Chapter 17 Exercises: Portfolio Construction and Risk Management

Conceptual Exercises

Exercise 1: Portfolio vs Single-Bet Thinking

You have a bankroll of $5,000 and have identified 10 prediction market opportunities, each with an estimated 55% true probability priced at $0.50. Compare the following two strategies:

• Strategy A: Bet your full Kelly fraction on one randomly chosen market.
• Strategy B: Split your allocation equally across all 10 markets (assume they are uncorrelated).

Calculate the expected return and standard deviation of returns for each strategy. Which would you choose and why?

Exercise 2: Correlation Bounds

Two binary events have marginal probabilities $p_X = 0.80$ and $p_Y = 0.20$.

a) Calculate the maximum and minimum feasible Pearson correlations between these events. b) If someone tells you the correlation is 0.90, explain why this is impossible. c) Find the joint probability $P(X=1, Y=1)$ that corresponds to the maximum feasible positive correlation.

Exercise 3: Why Correlations Matter for Risk

You hold two equally-weighted positions in markets priced at $0.50 each, with true probabilities of 60%.

a) If the events are independent ($\rho = 0$), what is the probability you lose on both? b) If the events are maximally positively correlated, what is the probability you lose on both? c) Calculate your portfolio expected return and variance in each case.

Exercise 4: Fractional Kelly Justification

Explain in your own words why fractional Kelly (using 25-50% of full Kelly) is especially important when managing a portfolio of prediction market bets, compared to a single bet. Address at least three distinct reasons.

Exercise 5: Diversification Across Event Types

A trader has the following portfolio allocation: - 60% in U.S. political markets - 20% in U.S. economic indicator markets - 15% in sports markets - 5% in entertainment markets

a) Why might this portfolio be less diversified than it appears? b) Suggest a reallocation that improves diversification. c) What is the tradeoff involved in forcing diversification if the trader's edge is concentrated in political markets?

Exercise 6: Platform Risk

You have $20,000 in prediction market capital spread as follows: - Platform A: $12,000 - Platform B: $5,000 - Platform C: $3,000

a) If Platform A experiences a sudden shutdown, what percentage of your capital is lost? b) What maximum per-platform allocation would you recommend and why? c) What factors other than capital amount should you consider when evaluating platform risk?

Exercise 7: Drawdown Psychology

Describe the four common psychological reactions to drawdowns listed in the chapter. For each one, provide a concrete example of how it might manifest in prediction market trading, and propose a specific countermeasure.

Exercise 8: Bankroll Tiers

A trader has $50,000 available for prediction market activity. Using the chapter's recommended tier structure, allocate this capital across Trading Capital, Reserve Capital, and Emergency Fund. For each tier, explain how and when the capital would be used.

Exercise 9: Rebalancing Triggers

A position that you originally bought at $0.40 (estimating true probability of 55%) has moved to a market price of $0.52. Your updated probability estimate is 56%.

a) What is the remaining edge? b) Should you keep, reduce, or close this position? Justify quantitatively. c) What transaction costs would be relevant to this decision?

Exercise 10: Stress Testing Mindset

Explain why stress testing should assume "impossible" scenarios. Reference at least one historical prediction market event where conventional wisdom about correlations or probabilities was dramatically wrong.

Calculation Exercises

Exercise 11: Binary Correlation Calculation

Events A and B have marginal probabilities $p_A = 0.65$ and $p_B = 0.40$. You estimate that $P(B=1 | A=1) = 0.50$.

a) Calculate $P(A=1, B=1)$. b) Calculate the Pearson correlation $\rho_{AB}$. c) Are these events positively or negatively correlated? Interpret the result.

Exercise 12: Two-Bet Portfolio Kelly

You have two simultaneous binary bets: - Bet 1: True probability 0.60, market price 0.50 - Bet 2: True probability 0.55, market price 0.45 - Correlation between events: $\rho = 0.3$

a) Compute the four joint outcome probabilities. b) Write the expected log growth function $G(f_1, f_2)$ using these probabilities. c) Argue qualitatively how the optimal $(f_1, f_2)$ would differ from the independent case.

