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# 金融代写|金融工程代写FINANCIAL ENGINEERING代写|FIN4520 Risk and Expected Return on a Portfolio

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## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Risk and Expected Return on a Portfolio

The expected return on a portfolio consisting of two securities can easily be expressed in terms of the weights and the expected returns on the components,
$$E\left(K_{V}\right)=w_{1} E\left(K_{1}\right)+w_{2} E\left(K_{2}\right) .$$
This follows at once from (5.2) by the additivity of mathematical expectation.
Example $5.5$
Consider three scenarios with the probabilities given below (a trinomial model).
Let the returns on two different stocks in these scenarios be as follows:
$\begin{array}{llcc}\text { Scenario } & \text { Probability } & \text { Return } K_{1} & \text { Return } K_{2} \ \omega_{1} \text { (recession) } & 0.2 & -10 \% & -30 \% \ \omega_{2} \text { (stagnation) } & 0.5 & 0 \% & 20 \% \ \omega_{3} \text { (boom) } & 0.3 & 10 \% & 50 \%\end{array}$
The expected returns on stock are
\begin{aligned} &E\left(K_{1}\right)=-0.2 \times 10 \%+0.5 \times 0 \%+0.3 \times 10 \%=1 \% \ &E\left(K_{2}\right)=-0.2 \times 30 \%+0.5 \times 20 \%+0.3 \times 50 \%=19 \% \end{aligned}

Suppose that $w_{1}=60 \%$ of available funds is invested in stock 1 and $40 \%$ in stock 2. The expected return on such a portfolio is
\begin{aligned} E\left(K_{V}\right) &=w_{1} E\left(K_{1}\right)+w_{2} E\left(K_{2}\right) \ &=0.6 \times 1 \%+0.4 \times 19 \%=8.2 \% . \end{aligned}

## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Risk and Expected Return on a Portfolio

A portfolio constructed from $n$ different securities can be described in terms of their weights
$$w_{i}=\frac{x_{i} S_{i}(0)}{V(0)}, \quad i=1, \ldots, n,$$
where $x_{i}$ is the number of shares of type $i$ in the portfolio, $S_{i}(0)$ is the initial price of security $i$, and $V(0)$ is the amount initially invested in the portfolio. It will prove convenient to arrange the weights into a one-row matrix
$$\boldsymbol{w}=\left[\begin{array}{llll} w_{1} & w_{2} & \cdots & w_{n} \end{array}\right] .$$
Just like for two securities, the weights add up to one, which can be written in matrix form as
$$1=\boldsymbol{u} \boldsymbol{w}^{T}$$
where
$$\boldsymbol{u}=\left[\begin{array}{llll} 1 & 1 & \cdots & 1 \end{array}\right]$$
is a one-row matrix with all $n$ entries equal to $1, \boldsymbol{w}^{T}$ is a one-column matrix, the transpose of $\boldsymbol{w}$, and the usual matrix multiplication rules apply. The attainable set consists of all portfolios with weights $\boldsymbol{w}$ satisfying (5.14), called the attainable portfolios.

Suppose that the returns on the securities are $K_{1}, \ldots, K_{n}$. The expected returns $\mu_{i}=E\left(K_{i}\right)$ for $i=1, \ldots, n$ will also be arranged into a one-row matrix
$$\boldsymbol{m}=\left[\begin{array}{llll} \mu_{1} & \mu_{2} & \cdots & \mu_{n} \end{array}\right] .$$

## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Risk and Expected Return on a Portfolio

$$E\left(K_{V}\right)=w_{1} E\left(K_{1}\right)+w_{2} E\left(K_{2}\right)$$

Scenario Probability Return $K_{1} \quad$ Return $K_{2} \omega_{1}$ (recession) $\quad 0.2 \quad-10 \% \quad-30 \% \omega_{2}\left(\right.$ stagnation) $\quad 0.5 \quad 0 \% \quad 20 \% \omega_{3}($ boom) $\quad 0.3 \quad 10 \%$

$E\left(K_{1}\right)=-0.2 \times 10 \%+0.5 \times 0 \%+0.3 \times 10 \%=1 \% \quad E\left(K_{2}\right)=-0.2 \times 30 \%+0.5 \times 20 \%+0.3 \times 50 \%=19 \%$

$$E\left(K_{V}\right)=w_{1} E\left(K_{1}\right)+w_{2} E\left(K_{2}\right) \quad=0.6 \times 1 \%+0.4 \times 19 \%=8.2 \% .$$

## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Risk and Expected Return on a Portfolio

$$w_{i}=\frac{x_{i} S_{i}(0)}{V(0)}, \quad i=1, \ldots, n$$

$$\mathrm{W}=\left[\begin{array}{llll} w_{1} & w_{2} & \cdots & w_{n} \end{array}\right]$$

$$1=u w^{T}$$

$$\mathrm{u}=\left[\begin{array}{llll} 1 & 1 & \cdots & 1 \end{array}\right]$$

$$\mathrm{m}=\left[\begin{array}{llll} \mu_{1} & \mu_{2} & \cdots & \mu_{n} \end{array}\right] \text {. }$$

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