frac{d}{di}\left[C_t(1+i)^{-t}\right] =-tC_t(1+i)^{-(t+1)}

\frac{d^2}{di^2}\left[C_t(1+i)^{-t}\right] =t(t+1)C_t(1+i)^{-(t+2)}

Adding these derivatives over all payment times gives the derivatives of total present value. Dividing the first derivative by present value and multiplying by $-(1+i)$ gives the stated duration formula. Dividing the second derivative by present value gives normalized convexity. In particular, the factor $t(t+1)$, rather than just $t^2$, comes directly from differentiating twice with respect to the effective annual rate.

After duration matching, the first-order terms in the Taylor expansions of asset and liability values cancel. If present values match as well, the leading local difference in surplus for a small rate change $h$ is $$ S(i_0+h)=\frac{1}{2}\left[A''(i_0)-L''(i_0)\right]h^2+o(h^2). $$ When the second-derivative difference is strictly positive, the quadratic term is positive for sufficiently small nonzero $h$. That is why the strict convexity comparison is the final local test. It does not guarantee a surplus for every possible size or shape of rate change.

Calculating Convexity After Matching

A portfolio may contain several assets. Its present value and derivatives are the sums of the corresponding values and derivatives of the assets. When assets are described by their present-value contributions, portfolio duration and convexity are averages weighted by those contributions. These are present-value weights, not weights based on the assets’ face amounts or on the sizes of individual future payments.

For example, if assets have present values $P_1$ and $P_2$ at the valuation rate and convexities $C_1$ and $C_2$, then portfolio convexity is $(P_1C_1+P_2C_2)/(P_1+P_2)$. If liability present value matches total asset present value, compare this portfolio convexity directly with liability convexity. Alternatively, calculate the unnormalized second derivatives and compare them; the conclusion is the same.

Worked Example: Two Assets Pass the Strict Test

At an effective annual valuation rate of 10%, a liability is a single payment of $121,000 at time 2. Two assets are zero-coupon payments: $55,000 at time 1 and $66,550 at time 3. Determine whether they match present value and duration and whether they satisfy the strict convexity condition.

Method: discount the cash flows, then compare second derivatives. The discount factor per year is $v=1/1.10$. The liability present value is $121{,}000v^2=121{,}000/1.21=\$100{,}000$.

The first asset has present value $55{,}000/1.10=\$50{,}000$. The second has present value $66{,}550/1.331=\$50{,}000$, since $1.10^3=1.331$. Total asset present value is therefore $100,000, equal to the liability present value.

The liability duration is 2, because all of its present value comes from time 2. The asset duration is the present-value-weighted average of times 1 and 3: $$ D_A=\frac{50{,}000(1)+50{,}000(3)}{100{,}000}=2. $$ Thus duration matches.

At 10%, the convexity contribution for a cash flow at time $t$, before present-value weighting, is $t(t+1)/(1.10)^2$. The liability convexity is $2(3)/1.21=6/1.21$. The asset convexity is $$ C_A=\frac{50{,}000(1)(2)+50{,}000(3)(4)}{100{,}000(1.21)} =\frac{2+12}{2(1.21)} =\frac{7}{1.21}. $$ Since $7/1.21>6/1.21$, asset convexity is strictly greater.

As a check using unnormalized derivatives, $A''=100{,}000(7/1.21)=\$578{,}512.40$ per rate unit squared, while $L''=100{,}000(6/1.21)=\$495{,}867.77$. The difference is positive, approximately $82,644.63 per rate unit squared. The portfolio passes the strict test. The common mistake avoided here is comparing only present value and duration and stopping before checking the second derivative.

A Matched Portfolio Can Still Fail

Matching present value and duration does not force the asset convexity to exceed the liability convexity. If liability cash flows are more dispersed in time than the matched assets’ cash flows, liability convexity can be larger. The strict inequality must be calculated; it cannot be inferred from the fact that the weights were chosen to match the first two quantities.

Worked Example: A Matched Portfolio Fails the Strict Test

At an effective annual rate of 10%, liabilities consist of $44,000 payable at time 1 and $58,564 payable at time 4. Assets are zero-coupon payments of $48,400 at time 2 and $53,240 at time 3. Test present value, duration, and strict convexity.

Method: compare the present-value-weighted convexities after verifying the matching conditions. The liability present values are $44{,}000/1.10=\$40{,}000$ at time 1 and $58{,}564/1.4641=\$40{,}000$ at time 4. Liability present value is $80,000. Its duration is $$ D_L=\frac{40{,}000(1)+40{,}000(4)}{80{,}000}=2.5. $$

For the assets, $48{,}400/1.21=\$40{,}000$ at time 2 and $53{,}240/1.331=\$40{,}000$ at time 3. Asset present value is also $80,000, and $$ D_A=\frac{40{,}000(2)+40{,}000(3)}{80{,}000}=2.5. $$ Both matching conditions hold.

The normalized asset convexity is $(6+12)/(2(1.21))=9/1.21$, using $t(t+1)=6$ at time 2 and 12 at time 3. The normalized liability convexity is $(2+20)/(2(1.21))=11/1.21$, using contributions 2 at time 1 and 20 at time 4. Thus $C_A<C_L$.

