Chapter 9 · Portfolio Choice and Dynamic Optimization
Part III · Optimization, Control, and Learning
Chapter at a glance
Modern Finance Laboratory · Module 9 · Week 9
Module 9 has four panels. The frontier panel accepts a capital-market-assumptions table, draws the frontier, capital market line, and tangency of Figure 9.1, and reports any candidate portfolio’s Sharpe gap to efficiency. The fragility panel attaches sliders to every mean input and animates the tangency weights as they move, then runs the resampling experiment of Figure 9.2(b) with the shrinkage, constraint, and 1/𝑁 defenses toggleable—each defense visibly taming the cloud. The Merton panel solves the dynamic problem live: a 𝛾 dial moves the constant weight along (9.4), an implied-𝛾 readout inverts any entered policy weight, and a horizon sli
Learning Outcome Statements
LOS 9.1 Represent preferences by expected utility, compute absolute and relative risk aversion and certainty equivalents, and deploy the CRRA family.
LOS 9.2 Derive the mean–variance frontier and prove two-fund separation and the tangency characterization with a riskless asset.
LOS 9.3 Diagnose the estimation-error fragility of optimized portfolios and apply the standard defenses.
LOS 9.4 Derive the beta-pricing relation from the tangency property and state the CAPM’s logic and limits.
LOS 9.5 Solve Merton’s dynamic portfolio problem by the martingale method, prove horizon irrelevance under CRRA, and translate policy weights into implied risk aversion.
LOS 9.6 Derive the Kelly criterion and the growth-rate parabola, and quantify the cost of over-betting.
Laboratory · Module 9 (book §9.11)
Module 9: The Allocation Laboratory
Course Website · Week 9
Guided experiment (supports LOS 9.2–9.6). (i) Reproduce the tangency weights 16.5/37.0/46.5 and the Sharpe gap of the 60/40 policy. (ii) Move the equity premium from 2% to 4% and record the equity weight’s path through zero; then engage noshorting plus shrinkage and record the tamed path. (iii) Set 𝛾 = 1.74 in the Merton panel and confirm the 60% weight; slide the horizon from 1 to 30 years and write one sentence on what happened. (iv) In the growth panel, read 𝑔 at 60%, Kelly, and twiceKelly; simulate the 70/30 proposal and report its median gain and tenth-percentile cost over twenty years.
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Exercises
Exercises are grouped A–E throughout the book: A concept checks, B computations, C proofs and extensions, D modeling and application, E laboratory. Starred exercises (∗) are on the advanced track.
Part A — Concept checks
Exercise 9.1. Two-fund separation says all investors hold the same risky portfolio. List the three assumptions doing the work (common inputs; mean–variance preferences or their justification; frictionless riskless borrowing), and for each, one real-world violation and its qualitative effect on the conclusion.
Exercise 9.2. A trustee says: “Stocks almost always beat bonds over twenty years, so long-horizon investors should hold more stocks.” Using Theorem 9.3 and the growth parabola, write the four-sentence reply that (a) concedes the probability claim, (b) explains why it does not imply the allocation claim under the theorem’s hypotheses, (c) names the hypothesis whose failure would rescue the argument, and (d) states what evidence that failure requires.
Exercise 9.3. Rank by their Kelly fractions, without computing: (i) premium 3%, volatility 17%; (ii) premium 6%, volatility 24%; (iii) premium 1.5%, volatility 8.5%. Then compute all three and explain why (i) and (iii) tie—what invariance of (9.6) does the tie expose?
Part B — Computations
Exercise 9.4. (The committee’s numbers, end to end.) With the chapter’s three-asset assumptions: (a) reproduce the tangency weights (16.5%, 37.0%, 46.5%) and Sharpe 0.225; (b) compute the 60/40 policy’s mean, volatility, and Sharpe; (c) find the frontier portfolio with the policy’s volatility and report the mean give-up; (d) rerun (a) with the equity premium at 2% and 4% and tabulate the swing.
Exercise 9.5. For CRRA 𝛾 = 3 and initial wealth 100: compute the certainty equivalent of terminal wealth that is lognormal with median growth 5% and log-volatility 12% over one year, and the risk premium (mean minus CE). Repeat for 𝛾 = 1 and 𝛾 = 10; comment on the pattern.
