Quant Puzzles 1: Portfolio Theory
Conceptual interview puzzles on portfolio theory
Every Monday, I’ll be releasing a batch of quant puzzles. These problems are typically expansions of well-known interview problems or drill-downs into key topics that are necessary for acing quant interviews. This week we’ll focus on portfolio theory - an underserved but frequently asked topic.
Detailed solutions are attached in a pdf at the bottom of the post.
Problems
Question 1: Time To Positive Returns
An investment strategy has an annualized expected return of 3% and an annualized volatility of 10%. Assume returns are normally distributed and independent from one year to the next. After how many years will the mean annualized returns of a portfolio be positive with at least 91.3% probability? What about the compounded returns?
[You are allowed to use code to solve this problem if necessary]
Question 2: Long or Short
Explain what alpha and beta are. Suppose for some time period, the returns of a benchmark investment vehicle is x. Stock A has a beta of 1.5, the returns of the stock are r.
Stock B has returns strictly equal to 1.5x. Your portfolio manager requires that you must go long one stock and short the other. Which stock do you go long and which stock do you go short?
Question 3: Adding an Asset To A Portfolio
The sharpe ratio of our portfolio is s1, the sharpe ratio of some stock is s2. The correlation of your portfolio and the stock is p.
What is the maximum correlation that the portfolio can have to the stock before adding the stock to our portfolio would have a negative impact on our sharpe ratio?
Hey,
I wanted to understand if there is an analytical way of solving the first question. Intuitively, I think it can be done by solving this equation for minimum t
cagr * t - 1.711 * vol * sqrt( t ) >0
which gives t as 32.55 years. 91.3% probability is 1.711 std