Econ 185 Fall 2004

The Keynesian IS/LM Model

Puts and Calls

Call (Put) - the right to buy (sell) a specified quantity at a specified date at a specified price (the strike price).

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Review

HW3

1. Calculate the expected value, variance, and standard deviation for the following possible asset values. "(a,b) for prob. (p,q)" means X takes on value a and b with probabilities p and q.
(a) (1,2,3) and (0,2,4) for prob. (1/3, 1/3, 1/3) [2 cases].
(b) (9,10,11), (8,10,12), and (0,10,20) for prob. (1/4, 1/2, 1/4) [3 cases].
(c) (1,2,3) and (0,2,4) for prob. (1/10, 8/10, 1/10) [2 cases].
(d) (0,1), (0,4), (9,11), (8,12), (0,20) for prob. (1/2, 1/2) [5 cases].

2. What general propositions do each of the parts (a) - (d) of the previous problem illustrate?

3. Suppose a junk bond is worth $1000 with probability 0.90 and $200 with probability 0.10. What would an investor for square root utility be willing to pay for this bond? Please explain why that figure is less that the mathematical expected value of the dollar payouts.

4. Suppose the risk-free interest rate of 10% can be obtained by buying government bonds. Stock in Company A, on the hand, sells today for $20 per share and has an expected value next year of $25, which is an increase of 25%. Unfortunately, the standard deviation of the stock's value next year is 8 so it is not without risk. What are the mean and standard deviation (for next year) of a portfolio consisting of $500 in government bonds and $500 in stock? How about the portfolio of $1000 in bonds and no stock and the portfolio of no bonds and $1000 in stock?

5. Using the information in the previous question, what are the mean and variance (for next year) of a portfolio consisting of $2000 in stock financed by $1000 of your money and $1000 borrowed from a bank at 10%?

6. Graph the results for the two previous questions using expected dollars on the vertical axis and standard deviation on the horizontal axis. (A sketch will be fine.) Should you get a straight line? Why or why not?

7. Prove or disprove the following conjecture. Allowing for various proportions of your wealth invested in two assets, a portfolio containing those two assets can be no safer than the safer asset of the two.

8. Draw two intertemporal substitution diagrams, each at least 4 inches by 4 inches. On one illustrate the income and substitution effects of an increase in interest rates for a saver. On the other illustrate the income and substitution effects of an increase in interest rates for a borrower.

9. Draw a production possibility frontier diagram that illustrates the motivation for saving for your retirement when you are in the 30-60 age range.

10. Suppose the one-year interest rate this year is 5%, but that rate is expected to increase to 7% next year? What current two-year rate would satisfy the expectations hypothesis?

11. Suppose the current one-year interest rate is 8% and the current two-year interest rate is 12%. You are going to receive $100 at the end of the fist year and you want to invest that sum over year two. Explain how you can lock in the forward interest rate implicit in the current term structure. What is that forward rate?

12. Suppose the difference between the one-year and two-year rates in the previous question is interpreted as a risk premium. What is the risk in locking in the forward rate?

13. Draw a diagram comparing the current yield curve to the ones on the handout.

The Term Structure of Interest Rates

The relationship between short-term and long-term interest rates depends on the risk of long-term assets. This link is a very important factor in how monetary policy determining short-term interest rates might hope to affect the economy through investment, which depends on long-term interest rates.

]]> The forward rate is a function of short-term and long-term interest rates.

The expectations hypothesis employs very similar algebra. Instead of calculating a forward rate, it assumes agents have an expectation of future short-term interest rates. Competition between short-term and long-term investments should then lead to a link between the current short-term and long-term rates.

The relation between short-term and long-term rates also includes some component that can be called a risk premium. In talking about the yield curve, the entire difference is often call the risk premium.

The risk that most concerns investors in long-term assets is, of course, the risk of inflation. There is no obvious process for distinguishing in the realized interest rates the difference, which long-term rates increase, between an increased risk premium due to the perceived risk of inflation and an increase in long-term rates due to expectations of higher future inflation and, hence, higher future short-term interest rates.

Mean-Variance Analysis

We construct a budget constraint from the aX+b rules for statistics, and we construct indifference curves from the utility-based analysis of risk.

Our mean-variance analysis diagram explains how people determine their optimal portfolio. Given that the parameters of the assets change every day with the arrival of news affecting the future, the diagram also explains why agents keep trading after they reach the optimal portfolio at one particular date.

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Homework Diagrams

Risk: Statitistics vs. Utility

We spent a few days (!) comparing the standard deviation/variance view of risk with the utility-based valuation of risk. The figures are based on the table in the handout.

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A closely related question is how much an agent will pay for insurance. The "insurance outcome" is less than the expected dollar outcome, but more than the value to the agent of the risky asset. Insurance companies can offer this deal because, by insuring a large number of agents, the insurance premium (the good outcome minus the insurance outcome) is greater than the good outcome minus the expected dollars.

Review of Some Statistics

We reviewed the definitions of expected value and variance. We also reviewd the "aX+b" rules that we will use to work with portfolios of securities.

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The next week we continued this discussion:

Statistics

Utility-Based Valuation of Risk

The fundamental basis for the value of risk could be curvature in agents utility functions.

]]> If the utility of $0 is 0, then there are three basic possibilities.

The log function would be an interesting choice, but we will use the square root function this semester.

The basic diagram showing the value of the risk inherent in an asset with two possible values.

Risk

We will study

1. Utility-Based Valuation of Risk

2. Mean-Variance Analysis

These are two views that are not entirely contradictory. The first emphasizes fundamental economics. The second emphasizes objective measures that might be obtained from historical statistics.

Investing in Education

What happens if you subsidize student loans?

Why is subsidizing education a bad idea? Why is it a good idea?

What if there is a limit to how much you can borrow?

Saving for Retirement

Progress Reports and Tables

Progress Report 3 is due before Halloween. You have to come see me in my office to get your term paper topic approved. Please note that the pdf file referenced below contains a form for PR3 and the Reference Report Form you should use to document your references.

]]> Term Paper Progress Reports and Forms (Please note that some of these pages, especially PR 2, are not as current as the revised printed handouts.)

Tables A, B, C, and D