[FRM] Statistical and Probability Foundations

2 분 소요

CH4_Statistical and probability foundations

One-day absolute price change: $D_t = P_t - P_{t-1}$
One-day relative return: $R_t = \frac{P_t - P_{t-1}}{P_{t-1}}$
One-day 로그수익률(or continuously-compounded return): $r_t = ln(1+R_t) = ln(\frac{P_t}{P_{t-1}}) = ln(P_t) - ln(P_{t-1}) = p_t - p_{t-1}$
Multiple day 수익률(시차 = k): $R_t(k) = \frac{P_t-P_{t-k}}{P_{t-k}}$
Multiple day gross return: $r_t(k) = ln[1+R_t(k)] = ln[(1+R_t)\cdot{}(1+R_{t-1})\cdot(1+R_{t-k-1}) = r_t + r_{t-1} + \cdots{} + r_{t-k-1}$
Percent and continuous compounding in aggregating returns

어떤 포트폴리오의 최초 평가가격이 $P_0$ 일때, 해당 포트폴리오의 다음 period 후의 평가가격은
- \[P_1 = w_1\cdot P_0\cdot e^{r_1}+w_2\cdot P_0 \cdot e^{r_2} + w_3\cdot P_0\cdot e^{r_3}\]
- \[r_p = ln(w_1\cdot{}P_0\cdot{}e^{r_1}+w_2\cdot{}P_0\cdot{}e^{r_2} + w_3\cdot{}P_0\cdot{}e^{r_3})\]
- \[P_1 = W_1\cdot{}P_0\cdot{}(1+r_1)+W_2\cdot{}P_0\cdot{}(1+r_2)+W_3\cdot{}P_0\cdot{}(1+r_3)\]
- 포트폴리오 수익률: $R_p = W_1\cdot{}r_1 + W_2\cdot{}r_2 + W_3\cdot{}r_3 = \frac{P_1 - P_0}{P_0}$

Random walk for single-price
- 어떤 자산의 $t$시점의 자산가격은 $P_t = \mu + P_{t-1} + \sigma \epsilon_t$, $\epsilon_t$ ~ $N(0,1)$
- 로그수익률은 $ln(P_t) = \mu + ln(P_{t-1}) + \sigma\epsilon_t$
- \[P_t = P_{t-1}exp(\mu+\sigma\epsilon_t)\]
Random walk for fixed-income
- 주식과는 다르게 채권은 가격(price)와 이자율(yield)가 있기 때문에, 이자율로 random walk를 계산
- \[y_t = \mu + y_{t-1} + \sigma\epsilon_t\]
Time dependent properties of the random walk model
- Homoskedastic, 평균과 분산이 일정(constant): $P_t = \mu + c\cdot{}P_{t-1}+\epsilon_t$
- Heteroskedastic, 평균과 분산이 변동(non-stationary): $P_t = \mu_t + P_{t-1} + \epsilon_t$
  - \[E_0[P_t|P_0] = P_t + \mu t\]
  - \[V_0[P_t|P_0] = \sigma^2 t\]

Autocorrelation
- Standard Correlation of $x,y$: $\rho_{xy} = \frac{\sigma^2_{xy}}{\sigma_x\sigma_y}$, $(\sigma^2_{xy} = E[(X-\mu_x)(Y-\mu_y)])$
- $k$th 수익률의 autocorrelation: $\rho_k = \frac{\sigma^2_{t,t-k}}{\sigma_t\sigma_{t-k}} = \frac{\sigma^2_{t,t-k}}{\sigma^2_t}$일때 sampling 수익률 $r_t,t=1, \cdots{},T$라면 autocorrelation의 추정값
  \[\hat{\rho}_k = \frac{\Sigma_{t=k+1}^T{(r_t-\bar{r})(r_{t-k}-\bar{r})/[t-(k-1)]}}{\Sigma_{t=1}^T(r_t-\bar{r})^2/(T-1)}\]
Box-Ljung Statistic for daily log price changes: autocorrelation 검증을 위한 통계량
- $H_0:$ time series is not autocorrelation
- \[BL(P) = T\cdot{}(T+2)\Sigma_{k=1}^P\frac{\rho_k^2}{T-k}\]
- 그리고, Box-Ljung 통계량은 카이제곱 분포를 따른다.
Multivariate extensions
- 두 자산 수익률 $r_{1t}, r_{2t}$에 대해 $\sigma^2_{12t} = E\{[r_{1t}-E(r_{1t})][r_{2t}-E(r_{2t})]\} = E(r_{1t}r_{2t})- E(r_{1t})E(r_{2t})$

수익률이 정규분포를 따를때
- PDF: $f(r_t) = \frac{1}{\sqrt{2\pi\sigma^2}}exp-(\frac{1}{2\sigma^2})(r_t-\mu)^2$
- Skewness: $s^3 = E[(r_t-\mu)^3]$
- Skewness coefficient: $r = \frac{E[(r_t-\mu)^3]}{\sigma^3}$
- Kurtosis(뾰족성): $s^4 = E[(r_t-\mu)^4]$
- Kurtosis coefficient: $k = E[(r_t-\mu)^4]/\sigma^4$
Percentiles for measuring market risk
- standardized return: $\tilde{r}_t = \frac{r_t-\mu_t}{\sigma_t}$
- \[Pr(\tilde{r}_t < -1.65)=Pr((r_t-\mu_t)/\sigma_t < -1.65) = Pr(r_t < -1.65\sigma_t+\mu_t)=5\%\]
Aggregation in normal model

포트폴리오 수익률 $r_{pt} = w_1r_{1t} + w_2r_{2t} + w_3r_{3t}$에 대해

$r_{1t} = \mu_1+\sigma_1 + \epsilon_{1t},r_{2t} = \mu_2+\sigma_2 + \epsilon_{2t},r_{3t} = \mu_3+\sigma_3 + \epsilon_{3t}$라면
\[\begin{bmatrix} \epsilon_{1t}\\ \epsilon_{2t}\\\epsilon_{3t}\\\end{bmatrix} = MVN \begin{pmatrix}\begin{bmatrix} 0\\0\\0\\\end{bmatrix},\begin{bmatrix}1 & \rho_{12t}&\rho_{13t}\\ \rho_{21t}&1&\rho_{23t}\\ \rho_{31t}&\rho{32t}&1\end{bmatrix}\end{pmatrix} = (\mu_t,R_t)\]
$r_t$는 오차항의 correlation matrix, $\mu_{pt} = w_1\mu_1 + w_2\mu_2 + w_3\mu_3,$
\[\sigma^2_{pt} = w_1^2 \sigma_{1t}^2 + w_2^2\sigma_{2t}^2 + w_3^2\sigma_{3t}^2 + 2w_1w_3\sigma^2_{13t} + 2w_2w_3\sigma^2_{23t}\]