## Learning Module 5 Due May 19

Objective

The objective of this learning module is to solidify our knowledge of derivative securities and to move into the area of portfolio diversification. Question 1 deals with put-call parity. This the relation between puts and calls that is enforced by arbitrage.

Question 2 deals with valuation. You are expected to be able to use the Black-Scholes option valuation program which is available in Java. With a given level of volatility and other inputs which are readily available, you determine what the option price should be. Of course, this valuation is critically dependent on your assessment of the future volatility of the underlying asset return. The question also explores implied volatility. In this case, you are given the option price (say from the newspaper) and other readily available inputs (strike price, time to expiration, underlying stock price, interest rate). You need to try different volatilities until the Java program delivers to you exactly the given option price. This is the implied volatility

Question 3 deals with diversification. In the first part, we have two securities that seem very similar (same mean and same variance) and we figure out what happens to the the portfolio variance when different assumptions are made on correlation. It is useful to look at the two asset efficient frontier program for this problem (though it does not allow you to input weights, you can visually see what is going on, just input some levels of means and variances, say 10% for the two assets' means and 20% for the two assets' variances).

In the second part, forces you to work out some numbers for portfolio variance. Given the data, use the formula for portfolio variance. I also recommend the Java program here to visualize what is going on.

The third part shows a variance-covariance matrix for three assets [remember the main diagonal are the variances and the off diagonal are covariances]. You are asked to calculate the correlations.

Question 4 explores the CAPM. In the first part, we use the CAPM delivers the expected rate of return for the security. The twist in this question is that we are given what the market expects the stock price to be in one year. Given the expected rate of return from the CAPM, we need to back out the stock price today.

The second part explores the idea of levering. Remember that the portfolio beta is just the weighted sum of individuals asset betas. We assume that the borrowing or lending has a beta of zero (although we could modify that assumption). Suppose a security had a beta of 1.00 and we had \$1 million to invest. Suppose we wanted a portfolio beta of 1.5. To do this, we borrow \$500,000 and invest \$1.5 million in the security. (Actually, this is just an example! I would want a diversified portfolio)

The third part explores what happens if risk changes. This is very important. As a firm takes on new projects and given that the expected rate of return on the firm's stock is a function of the underlying weighted average risk of its assets, we need to know what happens if risk changes.

1. Put-Call Parity

(a)You are given the following data. Intel's stock price is \$162.50. For calls and put options with an exercise price of \$160 and the same time to expiration (2 months), we have the following prices \$11.75 and \$8.125, respectively. There are no dividends. We can borrow or lend at the Treasury bill rate. A Treasury bill that pays \$1 million on the day the option expires is trading for \$992,968.75. [This delivers the e^-rT. So we don't need to worry about the interest rate or the time-to-maturity in days.] Show the options are price relatively correctly.

(b)Suppose instead of a price of \$8.125, the put is trading at \$9.125. All other information is identical. Show put call parity does not hold. Show how you might profit from this situation. Construct an arbitrage table. The table should have four columns: position in assets, cash flow today, cash flow at expiration if S*>K and cash flow at expiration if S* [Extra: what happens when you put your table positions into the advanced Java program.]

2. Derivative valuation

(a) You are given the following data. Intel's stock price is \$162.50. The call option with an exercise price of \$160 and 50 days to expiration has a price of \$11.75. The interest rate over this period is 5.151%. Using the Java program, try different volatilities until you hit exactly the market price of \$11.75.

(b)Suppose you believe that the true future volatility is really 55%. Is the option over or undervalued? What might you do to make a profit? Describe in words what you would do no calculations are necessary. Is this a pure "arbitrage" position (i.e. will you make profits for sure with no risk).

3. Diversification

Part I

Two securities, 1 and 2, have identical expected rates of return and variances of return, that is, E[R1] = E[R2] and Var(R1) = Var(R2) = .  Calculate the expected rate of return and the variance of the rate of return of a portfolio formed by investing 1/2 of one’s wealth in each of the two securities when:

(a) The rates of return of the two securities are perfectly positively correlated, that is, (b) The rates of return of the two securities are independent, that is, (c) The rates of return of the two securities are perfectly negatively correlated, that is, Note, you can use the Java program to visualize this but the program does not allow you to put in the weights.

Part II

Assume that you can either invest all of your wealth in one of two securities, 1 and 2, or some proportion of your wealth in each. The distribution of the rates of return have the following characteristics:

 Security Expected Return Standard Deviation 1 0.19 0.12 2 0.15 0.12

Assume that the returns of the two securities have zero correlation.

(a) Assume you place proportion w of your wealth in security 1 and 1-w of your wealth in security 2. Calculate the expected returns on the standard deviations of the following portfolios.

 Case w 1-w i 1.00 0.00 ii 0.75 0.25 iii 0.50 0.50 iv 0.25 0.75 v 0.00 1.00

(b) Plot your results from part (a) on a graph with the portfolio’s expected rate of return on the vertical axis and the standard deviation of the portfolio’s return on the horizontal axis. Sketch the expected return as a function of the volatility for the range of portfolio compositions where w ranges from 0 to 1.

Again, you can use the Java program to visualize this one - but you need to work the numbers.

Part III

Given the following variance/covariance matrix for assets A, B and C, calculate the three cross-correlations between the assets (i.e., )

 A B C A 0.20 0.17 0.12 B 0.17 0.25 0.07 C 0.12 0.07 0.30

I have a new Java program which does the three asset case and you can visualize how the efficient frontier if formed - by brute force!

4. Capital Asset Pricing Model

Part I

Suppose that the ABC company is expected to be worth \$100 per share one year from today. How much are you willing to pay for one share today if the risk-free rate is 8%, the expected rate of return on the market is E[rm]=17%, and the company’s beta is 1.9? Assume that no dividends are paid.

Part II

Suppose that the correlation coefficient between the rates of return on Knowlode Mutual Fund and the market portfolio is 0.7. The standard deviations of the rates of return are 0.25 for Knowlode and 0.20 for the market portfolio. How would you combine the Knowlode Fund and the riskless asset to obtain a portfolio with a relative systematic risk (beta) of 1.6?

Part III

The market price of a security is \$36, the security’s expected rate of return is 13%, the riskless rate of interest is 7% and the market risk premium is 8%. What will the security’s price be if the covariance of its rate of return with the market portfolio doubles?