Portfolio Scaling Under Redington Conditions

Why Portfolio Scale Matters

A portfolio can have the right shape of cash flows for immunization but the wrong size. Multiplying every asset cash flow by the same positive factor preserves the relative timing of those payments. It therefore preserves the portfolio’s duration and normalized convexity. But its present value and dollar derivatives change, and Redington immunization compares those dollar amounts with the liabilities.

The key distinction is between a portfolio measure divided by the portfolio’s present value and an unnormalized amount such as its second derivative. A positive scaling factor cancels from the first kind of measure; it does not cancel when the scaled assets are compared with fixed liabilities.

Definition: Let $A(i)$ be the present value of an asset portfolio and $L(i)$ the present value of its liabilities at effective annual rate $i$. At a valuation rate $i_0$, the portfolio satisfies the Redington conditions if $A(i_0)=L(i_0)$, $A'(i_0)=L'(i_0)$, and $A''(i_0)>L''(i_0)$. The first two conditions match present value and first derivative; the strict inequality gives the surplus positive second derivative.

In this tutorial, scaling assets by $c$ means replacing every asset cash flow $C_t$ by $cC_t$, where $c>0$. Liabilities remain unchanged unless the problem explicitly says they are scaled too. The positive-factor assumption matters: a negative factor would reverse cash-flow signs, and at $c=0$ the asset portfolio has no present value with which to define duration or normalized convexity.

How Scaling Changes Portfolio Measures

For dated asset cash flows $C_t$ paid at times $t$, valued using an effective annual rate $i$, write the present value as $A(i)$. Its first two derivatives with respect to $i$ are:

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

These formulas apply to a finite set of fixed cash flows, with $i>-1$. They follow by differentiating each present-value term. If every $C_t$ is multiplied by $c$, then the entire present-value function is multiplied by $c$, so its derivatives are multiplied by $c$ as well.

A_c(i)=cA(i),\qquad A_c'(i)=cA'(i),\qquad A_c''(i)=cA''(i).

For positive cash flows, Macaulay duration is the present-value-weighted average payment time, modified duration is $-A'(i)/A(i)$, and normalized convexity is $A''(i)/A(i)$. These ratios do not change when the numerator and denominator are both multiplied by $c$.

D_{\mathrm{Mac}}=\frac{\sum_t tC_t(1+i)^{-t}}{A(i)},\qquad D_{\mathrm{mod}}=-\frac{A'(i)}{A(i)} =\frac{D_{\mathrm{Mac}}}{1+i},\qquad \mathcal{C}=\frac{A''(i)}{A(i)}.

Here $\mathcal{C}$ denotes normalized convexity under the stated derivative convention. Some contexts use alternative convexity conventions, so use the definition given in a problem. Redington’s strict condition, however, is stated using the dollar second derivatives, $A''(i_0)$ and $L''(i_0)$, not merely a comparison of normalized convexities.

Key result: Positive scaling preserves an asset portfolio’s duration and normalized convexity, but it does not preserve its present value or dollar derivatives. When liabilities are fixed, scaling assets alone preserves all Redington conditions only if $c=1$.

Why the Normalized Measures Stay the Same

The invariance follows directly from the definitions. Scaling each cash flow multiplies every present-value weight by $c$. Thus the numerator and denominator of Macaulay duration both gain the same factor. The same cancellation applies to modified duration and normalized convexity.

D_{\mathrm{Mac},c} =\frac{\sum_t t(cC_t)(1+i)^{-t}}{\sum_t(cC_t)(1+i)^{-t}} =D_{\mathrm{Mac}}, \qquad \mathcal{C}_c=\frac{cA''(i)}{cA(i)}=\mathcal{C}.

This cancellation does not apply to matching assets against fixed liabilities. For example, if the original portfolio meets present-value and first-derivative matching, then $A(i_0)=L(i_0)$ and $A'(i_0)=L'(i_0)$. After scaling the assets, the two matching differences become:

A_c(i_0)-L(i_0)=(c-1)L(i_0),\qquad A_c'(i_0)-L'(i_0)=(c-1)L'(i_0).

For positive liability present value, the first difference is zero only when $c=1$. Therefore a scaled asset portfolio cannot preserve all the Redington conditions against unchanged liabilities unless it is left at its original scale. It is possible for the second-derivative inequality to remain true at other values of $c$, but that alone does not restore present-value and first-derivative matching.

