Analyze expectations, correlations, and investment strategies

Consider the following independent quantitative questions.

1. Stopping game with three outcomes

You play a game consisting of independent rounds. At the start your cumulative reward is 0. In each round, one of three things happens:

• With probability p1 , you gain a fixed amount a dollars , and the game continues to the next round.
• With probability p2 , the game stops immediately and you keep your accumulated reward .
• With the remaining probability 1 - p1 - p2 , the game ends immediately and your reward is reset to 0 (you lose everything earned so far).

Assume 0 < p1 + p2 < 1, p1 >= 0, p2 >= 0, and a > 0.

Question: Derive a closed-form expression for the expected payoff of this game at the time it eventually ends, in terms of p1, p2, and a.

2. Expected length of initially increasing run in a permutation

Let x1, x2, ..., xn be a random permutation of n distinct numbers, where all n! permutations are equally likely.

Define a random variable Y as the length of the longest strictly increasing prefix of the sequence, using the following rule:

• Y = k if
  • x1 < x2 < ... < xk , and
  • either k = n (the entire sequence is strictly increasing) or xk >= x(k+1) (the first position where the prefix stops being strictly increasing is between positions k and k+1 ).

Question: Find E[Y] as a function of n.

3. Why can a constant-dollar strategy make money?

A constant-dollar strategy in a market with one risky asset and cash is defined as:

• You target a fixed dollar amount W invested in the risky asset.
• After each price move, you rebalance : if the risky asset has gone up, you sell some shares to bring its dollar value back down to W ; if it has gone down, you buy more shares to bring it back up to W . The remainder stays in cash.

Assume a volatile market where the risky asset price fluctuates up and down over time, but has approximately zero long-term drift in expectation.

Question: Explain, both intuitively and with a simple numerical or probabilistic argument, why and under what conditions such a constant-dollar rebalancing strategy can generate a positive expected profit ("volatility harvesting") even when the asset's long-run expected price change is roughly zero.

4. Correlation constraints and correlation matrix properties

Let a, b, and c be random variables with correlations

• corr(a, b) = rho_ab ,
• corr(b, c) = rho_bc , where -1 <= rho_ab <= 1 and -1 <= rho_bc <= 1 .

1. Range of corr(a, c):
   • Let rho_ac = corr(a, c) . Use the fact that any correlation matrix must be positive semidefinite (PSD) to derive the feasible range of rho_ac in terms of rho_ab and rho_bc .
2. Correlation matrix properties:
   • List the key mathematical properties that any valid correlation matrix must satisfy (e.g., symmetry, diagonal elements, PSD, etc.).

5. Multi-year mean and variance of returns

Consider an investment strategy whose one-year simple return (not log return) is a random variable R with:

• Mean E[R] = mu .
• Standard deviation std(R) = sigma (so variance Var(R) = sigma^2 ).

Assume annual returns for different years are independent and identically distributed (i.i.d.) copies of R:

• R1, R2, ..., RT for T years.

Define:

• The T-year cumulative simple return as R_total = R1 + R2 + ... + RT (ignoring compounding, treating each year's return as additive P&L for simplicity).
• The average annual return over T years as R_avg = R_total / T .

Questions:

1. Find E[R_total] and Var(R_total) in terms of mu , sigma^2 , and T .
2. Find E[R_avg] and Var(R_avg) in terms of mu , sigma^2 , and T .
3. Briefly comment on how the variance of the average return behaves as T increases, assuming the i.i.d. model holds.