### 金融代写|利率建模代写Interest Rate Modeling代考|INTEREST RATE RISK

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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

## 金融代写|利率建模代写Interest Rate Modeling代考|Interest Rate and Discount Factor

This section introduces some fundamentals of interest rate theory, including discount factors, bond pricing, yield, forward rates, swap rates, and so on.
Discount factor
Let $r$ denote a simple interest rate for a period $[0, T]$. If we invest $1 \mathrm{unit}$ of currency today, this investment will be worth $1+r T$ at time $T$. Thus, a present value of 1 is economically equivalent to a value of $1+r T$ at time $T$. In other words, today’s $1 /(1+r T)$ has a future value of 1 at time $T$. Accordingly a present value of
$$\frac{C}{1+r T}$$
is equivalent to a future value of $C$ at time $T$.
The factor $1 /(1+r T)$ is called the discount factor, and $r$ is called the discount rate. The discount factor to time $T$ is denoted by $D(T)$, hence
$$D(T)=\frac{1}{1+r T}$$
Typically, the present value of a cash flow $C$ at time $T$ is defined as the result of discounting back the cash flow as $D(T) C$.

Next, let $r$ denote an annually compounded interest rate. If we repeatedly invest 1 at this rate $r$ for $n$ years, then the value of the investment would be $(1+r)^{n}$ at time $n$. The discount factor for $n$ years is thus formulated as
$$D(n)=\frac{1}{(1+r)^{n}}$$
Similarly, when $r$ denotes the semi-annually compounded interest rate, the discount factor for $n$ years is
$$D(n)=\frac{1}{(1+r / 2)^{2 n}}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Swap Rate and Forward LIBOR

This section introduces the fundamentals of the London Interbank Offered Rate (LIBOR) and swap rates, which are the most important interest rates in derivatives trading.
Interest rate swap
The most common example of a swap contract is a fixed-for-floating rate swap in a single currency. The floating rate is usually determined from the LIBOR, and the fixed rate is referred to as the swap rate. For example in a semi-annual swap, the six-month LIBOR rate is converted to a fixed rate.

Let $T_{i}, i=1, \cdots, n$ be a sequence of future days, and let $T_{n}$ be the maturity date of the swap. For simplicity, we assume that the notional principal is equal to 1 . In the swap contract, the payer of the fixed rate pays the swap rate and receives the forward LIBOR at every $T_{i}$. The receiver of the fixed rate pays the forward LIBOR and receives the swap rate at every $T_{i}$.

We denote by $S(n)$ a swap rate with maturity $T_{n}$, and by $L(i)$ a forward LIBOR for the period $\left[T_{i}, T_{i+1}\right] . D(i)$ denotes the discount factor to time $T_{i}$. For simplicity, we assume that each interval of the period $\left[T_{i}, T_{i+1}\right]$ is authorthe same length, $\delta>0$. That is, $T_{i+1}-T_{i}=\delta$ for all $i<n$. Note that the floating payment $\delta L(i)$ takes place at time $T_{i+1}$, and so its present value is given by $L(i) D(i+1)$. Since the present value of the fixed cash flows should be equal to that of the floating cash flows, we obtain
$$S(n) \sum_{i=0}^{n-1} \delta D(i+1)=\sum_{i=0}^{n-1} \delta D(i+1) L(i)$$
From (1.12), it holds that
$$\delta D(i+1) L(i)=D(i)-D(i+1)$$
Taking the sum of $(1.16)$ over $i=0, \cdots, n-1$, the right side of $(1.15)$ becomes
\begin{aligned} \delta \sum_{i=0}^{n-1} D(i+1) L(i) &=\sum_{i=0}^{n-1} D(i)-D(i+1) \ &=1-D(n) \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|Term Structure of Interest Rates

The yield to maturity of bonds can be observed in the bond market. The yield curve is a graph that plots time to maturity on the horizontal axis and the yield to maturity on the vertical axis, which visually exhibits the term structure of interest rates. The swap rate curve and the forward LIBOR curve are similarly exhibited.

For example, from (1.15) the swap rate $S(n)$ is represented by a series of forward LIBORs $L(i), i=1, \cdots, n-1$ as
$$S(n)=\frac{\sum_{i=0}^{n-1} \delta D(i+1) L(i)}{\sum_{i=0}^{n-1} \delta D(i+1)} .$$

This means that the swap rate consists of forward LIBORs $L(i)$ for $i<n$. This property is the same when considering the yield to maturity.

In contrast, the forward rate is the obtained interest rate of the corresponding future period. Therefore, the forward rate curve shows us the term structure of interest rates more clearly than the yield curve and the swap rate curve can. For this reason, the forward rate curve is often used to analyze the interest rate market. In particular, the forward rates derived from a term structure of interest rates are called the implied forward rate. For example,the “implied” forward rate can be practically observed in a government bond market or a swap market.

## 金融代写|利率建模代写Interest Rate Modeling代考|Interest Rate and Discount Factor

Letr表示一个时期的简单利率[0,吨]. 如果我们投资1在n一世吨今天的货币，这项投资将是值得的1+r吨有时吨. 因此，现值 1 在经济上等价于1+r吨有时吨. 换句话说，今天1/(1+r吨)未来值为 1吨. 因此，现值

C1+r吨

D(吨)=11+r吨

D(n)=1(1+r)n

D(n)=1(1+r/2)2n

## 金融代写|利率建模代写Interest Rate Modeling代考|Swap Rate and Forward LIBOR

dD(一世+1)大号(一世)=D(一世)−D(一世+1)

d∑一世=0n−1D(一世+1)大号(一世)=∑一世=0n−1D(一世)−D(一世+1) =1−D(n)

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