Deriving Portfolio Duration From Component Prices

Why Portfolio Duration Comes From Price Sensitivity

A portfolio’s price response to yield changes depends on the price response of every holding and on how much of each holding is owned. The central question is therefore not simply how to average the holdings’ durations. It is how the total market value changes when the relevant yield changes. Starting from that total-value change gives both the portfolio’s modified duration and the correct weights.

This derivation uses modified duration as a local measure of price sensitivity. The result applies to a small yield change under the stated assumptions; it is not an exact prediction for a large change. The discussion is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $P(y)$ be the price of an instrument as a differentiable function of its yield $y$, where $y$ is expressed as a decimal annual rate under the instrument’s stated yield convention. Its modified duration is $$ D_{\mathrm{mod}}=-\frac{1}{P(y)}\frac{dP(y)}{dy}. $$ It measures the instrument’s local proportional price sensitivity to a change in yield. A positive modified duration means that a small yield increase produces an approximate price decrease.

The derivative is taken with respect to the yield itself, not with respect to a number of basis points. For example, a 10-basis-point increase is a change of $0.001$ in decimal yield. The sign in the definition makes duration positive when price falls as yield rises.

Deriving the Portfolio Formula

Suppose a portfolio contains $k$ instruments. Let $n_j$ be the fixed number of units held in instrument $j$, and let $P_j(y)$ be the current price per unit under a common yield parameter $y$. The portfolio value is the sum of the current market values of its holdings:

V(y)=\sum_{j=1}^{k} n_jP_j(y).

The quantities held are fixed during the small yield change. Differentiating the sum gives the portfolio’s price sensitivity:

\frac{dV}{dy}=\sum_{j=1}^{k}n_j\frac{dP_j}{dy}.

For each component, the definition of modified duration can be rearranged as $\frac{dP_j}{dy}=-D_{\mathrm{mod},j}P_j$. Substituting that identity into the derivative of portfolio value gives:

\frac{dV}{dy} =-\sum_{j=1}^{k}n_jP_jD_{\mathrm{mod},j}.

Applying the definition of modified duration to the portfolio itself now yields the result. The market value of holding $j$ is $n_jP_j$, and its weight in the portfolio’s current value is $w_j=n_jP_j/V$.

D_{\mathrm{mod},\mathrm{portfolio}} =-\frac{1}{V}\frac{dV}{dy} =\frac{\sum_{j=1}^{k}n_jP_jD_{\mathrm{mod},j}}{V} =\sum_{j=1}^{k}w_jD_{\mathrm{mod},j}.

Theorem: If the holdings are fixed and all component prices respond to the same small yield shift, portfolio modified duration is the sum of component modified durations multiplied by their current market-value weights. For ordinary long holdings with positive prices, those weights are nonnegative and sum to 1.

The market-value weight appears because the derivative of total value contains each component’s dollar sensitivity, $n_jP_jD_{\mathrm{mod},j}$. A component’s duration alone says how sensitive its own price is proportionally; its current market value says how much that proportional sensitivity contributes to the whole portfolio. Face amount and original purchase price do not appear in the derivative of current portfolio value.

Assumptions and Interpretation

The formula is exact as a statement about the first derivative at the current yield, subject to its assumptions. Its use as an estimate of a finite price change is a first-order approximation. In particular, the component holdings must remain fixed while prices change, and each component duration must describe the same yield shock used for the portfolio.

Common yield shift: The formula applies directly when a single yield parameter changes for every component, or when the shock is a parallel shift and each component’s duration is defined for that shift.
Consistent conventions: Yield units, compounding conventions, and time units must be consistent with the component prices and durations. A duration calculated under one yield convention cannot be mixed unadjusted with one under a different convention.
Locally differentiable prices: The relevant component prices must have well-defined local derivatives with respect to the shift.
Fixed quantities: The formula describes the price effect on the existing holdings. It does not include trades, cash flows into or out of the portfolio, or changes in quantities during the shift.
Stable pricing rules: The component prices must respond under the pricing assumptions used to calculate their durations. If a yield change alters expected cash flows or triggers a different contractual outcome, a duration based on unchanged cash flows may not describe that response.

With nonparallel changes across a term structure, one scalar shift may not describe the movement. If each component has sensitivity to several distinct rates, the total change must instead be assembled from the component sensitivities to those rates and the changes in the rates. A single market-value-weighted duration is suitable only for the particular common shift for which its component durations were defined.

Worked Applications

Worked Example: Combining Two Component Sensitivities

A hypothetical portfolio holds two fixed-income investments. At the valuation yield, Investment A has a market value of $\$100{,}000$ and modified duration $4.0$. Investment B has a market value of $\$200{,}000$ and modified duration $1.5$. Both durations refer to the same annual yield parameter and the same small yield shift. Find the portfolio modified duration and estimate the value change for a 25-basis-point yield increase.

