FBE459_Homework2_solutions

FBE459_Homework2_solutions - UNIVERSITY OF SOUTHERN...

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UNIVERSITY OF SOUTHERN CALIFORNIA MARSHALL SCHOOL OF BUSINESS FBE 459 Financial Derivatives (P. Matos – Spring 08) Homework 2 (Solutions): 1. Suppose you have a portfolio of stocks which you want to hedge using S&P500 index futures. You regress weekly changes of the portfolio value on the S&P500 index futures price changes. The Excel results are: SUMMARY OUTPUT: Dependent Variable: CHANGES IN PORTFOLIO VALUE Regression Statistics: R Square = 0.857 Observations = 159 Coefficients Standard Error Intercept -0.0002 0.0032 S&P INDEX FUTURES CHANGES 1.35 0.112 1.a) What is the optimal hedge ratio? The optimal hedge ratio is 1.35, the slope (or beta) of the regression. This is the hedge ratio that minimizes the risk (variance) of the hedged position. 1.b) If each S&P500 futures is for the delivery of 150 times the index level, the current S&P futures stand at 1,050 and the portfolio’s value is $1.10 Million, how many futures contracts should you trade? Should you go long or short on the futures? The number of contracts to be shorted is 1.35 * [1,100,000] / [150*1,050] = 9.428571 The hedged position (Portfolio + 9.428571 short contracts) will then vary by (Portfolio) – 1.35x F
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This is the lowest variance attainable. And the beta (systematic risk) of this position will be 0. Illustrating it for a $1,100,000 portfolio: β Portfolio x [Portfolio Value] – 9.428571 β Futures x [Each Futures Value] = = β Portfolio x [ $1,100,000] – 9.428571 β Futures x [150 x $1,050] = = 1.35 x [$1,100,000] – 9.428571 x 1 x [150 x $1,050] = 0 1.c) What should the company do if it wants to reduce the beta of the portfolio to 0.675? To reduce the beta of the portfolio position to 0.675 (half of current 1.35 beta), a short position of 4.714286 contracts (i.e. half of the optimal hedge ratio) is required. The hedged position (Portfolio + 4.714286 short contracts) will then vary by (Portfolio) – 0.675x F This is the lowest variance attainable. And the beta (systematic risk) of this position will now be 0.675: β Portfolio x [Portfolio Value] – 4.714286 β Futures x [Each Futures Value] = = β Portfolio x [ $1,100,000] – 4.714286 β Futures x [150 x $1,050] = = 1.35 x [$1,100,000] – 4.714286 x 1 x [150 x $1,050] = 0.675 x [$1,100,000] 2. The 6-month zero interest rate is 6% and the 12-month zero interest rate is 7%. A bank
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This note was uploaded on 02/29/2008 for the course FBE 459 taught by Professor Matos during the Spring '08 term at USC.

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FBE459_Homework2_solutions - UNIVERSITY OF SOUTHERN...

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