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Exam FM · Convexity and Price Approximation · Tutorial 599 of 998

Aggregating Convexity Across A Portfolio

Learn how individual security convexities combine into portfolio convexity and how to use the result with portfolio duration to estimate a value change.

Intermediate 10 min read

What You'll Learn

  • Define portfolio convexity as the second price derivative relative to portfolio value.
  • Derive its market-value-weighted aggregation formula for fixed-cash-flow securities.
  • Apply a common additive yield shift when securities have different starting yields.
  • Calculate portfolio duration and convexity from security-level measures.
  • Estimate a portfolio’s percentage and dollar value change from a yield shock.
  • Avoid weighting by face value or treating different starting yields as identical.

Why Portfolio Convexity Is an Aggregate Measure

A portfolio’s price response depends on the price response of each security it contains. A security with a large market value has more influence on the portfolio’s total value change than a small holding; therefore, portfolio convexity is not generally the simple average of the securities’ convexities. The appropriate weights are each holding’s share of the portfolio’s current market value.

The aggregation result applies to fixed-cash-flow securities whose prices respond to the same yield shift. The securities may have different starting yields. What must be common is the change applied to those yields, not their initial yield levels.

Definition: Let \(P_j(s)\) be the price of security \(j\) after a common additive yield shift \(s\), and let \(n_j\) be the number of units held. The portfolio value is \(P_{\mathrm{port}}(s)=\sum_j n_jP_j(s)\). At the starting point \(s=0\), portfolio duration and convexity with respect to this shift are \(D_{\mathrm{port}}=-P'_{\mathrm{port}}(0)/P_{\mathrm{port}}(0)\) and \(C_{\mathrm{port}}=P''_{\mathrm{port}}(0)/P_{\mathrm{port}}(0)\).

For a long-only portfolio, define the market-value weight of security \(j\) by \(w_j=n_jP_j(0)/P_{\mathrm{port}}(0)\). These weights sum to 1. If \(D_j=-P'_j(0)/P_j(0)\) and \(C_j=P''_j(0)/P_j(0)\), then

$$ D_{\mathrm{port}}=\sum_j w_jD_j, \qquad C_{\mathrm{port}}=\sum_j w_jC_j. $$

These are weighted averages of the individual measures, with weights based on current market values. A holding’s face amount or original purchase cost is not the relevant weight.

Model and Derivation

Assume each security has fixed cash flows \(CF_{jk}\) at times \(t_{jk}\), measured in years, and is priced using annual effective yield. Security \(j\) starts at its own yield \(y_{j,0}\). Applying a common additive shift \(s\) gives the price function

$$ P_j(s)=\sum_k CF_{jk}(1+y_{j,0}+s)^{-t_{jk}}. $$

This setup allows, for example, one security to start at 4% and another at 6%. Under a shift of \(s=0.002\), their yields become 4.2% and 6.2%, respectively. It would be incorrect to replace their starting yields with a single common yield.

Differentiating the price function with respect to the common shift gives

$$ P'_j(0)=-\sum_k t_{jk}CF_{jk}(1+y_{j,0})^{-t_{jk}-1}, \qquad P''_j(0)=\sum_k t_{jk}(t_{jk}+1)CF_{jk}(1+y_{j,0})^{-t_{jk}-2}. $$

For the portfolio, the numbers of units \(n_j\) are fixed, so differentiation distributes across the sum:

$$ P'_{\mathrm{port}}(0)=\sum_j n_jP'_j(0), \qquad P''_{\mathrm{port}}(0)=\sum_j n_jP''_j(0). $$

Since \(P'_j(0)=-D_jP_j(0)\) and \(P''_j(0)=C_jP_j(0)\), substituting into the portfolio definitions gives

$$ D_{\mathrm{port}} =\frac{\sum_j n_jD_jP_j(0)}{\sum_j n_jP_j(0)} =\sum_j w_jD_j, \qquad C_{\mathrm{port}} =\frac{\sum_j n_jC_jP_j(0)}{\sum_j n_jP_j(0)} =\sum_j w_jC_j. $$

This derivation explains why market values, rather than face amounts, appear in the weights: the derivatives of portfolio value add in dollars, and each security’s derivative is its own price times its normalized duration or convexity.