Exercise 9.6. Using (9.4) with the chapter’s market: tabulate 𝜋 ∗ for 𝛾 ∈ {1, 2, 3, 5, 10}; find the implied 𝛾 of the 70/30 proposal and of a 40/60 conservative alternative; and compute each allocation’s growth rate from (9.5). 9.12 Exercises 215
Exercise 9.7. Verify the chapter’s growth arithmetic: 𝑔 at 60%, at Kelly, at half-Kelly, and at twice Kelly; the twenty-year median multiples; and the 70/30 proposal’s +0.09% growth gain and (using the lognormal tenth percentile) its −9% tenth-percentile cost.
Part C — Proofs and extensions
Exercise 9.8. (a) Show that CRRA utilities (9.1) have 𝑅(𝑤) ≡ 𝛾, and that 𝑤1−𝛾 − 1 = ln 𝑤. 𝛾→1 1−𝛾 lim (b) Derive the lognormal certainty-equivalent formula ln CE = 𝑚 + 12 (1 − 𝛾)𝑠2 from the Gaussian MGF, and identify the 𝛾 = 0 and 𝛾 = 1 specializations.
Exercise 9.9. Complete the proof of Theorem 9.1(i): solve the two constraint equations for 𝛼(𝑚), 𝛽(𝑚) explicitly in terms of the scalars 𝐴 = 1⊤ 𝚺 −1 1, 𝐵 = 1⊤ 𝚺 −1 𝝁, 𝐶 = 𝝁⊤ 𝚺 −1 𝝁; derive the frontier’s equation 𝜎 2 (𝑚) = ( 𝐴𝑚 2 − 2𝐵𝑚 + 𝐶)/( 𝐴𝐶 − 𝐵2 ); and identify the global minimum-variance portfolio.
Exercise 9.10. In Theorem 9.3’s Step 3, carry out the matching in detail: write ln 𝜉𝑇 explicitly, compute ln 𝑊𝑇∗ , integrate the constant-𝜋 wealth SDE, and equate coefficients; then verify the budget constraint fixes 𝜂 so that initial wealth is 𝑤0 , and compute the optimal expected utility as a function of (𝑤0 , 𝛾, 𝜆, 𝑟, 𝑇).
Exercise 9.11. ∗ (Kelly’s drawdowns.) For the full-Kelly investor, show that discounted optimal wealth 𝑒 −𝑔 ( 𝜋𝐾 )𝑡 𝑊𝑡 … is not the right martingale; instead show that (𝑤0 /𝑊𝑡 ) is a positive supermartingale under an appropriate measure, and use optional stopping (Theorem 5.8’s continuous version, Section 7.6∗ ) to prove P inf 𝑡 𝑊𝑡 ≤ 𝑤0 /2 = 1/2 exactly at full Kelly—wealth halves with probability one-half. Generalize to P(inf 𝑊 ≤ 𝑤0 /𝑘) = 1/𝑘.
Part D — Modeling and application
Exercise 9.12. Write the one-page memo answering the CIO’s first question: the tangency recommendation and its Sharpe gain; the fragility evidence of Figure 9.2 with the forty-point swing quantified; the four defenses with one sentence each; and your recommended process—how often inputs are re-estimated, which constraints bind, and what evidence would trigger a policy change.
Exercise 9.13. Write the two-page ratification memo for agenda item three: the implied 𝛾 of the incumbent policy and of the 70/30 proposal, with both translated into growth and tenth-percentile terms; the horizon-irrelevance verdict on the trustee’s argument, with the honest list of assumptions under which horizons would matter; the Kelly ceiling and where each candidate sits on the parabola; and your recommendation with 216 9 Portfolio Choice and Dynamic Optimization the single sentence the minutes should record as the committee’s explicit risk-preference statement.
Part E — Laboratory
Exercise 9.14. (Laboratory Module 9; supports LOS 9.2–9.6.) Perform the four-part guided experiment of Section 9.11. Submit: (a) the frontier screen with the policy’s Sharpe gap; (b) the fragility path through zero and its tamed version under defenses; (c) the Merton panel at 𝛾 = 1.74 with the horizon slider’s non-effect; (d) the growth readouts and the 70/30 simulation.
Exercise 9.15. (Course Website, Week 9, Notebook 9.) Extend the notebook: implement the out-of-sample horse race of Section 9.3 on simulated data—tangency (plug-in), tangency with shrinkage, constrained tangency, 1/𝑁, and the incumbent 60/40—over 500 twenty-year worlds; report Sharpe ratios with standard errors and the frequency with which each strategy wins. Three sentences on what the committee should conclude.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 9; they are not posted here. Problem-set files are on the Course Website, Week 9.