Price Baskets, Bond Cash Flows, and Measurement Certificates

Conceptual map

IV-15.01Basket measurement
IV-15.02Source fields and revaluation
IV-15.03Bond cash-flow value
IV-15.04Sensitivity records
IV-15.05Information budgets

1. A valuation record needs an environment

A stored scalar value supports exact revaluation only when it carries sufficient information for the admissible changes in prices, quantities, discount factors, and conventions.

For item prices $$p$$ and quantities $$q$$ , expenditure is $V(p,q)=p^{\top}q$ . A base-period basket records $q^{0}$ and transports it across prices through $p^{\top}q^{0}$ . A current-period basket uses $q^{1}$ . These answer different questions when quantities adjust. Units, item coverage, quality adjustment, taxes, weights, and reference date belong to the definition of the index.

Definition 1 · Measurement certificate

For a target family { $T_{e}:e\in \mathcal{E}$ }, a record $$C(x)$$ is a certificate over environments $\mathcal{E}$ if every target factors as $T_{e}(x)=g_{e}(C(x))$ . The environment class is part of the certificate claim.

2. Dated cash flows determine parallel-yield sensitivity

For deterministic cash flows $c_{i}$ at times $t_{i}\ge0$ and a continuously compounded flat yield $$y$$ ,

\[\begin{aligned}P(y)=\sum _{i}c_{i}e^{-yt_{i}}, \quad D(y)=-P^{\prime}/P=\sum _{i}w_{i}t_{i}, \\ K(y)=P^{\prime\prime}/P=\sum _{i}w_{i}t_{i}^{2},\end{aligned}\](1)

where $w_{i}=c_{i}e^{-yt_{i}}/P(y)$ . With nonnegative cash flows and positive price, the weights form a probability vector. The second-order parallel-shift approximation is $\Delta P/P\approx-D\Delta y+\tfrac{1}{2}K(\Delta y)^{2}$ .

Proposition 1 · Price and duration do not certify convexity

Even with nonnegative cash flows and a common valuation yield, equal price and duration can coexist with different convexity and different values after a nonzero parallel yield shift.

Proof. At $$y=0$$ , let Bond A pay 0.5 at time one and 0.5 at time three. Let Bond B pay one at time two. Both have price one and duration two. Equation (1) gives convexities five and four. Their prices after shift $\delta$ are $0.5e^{-\delta}+0.5e^{-3\delta}$ and $e^{-2\delta}$ , which differ for every nonzero $\delta$ because their equality would require $\cosh(\delta)=1$ . ∎

3. One additional moment changes the revaluation

For $\delta=0.1$ , Bond A is worth approximately 0.822827 and Bond B approximately 0.818731. Their common first-order approximation is 0.8. Adding convexity gives 0.825 for A and 0.82 for B. The approximation error remains because higher time moments enter the exponential.

A full vector of dated cash flows certifies revaluation under any declared deterministic discount curve. Price and parallel duration provide only a local first-order certificate. Key-rate duration requires exposures to curve nodes and a specified interpolation rule. Cash-flow optionality, default, inflation linkage, and state dependence make future payments endogenous and require an additional model.

4. Exactness ends at the declared environment class

Failure case · Duration transported across compounding conventions

A derivative with respect to a continuously compounded yield cannot be inserted unchanged into a discrete-yield approximation. The yield coordinate and payment frequency determine the derivative.

When price is zero, normalized duration and convexity are undefined. Negative cash flows can create negative weights, and duration can leave the range of payment dates. A payment at time zero has no yield sensitivity. At a kink created by an exercise rule, ordinary derivatives may fail to exist. For price baskets, missing items, entry and exit, unit changes, and substitution determine whether the same quantity vector remains meaningful. Equality of two index numbers at one environment supplies no transport result for another environment.

5. Implementation, exercises, and sources

Preserve source fields before aggregation: identifiers, units, quantities or cash flows, dates, currencies, tax treatment, quality adjustments, discount convention, and version. Recompute the scalar from those records, verify derivative formulas by finite differences, and state the environment class over which the record is claimed to be sufficient. Store residuals and reconciliation totals.

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Exercises

Reproduce the exact and second-order prices in Section 3.
Construct two price baskets with the same base expenditure and different responses to item-specific price changes.
Show how duration changes when the yield coordinate is $$r$$ with annual discount factor $(1+r)^{-t}$ .

Partial solutions

1. Substitute $\delta=0.1$ in the two exponential formulas and use $$P(0)=1$$ , $$D=2$$ , and $K\in\{5,4\}$ . 2. Put the same base expenditure on different items; an item-specific shock then separates the values. 3. Differentiation gives $-P^{\prime}(r)/P(r)=\sum w_{i}t_{i}/(1+r)$ , the modified-duration coordinate.

John C. Hull, Options, Futures, and Other Derivatives, sections on duration and convexity.Cash-flow sensitivity under yield changes.
W. Erwin Diewert (1976), “Exact and Superlative Index Numbers,” Journal of Econometrics 4, 115–145.Price-index aggregation and economic foundations.
Chae-Yeon Xon (2026), “It’s a Price! It’s a Value!: Economic Measurement Across Environments.”Related certificate and revaluation application.

6. Audit checkpoint

Source identifiers, units, quantities, cash flows, payment dates, valuation date, currency, tax and quality treatment, discount and compounding convention, curve interpolation, sign of cash flows, positive-price condition, optionality, default, index weights, item support, environment class, derivative coordinate, finite-difference check, reconciliation residual, and data vintage.

7. Scope boundary

The chapter covers deterministic basket aggregation and fixed-cash-flow sensitivity. Arbitrage-free term-structure modeling, credit risk, embedded options, and statistical index construction require separate treatment.