Investments
Homework1
Alexi Savov
Due at the start of Class 4
Topic 1: Financial Markets
1. You are a market maker in the stock of Gilead Sciences (GILD) and quote a bid of $102.25 and an ask of $102.50. Suppose that you have zero inventory.
- On Day 1, you receive market buy orders for 10,000 shares and market sell ordersfor 4,000 shares. How much do you earn on the 4,000 shares that you bought and sold? What is the value of your inventory at the end of the day?
(Hints: It is possible to have negative inventory. Further, there is more than one correct way to value your inventory, but please state what assumption your valuation is based on.)
- Before trading begins on Day 2, the company announces promising clinical trialresults for its drug remdesivir in treating COVID-19. Your quoted bid and ask jump to $110.25 and $110.50, respectively.
During Day 2, you receive market sell orders for 8,000 shares and buy orders for 2,000 shares. What is your total profit or loss over the two-day period? What is the value of your inventory at the end of Day 2?
- Discuss the following. What is a market maker’s objective? Is there anythingyou could have done during Day 1, consistent with a market maker’s objective, that would have improved your performance over the two-day period?
Topic 2: Performance Measures
- Suppose a 5-year zero-coupon Treasury bond with face value $1,000 has a 5% yield to maturity (annually compounded).
- What price does this bond sell for?
- Suppose another zero-coupon Treasury bond also has a 5% yield, but sells for $325.57. What is the maturity of this bond?
- Which of the following investments do you prefer?
- Purchase a zero-coupon bond, which pays $1,000 in ten years, for a price of $550. (b) Invest $550 for ten years in Chase at a guaranteed annual interest rate of 5.5%.
- Suppose you get for free one of following two securities: (a) an annuity that pays $10,000 at the end of each of the next 6 years; or (b) a perpetuity that pays $10,000 forever, but it does not begin until 10 years from now (i.e., the first cash payment from this perpetuity is 11 years from now).
Which security would you choose if the annual interest rate is 5%?
Does your answer change if the interest rate is 10%? Explain why or why not.
- Suppose a hedge fund manager earns 1% per trading day. There are 250 trading days per year. Answer the following questions:
- What will be your annual return on $100 invested in her fund if she allows youto reinvest in her fund the 1% you earn each day?
- What will be your annual return assuming she puts all of your daily earningsinto a zero-interest checking account and pays you everything earned at the end of the year?
- Can you summarize when it is proper to “annualize” using APR (annual percentage rate) versus EAR (effective annual rate)?
- Here are some alternative investments you are considering for one year. (i) Bank A promises to pay 8% on your deposits, compounded annually. (ii) Bank B promises to pay 8% on your deposits, compounded daily. Compare the effective annual rates (EARs) on these investments.
- (a) Suppose that you have purchased a 3-year zero-coupon bond with face value of $1000 and a price of $850. If you hold the bond to maturity, what is your annual rate of return?
(b) Now suppose you have purchased a 3-year bond with face value of $1,000, 7% annual coupon rate, and a price of $975. Assuming that you hold the bond to maturity, is the IRR greater or less than the return on the bond in part (a)?
- Excel Question. Download monthly S&P 500 data from January 1950 to today:
Go to http://finance.yahoo.com/, click S&P 500, click Historical Data, under Time
Period change the start date to 1950-01-01, under Frequency select Monthly, click Apply, click Download Data. (Note: the Adjusted Close column includes the value of dividends paid.)
- What is your best estimate for next month’s return?
- What would have been your annualized HPR if you invested as of the start ofthe index?
- In what month occurred the lowest monthly return? What happened?
Topic 3: Portfolio Management with 2 Risky Assets
9. The expected returns and standard deviation of returns for two securities are as follows:
| Security Z | Security Y | |
| Expected Return | 15% | 35% |
| Standard Deviation | 20% | 40% |
The correlation between the returns is 0.25.
- Calculate the expected return and standard deviations of the following portfolios:
- 100% in Z
- 75% in Z and 25% in Y iii. 50% in Z and 50% in Y iv. 25% in Z and 75% in Y
v. 100% in Y
- Draw the mean-standard deviation frontier.
- Which portfolios might be held by an investor who likes high mean and low standard deviation?
Topic 4: Portfolio Theory with a Riskless Assets
- Suppose that a fund that tracks the S&P 500 has mean E[Rm] = 16% and standard deviation sM = 10%, and suppose that the T-Bill rate is Rf = 8%. Answer the following questions about efficient portfolios:
- What is the expected return and standard deviation of a portfolio that is totally invested in the risk-free asset?
- What is the expected return and standard deviation of a portfolio that has 50% of its wealth in the risk-free asset and 50% in the S&P 500?
- What is the expected return and standard deviation of a portfolio that has 125% of its wealth in the S&P 500, financed by borrowing 25% of its wealth at the risk-free rate?
- What are the weights for investing in the risk-free asset and the S&P 500 that produce a standard deviation for the entire portfolio that is twice the standard deviation of the S&P 500? What is the expected return on that portfolio?
- Consider the following data:
| Expected Return | Standard Deviation | |
| Russell Fund | 16% | 12% |
| Windsor Fund | 14% | 10% |
| S&P 500 Fund | 12% | 8% |
The correlation between the returns on the Russell Fund and the S&P 500 Fund is 0.7. The T-Bill rate is 6%. Which of the following portfolios would you prefer to hold in combination with T-bills and why?
- Russell Fund
- Windsor Fund
- S&P 500 Fund
- A portfolio of 60% Russell Fund and 40% S&P 500 Fund.
- Excel Question. Use the portfolio optimizer posted on the class website. Find out what happens to the share of asset 3 in the optimal risky portfolio in the following cases. Explain in words why you think this happens.
- The expected return of asset 3 is increased.
- The standard deviation of asset 3 is increased. (c) The standard deviation of asset 2 is increased.
(d) The correlation between assets 2 and 5 is increased.
