I received the following problem:

Your’re to invest EUR 100 for five years in a portfolio of stocks delivering normally distributed returns with µ = 0.05 p.a. and σ = 0.1 p.a. For each EUR invested in the stock portfolio you can borrow up to EUR 0.5 at the risk-free rate of 0%. Borrowed money invested in the stock portfolio can again be used for borrowing money with the same conditions. For each yearly period you can choose the proportion of your wealth to invest in the stock portfolio vs the risk-free rate (including leverage). The probability of total wealth falling below EUR 70.0 must not exceed 1.0 % at the beginning of each period. Should total wealth fall below EUR 70.0 at any point the game ends and current total wealth (positive or negative) is final.

Write a function to simulate the portfolio value given the leverage strategy. Assuming a leverage strategy maximizing the expected total wealth after five years, what’s the expected total wealth after five years?

The problem translates to choosing to having a maximum possible leverage of 2x or less depending on the total current wealth. Given some Monte Carlo simulation, the optimal strategy seems to be to always use to maximum allowed leverage.

Also this seems to suggests so Optimal leverage for strategy with normal returns.



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