Worked Example: Scaling a Fully Matched Portfolio

At an effective annual valuation rate of $i_0=10\%$, a liability pays $121 at time 2. An asset portfolio pays $55 at time 1 and $66.55 at time 3. The asset portfolio is fully matched under the Redington conditions. The assets are then scaled by $c=1.2$, while the liability is unchanged. Calculate the scaled present value, first derivative, and second derivative, and assess the conditions.

Method. Value each dated cash flow at 10%, calculate the derivatives, and then multiply the asset amounts by 1.2. This method avoids treating an unchanged duration as if it implied unchanged dollar matching.

The discount factors are $v=1/1.1$, $v^2=1/1.21$, and $v^3=1/1.331$. The asset present value is:

$$ A(i_0)=55v+66.55v^3=50+50=100. $$

The liability present value is $L(i_0)=121v^2=100$. For the asset portfolio, the derivative contributions use the present-value weights 50 and 50:

A'(i_0)=-\frac{1(50)+3(50)}{1.1} =-\frac{200}{1.1} =-181.8182.

For the liability, the present-value weight is 100 at time 2, so:

L'(i_0)=-\frac{2(100)}{1.1}=-181.8182.

The second derivatives are:

A''(i_0)=\frac{1(2)(50)+3(4)(50)}{1.1^2} =\frac{700}{1.21} =578.5124, \qquad L''(i_0)=\frac{2(3)(100)}{1.21} =\frac{600}{1.21} =495.8678.

The original portfolio matches present value and first derivative, and its second-derivative surplus is $(700-600)/1.21=82.6446>0$. After scaling the assets by 1.2:

A_c(i_0)=1.2(100)=120,\quad A_c'(i_0)=1.2(-181.8182)=-218.1818,\quad A_c''(i_0)=1.2\left(\frac{700}{1.21}\right)=694.2149.

The scaled portfolio’s duration is still 2 years, its modified duration is $2/1.1=1.8182$ years, and its normalized convexity is $7/1.21=5.7851$. But against the unchanged liability, the present values differ by $20 and the first derivatives differ by $-36.3636$. The second-derivative surplus is $694.2149-495.8678=198.3471>0$. Thus the strict curvature inequality still holds, but the scaled portfolio does not satisfy all Redington conditions. The common mistake is to infer full immunization from unchanged duration and a positive second-derivative surplus alone.

When Does the Curvature Inequality Survive?

Suppose the original portfolio satisfies $A''(i_0)>L''(i_0)$, with both second derivatives positive. After scaling assets by $c>0$, the strict Redington curvature condition is:

cA''(i_0)>L''(i_0) \quad\Longleftrightarrow\quad c>\frac{L''(i_0)}{A''(i_0)}.

The ratio on the right is the threshold for retaining the curvature inequality, not a scaling factor that restores full immunization. At equality, the second derivatives are equal, so the required strict inequality fails. A factor above the threshold retains positive second-derivative surplus; present-value and first-derivative matching still require $c=1$ when liabilities remain fixed.

Worked Example: Finding the Curvature Threshold

Use the portfolio in the previous example, for which $A''(i_0)=700/1.21$ and $L''(i_0)=600/1.21$. Assets are scaled by $c=0.9$, with liabilities fixed. Determine whether the strict curvature inequality holds and whether the full Redington conditions hold.

Method. Compare the scaling factor with the curvature threshold, then check matching separately. This avoids mistaking a successful curvature test for a complete Redington match.

The threshold is:

\frac{L''(i_0)}{A''(i_0)} =\frac{600/1.21}{700/1.21} =\frac{6}{7} =0.8571\text{ (to four decimals)}.

Since $0.9>6/7$, the strict curvature condition holds. Directly, the scaled asset second derivative is $0.9(700/1.21)=630/1.21=520.6612$, which exceeds $L''(i_0)=600/1.21=495.8678$ by $30/1.21=24.7934$.

However, the scaled asset present value is $0.9(100)=90$, not 100. Its first derivative is $0.9(-181.8182)=-163.6364$, not $-181.8182$. The portfolio retains the curvature inequality but fails both matching equalities, so it does not satisfy the full Redington conditions. The cutoff $6/7$ answers only the curvature question.

Scaling Assets and Liabilities Together

There is a different result if a hypothetical transaction scales both sides by the same positive factor. If assets and liabilities are both multiplied by $c$, present-value equality and first-derivative equality remain equalities. The second-derivative surplus also becomes $c$ times its original positive amount, so it remains strictly positive.