Method. Add the component dollar sensitivities, then divide by total market value. This avoids averaging the durations without accounting for the size of each position.

Step 1: Find total market value.

V=\$100{,}000+\$200{,}000=\$300{,}000.

Step 2: Find each component’s derivative. From $\frac{dP}{dy}=-PD_{\mathrm{mod}}$, the dollar sensitivity of each holding is its market value times the negative of its modified duration:

\left(\frac{dV}{dy}\right)_A =-(\$100{,}000)(4.0)=-\$400{,}000, \qquad \left(\frac{dV}{dy}\right)_B =-(\$200{,}000)(1.5)=-\$300{,}000.

These derivatives are dollars per unit change in decimal yield. The portfolio derivative is their sum, $-\$400{,}000-\$300{,}000=-\$700{,}000$.

Step 3: Calculate portfolio modified duration.

D_{\mathrm{mod},\mathrm{portfolio}} =-\frac{-\$700{,}000}{\$300{,}000} =\frac{7}{3} =2.3333.

Step 4: Estimate the effect of the yield increase. A 25-basis-point increase is $\Delta y=0.0025$. Using the local first-order estimate:

\Delta V\approx \frac{dV}{dy}\Delta y =(-\$700{,}000)(0.0025) =-\$1{,}750.

The estimated portfolio value after the shift is $\$300{,}000-\$1{,}750=\$298{,}250$. Equivalently, the proportional estimate is $-2.3333(0.0025)$, or approximately $-0.5833\%$ of the original value. The two calculations agree to the displayed precision. This is an estimate for a finite change, not an exact repricing.

Worked Example: Deriving Portfolio Duration From Zero-Coupon Prices

A hypothetical portfolio contains two zero-coupon securities, each paying $\$1{,}000$ at maturity. There are two 2-year securities and one 1-year security. Each is priced at an effective annual yield of $10\%$, with annual compounding and no payments before maturity. Derive the portfolio modified duration directly from the component prices and their derivatives.

Method. Price each component from its single payment, differentiate with respect to the annual yield, and combine the resulting dollar sensitivities. The yield is stated as an effective annual rate, so the discount factor for $t$ years is $(1+y)^{-t}$.

Step 1: Price one 2-year security and find its price derivative. Its price function is $P_2(y)=1000(1+y)^{-2}$. Differentiating gives $P_2'(y)=-2000(1+y)^{-3}$. At $y=0.10$:

P_2(0.10)=\frac{1000}{1.1^2} =\frac{1000}{1.21} =\$826.4463, \qquad P_2'(0.10)=-\frac{2000}{1.1^3} =-\frac{2000}{1.331} =-\$1{,}502.6296.

Thus one 2-year security has modified duration $\frac{1502.6296}{826.4463}=1.8182$, which also follows from $2/1.1=1.8182$. The two securities together have value $2(\$826.4463)=\$1{,}652.8926$ and derivative $2(-\$1{,}502.6296)=-\$3{,}005.2592$.

Step 2: Price the 1-year security and find its derivative. Its price function is $P_1(y)=1000(1+y)^{-1}$, with derivative $P_1'(y)=-1000(1+y)^{-2}$. At $y=0.10$:

P_1(0.10)=\frac{1000}{1.1}=\$909.0909, \qquad P_1'(0.10)=-\frac{1000}{1.1^2} =-\frac{1000}{1.21} =-\$826.4463.

Step 3: Add component values and derivatives. The portfolio’s total current market value and derivative are:

V=\$1{,}652.8926+\$909.0909=\$2{,}561.9835, $$ $$ \frac{dV}{dy}=-\$3{,}005.2592-\$826.4463=-\$3{,}831.7055.

Step 4: Calculate portfolio modified duration.

D_{\mathrm{mod},\mathrm{portfolio}} =-\frac{-\$3{,}831.7055}{\$2{,}561.9835} =1.4956.

As a check, the component modified durations are $1.8182$ for the 2-year securities and $0.9091$ for the 1-year security. Weighting them by the current market values gives $\frac{(\$1{,}652.8926)(1.8182)+(\$909.0909)(0.9091)}{\$2{,}561.9835}\approx1.4956$. Both approaches agree. Intermediate dollar amounts are shown to four decimal places; the final duration is rounded to four decimal places.

Worked Example: Why Face-Value Weights Give the Wrong Result

A hypothetical portfolio holds 1,000 bonds of Type A, each with face amount $\$100$, current market price $\$96$, and modified duration $4.0$. It also holds 500 bonds of Type B, each with face amount $\$100$, current market price $\$108$, and modified duration $2.0$. The two durations use the same yield convention and common small yield shift. Find the portfolio modified duration and identify the error in using face-amount weights.

