# Fight Finance

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Which of the following quantities is commonly assumed to be normally distributed?

If a variable, say X, is normally distributed with mean $\mu$ and variance $\sigma^2$ then mathematicians write $X \sim \mathcal{N}(\mu, \sigma^2)$.

If a variable, say Y, is log-normally distributed and the underlying normal distribution has mean $\mu$ and variance $\sigma^2$ then mathematicians write $Y \sim \mathbf{ln} \mathcal{N}(\mu, \sigma^2)$.

The below three graphs show probability density functions (PDF) of three different random variables Red, Green and Blue.

Select the most correct statement:

The below three graphs show probability density functions (PDF) of three different random variables Red, Green and Blue.

Which of the below statements is NOT correct?

The below three graphs show probability density functions (PDF) of three different random variables Red, Green and Blue. Let $P_1$ be the unknown price of a stock in one year. $P_1$ is a random variable. Let $P_0 = 1$, so the share price now is $1. This one dollar is a constant, it is not a variable. Which of the below statements is NOT correct? Financial practitioners commonly assume that the shape of the PDF represented in the colour: A stock has an arithmetic average continuously compounded return (AALGDR) of 10% pa, a standard deviation of continuously compounded returns (SDLGDR) of 80% pa and current stock price of$1. Assume that stock prices are log-normally distributed.

In one year, what do you expect the mean and median prices to be? The answer options are given in the same order.

A stock has an arithmetic average continuously compounded return (AALGDR) of 10% pa, a standard deviation of continuously compounded returns (SDLGDR) of 80% pa and current stock price of $1. Assume that stock prices are log-normally distributed. In 5 years, what do you expect the mean and median prices to be? The answer options are given in the same order. Fred owns some Commonwealth Bank (CBA) shares. He has calculated CBA’s monthly returns for each month in the past 20 years using this formula: $$r_\text{t monthly}=\ln⁡ \left( \dfrac{P_t}{P_{t-1}} \right)$$ He then took the arithmetic average and found it to be 1% per month using this formula: $$\bar{r}_\text{monthly}= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( r_\text{t monthly} \right)} }{T} =0.01=1\% \text{ per month}$$ He also found the standard deviation of these monthly returns which was 5% per month: $$\sigma_\text{monthly} = \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( \left( r_\text{t monthly} - \bar{r}_\text{monthly} \right)^2 \right)} }{T} =0.05=5\%\text{ per month}$$ Which of the below statements about Fred’s CBA shares is NOT correct? Assume that the past historical average return is the true population average of future expected returns. Here is a table of stock prices and returns. Which of the statements below the table is NOT correct?  Price and Return Population Statistics Time Prices LGDR GDR NDR 0 100 1 50 -0.6931 0.5 -0.5 2 100 0.6931 2 1 Arithmetic average 0 1.25 0.25 Arithmetic standard deviation -0.6931 0.75 0.75 Here is a table of stock prices and returns. Which of the statements below the table is NOT correct?  Price and Return Population Statistics Time Prices LGDR GDR NDR 0 100 1 99 -0.010050 0.990000 -0.010000 2 180.40 0.600057 1.822222 0.822222 3 112.73 0.470181 0.624889 0.375111 Arithmetic average 0.0399 1.1457 0.1457 Arithmetic standard deviation 0.4384 0.5011 0.5011 If a stock's future expected continuously compounded annual returns are normally distributed, what will be bigger, the stock's or continuously compounded annual return? Or would you expect them to be ? If a stock's future expected effective annual returns are log-normally distributed, what will be bigger, the stock's or effective annual return? Or would you expect them to be ? If a stock's expected future prices are log-normally distributed, what will be bigger, the stock's or future price? Or would you expect them to be ? Fred owns some BHP shares. He has calculated BHP’s monthly returns for each month in the past 30 years using this formula: $$r_\text{t monthly}=\ln⁡ \left( \dfrac{P_t}{P_{t-1}} \right)$$ He then took the arithmetic average and found it to be 0.8% per month using this formula: $$\bar{r}_\text{monthly}= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( r_\text{t monthly} \right)} }{T} =0.008=0.8\% \text{ per month}$$ He also found the standard deviation of these monthly returns which was 15% per month: $$\sigma_\text{monthly} = \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( \left( r_\text{t monthly} - \bar{r}_\text{monthly} \right)^2 \right)} }{T} =0.15=15\%\text{ per month}$$ Assume that the past historical average return is the true population average of future expected returns and the stock's returns calculated above $(r_\text{t monthly})$ are normally distributed. Which of the below statements about Fred’s BHP shares is NOT correct? A risk manager has identified that their hedge fund’s continuously compounded portfolio returns are normally distributed with a mean of 10% pa and a standard deviation of 30% pa. The hedge fund’s portfolio is currently valued at$100 million. Assume that there is no estimation error in these figures and that the normal cumulative density function at 1.644853627 is 95%.

Which of the following statements is NOT correct? All answers are rounded to the nearest dollar.

A risk manager has identified that their pension fund’s continuously compounded portfolio returns are normally distributed with a mean of 5% pa and a standard deviation of 20% pa. The fund’s portfolio is currently valued at $1 million. Assume that there is no estimation error in the above figures. To simplify your calculations, all answers below use 2.33 as an approximation for the normal inverse cumulative density function at 99%. All answers are rounded to the nearest dollar. Which of the following statements is NOT correct? A risk manager has identified that their investment fund’s continuously compounded portfolio returns are normally distributed with a mean of 10% pa and a standard deviation of 40% pa. The fund’s portfolio is currently valued at$1 million. Assume that there is no estimation error in the above figures. To simplify your calculations, all answers below use 2.33 as an approximation for the normal inverse cumulative density function at 99%. All answers are rounded to the nearest dollar. Assume one month is 1/12 of a year. Which of the following statements is NOT correct?