Exercise 13: Position Sizing with Caps

You have a $10,000 bankroll and the following five opportunities (all independent):

Using half-Kelly with a 5% individual cap and a 10% category cap for Politics: a) Compute the raw half-Kelly fraction for each market. b) Apply the individual caps. c) Apply the Politics category cap. d) What is the total allocation?

Exercise 14: VaR Calculation

You hold three equally-weighted independent positions (each 5% of bankroll) at prices of $0.50 each, with true probabilities of 60%.

Enumerate all $2^3 = 8$ possible outcomes and calculate: a) The portfolio return for each outcome. b) The 87.5% VaR (i.e., the second-worst outcome). c) The Expected Shortfall at 87.5%.

Exercise 15: Drawdown Recovery Time

Your bankroll was $10,000 and has dropped to $7,500 (a 25% drawdown). Your strategy normally earns 3% per round, but you have scaled positions to 50% per your drawdown rules.

a) What is your effective expected return per round? b) How many rounds are needed to recover to $10,000? c) If each round is one week, how many months will recovery take?

Exercise 16: Diversification Ratio

A portfolio has three positions with weights $w = (0.4, 0.35, 0.25)$, individual volatilities $\sigma = (0.5, 0.4, 0.6)$, and correlation matrix:

$$\rho = \begin{pmatrix} 1.0 & 0.2 & 0.0 \\ 0.2 & 1.0 & -0.1 \\ 0.0 & -0.1 & 1.0 \end{pmatrix}$$

a) Calculate the portfolio volatility. b) Calculate the diversification ratio. c) Is this portfolio well-diversified? What would a perfectly uncorrelated portfolio's diversification ratio be?

Exercise 17: Risk of Ruin

A trader makes repeated identical bets with probability 0.55, paying even money (b=1), sizing at 5% of bankroll each bet.

a) What is the Kelly fraction for this bet? b) Is the trader over- or under-betting relative to Kelly? c) Estimate the approximate risk of ruin (bankroll reaching 0) using the formula $P(\text{ruin}) = (q/p)^{B/u}$ where B=100 units and u=5 units.

Exercise 18: Monte Carlo Sample Size

You run a Monte Carlo simulation with 1,000 scenarios and find a mean portfolio return of 3.5% with standard deviation 8%.

a) What is the standard error of the mean estimate? b) What is the 95% confidence interval for the true expected return? c) If you need the confidence interval to be within +/- 0.1%, how many simulations do you need?

Exercise 19: Correlation Impact on VaR

Consider a portfolio of 10 equally-weighted positions, each at price $0.50 with true probability 0.60.

a) If all positions are independent, what is the probability that 7 or more lose (i.e., 70%+ loss rate)? b) If all positions have pairwise correlation 0.5, qualitatively describe how this probability changes. c) Why does this make correlation estimation so critical for risk management?

Exercise 20: Bankroll Growth

A trader starts with $10,000 and deploys 50% of their bankroll across prediction market positions each round. Each round, the deployed capital earns an expected 4% return.

a) What is the expected bankroll after 1 round? b) What is the expected bankroll after 10 rounds? c) If the trader withdraws 50% of profits above $10,000 after every 5 rounds, what is the expected bankroll after 10 rounds?

Programming Exercises

Exercise 21: Correlation Matrix Builder

Write a Python function that takes a list of events with marginal probabilities and a dictionary of conditional probabilities, validates that all implied correlations are feasible (within bounds), and returns a valid correlation matrix. Raise a ValueError if any conditional probability implies an infeasible correlation.

