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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

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|PRESENT VALUES OF CASH FLOWS

In many compound interest problems, one is required to find the discounted present value of cash payments (or, as they are often called, cash flows) due in the future. It is important to distinguish between discrete and continuous payments.
Discrete Cash Flows
The present value of the sums $C_{t_1}, C_{t_2}, \ldots, C_{t_n}$ due at times $t_1, t_2, \ldots, t_n$ (where $0 \leq t_1<t_2<\ldots<t_n$ ) is, by Eq. 2.5.4,
$$c_{t_1} v\left(t_1\right)+c_{t_2} v\left(t_2\right)+\cdots+c_{t_n} v\left(t_n\right)=\sum_{j=1}^n c_{t_j} v\left(t_j\right)$$
If the number of payments is infinite, the present value is defined to be
$$\sum_{j=1}^{\infty} c_{t_j} \nu\left(t_j\right)$$
provided that this series converges, which it usually will, in practical problems.
The process of finding discounted present values may be illustrated as in Figure 2.6.1. The discounting factors $v\left(t_1\right), v\left(t_2\right)$, and $v\left(t_3\right)$ are applied to bring each cash payment “back to the present time”.

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|Continuously Payable Cash Flows

The concept of a continuously payable cash flow, although essentially theoretical, is important. For example, for many practical purposes, a pension that is payable weekly may be considered as payable continuously over an extended time period. Suppose that $T>0$ and that between times 0 and $T$ an investor will be paid money continuously, the rate of payment at time $t$ being $£ \rho(t)$ per unit time. What is the present value of this cash flow?

In order to answer this question, one needs to understand what is meant by the rate of payment of the cash flow at time $t$. If $M(t)$ denotes the total payment made between time 0 and time $t$, then, by definition,
$$\rho(t)=M^{\prime}(t) \quad \text { for all } t$$
where the prime denotes differentiation with respect to time. Then, if $0 \leq \alpha<\beta$ $\leq T$, the total payment received between time $\alpha$ and time $\beta$ is
\begin{aligned} M(\beta)-M(\alpha) &=\int_\alpha^\beta M^{\prime}(t) \mathrm{d} t \ &=\int_\alpha^\beta \rho(t) \mathrm{d} t \end{aligned}
The rate of payment at any time is therefore simply the derivative of the total amount paid up to that time, and the total amount paid between any two times is the integral of the rate of payment over the appropriate time interval.

Between times $t$ and $t+d t$, the total payment received is $M(t+d t)-M(t)$. If $d t$ is very small, this is approximately $M^{\prime}(t) d t$ or $\rho(t) d t$. Theoretically, therefore, we may consider the present value of the money received between times $t$ and $t+d t$ as $v(t) \rho(t) \mathrm{d} t$. The present value of the entire cash flow is then obtained by integration as
$$\int_0^T v(t) \rho(t) \mathrm{d} t$$
A rigorous proof of this result is given in textbooks on elementary analysis but is not necessary here; $\rho(t)$ will be assumed to satisfy an appropriate condition (e.g. that it is piecewise continuous).

# 金融数学代考

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|PRESENT VALUES OF CASH FLOWS

$$c_{t_1} v\left(t_1\right)+c_{t_2} v\left(t_2\right)+\cdots+c_{t_n} v\left(t_n\right)=\sum_{j=1}^n c_{t_j} v\left(t_j\right)$$

$$\sum_{j=1}^{\infty} c_{t_j} \nu\left(t_j\right)$$

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|Continuously Payable Cash Flows

$$\rho(t)=M^{\prime}(t) \quad \text { for all } t$$

$$M(\beta)-M(\alpha)=\int_\alpha^\beta M^{\prime}(t) \mathrm{d} t \quad=\int_\alpha^\beta \rho(t) \mathrm{d} t$$

$$\int_0^T v(t) \rho(t) \mathrm{d} t$$

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