3. Last year you took a long position on Citibank stock (C) future with maturity date April 2020 at a price of $34.8/share (100 shares per contract). The initial margin requirement is 12%. On the future contract maturity date, Citibank stock price rises to $44.5/share. What is the stock future price on maturity date? What is the return rate of your investment (Hint: how much is your investment)?

4. Assume a future contract on Treasury Bonds with a face value of $100,000 is purchased at 93. How many T-bonds will be delivered under the contract (what is T-bond face value)? If the same future contract is later sold at 94.56, what is the profit/loss from the price movement? If initial margin is 10%, what is the return rate of the investment?

The Saunders Investment Bank has the following financing outstanding. Debt: 51,000 bonds with a coupon rate of 4.7 percent and a current price quote of 106.3; the bonds have 13 years to maturity and a par value of $1,000. 17,100 zero coupon bonds with a price quote of 27.3, 29 years until maturity, and a par value of $10,000. Both bonds have semiannual compounding. Preferred stock: 146,000 shares of 3.8 percent preferred stock with a current price of $93 and a par value of $100. Common stock: 2,120,000 shares of common stock; the current price is $83 and the beta of the stock is 1.15. Market: The corporate tax rate is 21 percent, the market risk premium is 6.7 percent, and the risk-free rate is 3.1 percent. What is the WACC for the company?

1. Assume that Mr. Shulman owned 100 shares of Exxon Mobile stock and 250 shares of General Electric stock when he died on December 1. The highest selling price for Exxon Mobile on that day was 41.24; the lowest selling price was 40.40. The highest selling price for GE on that day was 39.35; the lowest selling price was 38.95. Calculate the fair market value of all of the stock as of December 1.

2. In December 2003, Mrs. Nichols gave gifts of $20,000 to her son, Bruce; $20,000 to her son and daughter-in-law, David and Cheryl; $15,000 to a favorite nephew; and $10,000 to each of her seven grandchildren. What amount must Mrs. Nichols report to the IRS on her 2003 gift-tax return? Please show your calculations.

The Carseat is a data set containing sales of child car seats at 400 different stores. Your task is to develop a regression model to predict sales and use this data to estimate your model.

Description of variables:
Sales: Unit sales (in thousands) at each location
CompPrice: Price charged by competitor at each location
Income: Community income level (in thousands of dollars)
Advertising: Local advertising budget for company at each location (in thousands of dollars)
Population: Population size in region (in thousands)
Price: Price company charges for car seats at each site
ShelveLoc: A factor with levels Bad, Good and Medium indicating the quality of the shelving location for the car seats at each site
Age: Average age of the local population
Education: Education level at each location
Urban: A factor with levels No and Yes to indicate whether the store is in an urban or rural location
US: A factor with levels No and Yes to indicate whether the store is in the US or not

a. Which variables are categorical variable? For each categorical variable, define the corresponding dummy variables. [1pt]

b. Develop a correlation matrix between all variables. Copy your correlation matrix here. [1pt]

c. Develop a regression model to predict sales and use all the data to estimate your model. Write the estimated equation here. What factors are significant predictors of sales? [1pt]

d. Use the first 300 rows of the data as training set to estimate your model and use the remaining 100 rows as test set to evaluate the out-of-sample prediction performance of your model. Use rMSE as the performance measure. What is the rMSE for your in-sample and out-of-sample prediction? [1pt]

Enter your answers in the empty code chunks.

Don't change anything in the chunk below, and make sure you run it before attempting any of the problems:

```{r message=FALSE, warning=TRUE}
library(tidyverse)
library(ggpubr)
set.seed(2018) # Belgium win 3rd place in the World Cup
```

# Basics

Calculate $\frac{(2+2)\times (3^2 + 5)}{(6/4)}$:

```{r basics1}
# your code here
```

Create a vector called "x" with the following values: 10, 15, 18, 20.

```{r basics2}
# your code here
```

Calculate the mean (`mean()`), median (`median()`) and standard deviation (`sd()`) of `x`.

```{r basics3}
# your code here
```

Draw two numbers from a standard normal distribution (`rnorm()`):

```{r basics4}
# your code here
```

Draw 100 numbers from a normal distribution with a mean of **7** and standard deviation of **4**, and store the output in an object called "x1":

```{r basics5}
# your code here
```

Draw 100 numbers from a normal distribution with a mean of **5** and standard deviation of **2**, and store the output in an object called "x2":

```{r basics6}
# your code here
```

Use `data.frame()` to combine `x1` and `x2` into a data frame called "df":

```{r basics7}
# your code here
```

