On a monthly credit card balance of $1000, a typical credit card company will only ask for a minimum payment of $20. Why do credit card companies do that?
Mathematics of Credit Card Debt
Suppose we do what the company wants and make only the minimum payment p every month against an initial balance of b. If the company charges monthly interest rate r, what is the balance after n months?
See if we can notice a pattern.
Balance after n months
A1. Looking at the pattern above, derive a general function, f(n,r,p,b), for the balance after n months. Hint: use summation notation ∑_(k=1)^nwhere applicable when deriving the function.
A2. If your credit card company charges a monthly interest rate of 2% (annually 24%) on an initial balance of $1000, and you make a monthly payment of $30, what is your balance after one year? That is, find the value of f(12,0.02,$30,$1000).
A3. Based on your answer in A2, how much did you end up paying in interest rate charges over a year?
A4. Use geometric progression properties to convert the general formula in A1 above to a functional form that excludes the summation notation. Hint: You want to replace the summation notation ∑_(i=1)^n with a ratio; see https://en.wikipedia.org/wiki/Geometric_progression, subsection titled Related Formulas.
A5. How many months would it take to pay off a balance of $1000 if you made $30 monthly payments while being charged 2% monthly interest?What if we double the payment to $60, do we cut the time in half?Hint: equate the function for the balance after n month to zero and solve for n.
A6. Plot the function derived in A5 in a two-dimensional coordinate system with n on the y-axis and p on the x-axis. Assume the initial balance of b=$1000, and monthly interest of r=0.02. Find the vertical asymptote of this function, that is, find the value p (monthly minimum payment on your credit card) such that the number of months required to pay off your credit card debt is equals to infinity (that is a monthly minimum payment that makes you forever indebted to your credit card provider!).
Question B- Stock Markets
TheCapital asset pricing model (CAPM) takes into account the stock’s sensitivity to non-diversifiable risk (also known as systematic risk or market risk), often represented by β in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset. CAPM shows that the cost of equity capital is determined only by beta. Despite it was invented in the 1960s, the CAPM still remains popular due to its simplicity and applicability in a variety of situations.It may be a good idea to check out Understanding Beta athttp://www.investopedia.com/video/play/understanding-beta/ .
The CAPM is a model for pricing an individual security or portfolio. The risk of a portfolio comprises systematic risk, also known as undiversifiable risk, and unsystematic risk which is also known as idiosyncratic risk or diversifiable risk. Systematic risk refers to the risk common to all securities—i.e. market risk. Unsystematic risk is the risk associated with individual assets. Unsystematic risk can be diversified away to smaller levels by including a greater number of assets in the portfolio (specific risks “average out”). The same is not possible for systematic risk within one market. Depending on the market, a portfolio of approximately 20securities would be sufficiently diversified.
The beta from a single factor model in the form
is a good approximation to the CAPM beta.
The basic idea is that stocks tend to move together, driven by the same economic forces (the market). Here, the dependent variable, r_i are percentage returns for stock i, and independent variable, r_m are percentage returns for a broad market index.
α_i is the intercept and β_i is the slope of the linear relationship between the stock returns and the market. ε_i are the residual returns that cannot be explained by the market fluctuation (this is your idiosyncratic or firm-specific fluctuations).
In Lecture 6 (file ASX200.xlsx), you were provided with the prices for 165 stocks as well as the S&P/ASX 200 Index (a benchmark for the Australian stock market) from January 1, 2013 to December 30, 2015.
Pick any 3 securities (full name, industry and sector information is provided in Stock Information tab in ASX200.xlsx file).
Convert your chosen security prices and the market index into percentage returns. For each asset/index, percentage returns are defined as (price in day (t) – price in day (t-1))/(price in day (t-1)). This will define your returns for the three stocks, r_i, and the market return r_m.
B1. Perform OLS regression for each stock separately and report regression outputs for the three models from Excel/Matlab including line fit plots and residual plots.
B2. For each stock, discuss the OLS assumptions and violations (if any) based on the results from B1.
B3. Discuss the estimated betas for your three stocks and their statistical significance. Are these betas in line with your expectations? Provide your reasoning. What does it mean if a stock has a beta equal to 1? What does it mean if a stock has a beta equal to zero?
B4. Discuss the measure of fit (R^2) of your regressions in B1. Are these R^2 in line with your expectations? Provide your reasoning. Note that R^2 gives the fraction of the variance of the dependent variable (the return on a stock/portfolio of stocks) that is explained by movements in the independent variable (the return on the market index).
B5. Construct an equally weighted portfolio consisting of your three chosen stocks (equally weighted portfolio returns are simply the average of individual stock returns in that portfolio, r_p=(r_1+r_2+r_3)/3) and find the portfolio beta. Report regression output (including line fit plots and residual plots), assess the OLS assumptions and violations (if any) and discuss the estimated portfolio beta and the measure of fit of your regression. How does the measure of fit for the portfolio compares with the measures of fit for your individual stocks? Comment on portfolio diversification effect using your R^2s.