Worked Example: Scaling Both Sides

At the same 10% valuation rate, the portfolio has asset and liability present values of 100, equal first derivatives of $-181.8182$, and second derivatives $700/1.21$ and $600/1.21$, respectively. A hypothetical adjustment scales every asset and liability cash flow by $c=1.25$. Check all Redington conditions.

Method. Apply the same factor to each dollar amount on both sides. This differs from scaling assets alone because neither matching equation is compared with an unchanged amount.

The scaled present values are $1.25(100)=125$ on both sides. The scaled first derivatives are $1.25(-181.8182)=-227.2727$ on both sides. The second derivatives are:

A_c''(i_0)=1.25\left(\frac{700}{1.21}\right)=723.1405,\qquad L_c''(i_0)=1.25\left(\frac{600}{1.21}\right)=619.8347.

Their difference is $1.25(100/1.21)=103.3058>0$. All three Redington conditions continue to hold. The common mistake is to apply this both-sides conclusion when only the assets were rescaled; the problem’s scaling instruction determines which amounts change.

Practical Checks and Common Traps

When a problem changes portfolio scale, separate the checks rather than relying on a single summary measure. The following sequence keeps the equalities and the strict inequality distinct.

Identify what is scaled.
Determine whether the factor applies to assets alone or to both assets and liabilities.

Update dollar amounts.
Multiply the affected present values and derivatives by the scaling factor.

Recheck all conditions.
Test present-value equality, first-derivative equality, and the strict second-derivative inequality separately.

Using duration as a dollar match. Duration is a ratio. Equal duration does not imply equal present value or equal first derivatives after scaling.
Comparing normalized convexities instead of dollar second derivatives. Redington’s curvature condition compares $A''(i_0)$ with $L''(i_0)$. A higher normalized convexity by itself does not establish that inequality.
Including equality at the curvature cutoff. If $c=L''(i_0)/A''(i_0)$, then $cA''(i_0)=L''(i_0)$; the strict condition is not met.
Ignoring what remains fixed. Scaling both sides preserves matching, but scaling assets against fixed liabilities generally does not.

Key takeaway: For positive $c$, scaling an asset portfolio leaves its duration and normalized convexity unchanged while multiplying its present value and dollar derivatives by $c$. Against fixed liabilities, the second-derivative inequality may survive when $c>L''(i_0)/A''(i_0)$, but all Redington conditions are preserved only at $c=1$. Scaling both assets and liabilities by the same positive factor preserves all three conditions.

Check Your Understanding

For each problem, distinguish portfolio ratios from the dollar quantities used in the Redington conditions.

A portfolio has $A(i_0)=80$, $A'(i_0)=-120$, and $A''(i_0)=240$. Fixed liabilities have $L(i_0)=80$, $L'(i_0)=-120$, and $L''(i_0)=200$. Assets alone are scaled by 1.1. Does the scaled portfolio satisfy all Redington conditions? Give the scaled asset present value, first derivative, and second derivative. (Answer: No; $88$, $-132$, and $264$. The curvature inequality holds, but the two matching equalities fail.)
A portfolio’s modified duration is 4.2 and normalized convexity is 22 at the valuation rate. Every asset cash flow is multiplied by 0.75. What are the new measures? (Answer: Modified duration $4.2$; normalized convexity $22$.)
At the valuation rate, an asset portfolio has $A''=350$ and fixed liabilities have $L''=280$. What is the minimum scaling threshold for the strict curvature condition, and does $c=0.8$ meet it? (Answer: The condition is $c>0.8$; $c=0.8$ does not meet the strict inequality.)
Assets and liabilities each initially have present value 150 and first derivative $-240$; their second derivatives are 500 and 440. Both sides are scaled by 0.6. State the three resulting Redington comparisons. (Answer: Present values $90=90$; first derivatives $-144=-144$; second derivatives $300>264$.)
A fully matched portfolio has positive liability present value, and assets alone are scaled by an unknown $c>0$. What value of $c$ preserves all Redington conditions, and why can a different value not work even if the curvature inequality remains strict? (Answer: $c=1$; otherwise asset and liability present values differ, so full matching fails regardless of the curvature comparison.)

What You'll Learn

Why Portfolio Scale Matters

How Scaling Changes Portfolio Measures

Why the Normalized Measures Stay the Same

Worked Example: Scaling a Fully Matched Portfolio

Worked Example: Scaling a Fully Matched Portfolio

When Does the Curvature Inequality Survive?

Worked Example: Finding the Curvature Threshold

Scaling Assets and Liabilities Together

Worked Example: Scaling Both Sides

Practical Checks and Common Traps

Check Your Understanding

Previous

Next