Exercise 22: Portfolio Kelly Optimizer

Implement a portfolio Kelly optimizer that: a) Takes a list of opportunities (true probability, market price) and a correlation matrix. b) Uses the Gaussian copula to generate correlated binary outcomes. c) Optimizes the log growth rate using scipy.optimize. d) Returns both full Kelly and half-Kelly allocations. e) Prints a comparison showing how correlation affects sizing.

Test it with three bets: two correlated political bets ($\rho = 0.6$) and one uncorrelated sports bet.

Exercise 23: Monte Carlo Simulator

Build a Monte Carlo simulation engine that: a) Simulates 10,000 scenarios of a 20-position portfolio. b) Computes the full distribution of portfolio returns. c) Calculates VaR (95% and 99%), Expected Shortfall, and probability of profit. d) Plots a histogram of returns with VaR lines marked. e) Runs with two different correlation assumptions (independent and moderately correlated) and compares results.

Exercise 24: Drawdown Simulator

Write a program that simulates 1,000 sequential paths of a prediction market portfolio over 52 weekly rounds. For each path: a) Start with $10,000 bankroll. b) Each round, deploy 50% of bankroll across 15 independent bets with 55% true prob at $0.50 price. c) Use half-Kelly sizing. d) Record the maximum drawdown experienced on each path.

Report the distribution of maximum drawdowns: mean, median, 95th percentile, and worst case.

Exercise 25: Risk Dashboard

Create a Python class that serves as a portfolio risk dashboard. It should: a) Accept a portfolio of current positions with estimated probabilities and market prices. b) Compute and display: expected return, VaR, CVaR, Sharpe ratio, diversification ratio. c) Run a stress test where all correlations spike to 0.5. d) Run a stress test where edge is halved. e) Output a formatted report summarizing all metrics and stress test results.

Exercise 26: Position Sizing Engine

Implement the full position sizing pipeline from Section 17.4: a) Kelly-based initial sizing. b) Fractional Kelly adjustment. c) Individual position caps. d) Correlated group caps. e) Platform caps. f) Aggregate deployment cap.

Test with a portfolio of 20 diverse opportunities and print the before/after comparison at each step.

Exercise 27: Bankroll Simulator

Write a simulation that compares three bankroll management strategies over 100 rounds: a) Aggressive: Full Kelly, deploy 90% of bankroll. b) Moderate: Half Kelly, deploy 60% of bankroll, withdraw 50% of profits monthly. c) Conservative: Quarter Kelly, deploy 40% of bankroll, withdraw all profits monthly.

Run 5,000 simulated paths for each strategy and compare: median final bankroll, probability of ruin (dropping below 10% of start), maximum drawdown distribution, and total amount withdrawn.

Exercise 28: Correlation Sensitivity Analysis

Write a program that takes a fixed portfolio and varies the average pairwise correlation from -0.2 to 0.8. For each correlation level: a) Generate the correlation matrix. b) Run Monte Carlo simulation. c) Record expected return, volatility, VaR, and Sharpe ratio. d) Plot all four metrics as a function of average correlation.

Discuss: at what correlation level does the portfolio stop being attractive?

Exercise 29: Dynamic Drawdown Manager

Implement a DrawdownMonitor class that: a) Tracks bankroll over time. b) Implements graduated position scaling (100%, 75%, 50%, 25%, 0%) based on drawdown thresholds. c) Simulates 100 rounds of trading with and without drawdown management. d) Compares final bankroll distributions and maximum drawdowns. e) Demonstrates that drawdown management reduces tail risk at the cost of median returns.

Exercise 30: Full Portfolio System

Build an integrated portfolio management system that combines all chapter concepts: a) Takes a universe of 50 potential opportunities. b) Estimates correlations based on event categories. c) Runs portfolio Kelly optimization with fractional Kelly. d) Applies all position sizing constraints. e) Runs Monte Carlo simulation on the final portfolio. f) Runs stress tests. g) Outputs a comprehensive report including all risk metrics.

This is a capstone exercise that ties together every section of the chapter.