Method. Calculate each position’s current market value before weighting its duration. The number of bonds and their prices determine how much each position contributes to current portfolio sensitivity.

Step 1: Calculate current market values.

V_A=1{,}000(\$96)=\$96{,}000, \qquad V_B=500(\$108)=\$54{,}000, $$ $$ V=\$96{,}000+\$54{,}000=\$150{,}000.

Step 2: Add the dollar sensitivities.

\left(\frac{dV}{dy}\right)_A =-(\$96{,}000)(4.0)=-\$384{,}000, \qquad \left(\frac{dV}{dy}\right)_B =-(\$54{,}000)(2.0)=-\$108{,}000.

Therefore, $\frac{dV}{dy}=-\$384{,}000-\$108{,}000=-\$492{,}000$, and:

D_{\mathrm{mod},\mathrm{portfolio}} =-\frac{-\$492{,}000}{\$150{,}000} =3.28.

Common mistake. Weighting by face amounts gives $\frac{100{,}000(4.0)+50{,}000(2.0)}{150{,}000}=3.3333$. This is wrong because the face amounts are not the current market values: Type A is priced below face, while Type B is priced above face. The correct weights are $\$96{,}000/\$150{,}000=0.64$ and $\$54{,}000/\$150{,}000=0.36$, giving $0.64(4.0)+0.36(2.0)=3.28$. The check matches the derivative calculation.

Common Traps and Limits

The derivation makes the most frequent calculation errors easier to spot. The portfolio sensitivity is a sum of dollar sensitivities, while the portfolio duration is that sum divided by current total market value. Reversing those steps, or using a weight unrelated to current value, changes the result.

Using a simple average: An unweighted average treats a small position as if it contributed as much dollar sensitivity as a large position. Use current market-value weights instead.
Using face or book values: These values do not measure the current dollar exposure to the yield shift. Use current market prices multiplied by the units held.
Omitting the negative sign: For a conventional positive-duration holding, $\frac{dP}{dy}$ is negative. Portfolio modified duration is the negative of the portfolio derivative divided by portfolio value.
Mixing yield units: A derivative with respect to decimal yield is per unit change in that yield. Convert a basis-point change to decimal form before estimating a value change.
Treating a local estimate as exact: Modified duration uses the first derivative. For a non-negligible shift, the actual repriced value can differ from the linear estimate.

For a portfolio containing short positions, a component’s market value is signed: a short position contributes negative value. The algebraic identity still follows from differentiating total value, provided portfolio value is nonzero and the component sensitivities are defined for the same shift. In that case the weights need not all be positive or lie between 0 and 1. The usual interpretation of portfolio duration as an average between the component durations applies to ordinary long positions, not automatically to portfolios with offsetting signed positions.

Key takeaway: Start with total portfolio value, differentiate it, and use each component’s identity $\frac{dP_j}{dy}=-P_jD_{\mathrm{mod},j}$. This yields a portfolio modified duration weighted by current market values because current values scale the component price sensitivities.

Check Your Understanding

For each problem, use the stated common yield shift and fixed holdings. Treat any estimated finite price change as a first-order modified-duration estimate.

A portfolio has two positions: $\$40{,}000$ at modified duration $2.5$, and $\$60{,}000$ at modified duration $1.2$. Find the portfolio modified duration and its estimated dollar sensitivity per unit increase in decimal yield. (Answer: modified duration $1.72$; derivative $-\$172{,}000$ per unit yield.)
A fund owns 200 units priced at $\$98$ each with modified duration $3.0$, and 100 units priced at $\$104$ each with modified duration $1.5$. Calculate the total market value and portfolio modified duration. (Answer: market value $\$30{,}000$; modified duration $2.48$.)
Two positions have market values $\$250{,}000$ and $\$150{,}000$. Their price derivatives with respect to the same decimal yield are $-\$600{,}000$ and $-\$150{,}000$, respectively. Find the portfolio modified duration and the estimated value change for a 20-basis-point yield increase. (Answer: modified duration $1.875$; estimated change $-\$1{,}500$.)
A portfolio holds $\$80{,}000$ in a position with modified duration $4.0$ and $\$120{,}000$ in one with modified duration $1.5$. Find its modified duration, then estimate the value change for a 10-basis-point increase in yield. (Answer: modified duration $2.5$; estimated change $-\$500$.)
A portfolio has a long position worth $\$120{,}000$ with modified duration $2.0$, and a short position with signed market value $-\$20{,}000$ and modified duration $5.0$. Find the net portfolio value and its modified duration under the same yield shift. (Answer: net value $\$100{,}000$; modified duration $1.4$.)

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