Stopping Game with Three Outcomes

You play a game with independent rounds. Your cumulative reward starts at 0.

In each round, one of three events occurs:

• With probability p1, you gain a fixed amount a dollars, and the game continues.
• With probability p2, the game stops immediately, and you keep your accumulated reward.
• With probability 1 - p1 - p2, the game ends immediately, and your reward is reset to 0.

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.

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, according to 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.

Why Can a Constant-Dollar Strategy Make Money?

A constant-dollar strategy in a market with one risky asset and cash involves targeting a fixed dollar amount W invested in the risky asset. After each price move, you rebalance: if the risky asset's value increases, you sell shares to return its value to W; if it decreases, you buy more shares to reach W. The remaining capital is held in cash.

Assume a volatile market where the risky asset price fluctuates but has approximately zero long-term drift in expectation.

Question: Explain, 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.

Correlation Constraints and Correlation Matrix Properties

Let a, b, and c be random variables with correlations corr(a, b) = rho_ab and 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.

Multi-Year Mean and Variance of Returns

Consider an investment strategy whose one-year simple return R has mean E[R] = mu and standard deviation std(R) = sigma (variance Var(R) = sigma^2). Assume annual returns R1, R2, ..., RT for T years are independent and identically distributed (i.i.d.) copies of R.

Define:

• The T-year cumulative simple return as R_total = R1 + R2 + ... + RT (additive P&L).
• 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.