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Markov Chain
- State Transition Matrix
- Stationary distribution
- Properties:
- Classes: a set of states communicate with each other and do not communicate with the states outside the class
- Period: the greatest common divisor of $p_{ii}(n) > 0 $ period is a class property, all states should have same period
- Aperiodic: If the period of the state is one.
- A matrix P is stochastic if every row is a distribution, i.e. every row sums up to one
Queueing problem
- Arrival rate $\lambda$ per hour
- Service rate $\mu$ per hour
- Number of servers $s$
- Utilization rate $\rho = \lambda/s\mu$ the fraction of time we expect the service facility to be busy
- Steady state: The state N(t) of a queueing system is the number of customers in the system at time t. The system is said to be steady state if $P(N(t)=n)$ stops changing with t anymore. Only if $\rho < 1$ the system will have steady state.
MM1 Queueing problem
- Average queue length: $L_q$ = \frac{\rho^2}{1-\rho}
- Average waiting time in the queue: $W_q$ = L_q / \lambda$
EOQ
\( Q = \sqrt{2D\frac{C_o}{C_H}} \) D is the demand, $C_o$ is the fixed cost, $C_H$ is the holding cost
Newsvendor Problem
- Underage cost $C_u$: the profit could have earned with under-ordered, - profit
- Overage cost $C_o$: the money lost by over-ordered
- Optimal purchasing amount: F(Q) = \frac{C_u}{C_u + C_o}
- For example if a distribution of demand is Gaussian:
- \( \Phi(\frac{Q-\mu}{\sigma}) = \frac{C_u}{C_u + C_o} \)
Regression
- Correlation Coefficient r:
- Represent the linear relationship between two variables $X$ and $Y$, $-1\le r \le 1 $.
- If r close to 1, $X$ and $Y$ are strongly positively correlated, if close to -1, strongly negatively correlated.
- If close to 0, then they are weakly correlated or not correlated.
- \( r = \frac{\textrm{Cov}(X,Y)}{S_X, S_Y} \) where Covariance between X,Y and standard deviations are represented by $\textrm{Cov}(X,Y)$, $S_X, S_Y$ respectively
- R-square statistics:
- Proportion of variation in variable Y can be explained by the regression
- R is between 0 and 1
- If close 1, the regression line fits the data well.
- $R^2 = \frac{\textrm{RSS}}{\textrm{TSS}} = 1 - \frac{\textrm{ESS}}{\textrm{TSS}}$ where RSS: regression sum of squares ESS: error sum of squares TSS: total sum of squares
- Coefficient/slope for a single variable:
- Represents the change of the dependent variable Y with unit change in that single variable with other variables’ values fixed
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It holds that $r^2$ = $R^2$ and the slop $b = r \frac{S_Y}{S_X}$
- Assessing the performance of a regression problem:
- R-square statistics, the larger, the better fittness
- The standard error
- The t-statistics/p-statistics/confidence interval for each independent variable coefficient (slope). The larger the t-statistics, the more significant the regression model.
- We need to pay attention to multi-collinearity, which states that two independent variables are highly correlated and may give an incorrect impression that either of the two independent variables are not true drivers for the dependent variable.
- the error plot, the sign of the slope, sample size, other key drivers for the dependent variable
Forecasting
- Times Series Forecasting in the databook is Winter’s multiplicative exponential smoothing:
- Base $E_t$
- Trend $T_t$
- Seasonality $S_t$
- Prediction equation: \( F_{t+k} =(E_t + kT_t)S_{t+k-c} \)
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Difference between Winter’s multiplicative/additive exponential smoothing is that the division sign becomes the minus sign.
- For additive, the prediction equation is: \( F_{t+k} =E_t + kT_t + S_{t+k-c} \)
- When the seasonality factor is not considered, both of them becomes exponential smoothing method with trend
Portfolio Management
Market portfolio
- A portfolio is efficient if no other portfolio is better than this in both return and risk.
- The market portfolio is the one with the highest Sharp ratio.
- Sharp ratio is defined as: \( \frac{r - r_f}{\sigma - 0} = \frac{r - r_f}{\sigma} \) r is the return of the portfolio, $\sigma$ is the risk. $r_f$ is the risk free return, $\sigma_f = 0$ where risk is zero.
Capital Asset Pricing Model
- The capital market line relates the expected rate of return of an efficient portfolio to its standard deviation, but it does not show the expected rate of return of an individual asset relates to its individual risk.
- If the market portfolio M is efficient, then the expected rate of return $r_i$ of any asset i satisfies: \( r_i - r_f = \beta_i(r_m - r_f) \)
- This allows us to determine a theoretically appropriate required rate of return of an asset in order to make decisions about including the asset to a portfolio.
