QuestionNormal distribution: Given expected return $\bar{r} = 18.9\%$ and $C V = 0.75$ , find: a. $\sigma_{r}$ , b. ranges for $68\%, 95\%, 99\%$ , c. draw the distribution.

Studdy Solution

STEP 1

Assumptions1. The rates of return associated with a given asset investment are normally distributed. . The expected return, $\bar{r}$ , is $18.9 \%$ .
3. The coefficient of variation, $CV$ , is0.75.

STEP 2

First, we need to find the standard deviation of returns, $\sigma_{r}$ . We can do this by multiplying the expected return by the coefficient of variation.
$\sigma_{r} = \bar{r} \times CV$

STEP 3

Now, plug in the given values for the expected return and the coefficient of variation to calculate the standard deviation.
$\sigma_{r} =18.9\% \times0.75$

STEP 4

Convert the percentage to a decimal value.
$18.9\% =0.189$ $\sigma_{r} =0.189 \times0.75$

STEP 5

Calculate the standard deviation of returns.
$\sigma_{r} =0.189 \times0.75 =0.14175$

STEP 6

Now that we have the standard deviation, we can calculate the range of expected return outcomes associated with the following probabilities of occurrence (1) $68 \%$ , (2) $95 \%$ , (3) $99 \%$ .
For a normal distribution, these probabilities correspond to the following multiples of the standard deviation (1) $\pm1\sigma$ , (2) $\pm2\sigma$ , (3) $\pm3\sigma$ .

STEP 7

Calculate the range of expected return outcomes for each probability.
1. $68\%$ probability $\bar{r} \pm1\sigma_{r}$
2. $95\%$ probability $\bar{r} \pm2\sigma_{r}$
3. $99\%$ probability $\bar{r} \pm3\sigma_{r}$

STEP 8

Plug in the values for the expected return and the standard deviation to calculate the range of expected return outcomes.
1. $68\%$ probability $0.189 \pm1 \times0.14175$
2. $95\%$ probability $0.189 \pm2 \times0.14175$
3. $99\%$ probability $0.189 \pm3 \times0.14175$

STEP 9

Calculate the range of expected return outcomes for each probability.
. $68\%$ probability $.189 \pm.14175 = [.04725,.33075]$
2. $95\%$ probability $.189 \pm2 \times.14175 = [-.0945,.4725]$
3. $99\%$ probability $.189 \pm3 \times.14175 = [-.23625,.61425]$

STEP 10

The probability distribution associated with these findings can be drawn as a normal distribution curve with mean $\bar{r}$ and standard deviation $\sigma_{r}$ . The ranges calculated above represent the intervals within which the return will fall with $68\%$ , $95\%$ , and $99\%$ probability respectively.

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