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# 金融代写|金融工程代考Financial Engineering代写|CMSE11471 Value at Risk

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## 金融代写|金融工程代考Financial Engineering代写|Value at Risk

Let us present the basic idea using a simple example. We buy a share of stock for $S(0)=100$ dollars to sell it after one year. The selling price $S(1)$ is random. We shall suffer a loss if $S(1)<100 \mathrm{e}^{r}$, where $r$ is the risk-free rate under continuous compounding. (The purchase can either be financed by a loan, or, if the initial sum is already at our disposal, we take into account the foregone opportunity of a risk-free investment.) What is the probability of a loss being less than a given amount, for example,
$$P\left(100 \mathrm{e}^{r}-S(1)<20\right)=?$$
Let us reverse the question and fix the probability, $95 \%$ say. Now we seek an amount such that the probability of a loss not exceeding this amount is $95 \%$. This is referred to as Value at Risk at $95 \%$ confidence level and denoted by VaR. (Other confidence levels can also be used.) So, VaR is an amount such that
$$P\left(100 \mathrm{e}^{r}-S(1)<\mathrm{VaR}\right)=95 \% .$$
It should be noted that the majority of textbooks neglect the time value of money in this context, stating the definition of $\mathrm{VaR}$ only for $r=0$.

## 金融代写|金融工程代考Financial Engineering代写|Case Study

We shall discuss a number of ways in which VaR can be managed with the aid of derivative securities. The methods will be illustrated by a simple example of business activity.
Case $9.1$
A company manufactures goods in the UK for sale in the USA. The investment to start production is 5 million pounds. Additional funds can be raised by borrowing British pounds at $16 \%$ to finance a hedging strategy. The rate of return demanded by investors, bearing in mind the risk involved, is $25 \%$. The sales are predicted to generate 8 million dollars at the end of the year. The manufacturing costs are 3 million pounds per year. The interest rate is $8 \%$ for dollars and $11 \%$ for pounds. The current rate of exchange is $1.6$ dollars to a pound. The volatility of the logarithmic return on the rate of exchange is estimated at $15 \%$. The company pays $20 \%$ tax on earnings.

First note that to satisfy the expectations of investors the company should be able to achieve a profit of $1.25$ million pounds a year to pay the dividend. A lower profit would mean a loss. The profit depends on the rate of exchange $d$ at the end of the year, hence some risk emerges. (We assume that the other values will be as predicted.)
To begin with, suppose that no action is taken to manage the risk.

## 金融代写|金融工程代考FINANCIAL ENGINEERING代写|VALUE AT RISK

Thepurchasecaneitherbefinancedbyaloan, or, iftheinitialsumisalreadyatourdisposal, wetakeintoaccountthe foregoneopportunityofarisk – freein

$P\left(100 \mathrm{e}^{r}-S(1)<20\right)=?$

Otherconfidencelevelscanalsobeused.因此，VaR 是一个数量，使得
$P\left(100 \mathrm{e}^{r}-S(1)<\mathrm{VaR}\right)=95 \% .$

## MATLAB代写

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