Chapter 1 : Markowitz, Mean-Variance Portfolio Theory

There is a risk that we measure financial risks wrong.

Expected return Portfolio E(Rp) :

Arithmetic E(Rp) = Σ(Probability * return) = 1/N * Σ(return)

Geometric E(Rp) : Terminal value = (1+R1)*(1+R2)*.....*(1+RN) gE(Rp) = Terminal value ^(1/N) – 1 or gE(Rp) = Arithmetic E(Rp) – ½ * Variance of the distribution

Remark : the difference arises from the asymetric effect of positive and negative rates of returns on the terminal value of the portfolio. The larger the swings in rates of return, the greater the discrepancy between the arithmetic and geometric averages.

Variance (population) = Σ ( P(s) * [ R(s) – E(Rp) ] ^2 )

Historical data with n observations : Variance (sample) = 1/(N-1) * Σ [R(s) - Sample Arithmetic Avg]^2

Standard deviation = square root of Variance

Risk premium = E(Rportfolio) - Rf

Standard deviation of excess return = SD(portfolio)

Sharpe ratio portfolio = Risk premium / Standard Deviation of excess retrun

Holding Period Returns (HPR) = (Ending price of a share – Beginning price + Cash dividend)/Beginning price

Excess return = HPR – Risk-Free Return (RFR)

Forward scenario analysis : determine a set of relevant scenarios and associated investment outcome, assign probabilities, and compute risk premium and standard deviation.

Time series : we must find from past HPRs of portfolios and assets, the probability distribution, or some characteristics of it such as expected return and standard deviation. Will a security or a portfolio repeat the same performance than the previous year?

The normal distribution : used to price many option contracts. M +- 1SD = 68,26%

M +- 2SD = 95,44% M +- 3SD = 99,74%

If the normal distribution does not apply use an event tree with the probabilities and the different possible outcomes.

Excel : NORMDIST(cutoff; mean; standard deviation; TRUE) = NORMSDIST (-Mean / Standard deviation) cutoff can be for example, the probability of an outcome below 0 : cutoff = 0.

Potential deviations from normality :

1. Skew (measure of asymetry) = E[R(s) – E(Rp)]^3 / SD^3

Normal distribution : skew = 0 : mass under both tails are equal

Positively skewed : more mass in the right tail

Negatively skewed distribution: more mass in the left tail

1. Kurtosis (measure of the degree of fat tails) = E[R(s) – E(Rp)]^4 / SD^4 -3

When the tails of a distribution are fat, there is more probability mass in the tails of the distribution than predicted by the normal distribution, and so less probability mass near the center of the distribution.SD will underestimate the likelihood of extreme events.

US small and large stocks have fat tails, long term Treasury bonds have a little of mass under tails, T-bills is almost like a normal distribution.

To be able to use normal distribution, you need to have a large portfolio with multiple data. Otherwise, use the binomial tree (event tree).

Chapter 2 : Risk tolerance and Asset Allocation

Unsystematic risk : risk that can be eliminated through diversification. Also called unique risk, or diversifiable risk, firm-specific risk, or indiosyncratic risk.

Systematic risk : the risk that remains after diversification. Also called market risk, or nondiversifiable risk.

Portfolio risk falls rapidly as the number of stocks included in the portfolio increases. We estimate that a portfolio must contain around 20 – 25 securities to eliminate the unsystematic risk.

Efficient diversification : risky portfolios to provide the lowest possible risk for any given level of expected return (or vice versa).

Expected return of a portfolio : E(Rp) = W(debt) * E(Rdebt) + W(equity) * E(Requity)

Variance of a portfolio = W(debt)^2 * Var(debt) + W(equity)^2 * Var(equity) + 2 * W(debt) * W(equity) * Cov(R(d),R(e))

=> Cov(R(s),R(s)) = Var(s) so possibility to rewrite the Var formula with Cov

=> Cov(R(debt),R(equity)) = P(debt,equity) *