not just few big events in stories, which is not helpful to understand the crisis, while the accumulation of many little events according to laws of probability are more important
some basic concepts like variance, etc., and idiosyncratic risk, systematic risk
interpretations of the crisis:
- failure of independence
- a tendency for outliers or fat-tailed distributions
1. Basic concept: Return（投资[净]回报[率]？）
- Return = $\frac{Price_{t+1} - Price_{t} + Dividend_{t}}{Price_{t}}$
- $Price_{t+1} - Price_{t}$: capital gain
- $Dividend_{t}$: 利息
- [Assumption] cannot lose money more than you put in
- [Range] $[-1, +\infty]$
- Gross Return = 1 + Return = $\frac{Price_{t+1} + Dividend_{t}}{P_{t}}$
- [Range] $[0, +\infty]$
2. Statistical concepts
2.1 Central tendency: Average, Mean
- Arithmetic mean = $\frac{\sum_{i=1}^{n}{x_i}}{n}$
- NOT useful for evaluate investment success
- "if the guy wipes you out, whatever else is done after that doesen't make sense"???
- Geometric mean = $({\prod_{i=1}{n}{x_i}}){\frac{1}{n}}$
- use
gross returnas the $x_i$ - make sense for evaluate (gross) return
- use
2.2 Risk: Variance, Standard Deviation
- $var(x) = \sum_{i=1}^{n}{P(x=x_{i}) * (x - \mu_{x})}$
- $s(x) = \frac{\sum_{i=1}^{n}{(x_i - \overline{x})^{2}}}{n}$
- s(x) is the estimate of var(x), or sample variance
- sometimes divided by
n-1, which is unbiased estimate(estimate for sample variance = population variance)
- $cov(x, y) = covar(x, y) = \frac{\sum_{i=1}^{n}{(x_i - \overline{x})(y_i - \overline{y})}}{n}$
- $cov(x, y) > 0$ when x and y move towards the same direction together, $cov(x, y) < 0$ when they move the opposite diretion together, $cov(x, y) = 0$ when they are unrelated
- $var(x+y) = var(x) + var(y) + 2cov(x, y)$
- correlation = $\rho = \frac{cov(x, y)}{s(x)s(y)}$
- [Range] $[-1, 1]$
Qustion:
Are the shocks that affected the markets independent, or somehow related?
2.3 Law of Large numbers(LLN)
3. Core concept of finance and insurance: Value of Risk(VaR)
"5% prob to lose $10 billion a year"
what brought us the crisis is the calculations are TOO OPTIMISTIC!(many people calculate the VaR together)
Return of some company = market returns + idiosyncratic returns
market returns = beta * (return of S&P 500), where beta can be gotton using least squares applied on scatter plots, whose axes are the S&P 500 and this company
4. Fat-tailed distribution, instead of normal distribution
people assumes that the price of stocks behave like normal distribution, while in fact sometimes the assumptions fails and it behave like the fat-tailed distribution, e.g. Cauchy distribution, where the probalbility is higher of being far out than it is in normal distribution
such situations really happened in history
5. RECAP
Independence leads to law of large numbers, and it leads to some sort of stability, either independence throught time or across stocks.
fat-tailed distribution matters, which will fool you so that you get big incredible shocks you thought couldn't happen.
