Investing: Inverse ETFs (Exchange Traded Funds) Inverse ETFs, sometimes referred to as “bear market” or “short” funds, are designed to deliver the opposite of the performance of the index or category they track and so can be used by traders to bet against the stock market. Exercises 55–56 are based on the following table, which shows the performance of three such funds as of August 5, 2015:^{18}
Year-to-Date Loss (%) | |
MYY (ProShares Short Midcap 400) | 6 |
SH (ProShares Short S&P 500) | 5 |
REW (ProShares UltraShort Technology) | 7 |
You invested a total of $6,000 in the three funds at the beginning of 2011, including an equal amount in MYY and SH. Your total year-to-date loss amounted to $360. How much did you invest in each of the three funds?
To calculate: The amount investment in each of the 3 funds if the total amount invested is
Given Information:
The total amount invested is
The performance of the 3 funds are given as,
Year-to-Date Loss (%) | |
MYY (ProShares Short Midcap 400) | 6 |
SH (ProShares Short S&P 500) | 5 |
REW (ProShares UltraShort Technology) | 7 |
Formula used:
The matrix equation is,
Where, A is the matrix that has entries equal to the coefficients of the left hand side of the system of equations and X is the column matrix of the unknowns in the system and B is the column matrix and has entries equal to the coefficients of the right hand side of the system of equations.
The matrix property,
Two matrices A and B are equal if they have same dimensions and their corresponding entries are equal.
For matrices A with dimension
Calculation:
Consider the performance of the 3 funds,
Year-to-Date Loss (%) | |
MYY (ProShares Short Midcap 400) | 6 |
SH (ProShares Short S&P 500) | 5 |
REW (ProShares UltraShort Technology) | 7 |
Let x be the amount invested in MYY, y be the amount in SH and z be the amount in REW.
Since, the total investment is
Thus,
Since, the amount invested in MYY and SH are equal.
Thus,
Since, the total year-to-date loss is
Thus,
Therefore, the system of linear equations is,
Thus,
Hence, the matrix equation is
Solve the equation for X.
Consider the equation
Multiply both sides on the left of the equation by
As
Substitute
Thus, the equation is
Consider the matrix
Since, number of rows in
Thus, dimension of
Thus, the augmented matrix is,
Row reduce the matrix
Perform the row operation
Perform the row operations
Perform the row operations
Perform the row operations
Therefore,
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