Assume the Efficient Market Hypothesis.


How to maximize returns while minimizing risk [=volatility =std. dev.]? → Since risk/return is proportional to each other, you choose one optimization goal.

  • Single Stock (See Dividend Discount Model)
  • Price of stock at time :
  • Return of stock at time :
  • Expected Return of stock :
  • Volatility (=Risk) of Stock :

Two-stock Portfolios

  • Covariance of stocks :
  • Correlation of stock : such that
  • Weights for
  • Portfolio Expected Return:
  • Portfolio Variance:
  • Minimum Variance Portfolio:

N-Stock Portfolio

  • Known information: are all known
  • Multiple Stocks (=Portfolio)
    • Position of portfolio at time (=amount invested):
    • Portfolio Return at time
    • Portfolio Expected Return at time :
    • Portfolio Variance:

def. Feasible Portfolios. Set of tuples given we have many assets weighted .

  • ← Short-selling is allowed
  • Lower the correlation [= the more negatively correlated] the better diversification is.
    • When two stocks have perfectly negative correlation , the efficient frontier touches the -axis (=)Untitled

Efficient Frontier

  • “Minimize the variance such that the sum of weights are and the portfolio return is (a constant)”
  • Key Assumption: neither the returns or risks are identical, and there is no perfect correlation
  • Minimum for given return level at:
  • Efficient Frontier:
  • …where are scalars: