The current gold price is $**700**, gold storage costs are **2**% pa and the risk free rate is **10**% pa, both with **continuous compounding**.

What should be the **3** year gold futures price?

A **2**-year futures contract on a stock paying a continuous dividend yield of **3**% pa was bought when the underlying stock price was $**10** and the risk free rate was **10**% per annum with **continuous compounding**. Assume that investors are risk-neutral, so the stock's total required return is the risk free rate.

Find the forward price ##(F_2)## and value of the contract ##(V_0)## initially. Also find the value of the contract in 6 months ##(V_{0.5})## if the stock price rose to $**12**.

An equity index is currently at **5,000** points. The **2** year futures price is **5,400** points and the total required return is **8**% pa with continuous compounding. Each index point is worth $**25**.

What is the implied continuous dividend yield as a continuously compounded rate per annum?

A stock is expected to pay a dividend of $**5** per share in **1** month and $**5** again in **7** months.

The stock price is $**100**, and the risk-free rate of interest is **10**% per annum with continuous compounding. The yield curve is flat. Assume that investors are risk-neutral.

An investor has just taken a **short** position in a **one** year forward contract on the stock.

Find the forward price ##(F_1)## and value of the contract ##(V_0)## initially. Also find the value of the short futures contract in 6 months ##(V_\text{0.5, SF})## if the stock price fell to $**90**.

A $**100** stock has a continuously compounded expected **total** return of **10**% pa. Its **dividend** yield is **2**% pa with continuous compounding. What do you expect its price to be in **one** year?

A $**100** stock has a continuously compounded expected **total** return of **10**% pa. Its **dividend** yield is **2**% pa with continuous compounding. What do you expect its price to be in **2.5** years?

An equity index is currently at **5,200** points. The **6** month futures price is **5,300** points and the total required return is **6**% pa with continuous compounding. Each index point is worth $25.

What is the implied dividend yield as a continuously compounded rate per annum?

An equity index is currently at **4,800** points. The **1.5** year futures price is **5,100** points and the total required return is **6**% pa with continuous compounding. Each index point is worth $25.

What is the implied dividend yield as a continuously compounded rate per annum?

**Question 691** continuously compounding rate, effective rate, continuously compounding rate conversion, no explanation

A bank quotes an interest rate of **6**% pa with quarterly compounding. Note that another way of stating this rate is that it is an annual percentage rate (APR) compounding discretely every 3 months.

Which of the following statements about this rate is **NOT** correct? All percentages are given to 6 decimal places. The equivalent:

**Question 707** continuously compounding rate, continuously compounding rate conversion

Convert a **10**% effective annual rate ##(r_\text{eff annual})## into a continuously compounded annual rate ##(r_\text{cc annual})##. The equivalent continuously compounded annual rate is:

**Question 708** continuously compounding rate, continuously compounding rate conversion

Convert a **10**% continuously compounded annual rate ##(r_\text{cc annual})## into an effective annual rate ##(r_\text{eff annual})##. The equivalent effective annual rate is:

Which of the following interest rate quotes is **NOT** equivalent to a **10**% effective annual rate of return? Assume that each year has 12 months, each month has 30 days, each day has 24 hours, each hour has 60 minutes and each minute has 60 seconds. APR stands for Annualised Percentage Rate.

**Question 710** continuously compounding rate, continuously compounding rate conversion

A continuously compounded **monthly** return of 1% ##(r_\text{cc monthly})## is equivalent to a continuously compounded **annual** return ##(r_\text{cc annual})## of:

**Question 711** continuously compounding rate, continuously compounding rate conversion

A continuously compounded **semi-annual** return of **5**% ##(r_\text{cc 6mth})## is equivalent to a continuously compounded **annual** return ##(r_\text{cc annual})## of:

**Question 719** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate

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. The graph below summarises this information and provides some helpful formulas.

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

**Question 720** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate

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 median and mean prices to be? The answer options are given in the same order.

**Question 721** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate

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:

He also found the standard deviation of these monthly returns which was **5**% 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.

**Question 722** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate

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 | 0.9802 | 1.0607 | 1.0607 | |

**Question 723** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate

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 | |

**Question 779** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate

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:

He also found the standard deviation of these monthly returns which was **15**% 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?

**Question 790** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate, log-normal distribution, VaR, confidence interval

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.

**Question 791** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate, log-normal distribution, VaR, confidence interval

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?

**Question 792** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate, log-normal distribution, confidence interval

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?

**Question 907** continuously compounding rate, return types, 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 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 925** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate, no explanation

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:

**Question 926** mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate

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?

**Question 927** mean and median returns, mode return, return distribution, arithmetic and geometric averages, continuously compounding rate

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?

**Question 928** mean and median returns, mode return, return distribution, arithmetic and geometric averages, continuously compounding rate, no explanation

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.

**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} ##

**Question 929** standard error, mean and median returns, mode return, return distribution, arithmetic and geometric averages, continuously compounding rate

**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?