ARE271 Financial Economics
Problem Set 1
Instructions: The problem set is due on Friday of week 3 at 4pm. Please write up your answers electronically. You may supplement this with hand-written answers, which you can scan and upload to canvas. All submissions must be made electronically.
For analytical questions make sure to show your work (e.g. formulas and calculations). You are encouraged to work in small groups of 2-3 people to discuss the problem set but you must write up answers individually, and have to list the names of the group members on your problem set.
1. Write a short one-paragraph summary in your own words of the FT article on the yield curve that is in the folder “Articles” on canvas. What things did you ind most interesting? Briely discuss what happened to interest rates since the article was written.
2. Calculate some numbers using the data in the spreadsheet PS1 posted on canvas. I would advise doing all of this in Excel. If you are more comfortable using another program, that is ine, too, of course.
(a) Calculate the full-sample means and standard deviations of excess returns for each of the series. Report annualized numbers, so multiply by 12 or p12.
(b) Calculate full-sample Sharpe ratios for all risky assets. Use the annualized num-bers you calculated in (a).
(c) (OPTIONAL) Calculate geometric average returns for risk-free, market and small stocks. Speciically, if starting with USD 1 at the beginning of the sample, cal- culate the value at the end of the sample. Use the approximation from class to compare geometric and arithmetic averages. How well does the approximation do?
3. Assume that there is a representative agent (i.e. only one individual) with timesepara- ble CRRA (power or log) utility of consumption in the economy. There are two periods; next period there are two states of the world. The total endowment in the economy is equal to {100,90,110}, listed as endowment today followed by the two states next period. The probabilities for the two states are {0.2,0.8}. The representative agent has a discount factor of β = 1 and constant relative risk aversion = 1, i.e. log util- ity. A full set of Arrow-Debreu securities is trading in the market so that markets are complete.
(a) What is the ratio for the two states of the world? Keep in mind that con-sumption is equal to output (there is one agent who consumes the endowment.)
(b) Multiplying by the discount factor β what is the stochastic discount factor equal to for each state? What are the state prices?
(c) Which state is the bad state and which is the good state? Why? Now assume that there are two trees (assets) in the economy. One pays of {10,10} and one pays {6,11}.
(d) The irst tree pays the same amount in both states of the world. What might you call this asset?
(e) (OPTIONAL) What is the value (price) of the trees at time zero? What is the expected payof of the two trees?
(f) (OPTIONAL) Is the price of the second tree higher or lower than the irst tree? Why? Briely discuss and explain.