FIN311: If the Current Dividend is D0, show Clearly that if the Annual Dividends are Growing at a rate of g: Mathematics and Programming for Finance Assignment, SUSS, Singapore

Question 1
If the current dividend is D0, show clearly that if the annual dividends are growing at a rate of g, the intrinsic value of the equity, V may be formulated as

where rf and rm are the return of the risk-free asset and the market portfolio respectively. The equity CAPM beta is given as β.

Question 2
Figure 1 records the transaction data of a depositor for a given month.

Figure 1: Python inputs to the function calculate_daily_balance. The depositor had transactions on various dates in January and his balance was $1450.32 before the 1-Jan2020.

You are trying to evaluate the daily balance for a given month of the client in the
following format:

There was 0 days where his daily bank balance was $1450.32 in January.

There were 12 days (1-Jan to 12-Jan) where his daily bank balance was $1455.42 in January.

There were 2 days (13-Jan to 14-Jan) where his daily bank balance was $1443.12 in January.

There were 12 days (15-Jan to 26-Jan) where his daily bank balance was $1528.82 in January.

There were 5 days (27-Jan to 31-Jan) where his daily bank balance was $1473.37 in January.

The user-defined function given in Figure 2 was used to output the above information,

i.e. two lists that provide the number of days where a particular daily bank balance is observed.

The output of the user-defined function is expected to be
[1450.32, 1455.42, 1443.12, 1528.82, 1473.37] and [0, 12, 2, 12, 5]

However, they are two errors that appear from Line 6 onwards in the code.

Inspect and identify the two (2) errors by stating the line number where the error occurs. Provide the correct implementation of the errors.

There is no need to reproduce the entire code implementation in your answer

Question 3
If you are bearish about the stock market, for a fixed investment amount, propose which one (1) of the following strategies will maximize your return if you are correct.
(a) Long an OTM put option
(b) Long an ATM put option
(c) Long an OTM call option
(d) Long an ATM call option
Explain and justify your choice clearly.

Question 4
The returns of Stocks A and B are assumed to follow a Normal distribution. The returns of A have a mean and standard deviation of 5% and 2% respectively and the returns of B have a mean and standard deviation of 0.5% and 1% respectively.

The correlation between the returns of A and B is 38%.

Using the given inputs in Figure 3, develop Python code that can be used to evaluate the probability that an equal-weighed portfolio of A and B has a return greater than 5%. In your code, include a print statement that reports the result to an accuracy of three (3) decimal points in percentage (e.g. 32.564%).

The document clearly, the appropriate Python function(s) and module(s) that are required, and there is no need to report the eventual value coming out from your implementation.

SECTION B (Total 60 marks)
You must answer ALL the questions in this section and show your working/reasoning in full.
You are not required to provide Python code in your submitted answers unless it is explicitly required in the question.

Question 5
(a) Describe two (2) factors that will affect the prices of options. How does the price of the options change with these factors?

(b) An investor has a portfolio consisting of 1000 units of Stock X, purchased at a price of $25. His maximum investment horizon is 1 year and he intends to exit the position if his price target of $55 is achieved. If his price target of $55 is not achieved in a year’s time, he will still exit the position regardless of the price.

6 months have elapsed and the price of X has appreciated to $35. However, in the next 6 months, it is expected market volatility would increase due to potential US-China tensions.

Distinguish a combination of options, based on his trading strategy, that the investor can set up to protect his existing profits in the coming 6 months. Explain your answers clearly.

(c) Given that the volatility of X is 48% and the risk-free rate is 0.5%. Evaluate the amount of money, to the nearest dollars, that is required to set up the position suggested in (b).

Question 6
You are required to provide the code for this entire question. Only when requested, you are required to write down the result coming from the code in your examination script.

A trader hypothesizes that n consecutive days of price increase in a stock is a bullish signal to acquire the stock and he would like to test his hypothesis by carrying out some back-testing on the validity of this trading signal.

You are helping the trader to analyze and test his strategy with the inputs given in Figure4

You are recommended to make good use of available python packages and functions to:

(a) Generate a list of daily dates stipulated by start_date and end_date (both dates inclusive) and some random price data to begin the analysis. The random stock prices are to be integers bounded by the values specified in min_stock_prices and max_stock_prices

(b) Save the random price data in a pandas’ data frame with the list of daily dates as the index. The price data is to be saved as a column titled price. Hence, calculate the daily return of the price data and include it in the same a data frame in a column titled return.

(c) Determine and write down the list of dates where there are 3 consecutive days of the price increase.

(d) If 3 consecutive days of price increase is observed, the trader will acquire the stock on that day and sell it the next day. With the initial wealth given in Figure 4, write down the wealth of the trader on end_date.

You can assume fractional holding of shares is allowed. Report the wealth of the trader specified on the end_date in two (2) decimal points (e.g. $23.45).

Question 7
You are required to provide the code for this entire question. Only when requested, you are required to write down the result coming from the code in your examination script.

(a) There is no central decision-making authority to vet if a user is a malicious and anyone is welcome to join Bitcoin network. Despite this, the Bitcoin network is able to operate well. Explain why is the Bitcoin network able to operate without a decision-maker.

(b) The Bitcoin network is updated whenever a cryptographic puzzle is solved. This happens when the puzzle satisfied the difficulty stipulated by the Bitcoin network, i.e. the number of leading zeros in the puzzle has to meet the minimum number of zeros stipulated in the value of difficulty.

However, the value given in difficulty does not give the number of zeros required directly. It has to be processed to determine the number of zeros that are required from the puzzle.

To determine the number of zeros, the value in difficulty is first converted into its 16-bits binary equivalent and the binary bits are grouped into a size of 4. The number of zeros is then found by counting the number of consecutive groups, starting from the left, that have all-zeros in their group.

Two examples of the algorithm, using the difficulty of 88 and 1024 are analyzed in
Figure 5.

To convert the binary equivalent of difficulty to its 16-bits binary equivalent, simply prepend an appropriate number of zeros so that there are a total of 16- bits1
.
Implement a function num_of_zeros(difficulty) that will return the number of zeros that are required to satisfy the difficulty stipulated by the network.

Hint: You can make use of the function bin(x)[2:] to convert a number x to its binary equivalent.

(c) Hence, with the use of suitable code together with your function in (b), compute the number of zeros required when the difficulty is 2389, 1089, 724, 555, 24, and 8.