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Analyze expectations, correlations, and investment strategies

SquarepointData Scientist · Technical Screen
HardData ScientistTechnical ScreenMachine Learning

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This problem tests your understanding of stochastic processes in finance, the linear algebraic requirements for covariance matrices, and optimal decision-making under uncertainty.

Part 1: Correlation Matrices

Suppose you are given a symmetric matrix M of size n x n where all diagonal elements are 1, and all off-diagonal elements are equal to r.

  1. For what range of r is M positive semidefinite?
  2. Explain the intuition for why a correlation matrix must be positive semidefinite in terms of the linear independence of random variables.

Part 2: Optimal Stopping

Consider a sequence of n independent random variables X_1, X_2, ..., X_n drawn from a continuous uniform distribution U(0, 1). You observe the variables one by one. After observing X_i, you must decide whether to stop and accept the value or discard it and move to X_{i+1}. If you discard all n-1 variables, you must accept X_n.

Define a strategy to maximize the expected value of your accepted variable. What is the expected value for a given n?

Part 3: Volatility Harvesting

Assume an asset price S_t follows a geometric Brownian motion with no drift (drift mu = 0) and volatility sigma. You rebalance a portfolio consisting of this asset and cash to maintain a constant weight w in the asset.

Show that the portfolio value V_t exhibits a positive drift purely due to the rebalancing process (volatility harvesting). Express the growth rate of the portfolio as a function of w and sigma.

Hints

Hint 1

For the matrix, consider the eigenvalues of a matrix with constant off-diagonal elements; recall that for a positive semidefinite matrix, all eigenvalues must be non-negative.

Hint 2

For the stopping problem, use dynamic programming or backward induction, starting from the last step where the expected value is 0.5.

Hint 3

For volatility harvesting, use Ito's Lemma on the log of the portfolio value and identify the term involving the second-order derivative (the convexity adjustment).

Follow-up questions

  • How does the existence of a non-zero drift in Part 3 affect the optimal rebalancing frequency?
  • If the variables in Part 2 were not independent but followed a known Markov process, how would your strategy change?
  • In a real-world scenario, what are the primary transaction cost constraints that prevent continuous rebalancing to capture volatility?