Condition: The aggregation formula uses the same scalar yield shift for all security prices and fixed cash flows. If yields move by different amounts for different securities, this one-shift formula does not by itself describe the portfolio’s response. Also, if positions include shorts, use signed market values in the derivative ratios; the resulting weights need not be positive or sum as a long-only set of weights.

Worked Example: Aggregating Two Securities with Different Starting Yields

Worked Example: Aggregating Two Securities with Different Starting Yields

A portfolio holds two units of a one-year zero-coupon security with a maturity payment of $1,040 and three units of a two-year zero-coupon security with a maturity payment of $1,123.60. Their annual effective starting yields are 4% and 6%, respectively. Calculate portfolio duration and convexity with respect to a common additive yield shift, then estimate the value change for a 50-basis-point decrease in both yields.

Method and assumptions. Treat the cash flows as fixed, use annual effective yield and time in years, and apply the same shift to each security’s own starting yield. A zero-coupon security with payment \(F\), maturity \(t\), and starting yield \(y\) has price \(F(1+y)^{-t}\), duration \(t/(1+y)\), and convexity \(t(t+1)/(1+y)^2\).

One-year security. Its price per unit is \(1040/1.04=1000\), so the holding’s market value is \(2(1000)=\$2{,}000\). Its duration is \(1/1.04=0.961538\), and its convexity is \(2/(1.04)^2=1.849112\).

Two-year security. Its price per unit is \(1123.60/(1.06)^2=1000\), so the holding’s market value is \(3(1000)=\$3{,}000\). Its duration is \(2/1.06=1.886792\), and its convexity is \(2(3)/(1.06)^2=5.339267\).

The total portfolio value is \(\$2{,}000+\$3{,}000=\$5{,}000\). The market-value weights are \(2000/5000=0.4\) and \(3000/5000=0.6\). Therefore,

$$ D_{\mathrm{port}} =0.4(0.961538)+0.6(1.886792) =1.516691, $$

and

$$ C_{\mathrm{port}} =0.4(1.849112)+0.6(5.339267) =3.943205. $$

A 50-basis-point decrease is \(\Delta s=-0.005\). The duration-convexity estimate of the proportional value change is

$$ \frac{\Delta P_{\mathrm{port}}}{P_{\mathrm{port}}} \approx -D_{\mathrm{port}}\Delta s+\frac12C_{\mathrm{port}}(\Delta s)^2 =-(1.516691)(-0.005)+\frac12(3.943205)(0.005)^2 =0.007632744. $$

The estimated dollar change is \(5000(0.007632744)=\$38.16\), to the nearest cent. The estimated portfolio value after the shift is \(\$5{,}038.16\). The result is consistent with a yield decrease producing a value increase. The common mistake avoided here is discounting both securities at one shared starting yield; the shift is common, but the starting yields are not.

Using Portfolio Convexity in a Value Estimate

Once portfolio duration and convexity have been aggregated, the usual second-order price approximation can be applied to the portfolio. For a sufficiently small common yield shift \(\Delta s\),

$$ \frac{\Delta P_{\mathrm{port}}}{P_{\mathrm{port}}} \approx -D_{\mathrm{port}}\Delta s +\frac12C_{\mathrm{port}}(\Delta s)^2. $$

The first term estimates the linear price response; the second is the convexity adjustment. The shift must be expressed as a decimal rate change: 25 basis points is \(0.0025\), not 25 or \(0.25\). The approximation is an estimate, not an exact repricing, and its accuracy depends on the size of the shift and the behavior of the securities’ prices.

Worked Example: Estimating a Portfolio Value Change

A fixed-cash-flow portfolio holds 10 units of Security A, each currently worth $900, and 10 units of Security B, each worth $1,100. Their durations with respect to the same additive yield shift are 3.2 and 5.0; their convexities are 14 and 25. Estimate the portfolio’s dollar value change if both yields increase by 25 basis points.

Method. First calculate each holding’s market value, then use those values to weight duration and convexity. This avoids treating equal unit counts as equal market exposures.

