### 统计代写|金融统计代写Mathematics with Statistics for Finance代考|Some Basic Math

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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|金融统计代写Mathematics with Statistics for Finance代考|LOGARITHMS

In mathematics, logarithms, or logs, are related to exponents, as follows:
$$\log {b} a=x \Leftrightarrow a=b^{x}$$ We say, “The log of $a$, base $b$, equals $x$, which implies that $a$ equals $b$ to the $x$ and vice versa.” If we take the log of the right-hand side of Equation $1.1$ and use the identity from the left-hand side of the equation, we can show that: $$\log {b}\left(b^{x}\right)=x$$
Taking the log of $b^{x}$ effectively cancels out the exponentiation, leaving us with $x$.

An important property of logarithms is that the logarithm of the product of two variables is equal to the sum of the logarithms of those two variables. For two variables, $X$ and $Y$ :
$$\log {b}(X Y)=\log {b} X+\log {b} Y$$ Similarly, the logarithm of the ratio of two variables is equal to the difference of their logarithms: $$\log {b}\left(\frac{X}{Y}\right)=\log {b} X-\log {b} Y$$

If we replace $Y$ with $X$ in Equation 1.3, we get:
$$\log {b}\left(X^{2}\right)=2 \log {b} X$$
We can generalize this result to get the following power rule:
$$\log {b}\left(X^{n}\right)=n \log {b} X$$
In general, the base of the logarithm, $b$, can have any value. Base 10 and base 2 are popular bases in certain fields, but in many fields, and especially in finance, $e$, Euler’s number, is by far the most popular. Base $e$ is so popular that mathematicians have given it its own name and notation. When the base of a logarithm is $e$, we refer to it as a natural logarithm. In formulas, we write:
$$\ln (a)=x \Leftrightarrow a=e^{x}$$
From this point on, unless noted otherwise, assume that any mention of logarithms refers to natural logarithms.

Logarithms are defined for all real numbers greater than or equal to zero. Figure $1.1$ shows a plot of the logarithm function. The logarithm of zero is negative infinity, and the logarithm of one is zero. The function grows without bound; that is, as $X$ approaches infinity, the $\ln (X)$ approaches infinity as well.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|LOG RETURNS

One of the most common applications of logarithms in finance is computing log returns. Log returns are defined as follows:
$$r_{t} \equiv \ln \left(1+R_{t}\right) \quad \text { where } \quad R_{t}=\frac{P_{t}-P_{t-1}}{P_{t-1}}$$
Here $r_{t}$ is the log return at time $t, R_{t}$ is the standard or simple return, and $P_{t}$ is the price of the security at time $t$. We use this convention of capital $R$ for simple returns and lowercase $r$ for log returns throughout the rest of the book. This convention is popular, but by no means universal. Also, be careful: Despite the name, the $\log$ return is not the $\log$ of $R_{t}$, but the $\log$ of $\left(1+R_{t}\right)$.

For small values, log returns and simple returns will be very close in size. A simple return of $0 \%$ translates exactly to a log return of $0 \%$. A simple return of $10 \%$ translates to a log return of $9.53 \%$. That the values are so close is convenient for checking data and preventing operational errors. Table $1.1$ shows some additional simple returns along with their corresponding log returns.As long as $R$ is small, the second term on the right-hand side of Equation $1.9$ will be negligible, and the log return and the simple return will have very similar values.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|COMPOUNDING

Log returns might seem more complex than simple returns, but they have a number of advantages over simple returns in financial applications. One of the most useful features of log returns has to do with compounding returns. To get the return of a security for two periods using simple returns, we have to do something that is not very intuitive, namely adding one to each of the returns, multiplying, and then subtracting one:
$$R_{2, t}=\frac{P_{t}-P_{t-2}}{P_{t-2}}=\left(1+R_{1, t}\right)\left(1+R_{1, t-1}\right)-1$$
Here the first subscript on $R$ denotes the length of the return, and the second subscript is the traditional time subscript. With log returns, calculating multiperiod returns is much simpler; we simply add:
$$r_{2, t}=r_{1, t}+r_{1, t-1}$$
By substituting Equation $1.8$ into Equation $1.10$ and Equation 1.11, you can see that these are equivalent. It is also fairly straightforward to generalize this notation to any return length.

ln⁡(一种)=X⇔一种=和X

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|LOG RETURNS

r吨≡ln⁡(1+R吨) 在哪里 R吨=磷吨−磷吨−1磷吨−1

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|COMPOUNDING

R2,吨=磷吨−磷吨−2磷吨−2=(1+R1,吨)(1+R1,吨−1)−1

r2,吨=r1,吨+r1,吨−1

## 广义线性模型代考

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## MATLAB代写

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