Chapter 7, Problem 29PS

Summary Introduction

(a)

To calculate:

The expected profit and standard deviation where ten stocks have an positive alpha of$3\%$and ten have negative alpha of$-3\%$when 20 stocks data studied by analyst and beta of all the stock in the economy is$1$. Firms have a standard deviation of$30\%$.

Introduction:

Expected return: It is the gain or loss to an investor which is expected on an investment and known as expected rate of return.

Standard deviation: It is historical volatility. It's applied to the annual rate of return to quantify the investment volatility.

Answer to Problem 29PS

Expected profit is$\$60,000$

Standard deviation of return is$\$1,34,164.08$

Explanation of Solution

Given:

Beta of all stock is$1$

Alpha of$10$stock is$3\%$

Alpha of$10$stock is$-3\%$

Investors invest$\$1$million for buying and selling positive and negative alpha stock respectively.

  $\text{Expected return}=\text{Inv}\left[{\text{a}}_{\text{1}}\text{+}\left({\text{w×R}}_{\text{m}}\right)\right]\text{-Inv}\left[{\text{a}}_{\text{2}}\text{+}\left({\text{w×R}}_{\text{m}}\right)\right]$

Where:

${\text{a}}_{\text{1}}$is first 10 stock's positive alpha i.e.$3\%$

${\text{a}}_2$is next 10 stock's negative alpha i.e.$-3\%$

w is weight measurement of each stock i.e.$1$

  ${\text{R}}_{\text{m}}$is Market return.

So expected return by using above formula

  $\begin{array}{c}\text{Expected Return}=\$10,00,000\left[0.03+\left(1\times {\text{R}}_{\text{m}}\right)\right]-\$10,00,000\left[-0.03+\left(1\times {\text{R}}_{\text{m}}\right)\right]\\ =\$10,00,000\left[0.03+{\text{R}}_{\text{m}}\right]-\$10,00,000\left[-0.03-{\text{R}}_{\text{m}}\right]\\ =\$10,00,000\left[0.03+{\text{R}}_{\text{m}}+0.03-{\text{R}}_{\text{m}}\right]\\ =\$10,00,000\times 0.06\\ =\$60,000\end{array}$

Standard deviation for return as follows

  $\begin{array}{c}\text{Standard Deviation}=\sqrt{\text{No}\text{. of stocks}{\left[\left(\frac{\text{Total invest}}{10}\right)\text{Std}\text{. deviation}\right]}^2}\\ =\sqrt{20{\left[\left(\frac{\$10,00,000}{10}\right)\times 0.30\right]}^2}\\ =\sqrt{20{\left[\$30,000\right]}^2}\\ =\sqrt{\$18,00,00,00,000}\\ =\$1,34,164.08\end{array}$

The risk exposure on positive and negative alpha will be eliminated but the variance is not zero. So the portfolio is not diversified.

Summary Introduction

(b)

To calculate:

The Expected profit and standard deviation where ten stocks have an positive alpha of$3\%$and ten have negative alpha of$-3\%$when 50 stocks data studied by analyst. And beta of all the stock in the economy is$1$. Firms have a standard deviation is$30\%$.

Introduction:

Expected return: It is the gain or loss to an investor which is expected on an investment and known as expected rate of return.

Standard deviation: It is historical volatility. It's applied to the annual rate of return to quantify the investment volatility.

Answer to Problem 29PS

Expected profit is$\$60,000$

Standard deviation of return is$\$84,852.81$

Explanation of Solution

Given:

Beta of all stock is$1$

Alpha of$25$stock is$3\%$

Alpha of$25$stock is$-3\%$

  $\text{Expected return}=\text{Inv}\left[{\text{a}}_{\text{1}}\text{+}\left({\text{w×R}}_{\text{m}}\right)\right]\text{-Inv}\left[{\text{a}}_{\text{2}}\text{+}\left({\text{w×R}}_{\text{m}}\right)\right]$

Where:

${\text{a}}_{\text{1}}$is first 10 stock's positive alpha i.e.$3\%$

${\text{a}}_2$is next 10 stock's negative alpha i.e.$-3\%$

w is weight measurement of each stock i.e.$1$

  ${\text{R}}_{\text{m}}$is Market return.

