### 统计代写|应用时间序列分析代写applied time series analysis代考|Financial Time Series and Their Characteristics

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|应用时间序列分析代写applied time series anakysis代考|ASSET RETURNS

Most financial studies involve returns, instead of prices, of assets. Campbell, Lo, and MacKinlay (1997) give two main reasons for using returns. First, for average investors, return of an asset is a complete and scale-free summary of the investment opportunity. Second, return series are easier to handle than price series because the former have more attractive statistical properties. There are, however, several definitions of an asset return.

Let $P_{t}$ be the price of an asset at time index $t$. We discuss some definitions of returns that are used throughout the book. Assume for the moment that the asset pays no dividends.
One-Period Simple Return
Holding the asset for one period from date $t-1$ to date $t$ would result in a simple gross return
$$1+R_{t}=\frac{P_{t}}{P_{t-1}} \quad \text { or } \quad P_{t}=P_{t-1}\left(1+R_{t}\right)$$
The corresponding one-period simple net return or simple return is
$$R_{t}=\frac{P_{t}}{P_{t-1}}-1=\frac{P_{t}-P_{t-1}}{P_{t-1}}$$

## 统计代写|应用时间序列分析代写applied time series anakysis代考| Multiperiod Simple Return

Holding the asset for $k$ periods between dates $t-k$ and $t$ gives a $k$-period simple gross return
\begin{aligned} 1+R_{t}[k]=\frac{P_{t}}{P_{t-k}} &=\frac{P_{t}}{P_{t-1}} \times \frac{P_{t-1}}{P_{t-2}} \times \cdots \times \frac{P_{t-k+1}}{P_{t-k}} \ &=\left(1+R_{t}\right)\left(1+R_{t-1}\right) \cdots\left(1+R_{t-k+1}\right) \ &=\prod_{j=0}^{k-1}\left(1+R_{t-j}\right) \end{aligned}
Thus, the $k$-period simple gross return is just the product of the $k$ one-period simple gross returns involved. This is called a compound return. The $k$-period simple net return is $R_{t}[k]=\left(P_{t}-P_{t-k}\right) / P_{t-k}$ –

In practice, the actual time interval is important in discussing and comparing returns (e.g., monthly return or annual return). If the time interval is not given, then it is implicitly assumed to be one year. If the asset was held for $k$ years, then the annualized (average) return is defined as
$$\text { Annualized }\left{R_{t}[k]\right}=\left[\prod_{j=0}^{k-1}\left(1+R_{t-j}\right)\right]^{1 / k}-1 \text {. }$$
This is a geometric mean of the $k$ one-period simple gross returns involved and can be computed by
$$\text { Annualized }\left{R_{t}[k]\right}=\exp \left[\frac{1}{k} \sum_{j=0}^{k-1} \ln \left(1+R_{t-j}\right)\right]-1 \text {, }$$
where $\exp (x)$ denotes the exponential function and $\ln (x)$ is the natural logarithm of the positive number $x$. Because it is easier to compute arithmetic average than geometric mean and the one-period returns tend to be small, one can use a first-order Taylor expansion to approximate the annualized return and obtain
$$\text { Annualized }\left{R_{t}[k]\right} \approx \frac{1}{k} \sum_{j=0}^{k-1} R_{t-j}$$
Accuracy of the approximation in Eq. (1.3) may not be sufficient in some applications, however.

