Negative Cash Flows and Convexity Diagnostics

Why Cash-Flow Signs Matter

Duration and convexity summarize how present value changes as the valuation rate changes. For a stream of positive payments, the familiar intuition is that present value decreases as the rate rises and has positive curvature. But a net cash-flow stream may include receipts and outflows, or a payment that occurs only if an event happens. In that setting, the derivative formulas still apply, but the signs of their terms matter: neither decreasing present value nor positive convexity is guaranteed.

This distinction matters in Redington immunization. At the valuation rate, assets and liabilities can have equal present values and equal first derivatives, while the second derivative of the surplus determines its local behavior. Signed cash flows can contribute in opposite directions to that second derivative. A convexity comparison based only on the fact that a portfolio has been matched can therefore be misleading.

Definition: Let $C_t$ be the signed cash flow at time $t$, with receipts positive and payments negative. At an effective annual valuation rate $i>-1$, the present value is $P(i)=\sum_t C_t(1+i)^{-t}$. When $P(i)\ne 0$, define the modified-duration ratio $D(i)=-P'(i)/P(i)$ and the convexity ratio $K(i)=P''(i)/P(i)$. These ratios are not generally positive for signed cash flows. If $P(i)=0$, the ratios are undefined, although the derivatives themselves remain meaningful.

Derivatives for Signed Cash Flows

Assume first that the cash-flow amounts and payment times are fixed as the valuation rate changes. Differentiating each term of the finite present-value sum gives

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

The factors $t$ and $t(t+1)$ are nonnegative for payment times $t\ge 0$, but the cash flow $C_t$ can have either sign. Thus, a negative cash flow contributes positively to $P'(i)$ and negatively to $P''(i)$; a positive cash flow contributes negatively to $P'(i)$ and positively to $P''(i)$. The sign of the total derivative depends on the balance of these signed contributions.

A contingent payment can be included through its expected amount when its probability is independent of the valuation rate. For example, a payment of amount $B$ at time $t$, made with probability $q$, contributes expected cash flow $-qB$ to $C_t$. If the event probability itself changes with the valuation rate, the expected cash flow is rate-dependent, and differentiating as though it were fixed omits derivative terms. This tutorial uses fixed cash flows or rate-independent event probabilities.

Key result: For signed cash flows, $D=-P'/P$ and $K=P''/P$ are algebraic ratios, not guarantees of positive duration or convexity. When comparing assets and liabilities for a local Redington test, work with the surplus derivatives directly rather than inferring their signs from the labels “duration” or “convexity.”

Why the Convexity Term Controls Local Surplus

Let $S(i)$ be the present value of assets minus the present value of liabilities. Suppose the valuation rate is $i_0$, and the portfolio is matched there: $S(i_0)=0$. First-order matching also requires $S'(i_0)=0$. For a small parallel change $h$, Taylor expansion gives

S(i_0+h) = S(i_0)+hS'(i_0)+\frac{h^2}{2}S''(i_0)+o(h^2).

Under the two matching conditions, the first two terms vanish. If $S''(i_0)>0$, then for sufficiently small nonzero $h$, the leading term is positive, so the surplus is locally positive on either side of the matched rate. This is the local second-order condition. It does not by itself prove stability for every possible rate change.

If $S$ is represented by signed net cash flows $C_t$, then $S''(i_0)=\sum_t t(t+1)C_t(1+i_0)^{-t-2}$. Positive and negative net flows both enter that sum. In particular, a negative net flow at a later time can outweigh earlier positive flows and make the surplus curvature negative.

Form the signed stream.
Write each receipt as positive and each payment as negative; for a rate-independent contingent payment, use its expected amount.

Calculate present value and derivatives.
Use the signed cash flows in $P(i)$, $P'(i)$, and $P''(i)$, retaining the leading minus sign in the first derivative.

Interpret ratios only when valid.
Divide by $P(i)$ to form duration and convexity ratios only if $P(i)\ne0$; otherwise inspect the derivatives directly.

Test local surplus conditions.
At a Redington match, check equal value, zero first derivative, and strictly positive second derivative of assets minus liabilities.

Worked Applications

Worked Example: A Negative Cash Flow Reverses the Curvature

A net stream receives $100 at the end of year 1 and pays $50 at the end of year 2. At an effective annual valuation rate of 10%, calculate present value, modified-duration ratio, and convexity ratio. State whether the present value is locally convex or concave.

Method and sign convention. Use $C_1=100$, $C_2=-50$, and $i=0.10$. The year-2 outflow must remain negative in every derivative sum.

Present value.

P=\frac{100}{1.1}-\frac{50}{1.1^2} =90.9091-41.3223 =49.5868.

First derivative. The year-1 receipt contributes $-100/(1.1)^2=-82.6446$. The year-2 outflow contributes $-2(-50)/(1.1)^3=+75.1315$. Therefore,

P'=-\frac{100}{1.1^2}+\frac{100}{1.1^3} =-82.6446+75.1315 =-7.5131.

The modified-duration ratio is $D=-P'/P=7.5131/49.5868=0.1515$.

Second derivative. The year-1 contribution is $1(2)(100)/(1.1)^3=150.2630$. The year-2 contribution is $2(3)(-50)/(1.1)^4=-204.9040$. Thus,

P''=\frac{200}{(1.1)^3}-\frac{300}{(1.1)^4} =150.2630-204.9040 =-54.6411, \qquad K=\frac{P''}{P}=\frac{-54.6411}{49.5868}=-1.1019.