The holding values are \(10(900)=\$9{,}000\) and \(10(1100)=\$11{,}000\), for a total of \(\$20{,}000\). Their weights are \(9000/20000=0.45\) and \(11000/20000=0.55\). Thus,

$$ D_{\mathrm{port}}=0.45(3.2)+0.55(5.0)=1.44+2.75=4.19, \qquad C_{\mathrm{port}}=0.45(14)+0.55(25)=6.30+13.75=20.05. $$

The yield shift is \(\Delta s=0.0025\). The proportional estimate is

$$ -(4.19)(0.0025)+\frac12(20.05)(0.0025)^2 =-0.010475+0.00006265625 =-0.01041234375. $$

The estimated dollar change is \(\$20{,}000(-0.01041234375)=-\$208.25\), to the nearest cent. The portfolio value is estimated to fall by $208.25. The common mistake avoided is weighting by unit count; the units have different market values, so they do not contribute equally.

Market Value, Not Face Value

Face amount can be useful for calculating a security’s cash flows, but it does not determine how strongly that security affects the current portfolio price. If two bonds have the same face amount but one trades at a premium and the other at a discount, their market-value weights differ. The following example makes the distinction explicit.

Worked Example: Correcting Face-Value Weights

A portfolio holds $10,000 face amount of Bond A, priced at $96 per $100 face, and $10,000 face amount of Bond B, priced at $104 per $100 face. Their durations are 2 and 4, and their convexities are 6 and 18. Estimate the dollar value change for a 50-basis-point decrease in the common yield shift.

Method. Convert each face amount to current market value before calculating the weights. The prices are quoted per $100 face, so Bond A is worth \(10000(96/100)=\$9{,}600\) and Bond B is worth \(10000(104/100)=\$10{,}400\). Total market value is \(\$20{,}000\), giving weights \(0.48\) and \(0.52\).

The portfolio measures are

$$ D_{\mathrm{port}}=0.48(2)+0.52(4)=3.04, \qquad C_{\mathrm{port}}=0.48(6)+0.52(18)=12.24. $$

For \(\Delta s=-0.005\), the estimated proportional change is

$$ -(3.04)(-0.005)+\frac12(12.24)(0.005)^2 =0.0152+0.000153 =0.015353. $$

The estimated dollar increase is \(\$20{,}000(0.015353)=\$307.06\), to the nearest cent. Equal face amounts would give incorrect equal weights: the premium and discount make the current market values unequal.

Common Mistakes and Practical Checks

  • Using one starting yield for all securities. When starting yields differ, write each price using its own \(y_{j,0}\), then apply the common shift \(s\) to each.
  • Weighting by face amount, units, or purchase cost. Use current holding market values \(n_jP_j(0)\).
  • Taking an unweighted average of individual convexities. That is correct only when all holdings have equal market value.
  • Applying a basis-point shock in the wrong units. Convert basis points to a decimal rate change before using the approximation.
  • Using the formula for nonparallel yield changes without qualification. The aggregation shown here is with respect to one shared scalar shift. Different yield movements require a different sensitivity analysis.

For a long-only portfolio of fixed-cash-flow securities, the weights are nonnegative and sum to 1. Portfolio convexity therefore lies between the smallest and largest individual convexities, provided all measures use the same shift convention. This is a useful check on arithmetic: a weighted average cannot exceed that range. The same reasoning applies to duration.

The aggregation result is useful because it turns security-level analysis into a portfolio-level estimate without repricing every cash flow separately for a small common yield movement. Its reliability depends on consistent yield-shift definitions and correctly measured current market values. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

Use market-value weights and, where requested, the duration-convexity approximation.

  1. A portfolio has market values of $6,000 and $14,000 in two fixed-cash-flow securities with convexities 9 and 16. Find portfolio convexity. (Answer: 13.9)
  2. A portfolio holds securities worth $8 million and $12 million. Their durations are 2.5 and 4.0, and their convexities are 7 and 15. Find portfolio duration and convexity. (Answer: \(D_{\mathrm{port}}=3.4\), \(C_{\mathrm{port}}=11.8\))
  3. Using the portfolio in Question 2, estimate the dollar value change for a yield decrease of 40 basis points. (Answer: an estimated increase of $273,888)
  4. Two securities have starting yields of 3.5% and 5.5%. Under a common additive shift of \(+0.003\), what yields should be used to revalue them? (Answer: 3.8% and 5.8%)
  5. A long position has market value $10 million and convexity 12. A short position has market value magnitude $2 million and convexity 20. Using signed positions, calculate net portfolio convexity relative to the net market value. (Answer: 10)

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