The unnormalized second derivatives confirm the result: $A''=80{,}000(9/1.21)=\$595{,}041.32$, while $L''=80{,}000(11/1.21)=\$727{,}272.73$. Their difference is approximately $-\$132{,}231.40$ per rate unit squared. The portfolio fails the strict condition. The mistake avoided is assuming that first-order neutrality itself establishes a local surplus minimum.

Present-Value Weights and a Three-Cash-Flow Liability

The convexity test is often easiest when each portfolio is represented by its present-value contributions at the valuation rate. Those contributions sum to the portfolio present value, and their weighted average payment times give duration. The next example uses that approach to compare a liability with two assets.

Worked Example: Comparing a Two-Asset Portfolio with Several Liability Payments

At an effective annual valuation rate of 10%, a liability has present-value contributions of $20,000 at time 1, $45,000 at time 2, and $35,000 at time 4. Two zero-coupon assets have present-value contributions of $50,000 at time 1 and $50,000 at time 4. Determine whether the assets match present value and duration and satisfy the strict convexity condition.

Method: use present-value weights for both portfolios. Liability present value is $20{,}000+45{,}000+35{,}000=\$100{,}000$; asset present value is $50{,}000+50{,}000=\$100{,}000$. Their durations are $$ D_L=\frac{20{,}000(1)+45{,}000(2)+35{,}000(4)}{100{,}000} =0.20+0.90+1.40=2.50 $$ and $$ D_A=\frac{50{,}000(1)+50{,}000(4)}{100{,}000}=2.50. $$ Present value and duration both match.

For the assets, the present-value-weighted $t(t+1)$ average is $(2+20)/2=11$. For the liabilities, it is $$ 0.20(2)+0.45(6)+0.35(20)=0.40+2.70+7.00=10.10. $$ Dividing each by $(1.10)^2=1.21$, asset convexity is $11/1.21$, while liability convexity is $10.10/1.21$. Therefore the strict inequality holds.

In unnormalized terms, $A''=100{,}000(11/1.21)=\$909{,}090.91$, and $L''=100{,}000(10.10/1.21)=\$834{,}710.74$. The difference is approximately $74,380.17 per rate unit squared, which is positive. The asset payments implied by the stated present values are $55,000 at time 1 and $73,205 at time 4; the liability payments are $22,000 at time 1, $54,450 at time 2, and $51,243.50 at time 4. Discounting these amounts at 10% recovers the stated present values. The common mistake avoided is weighting the convexity terms by the future payment amounts instead of their present values.

Exam Traps and Limits of the Test

The strict inequality is a local test at a specified rate, under a specified cash-flow and rate-change model. It is not a promise that the surplus remains positive after an arbitrarily large rate movement. It also does not test nonparallel changes in a yield curve. If cash flows are rate-dependent, or if an instrument has options that alter its payments, the fixed-cash-flow formulas above are not sufficient without modeling those effects.

Using the wrong convexity factor. For effective annual rate $i$ and time $t$, the second-derivative factor is $t(t+1)/(1+i)^2$. Replacing $t(t+1)$ with $t^2$ does not give this second derivative.
Mixing rate conventions. Asset and liability values, durations, and convexities must use the same valuation rate and compounding convention.
Using face-value weights. Portfolio averages use present-value weights at the valuation rate, not face amounts or unadjusted payment amounts.
Checking only approximate convexity numbers. If present values do not match, comparing normalized convexities alone is not a substitute for checking $A''-L''>0$. In a Redington test, establish the value and duration matches first.
Rounding too early. Retain sufficient precision through the convexity comparison, especially when the two values are close. Round only the final reported result.

The decision sequence is therefore: calculate asset and liability present values; verify equality; calculate and verify duration equality; then compare second derivatives or consistently normalized convexities. A positive difference establishes the strict Redington condition at the valuation rate; a zero or negative difference does not.

Check Your Understanding

Assume fixed positive cash flows and a parallel change in the stated effective annual valuation rate for each problem.

At 10%, a liability is $121,000 at time 2. Two assets pay $55,000 at time 1 and $66,550 at time 3. Do they match present value and duration, and does the strict convexity condition hold? (Answer: Present value is $100,000 on each side, duration is 2 on each side, and the strict condition holds because $7/1.21>6/1.21$.)
At 5%, liabilities have present-value contributions of $60,000 at time 1 and $40,000 at time 3. Assets are zero-coupon payments at times 1 and 4, with present-value contributions chosen to match liability present value and duration. Does the strict condition hold? (Answer: Asset present-value weights are $11/15$ at time 1 and $4/15$ at time 4; duration is 1.8, and asset convexity exceeds liability convexity by $0.8/(1.05)^2$, so it holds.)
At 10%, assets have equal present-value contributions at times 2 and 4. Liabilities have present-value weights 0.25 at time 1, 0.50 at time 3, and 0.25 at time 5. The total present value is the same for both sides. Do they match duration, and does the strict condition hold? (Answer: Both durations are 3; asset convexity is $13/1.21$, liability convexity is $14/1.21$, so the strict condition fails.)
At 8%, a matched asset portfolio has present value $250,000 and normalized convexity 12.4; its liability has the same present value and normalized convexity 11.9. What is $A''-L''$, and does the strict condition hold? (Answer: $250{,}000(12.4-11.9)=\$125{,}000$ per rate unit squared; it is positive, so the strict condition holds.)

What You'll Learn