FE Problem Set #07
Team Latte Mar 02, 2006
Problem 1:
A fund manager who is long a certain stock notices the following pattern in the stock price:
Weeks 
Stock Price (in US$) 
1 
100.00 
2 
101.00 
3 
108.50 
4 
106.50 
5 
101.00 
6 
98.5 
7 
97.5 
8 
98.25 
9 
99.00 
10 
103.50 
11 
101.50 
12 
96.00 
With this very limited information (data) about the stock price movement and in the absence of any other information what can he conclude about the meanreversion nature (or the lack of it) of the stock. Please outline your method of analysis (algorithm or reasoning/rationale/argument).
( Hint: Try finding the serial dependence in the above time series and do a runs test )
Problem 2
You are a fund manager with no knowledge of derivatives pricing using BlackScholes model or any other model. You want to invest $ 100 in a stock which is trading at $1.00 per share and want to protect your investment such that the floor is $95, i.e. the value of your investment never falls below this level. To insure your portfolio – investment in the stock – you need to follow a protective (synthetic) put option strategy and thereby you need to value a synthetic put on this stock. You have no other information about the stock (such as volatility) except that the stock pays no dividend and the risk free interest rate in the market is 8%. How will you value the synthetic put option?
( Hint: A put option is equivalent to selling a risky asset short and lending money at the risk free rate. With this equivalence relation in mind you can create a binomial model of the stock price and solve two simultaneous equations in two unknowns ).
Problem 3
Consider a three asset portfolio. An quantitative analyst estimates the correlation between the three assets using historical time series. He finds that the correlation between asset 1 and 2 and the correlation between asset 1 and 3 are both equal to 1 (minus one). The correlation between asset 2 and 3 is 0.25. The correlation matrix is given by
If all positions (amount invested in the asset) and volatilities (of the three assets) are equal to one then is the above correlation matrix a valid one, i.e. is the above correlation matrix possible and stable or is it nonsensical?
( Hint: Use the portfolio volatility formula and the fact that a valid asset correlation matrix should be positive semidefinite to test the above.)
Problem 4
A thousand fund managers are chosen at random by a fund of funds investor who wants to evaluate the performance of these fund managers over a period of one year. Every month the portfolios of these fund managers are evaluated. At the end of the month if the return is positive then the fund manager stays otherwise (if the return is negative) he is thrown out of the list. Assuming that the returns are totally random and are always either positive or negative (i.e. it cannot remain same between two months) approximately how many fund managers will be left by the end the year? Is this a good way of evaluating the fund managers' performance? Discuss.
( Hint: Relate this problem to a coin tossing problem )
Problem 5
A fund manager wants to invest a certain amount in two asset classes, stocks and corporate bonds. Stocks have an expected return of 12% and volatility of 19% whereas bonds have an expected return of 6% and volatility of 11%. They are correlated with each other by 0.32. Is it possible for the fund manager to make asst allocation decision with only the above information? And if so, then what proportion of the funds should the fund manager invest in bonds? Explain your answer clearly.
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