week 7

• Risk


  Variability of return, if we are certain of the return there is no risk
• Risk attitudes


  • Risk loving (or seeking)
  • Risk neutral
  • Risk averse
• Rational people would want (or expected) to be rewarded for risk they take on
• High risk


  High return
• Low risk


  Low return
• Treasury bills


  No risk of default (safe heaven - as safe as an investment as one can make), Short maturity means they are relatively stable
• Government bonds


  Price fluctuates as interest rates vary, Bond prices fall as interest rates rise and rise as interest rates fall
• Common stocks


  An investor who shifts from bonds to common stocks share in ALL the ups and downs of the issuing companies
• Average annual real rate of return for treasury bills in the US 1900-2000 was 1.1%
• Average annual real rate of return for long-term government bonds in the US 1900-2000 was 2.4%
• Average annual real rate of return for common stocks in the US 1900-2000 was 8.5%
• Average annual standard deviation (variance) for treasury bills in the US 1900-2000 was 2.8% (7.8%)
• Average annual standard deviation (variance) for long-term government bonds in the US 1900-2000 was 8.1% (66.4%)
• Average annual standard deviation (variance) for common stocks in the US 1900-2000 was 19.8% (391.5%)
• Normal distribution


  A large enough sample drawn from a normal distribution looks like a 'bell-shaped' curve
• Approximately 68.26% of observations will fall within ±1 standard deviation of the mean in a normal distribution
• Approximately 95.44% of observations will fall within ±2 standard deviations of the mean in a normal distribution
• Approximately 99.74% of observations will fall within ±3 standard deviations of the mean in a normal distribution
• Mean (or expected value)


  A measure of the return associated with an investment
• Variance (s^2)


  A measure of the risk associated with an investment
• Given a normal distribution, the mean and variance provide good measures of the return and risk associated with any investment
• Holding a diversified portfolio reduces risk as measured by variance
• The expected return of a portfolio is the weighted average of the expected returns of the individual assets
• The variance of a portfolio depends on the variances of the individual assets and the covariances between them
• Diversification reduces portfolio risk because assets with low or negative covariances are included in the portfolio
• Portfolio


  Need to specify (i) mean, (ii) variance and (iii) covariance of the individual stocks
• It can be shown that risk, as measured by variance, is reduced by holding a diversified portfolio
• How and why risk is reduced by holding a diversified portfolio


  Let's look at the (basic) mathematics of the portfolio theory!
• Two asset portfolios


  Expected return on portfolio: E(Rp) = x1E(R1) + x2E(R2)

  Portfolio risk: σp^2 = x1^2σ1^2 + x2^2σ2^2 + 2x1x2cov(R1,R2)
• Variance vs. Covariance


  Variance and standard deviation measure the variability of individual stocks

  Covariance and correlation measure how two random variables are related
• If ρ12= 0 the portfolio risk is given by the sum of the asset variances only

  If ρ12= 1 there is no risk reduction effect from holding both assets

  If ρ12= -1 maximum risk reduction effect
• What is true for a two-assets portfolio is also true to a multiple asset portfolios: by investing in assets which have a correlation coefficient less than 1 the overall risk can be reduced
• Diversifiable (non-systematic) (firm-specific) risk


  Risk that can be reduced, if not eliminated, by increasing the number of investments in your portfolio (i.e., by being diversified)
• Non-diversifiable (systematic) (market) risk


  Risk that cannot be eliminated by diversification
• Combining stocks into portfolios can reduce standard deviation below the level obtained from a simple weighted average calculation
• By changing the proportion of funds invested in stocks, one can change the risk-return characteristics of a portfolio
• Efficient portfolios


  Provide the highest return for a given level of risk or least risk for given level of returns
• Efficient frontier


  The set of efficient portfolios that provide the highest return for a given level of risk
• Optimal portfolio with a risk-free asset


  The capital allocation line with the steepest slope (highest Sharpe ratio) is the optimal portfolio
• Sharpe ratio


  The ratio of reward-to-volatility