Which of the following statements about probability distributions is NOT correct?

Who was the first theorist to endorse the maximisiation of the geometric average gross discrete return for investors (not gamblers) since it gave a "...portfolio that has a greater probability of being as valuable or more valuable than any other significantly different portfolio at the end of n years, n being large"?

(a) Daniel Bernoulli.

Question 906  effective rate, return types, net discrete return, return distribution, price gains and returns over time

For an asset's price to double from say $1 to$2 in one year, what must its effective annual return be? Note that an effective annual return is also called a net discrete return per annum. If the price now is $P_0$ and the price in one year is $P_1$ then the effective annul return over the next year is:

$$r_\text{effective annual} = \dfrac{P_1 - P_0}{P_0} = \text{NDR}_\text{annual}$$

For an asset's price to double from say $1 to$2 in one year, what must its continuously compounded return $(r_{CC})$ be? If the price now is $P_0$ and the price in one year is $P_1$ then the continuously compounded return over the next year is:

$$r_\text{CC annual} = \ln{\left[ \dfrac{P_1}{P_0} \right]} = \text{LGDR}_\text{annual}$$

Question 908  effective rate, return types, gross discrete return, return distribution, price gains and returns over time

For an asset's price to double from say $1 to$2 in one year, what must its gross discrete return (GDR) be? If the price now is $P_0$ and the price in one year is $P_1$ then the gross discrete return over the next year is:

$$\text{GDR}_\text{annual} = \dfrac{P_1}{P_0}$$

Who was the first theorist to propose the idea of ‘expected utility’?

The arithmetic average and standard deviation of returns on the ASX200 accumulation index over the 24 years from 31 Dec 1992 to 31 Dec 2016 were calculated as follows:

$$\bar{r}_\text{yearly} = \dfrac{ \displaystyle\sum\limits_{t=1992}^{24}{\left( \ln⁡ \left( \dfrac{P_{t+1}}{P_t} \right) \right)} }{T} = \text{AALGDR} =0.0949=9.49\% \text{ pa}$$

$$\sigma_\text{yearly} = \dfrac{ \displaystyle\sum\limits_{t=1992}^{24}{\left( \left( \ln⁡ \left( \dfrac{P_{t+1}}{P_t} \right) - \bar{r}_\text{yearly} \right)^2 \right)} }{T} = \text{SDLGDR} = 0.1692=16.92\text{ pp pa}$$

Assume that the log gross discrete returns are normally distributed and that the above estimates are true population statistics, not sample statistics, so there is no standard error in the sample mean or standard deviation estimates. Also assume that the standardised normal Z-statistic corresponding to a one-tail probability of 2.5% is exactly -1.96.

Which of the following statements is NOT correct? If you invested $1m today in the ASX200, then over the next 4 years: The arithmetic average continuously compounded or log gross discrete return (AALGDR) on the ASX200 accumulation index over the 24 years from 31 Dec 1992 to 31 Dec 2016 is 9.49% pa. The arithmetic standard deviation (SDLGDR) is 16.92 percentage points pa. Assume that the log gross discrete returns are normally distributed and that the above estimates are true population statistics, not sample statistics, so there is no standard error in the sample mean or standard deviation estimates. Also assume that the standardised normal Z-statistic corresponding to a one-tail probability of 2.5% is exactly -1.96. If you had a$1 million fund that replicated the ASX200 accumulation index, in how many years would the median dollar value of your fund first be expected to lie outside the 95% confidence interval forecast?

The arithmetic average continuously compounded or log gross discrete return (AALGDR) on the ASX200 accumulation index over the 24 years from 31 Dec 1992 to 31 Dec 2016 is 9.49% pa.

The arithmetic standard deviation (SDLGDR) is 16.92 percentage points pa.

Assume that the log gross discrete returns are normally distributed and that the above estimates are true population statistics, not sample statistics, so there is no standard error in the sample mean or standard deviation estimates. Also assume that the standardised normal Z-statistic corresponding to a one-tail probability of 2.5% is exactly -1.96.

If you had a $1 million fund that replicated the ASX200 accumulation index, in how many years would the mean dollar value of your fund first be expected to lie outside the 95% confidence interval forecast? The arithmetic average continuously compounded or log gross discrete return (AALGDR) on the ASX200 accumulation index over the 24 years from 31 Dec 1992 to 31 Dec 2016 is 9.49% pa. The arithmetic standard deviation (SDLGDR) is 16.92 percentage points pa. Assume that the log gross discrete returns are normally distributed and that the above estimates are true population statistics, not sample statistics, so there is no standard error in the sample mean or standard deviation estimates. Also assume that the standardised normal Z-statistic corresponding to a one-tail probability of 2.5% is exactly -1.96. If you had a$1 million fund that replicated the ASX200 accumulation index, in how many years would the mode dollar value of your fund first be expected to lie outside the 95% confidence interval forecast?

Note that the mode of a log-normally distributed future price is: $P_{T \text{ mode}} = P_0.e^{(\text{AALGDR} - \text{SDLGDR}^2 ).T}$

The arithmetic average continuously compounded or log gross discrete return (AALGDR) on the ASX200 accumulation index over the 24 years from 31 Dec 1992 to 31 Dec 2016 is 9.49% pa.

The arithmetic standard deviation (SDLGDR) is 16.92 percentage points pa.

Assume that the data are sample statistics, not population statistics. Assume that the log gross discrete returns are normally distributed.

What is the standard error of your estimate of the sample ASX200 accumulation index arithmetic average log gross discrete return (AALGDR) over the 24 years from 1992 to 2016?