The present value is locally concave because $P''<0$. Check: the outflow contributes negatively to the signed first-moment sum, so it contributes positively to $P'$; it contributes negatively to $P''$. The common mistake is to treat the outflow as a positive payment when calculating convexity, which would reverse its contribution and produce the wrong curvature.

Worked Example: A Rate-Independent Contingent Payment

A contract receives $60 at the end of year 1 and pays $100 at the end of year 2 if a specified event occurs. The event probability is 0.20 and is independent of the valuation rate. At $i=0$, find expected present value, the signed slope-to-value ratio $P'/P$, modified-duration ratio, and convexity ratio.

Method. Replace the contingent payment by its expected amount, retaining the negative sign: the expected year-2 cash flow is $-100(0.20)=-20$. The expected stream is therefore $C_1=60$, $C_2=-20$.

At $i=0$, each discount factor is 1, so $P=60-20=40$. The first derivative is

$$ P'=-1(60)-2(-20)=-60+40=-20. $$

Hence the signed slope-to-value ratio is $P'/P=-20/40=-0.5$, and the modified-duration ratio under the convention $D=-P'/P$ is $0.5$. The second derivative is

P''=1(2)(60)+2(3)(-20)=120-120=0, \qquad K=\frac{P''}{P}=0.

Check: $P=40$ is nonzero, so both ratios are defined. The slope ratio is negative while the modified-duration ratio is positive because the latter includes a leading minus sign. The common trap is to report $P'/P$ as modified duration without applying that sign convention, or to omit the negative sign on the expected payment.

Worked Example: Redington Conditions with a Negative Net Cash Flow

At a base rate of $i_0=0$, assets pay $50 at the end of year 1 and $50 at the end of year 3. A liability of $100 is due at the end of year 2. Treat receipts as positive and the liability as negative. Verify the value and first-derivative match, then calculate the surplus second derivative and explain the local conclusion.

Method. The net cash flows are $C_1=50$, $C_2=-100$, and $C_3=50$. At zero interest, present value is the sum of the signed amounts:

$$ S(0)=50-100+50=0. $$

The first derivative at zero is the negative time-weighted sum:

S'(0)=-\bigl(1(50)+2(-100)+3(50)\bigr) =-(50-200+150)=0.

The second derivative is

$$ S''(0)=1(2)(50)+2(3)(-100)+3(4)(50) =100-600+600=100. $$

Thus the value and first derivative match, and the surplus has strictly positive second derivative. By the Taylor argument, the surplus is positive for sufficiently small nonzero rate changes. Check: the negative year-2 net flow contributes $-600$ to the second derivative, but the year-3 receipt is large enough to leave a positive total. The common mistake is to ignore the negative middle flow and infer the result from the positive asset cash flows alone.

Contingency, Ratios, and the Limits of the Test

Expected cash flows are appropriate for these calculations when the event probabilities are fixed with respect to the valuation rate and present value is being calculated on an expected-value basis. This is a mathematical valuation assumption, not a claim that the actual payment will equal its expectation. If a payment is random, realized cash flows can differ from expected cash flows; if its probability depends on the rate, the rate dependence must be included before differentiating.

Duration and convexity ratios can also obscure what is happening in a signed portfolio. If $P>0$, the sign of $K$ agrees with the sign of $P''$; if $P<0$, it does not. If $P=0$, a ratio such as $P''/P$ cannot be used at all. For an immunization test, the surplus derivatives are often the clearest quantities because the target is a surplus of zero at the matched rate.

A positive second derivative establishes a local second-order result only under the stated parallel-rate-change model and matching conditions. It does not guarantee positive surplus for a large rate movement, nor does it establish stability under a change in the shape of the yield curve. Those questions require additional analysis beyond this local diagnostic.

Common Mistakes

Dropping the sign of an outflow. A negative $C_t$ contributes negatively to $P''$, not positively.
Forgetting the leading minus sign in $P'$. A negative cash flow contributes positively to $P'$, even though its signed first-moment contribution is negative.
Assuming convexity is positive. This follows term by term for nonnegative cash flows, not for a signed stream.
Confusing the slope ratio and modified duration. $P'/P$ and $-P'/P$ have opposite signs.
Dividing by zero present value. Derivative ratios are undefined when $P=0$; use the derivatives themselves.
Treating a local condition as a global guarantee. Positive surplus curvature supports stability for sufficiently small changes, not automatically for every change.

Key takeaway: Preserve cash-flow signs through every derivative calculation. For signed or contingent streams, inspect $P'$, $P''$, and especially the surplus derivatives directly; do not rely on positive-cash-flow duration or convexity intuition without checking its assumptions.

Check Your Understanding

Use receipts-positive, payments-negative signs. Unless stated otherwise, treat cash flows as fixed and use $i=0$.

A net stream receives $70 at year 1 and pays $20 at year 2. Find $P$, $P'$, and $P''$. (Answer: $P=50,\ P'= -30,\ P''=20$.)
A contract receives $40 at year 1 and pays $60 at year 2 with probability 0.25, independent of the valuation rate. Find expected $P$, $P'/P$, modified-duration ratio, and convexity ratio. (Answer: $P=25,\ P'/P=-0.4,\ D=0.4,\ K=-0.4$.)
A signed stream has cash flows $+30$ at year 1 and $-10$ at year 3. Find its present value, modified-duration ratio, and convexity ratio. (Answer: $P=20,\ D=0,\ K=-3$.)
Assets pay $40 at year 1 and $40 at year 3; a liability of $80 is due at year 2. At zero interest, verify the value and first-derivative matches and calculate surplus curvature. (Answer: $S=0,\ S'=0,\ S''=80>0$, so the surplus is locally positive for sufficiently small nonzero rate changes.)

What You'll Learn