# Question A – Credit Card Mathematics

Introduction

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  every month against an initial balance of .  If the company charges monthly interest rate , what is the balance after  months?

See if we can notice a pattern.

 Balance after  months

A1. Looking at the pattern above, derive a general function, , for the balance after  months. Hint: use summation notation  where 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 .

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  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  month to zero and solve for .

A6. Plot the function derived in A5 in a two-dimensional coordinate system with  on the -axis and  on the -axis. Assume the initial balance of , and monthly interest of . Find the vertical asymptote of this function, that is, find the value  (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

Introduction

The Capital 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  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 at  http://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 20 securities 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,  are percentage returns for stock , and independent variable,  are percentage returns for a broad market index.

is the intercept and  is the slope of the linear relationship between the stock returns and the market.  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.

1. Pick any 3 securities (full name, industry and sector information is provided in Stock Information tab in ASX200.xlsx file).

1. Convert your chosen security prices and the market index into percentage returns. For each asset/index, percentage returns are defined as . This will define your returns for the three stocks, , and the market return .

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 ( ) of your regressions in B1. Are these  in line with your expectations? Provide your reasoning. Note that    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, ) 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 s.

# Question C  – Population Survey

Introduction

Each month the Bureau of Labor Statistics in the U.S. Department of Labor conducts the “Current Population Survey” (CPS), which provides data on labor force characteristics of the population, including the level of employment, unemployment, and earnings.  Approximately 65,000 randomly selected U.S. households are surveyed each month.  The sample is chosen by randomly selecting addresses from a database comprised of addresses from the most recent decennial census augmented with data on new housing units constructed after the last census.  The exact random sampling scheme is rather complicated (first small geographical areas are randomly selected, then housing units within these areas randomly selected).

The file cps08.xlsx contains the data for 2008 (from the March 2009 survey).  These data are for full-time workers, defined as workers employed more than 35 hours per week for at least 48 weeks in the previous year.  Data are provided for workers whose highest educational achievement is (1) a high school diploma, and (2) a bachelor’s degree.

Series in Data Set:

FEMALE:        1 if female; 0 if male

YEAR:             Year

AHE    :           Average Hourly Earnings

BACHELOR:   1 if worker has a bachelor’s degree; 0 if worker has a high school degree

Use the data in cps08.xlsx to answer the following questions:

C1. Run a regression of average hourly earnings (AHE) on age (Age). Report Excel/Matlab output. What is the estimated intercept? What is the estimated slope?

C2. Run a regression of AHE on Age, gender (Female), and education (Bachelor). Report Excel/Matlab output. What is the estimated effect of Age on earnings? Construct a 95% confidence interval for the coefficient on Age in the regression.

C3. Are the results from the regression in C2 substantively different from the results in C1 regarding the effects of Age and AHE? Does the regression in C1 seem to suffer from omitted variable bias?

C4. Bob is a 26-year-old male worker with a high school diploma. Predict Bob’s earnings using the estimated regression in C2. Alexis is a 30-year-old female worker with a college degree. Predict Alexis’s earnings using the regression.

C5. Compare the fit of the regression in C1 and C2 using the regression standard errors,  and . Why are the  and  so similar in regression C2?

C6. Are gender and education determinants of earnings? Test the null hypothesis that Female can be deleted from the regression. Test the null hypothesis that Bachelor can be deleted from the regression. Test the null hypothesis that both Female and Bachelor can be deleted from the regression.

C7. A regression will suffer from omitted variable bias when two conditions hold. What are these two conditions? Do these conditions seem to hold here?

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