So expected return by using above formula

  $\begin{array}{c}\text{Expected Return}=\$10,00,000\left[0.03+\left(1\times {\text{R}}_{\text{m}}\right)\right]-\$10,00,000\left[-0.03+\left(1\times {\text{R}}_{\text{m}}\right)\right]\\ =\$10,00,000\left[0.03+{\text{R}}_{\text{m}}\right]-\$10,00,000\left[-0.03-{\text{R}}_{\text{m}}\right]\\ =\$10,00,000\left[0.03+{\text{R}}_{\text{m}}+0.03-{\text{R}}_{\text{m}}\right]\\ =\$10,00,000\times 0.06\\ =\$60,000\end{array}$

Standard deviation is calculated as under:

  $\begin{array}{c}\text{Standard Deviation}=\sqrt{\text{No}\text{. of stocks}{\left[\left(\frac{\text{Total invest}}{25}\right)\text{Std}\text{. deviation}\right]}^2}\\ =\sqrt{50{\left[\left(\frac{\$10,00,000}{25}\right)\times 0.30\right]}^2}\\ =\sqrt{50{\left[\$12,000\right]}^2}\\ =\sqrt{\$7,20,00,00,000}\\ =\$84,852.81\end{array}$

Summary Introduction

(c)

To calculate:

The expected profit and standard deviation where ten stocks have an positive alpha of$3\%$and ten have negative alpha of$-3\%$when 50 stocks data studied by analyst and beta of all the stock in the economy is$1$. Firms have a standard deviation is$30\%$.

Introduction:

Expected return: It is the gain or loss to an investor which is expected on an investment and known as expected rate of return.

Standard deviation: It is historical volatility. It's applied to the annual rate of return to quantify the investment volatility.

Answer to Problem 29PS

Expected profit is$\$60,000$

Standard deviation of return is$\$60,000$

Explanation of Solution

Given:

Beta of all stock is$1$

Alpha of$50$stock is$3\%$

Alpha of$50$stock is$-3\%$

  $\text{Expected return}=\text{Inv}\left[{\text{a}}_{\text{1}}\text{+}\left({\text{w×R}}_{\text{m}}\right)\right]\text{-Inv}\left[{\text{a}}_{\text{2}}\text{+}\left({\text{w×R}}_{\text{m}}\right)\right]$

Where:

${\text{a}}_{\text{1}}$is first 10 stock's positive alpha i.e.$3\%$

${\text{a}}_2$is next 10 stock's negative alpha i.e.$-3\%$

w is weight measurement of each stock i.e.$1$

  ${\text{R}}_{\text{m}}$is Market return.

So expected return by using above formula

  $\begin{array}{c}\text{Expected Return}=\$10,00,000\left[0.03+\left(1\times {\text{R}}_{\text{m}}\right)\right]-\$10,00,000\left[-0.03+\left(1\times {\text{R}}_{\text{m}}\right)\right]\\ =\$10,00,000\left[0.03+{\text{R}}_{\text{m}}\right]-\$10,00,000\left[-0.03-{\text{R}}_{\text{m}}\right]\\ =\$10,00,000\left[0.03+{\text{R}}_{\text{m}}+0.03-{\text{R}}_{\text{m}}\right]\\ =\$10,00,000\times 0.06\\ =\$60,000\end{array}$

Standard deviation is calculated as under:

  $\begin{array}{c}\text{Standard Deviation}=\sqrt{\text{No}\text{. of stocks}{\left[\left(\frac{\text{Total invest}}{25}\right)\text{Std}\text{. deviation}\right]}^2}\\ =\sqrt{100{\left[\left(\frac{\$10,00,000}{50}\right)\times 0.30\right]}^2}\\ =\sqrt{100{\left[\$6,000\right]}^2}\\ =\sqrt{\$3,60,00,00,000}\\ =\$60,000\end{array}$