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Problem 1.1 An investor holds two portfolios of assets (bonds and stocks) with the following number of shares allocated in each of the assets. These are represented in column vectors in the following form of (assets, number of shares). PortfolioA =  2 4  PortfolioB =  1 8  Find the total assets held in the two portfolios. Problem 1.2 From Problem 1.1, eliminate the bonds of portfolio A from the total assets held in the two portfolios. Problem 1.3 The prices of bonds and stocks for five different weeks are contained in the following matrix in the form of (weeks, assets).       30 18 32 25 36 21 20 20 35 27       Suppose the investor holds two portfolios of assets (bonds and stocks) each with a different composition of the assets, represented in the following matrix (assets, portfolios).  30 18 32 25  Determine the value of the two portfolios on each of the five weeks. Problem 1.4 From Problem 1.1, what is the fraction of each asset in portfolio A from the total assets held in the two portfolios? Problem 1.5 From Problem 1.1, what is the asset composition of portfolio A if we increase its size six times? Problem 1.6 Solve for the following system of equations by using matrices: 3x + 4y − 6z − 9w = 15 2x − y + w = 2 y + z + w = 3 x + y + z = 1 Problem 1.7 Solve the equations AAXX = BB, where AA =   13 −8 −3 −8 10 −1 −3 −1 11   BB =   20 −5 0   XX =   x y z   1 Problem 1.8 There are two securities in a portfolio, bonds and stocks, which provide annual cash payments of $100 and $60 per unit, based on today’s state of the economy. If the economy slows down, their payments would be $100 and $20. An investor holds 20 units of bonds and 10 units of stocks. The investor’s receipts equal cash payments time units. The payments he will receive from the portfolio for each possible economic state are $2,600 (if the economy remains flat) and $2,200 (if the economy slows down). (a) Formulate the problem as two linear equations of the form a1x + a2y = b where a1, a2 and b are constants and x and y are variables. (b) Represent this system with matrix algebra as an equation involving three matrices in the form AAAXXX = BBB. (c) Given these portfolio characteristics, how many units of each security should the investor hold to receive $8,000 if the economy remains flat and $6,000 if the economy slows down? Show the two equivalent solutions to this system.

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## Problem 1.1
PortfolioA <- matrix(c(2, 4), nrow = 2, ncol = 1, byrow = TRUE)
PortfolioB <- matrix(c(1, 8), nrow = 2, ncol = 1, byrow = TRUE)
total_assets <- PortfolioA + PortfolioB

## Problem 1.2
bondsA <- matrix(c(2, 0), nrow = 2, ncol = 1, byrow = TRUE)...

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