Use the `$` indexing method to calculate:

* the mean of the first column of `df`:

```{r basics8}
# your code here
```

* the standard deviation of the second column of `df`:

```{r basics9}
# your code here
```

Use `head()` to print the first **3** rows of `df`:

```{r basics10}
# your code here
```

# `dplyr`

Let's work with the data set `diamonds`:

```{r data}
data(diamonds)
head(diamonds)
```

Calculate the average price of a diamond:

```{r dplyr1}
# your code here
```

Use `group_by()` to group diamonds by **color**, then use `summarise()` to calculate the average price *and* the standard deviation in price **by color**:

```{r dplyr2}
# your code here
```

Use `group_by()` to group diamonds by **cut**, then use `summarise()` to count the number of observations **by cut**:

```{r dplyr3}
# your code here
```

Use `filter()` to remove observations with a depth greater than 62, then use`group_by()` to group diamonds by **clarity**, then use `summarise()` to find the maximum price of a diamond **by clarity**:

```{r dplyr4}
# your code here
```

Use `mutate()` and `log()` to add a new variable to the data called "log_price":

```{r dplyr5}
# your code here
```

# `ggplot2`

Continue using `diamonds`.

Use `geom_histogram()` to plot a histogram of prices:

```{r ggplot1}
# your code here
```

Use `geom_density()` to plot the density of *log prices* (the variable you added to the data frame):

```{r ggplot2}
# your code here
```

Use `geom_point()` to plot carats against log prices (i.e. carats on the x-axis, log prices on the y-axis):

```{r ggplot3}
# your code here
```

Use `stat_summary()` to make a bar plot of **average** cut:

Same as above but change the theme to `theme_classic()`:

```{r ggplot4}
# your code here
```

Finally,

* create a bar plot for **average** color and assign it to the object "plot_color";
* create a scatter plot for depth against log pricse (depth on x-axis, log prices on y-axis) and assign it to the object "plot_depth";
* use `ggarrange` from `ggpubr` to combine `plot_color` and `plot_depth` into a single plot with automatic labels.

```{r ggplot5}

```

# Inference

Use `t.test()` to test the following hypothesis on *log price*:

$$
H_0: \mu = 8 \\
H_A: \mu \neq 8
$$

Gladstone Company tracks the number of units purchased and sold throughout each accounting period but applies its inventory costing method at the end of each period, as if it uses a periodic inventory system. Assume its accounting records provided the following information at the end of the annual accounting period, December 31.

Assuming that for Specific identification method (item 1d) the March 14 sale was selected two-fifths from the beginning inventory and three-fifths from the purchase of January 30. Assume that the sale of August 31 was selected from the remainder of the beginning inventory, with the balance from the purchase of May 1.

Required:

1. Compute the amount of goods available for sale, ending inventory, and cost of goods sold at December 31 under each of the following inventory costing methods: (Round intermediate calculations to 2 decimal places and final answers to the nearest whole dollar amount.)

   	Amount of Goods Available for Sale	Ending Inventory	Cost of Goods Sold
   Last-in, first-out
   Weighted average cost
   First-in, first-out
   Specific identification

‘Lady tasting tea’ is a randomized experiment devised by Ronald Fisher and reported in his book The Design of Experiments (1935). The lady in question (Muriel Bristol) claimed that she can tell whether the tea or the milk was added first to a cup. Fisher proposed to give her eight cups, four of each variety, in random order. One could then ask what the probability was for her getting the specific number of cups she identified correct, but just by chance. In this problem we are going to modify and simulate ‘The Lady tasting tea’ experiment. Suppose that you are going to randomly selected a student and conduct the similar experiment; 10 cups (instead of 8) of tea of which some have milk added first and others have milk added last. We are going to assume that this student, unlike Ms Bristol, does not have the skill to tell if the milk was added first or last.

1. (Lady tasting tea Problem) What is the probability that the student correctly guesses if a given cup of tea has milk added first or last?

2. (Lady tasting tea Problem) For this student, what is the expected number of correct guesses out of 10? What about incorrect guesses?

3. (Lady tasting tea Problem) Would you be surprised if the student guesses 6 out of 10 cups correctly (60% correct rate)? Would you conclude that this student has the skill to tell if the milk was added first or last? Or would you conclude that he/she was just “lucky”?

4. (Lady tasting tea Problem) Suppose that you prepare 100 cups of tea instead of 10. Would you be surprised if the student guesses 60 out of 100 cups correctly (still 60% correction rate)? Would you conclude that this student has the skill to tell if the milk was added first or last? Or would you conclude that he/she was just “lucky”?