# 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&amp;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?

BuyFind

### Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
Publisher: Cengage Learning
ISBN: 9781337274203
BuyFind

### Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
Publisher: Cengage Learning
ISBN: 9781337274203

#### Solutions

Chapter
Section
Chapter 5.3, Problem 56E
Textbook Problem

## 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?

Expert Solution
To determine

To calculate: The amount investment in each of the 3 funds if the total amount invested is $6,000 at beginning of 2011 where the amount is equal in SH and MMY and the total year-to-date loss is$360.

### Explanation of Solution

Given Information:

The total amount invested is $6,000 at beginning of 2011 where the amount is equal in SH and MMY and the total year-to-date loss is$360.

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,

AX=B

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, AA1=I and A1A=I, where I is the n×n identity matrix.

Two matrices A and B are equal if they have same dimensions and their corresponding entries are equal.

For matrices A with dimension m×n and B with dimension n×k, the product AB is the matrix of dimension m×k and ijth entry of AB is the sum of product of corresponding entries of row i of A and column j of B.

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 $6,000. Thus, x+y+z=6,000. Since, the amount invested in MYY and SH are equal. Thus, xy=0. Since, the total year-to-date loss is$360 and the year-to-date loss from the table is 0.06 in MYY, 0.05 in SH and 0.07 in REW.

Thus, 0.06x+0.05y+0.07=360.

Therefore, the system of linear equations is,

x+y+z=6,000xy=00.06x+0.05y+0.07z=360

Thus, A=[1111100.060.050.07], X=[xyz] and B=[6,0000360].

Hence, the matrix equation is [1111100.060.050.07][xyz]=[6,0000360].

Solve the equation for X.

Consider the equation AX=B.

Multiply both sides on the left of the equation by A1.

A1AX=A1B

As AA1=I and A1A=I, where I is the n×n identity matrix.

Substitute I for A1A in A1AX=A1B and simplify.

A1AX=A1BIX=A1BX=A1B

Thus, the equation is [xyz]=[1111100.060.050.07]1[6,0000360].

Consider the matrix [1111100.060.050.07].

Since, number of rows in [1111100.060.050.07] is 3 and number of columns is 3.

Thus, dimension of [1111100.060.050.07] is 3×3.

Thus, the augmented matrix is, [1111001100100.060.050.07001].

Row reduce the matrix [1111001100100.060.050.07001].

Perform the row operation R3R30.06R1 and R2R2R1,

[1111001100100.060.050.07001][11110002111000.010.010.0601]

Perform the row operations R12R1+R2 and R32R30.01R2,

[11110002111000.010.010.0601][201110021110000.030.110.012]

Perform the row operations R1R1(1003)R3, R2R2+(1003)R3 and R3(1003)R3

[201110021110000.030.110.012][200143432003020143232003001113132003]

Perform the row operations R1(12)R1 and R1(12)R1,

[200143432003020143232003001113132003][1007323100301073131003001113132003]

Therefore,

[1111100

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