Portfolio Diversification

1
Chapter 12

• Portfolio Diversification

2
Return and Risk

• Two assets case
• Weights are the proportion of money invested in
  each asset.

3
Implications

• Changing weights in each asset will change the
  risk and return of the portfolio.
• Examples
• E(R1)0.08, ?10.15
• E(R2)0.15, ?20.25
• The diversification benefits depend on the
  correlation coefficient between the two assets
  (?12).

4
Implications

• Examples
• E(R1)0.08, ?10.15
• E(R2)0.15, ?20.25
• Consider the cases where ?12 is -1, 0, 0.5 and 1.
• Lets say w1 62.5, and w2_____
• E(Rp)62.50.0837.50.150.10625
• ?p0, 0.132583, 0.16238, 0.18750 when ?12 is
  -1, 0, 0.5 and 1, respectively.

5
Two Assets
6
Return and Risk

• General Case with Multiple Assets

7
Return and Risk

• General Case with Multiple Assets
• The number of different correlation terms is
  N(N-1)/2.
• e.g., 2 assets (A B), N2, 2(2-1)/21, one
  correlation term
• 3 Assets (A, B, C), N3, 3 different terms
• 4 Assets (A, B, C, D), N4, 6 different terms
• As number of assets increases in a portfolio, so
  does the number of correlation terms. The
  calculation of the variance of the portfolio
  becomes more difficult.
• Use of Matrix in Excel to simplify the
  calculation.

8
Return and Risk

• Matrix Basics
• Row matrix (vector) A(a1 a2)
• Column matrix (vector) B
• Square Matrix
• Number of rows equals number of columns
• Example of square matrix variance-covariance
  matrix

9
Return and Risk

• Matrix Basics
• Matrix transposition
• Change the rows (columns) in a matrix to columns
  (rows)
• e.g., A(1 2), its transpose AT
• B its transpose BT(0.8 0.2)
• exercise
• Transpose of a 3X2 matrix

10
Return and Risk

• Matrix Multiplication.
• Example
• A(0.6 0.4) B
• Here, A could represent the portfolio weights on
  two assets in a portfolio, and B could represent
  the returns on these two assets. AB gives
  return on this portfolio.
• When multiplying two matrices (AB), the number
  of columns in matrix A must be the same as the
  number of rows in matrix B.

11
Matrix in Excel

• Excel Functions
• Transposition
• TRANSPOSE()
• Multiplication
• MMult
• e.g., MMULT(A1C1, D1D3)
• Where A1C2 refers to a 1X3 matrix (row vector)
  and D1D3 refers to a 3X1 matrix (column vector).
  

12
Application of Matrix in Portfolio Optimization

• Portfolio return
• A row vector storing portfolio weights
• Multiply by
• A column vector storing portfolio return
• Or the transpose of a row vector storing
  portfolio return

13
Application of Matrix in Portfolio Optimization

• Portfolio Variance
• A row vector storing portfolio weights
• Multiply by
• The variance-covariance matrix
• Multiply by
• the transpose of the row vector storing portfolio
  weights

14
Portfolio Optimization

• Risk averse investors prefer higher return, lower
  risk.
• Optimal portfolio is the one that provides
  highest returns for a given level of risk
• In Excel, solver could help finding the optimal
  portfolio.