### 统计代写|应用时间序列分析代写applied time series analysis代考|Portfolio Return

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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|Portfolio Return

The simple net return of a portfolio consisting of $N$ assets is a weighted average of the simple net returns of the assets involved, where the weight on each asset is the percentage of the portfolio’s value invested in that asset. Let $p$ be a portfolio that places weight $w_{i}$ on asset $i$, then the simple return of $p$ at time $t$ is $R_{p, t}=$ $\sum_{i=1}^{N} w_{i} R_{i t}$, where $R_{i t}$ is the simple return of asset $i$.

The continuously compounded returns of a portfolio, however, do not have the above convenient property. If the simple returns $R_{i t}$ are all small in magnitude, then we have $r_{p, t} \approx \sum_{i=1}^{N} w_{i} r_{i t}$, where $r_{p, t}$ is the continuously compounded return of the portfolio at time $t$. This approximation is often used to study portfolio returns.

If an asset pays dividends periodically, we must modify the definitions of asset returns. Let $D_{t}$ be the dividend payment of an asset between dates $t-1$ and $t$ and $P_{t}$ be the price of the asset at the end of period $t$. Thus, dividend is not included in $P_{t}$. Then the simple net return and continuously compounded return at time $t$ become
$$R_{t}=\frac{P_{t}+D_{t}}{P_{t-1}}-1, \quad r_{t}=\ln \left(P_{t}+D_{t}\right)-\ln \left(P_{t-1}\right)$$

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

Excess return of an asset at time $t$ is the difference between the asset’s return and the return on some reference asset. The reference asset is often taken to be riskless, such as a short-term U.S. Treasury bill return. The simple excess return and log excess return of an asset are then defined as
$$Z_{t}=R_{t}-R_{0 t}, \quad z_{t}=r_{t}-r_{0 t}$$
where $R_{0 t}$ and $r_{0 t}$ are the simple and log returns of the reference asset, respectively. In the finance literature, the excess return is thought of as the payoff on an arbitrage portfolio that goes long in an asset and short in the reference asset with no net initial investment.

Remark: A long financial position means owning the asset. A short position involves selling asset one does not own. This is accomplished by borrowing the asset from an investor who has purchased. At some subsequent date, the short seller is obligated to buy exactly the same number of shares borrowed to pay back the lender.

Because the repayment requires equal shares rather than equal dollars, the short seller benefits from a decline in the price of the asset. If cash dividends are paid on the asset while a short position is maintained, these are paid to the buyer of the short sale. The short seller must also compensate the lender by matching the cash dividends from his own resources. In other words, the short seller is also obligated to pay cash dividends on the borrowed asset to the lender; see Cox and Rubinstein (1985).

## 统计代写|应用时间序列分析代写applied time series anakysis代考| Summary of Relationship

The relationships between simple return $R_{t}$ and continuously compounded (or log) return $r_{t}$ are
$$r_{t}=\ln \left(1+R_{t}\right), \quad R_{t}=e^{r_{t}}-1 .$$
Temporal aggregation of the returns produces
\begin{aligned} 1+R_{t}[k] &=\left(1+R_{t}\right)\left(1+R_{t-1}\right) \cdots\left(1+R_{t-k+1}\right), \ r_{t}[k] &=r_{t}+r_{t-1}+\cdots+r_{t-k+1} \end{aligned}
If the continuously compounded interest rate is $r$ per annum, then the relationship between present and future values of an asset is
$$A=C \exp (r \times n), \quad C=A \exp (-r \times n) .$$

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

R吨=磷吨+D吨磷吨−1−1,r吨=ln⁡(磷吨+D吨)−ln⁡(磷吨−1)

## 统计代写|应用时间序列分析代写applied time series anakysis代考| Summary of Relationship

r吨=ln⁡(1+R吨),R吨=和r吨−1.

1+R吨[ķ]=(1+R吨)(1+R吨−1)⋯(1+R吨−ķ+1), r吨[ķ]=r吨+r吨−1+⋯+r吨−ķ+1

## 广义线性模型代考

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## MATLAB代写

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