## 1.7 Problems: Return Calculations

**Exercise 1.1**Consider the following (actual) monthly adjusted closing price data for Starbucks stock over the period December 2004 through December 2005

Date | Price |
---|---|

December, 2004 | $31.18 |

January, 2005 | $27.00 |

February, 2005 | $25.91 |

March, 2005 | $25.83 |

April, 2005 | $24.76 |

May, 2005 | $27.40 |

June, 2005 | $25.83 |

July, 2005 | $26.27 |

August, 2005 | $24.51 |

September, 2005 | $25.05 |

October, 2005 | $28.28 |

November, 2005 | $30.45 |

December, 2005 | $30.51 |

- Using the data in the table, what is the simple monthly return between the end of December 2004 and the end of January 2005? If you invested $10,000 in Starbucks at the end of December 2004, how much would the investment be worth at the end of January 2005?
- Using the data in the table, what is the continuously compounded monthly return between December 2004 and January 2005? Convert this continuously compounded return to a simple return (you should get the same answer as in part 1).
- Assuming that the simple monthly return you computed in part (1) is the same for 12 months, what is the annual return with monthly compounding?
- Assuming that the continuously compounded monthly return you computed in part (2) is the same for 12 months, what is the continuously compounded annual return?
- Using the data in the table, compute the actual simple annual return between December 2004 and December 2005. If you invested $10,000 in Starbucks at the end of December 2004, how much would the investment be worth at the end of December 2005? Compare with your result in part (3).
- Using the data in the table, compute the actual annual continuously compounded return between December 2004 and December 2005. Compare with your result in part (4). Convert this continuously compounded return to a simple return (you should get the same answer as in part 5).

**Exercise 1.2**Consider a one month investment in two Northwest stocks: Amazon and Costco. Suppose you buy Amazon and Costco at the end of September at \(P_{A,t-1}=\$38.23\), \(P_{C,t-1}=\$41.11\) and then sell at the end of the October for \(P_{A,t}=\$41.29\) and \(P_{C,t}=\$41.74\). (Note: these are actual closing prices for 2004 taken from Yahoo!)

- What are the simple monthly returns for the two stocks?
- What are the continuously compounded returns for the two stocks?
- Suppose Costco paid a $0.10 per share cash dividend at the end of October. What is the monthly simple total return on Costco? What is the monthly dividend yield?
- Suppose the monthly returns on Amazon and Costco from question (1) above are the same every month for 1 year. Compute the simple annual returns as well as the continuously compounded annual returns for the two stocks.
- At the end of September 2004, suppose you have $10,000 to invest in Amazon and Costco over the next month. If you invest $8000 in Amazon and $2000 in Costco, what are your portfolio shares, \(x_{A}\) and \(x_{C}\)?
- Continuing with part 5, compute the monthly simple return and the monthly continuously compounded return on the portfolio. Assume that Costco does not pay a dividend.

**Exercise 1.3**Consider a 60-month (5 year) investment in two assets: the Vanguard S&P 500 index (VFINX) and Apple stock (AAPL). Suppose you buy one share of the S&P 500 fund and one share of Apple stock at the end of January, 2010 for \(P_{VFINF,t-60}=\$89.91\), \(P_{AAPL,t-60}=\$25.88\), and then sell these shares at the end of January, 2015 for \(P_{VFINX,t}=\$184.2\), \(P_{AAPL,t}=\$116.7\). (Note: these are actual adjusted closing prices taken from Yahoo!). In this question, you will see how much money you could have made if you invested in these assets right after the financial crisis.

- What are the simple 60-month (5-year) returns for the two investments?
- What are the continuously compounded 60-month (5-year) returns for the two investments?
- Suppose you invested $1,000 in each asset at the end of January,
- How much would each investment be worth at the end of January, 2015?
- What is the compound annual return on the two 5 year investments?
- At the end of January, 2010, suppose you have $1,000 to invest in VFINX and AAPL over the next 60 months (5 years). Suppose you purchase $400 worth of VFINX and the remainder in AAPL. What are the portfolio weights in the two assets? Using the results from parts 1. and 2., compute the 5-year simple and continuously compounded portfolio returns.

**Exercise 1.4**Consider an investment in a foreign stock (e.g., a stock trading on the London stock exchange) by a U.S. national (domestic investor). The domestic investor takes U.S. dollars, converts them to the foreign currency (e.g. British Pound) via the exchange rate (price of foreign currency in U.S. dollars) and then purchases the foreign stock using the foreign currency. When the stock is sold, the proceeds in the foreign currency must then be converted back to the domestic currency. To be more precise, consider the information in the table below:

Time | Cost of 1 Pound | Value of UK Shares | Value in U.S. \(\$\) |
---|---|---|---|

0 | \(\$1.50\) | \(£40\) | \(1.5\times40=60\) |

1 | \(\$1.30\) | \(£45\) | $ 1.345=58.5$ |

- Compute the simple rate of return, \(R_{e}\), from the prices of foreign currency. This is the return to the domestic investor of investing in the foreign currency.
- