Chapter 1 · Alternative Assets and the Limits of Classical Financial Engineering

Chapter at a glance

The chapter defines an alternative asset as a stochastic cash-flow claim, introduces the five classifying properties (A1)–(A5), contrasts traded price systems with non-traded claims, derives the first valuation interval, states the master SDF valuation formula, introduces the two-filtration view of smoothed observation, and hands the reader the book’s first laboratory: the liquidity-adjusted stochastic DCF sandbox, whose output is a valuation distribution, not a number.

Learning Outcome Statements

LOS 1.1 — Define an alternative asset as a stochastic cash-flow claim and classify the principal alternative asset classes according to their mathematical features: tradability, observability, cash-flow structure, and horizon.

LOS 1.2 — Formulate the mathematical distinction between a traded price system and a non-traded cash-flow claim, and explain why the gains-from-trade calculus of classical financial engineering does not apply directly to the latter.

LOS 1.3 — Derive, in a one-period setting, the interval of arbitrage-free values for a non-replicable claim, and interpret the interval bounds as super- and sub-replication prices.

LOS 1.4 — Formulate the valuation of a private-market claim under a stochastic discount factor and decompose the resulting expected return into risk-premium components, including an illiquidity component.

LOS 1.5 — Distinguish the full-information filtration from the observed filtration generated by appraisals and transactions, and explain how sparse observation and smoothing distort measured risk.

LOS 1.6 — Evaluate, for a given alternative asset, which of the classical assumptions (completeness, continuous tradability, continuous observation, linear pricing) fail, and identify the mathematical tool developed in this book that addresses each failure.

LOS 1.7 — Simulate a first liquidity-adjusted stochastic discounted cash-flow valuation in the companion webapp and interpret the resulting valuation distribution.

Laboratory (book §1.9)

Module: Taxonomy · Cash-Flow Visualization · Liquidity-Adjusted Valuation Sandbox — open in the Laboratory

Suggested experiments E1–E4 (assignable before any theory): E1 from number to distribution; E2 covariation and premia (vary ρ); E3 liquidity as timing risk (slow ν_{I→Liq}); E4 the opening problem, first pass. Validation checks V1–V4 must accompany every laboratory report.

Downloads: Python notebook · Excel workbook · Slides

Exercises

Solutions are distributed to instructors with the Instructor’s Solutions Manual; they are not posted here.

Conceptual Problems

1.1 For each of the following, state which properties (A1)–(A5) of Definition 1.2 hold and which fail, with justification: (a) a listed REIT share; (b) a direct commercialproperty holding; (c) an interval fund holding private credit; (d) a music-royalty stream; (e) a bilateral litigation-finance contract; (f) gold futures.

1.2 Explain, without formulas, why the value gap between claims (i) and (ii) in Example 1.3 must be random and state dependent rather than a constant percentage, and give two economic states in which the gap should widen.

1.3 An allocator states: “Private equity has equity-like returns with half the volatility.” Using Section 1.4, identify the measurement errors embedded in this statement and the filtration with respect to which each moment is implicitly computed.

1.4 A valuation report adds a 300 basis-point illiquidity premium to the discount rate and applies a 15% marketability discount to the terminal value. Using point (ii) of Section 1.7, explain the double-counting risk and propose a discipline for locating the friction in exactly one model component.

Mathematical Problems

1.5 Prove that the set Ψ of state-price vectors in Section 1.5 is convex, and that {𝜓 ⊤ 𝑋 : 𝜓 ∈ Ψ} is an interval. Exhibit a market ( 𝑝, 𝐴) with 𝐽 = 3, 𝑑 = 1 and a claim 𝑋 for which the valuation interval has length exceeding 50% of its midpoint.

1.6 Complete the duality step in the proof of Proposition 1.5: show that the linear program defining 𝜋(𝑋) has dual sup{𝜓 ⊤ 𝑋 : 𝜓 ≥ 0, 𝐴⊤ 𝜓 = 𝑝} and that strong duality applies.

1.7 In the smoothing model (1.3) with latent log returns i.i.d. with variance 𝜎 2 and 𝜀 𝑛 = 0, compute the variance and first-order autocorrelation of reported returns as functions of 𝛼, and the factor by which a variance estimated from reported returns understates 𝜎 2 .

1.8 Let 𝑀 be an SDF with 𝑀𝑡 /𝑀𝑡 = −𝑟𝑡 − 𝜆𝑊𝑡 (𝑟, 𝜆 constant) and let the claim pay the single lump sum 𝐶𝑇 = exp(𝜇𝑇 + 𝜎𝑊𝑇 − 12 𝜎 2𝑇) at 𝑇. Compute 0 from (1.6) in closed form and identify the risk premium as a covariation term. Verify that setting 𝜆 = 0 recovers actuarial discounting.

Computational Problems

1.9 Implement Algorithm 1.9 (or use the webapp module) and reproduce validation checks one and two of Section 1.9. Report Monte Carlo standard errors.

1.10 Using the sandbox, produce the risk-source decomposition of the valuation distribution for the private-credit template (cash-flow risk, rate risk, timing risk, liquidity risk) and present the results as a table of quantiles. Comment on which risk dominates the left tail.

1.11 Simulate the smoothing recursion (1.3) on top of a latent geometric Brownian value path for 𝛼 ∈ {0.25, 0.5, 0.75, 1}, estimate reported-return volatility and correlation with a correlated public index, and tabulate the distortion as a function of 𝛼. Research Extensions

Extension Problem

1.12 The interval of Proposition 1.5 treats all state-price vectors symmetrically. Formulate at least two economically motivated refinements of Ψ (e.g. bounds on the SDF’s variance in the spirit of good-deal restrictions, or factor-structure restrictions) and characterize the resulting narrower valuation intervals in the finite-state model. Anticipates Chapter 7.

Research Problem

1.13 Propose a formal definition of “degree of incompleteness” of a market relative to a claim 𝑋 (e.g. via the norm of the projection residual of 𝑋 on the traded span under a suitable inner product) and investigate how the valuation-interval width of Proposition 1.5 scales with it. Chapter Summary This chapter defined the object of the book. An alternative asset is a stochastic cashflow claim (𝐷, 𝜏) living in a market environment characterized by restricted tradability, sparse and smoothed observation, incompleteness, contractual nonlinearity, and endogenous timing (Definitions 1.2 and 1.2). Mathematically, the classical identification of value with replication cost fails because the gains process (1.1) cannot be formed in the claim; valuation must instead be built as an operator on cash-flow processes (Definition 1.3). In the simplest incomplete one-period market, no-arbitrage restricts the value of a non-replicable claim only to an interval whose endpoints are sub- and super-replication prices (Proposition 1.5); selecting a value inside the interval requires preferences, penalized measure families, or discipline on the stochastic discount factor. The SDF