## 统计代写|应用时间序列分析代写applied time series anakysis代考| Continuous Compounding

Before introducing continuously compounded return, we discuss the effect of compounding. Assume that the interest rate of a bank deposit is $10 \%$ per annum and the initial deposit is $\$ 1.00$. If the bank pays interest once a year, then the net value of the deposit becomes$\$1(1+0.1)=\$ 1.1$one year later. If the bank pays interest semi-annually, the 6 -month interest rate is$10 \% / 2=5 \%$and the net value is$\$1(1+0.1 / 2)^{2}=\$ 1.1025$after the first year. In general, if the bank pays interest$m$times a year, then the interest rate for each payment is$10 \% / \mathrm{m}$and the net value of the deposit becomes$\$1(1+0.1 / m)^{m}$ one year later. Table $1.1$ gives the results for some commonly used time intervals on a deposit of $\$ 1.00$with interest rate$10 \%$per annum. In particular, the net value approaches$\$1.1052$, which is obtained by $\exp (0.1)$ and referred to as the result of continuous compounding. The effect of compounding is clearly seen.
In general, the net asset value $A$ of continuous compounding is
$$A=C \exp (r \times n)$$
where $r$ is the interest rate per annum, $C$ is the initial capital, and $n$ is the number of years. From Eq. (1.4), we have
$$C=A \exp (-r \times n)$$
which is referred to as the present value of an asset that is worth $A$ dollars $n$ years from now, assuming that the continuously compounded interest rate is $r$ per annum.
Continuously Compounded Return
The natural logarithm of the simple gross return of an asset is called the continuously compounded return or $\log$ return:
$$r_{t}=\ln \left(1+R_{t}\right)=\ln \frac{P_{t}}{P_{t-1}}=p_{t}-p_{t-1}$$
where $p_{t}=\ln \left(P_{t}\right)$. Continuously compounded returns $r_{t}$ enjoy some advantages over the simple net returns $R_{t}$. First, consider multiperiod returns. We have

\begin{aligned} r_{t}[k] &=\ln \left(1+R_{t}[k]\right)=\ln \left[\left(1+R_{t}\right)\left(1+R_{t-1}\right) \cdots\left(1+R_{t-k+1}\right)\right] \ &=\ln \left(1+R_{t}\right)+\ln \left(1+R_{t-1}\right)+\cdots+\ln \left(1+R_{t-k+1}\right) \ &=r_{t}+r_{t-1}+\cdots+r_{t-k+1} . \end{aligned}
Thus, the continuously compounded multiperiod return is simply the sum of continuously compounded one-period returns involved. Second, statistical properties of log returns are more tractable.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|ASSET RETURNS

1+R吨=磷吨磷吨−1 或者 磷吨=磷吨−1(1+R吨)

R吨=磷吨磷吨−1−1=磷吨−磷吨−1磷吨−1

## 统计代写|应用时间序列分析代写applied time series anakysis代考| Multiperiod Simple Return

1+R吨[ķ]=磷吨磷吨−ķ=磷吨磷吨−1×磷吨−1磷吨−2×⋯×磷吨−ķ+1磷吨−ķ =(1+R吨)(1+R吨−1)⋯(1+R吨−ķ+1) =∏j=0ķ−1(1+R吨−j)

\text { 年化 }\left{R_{t}[k]\right}=\left[\prod_{j=0}^{k-1}\left(1+R_{tj}\right)\right] ^{1 / k}-1 \文本{。}\text { 年化 }\left{R_{t}[k]\right}=\left[\prod_{j=0}^{k-1}\left(1+R_{tj}\right)\right] ^{1 / k}-1 \文本{。}

\text { 年化 }\left{R_{t}[k]\right}=\exp \left[\frac{1}{k} \sum_{j=0}^{k-1} \ln \left( 1+R_{tj}\right)\right]-1 \text {, }\text { 年化 }\left{R_{t}[k]\right}=\exp \left[\frac{1}{k} \sum_{j=0}^{k-1} \ln \left( 1+R_{tj}\right)\right]-1 \text {, }

\text { 年化 }\left{R_{t}[k]\right} \approx \frac{1}{k} \sum_{j=0}^{k-1} R_{tj}\text { 年化 }\left{R_{t}[k]\right} \approx \frac{1}{k} \sum_{j=0}^{k-1} R_{tj}

## 统计代写|应用时间序列分析代写applied time series anakysis代考| Continuous Compounding

C=一种经验⁡(−r×n)

r吨=ln⁡(1+R吨)=ln⁡磷吨磷吨−1=p吨−p吨−1

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## MATLAB代写

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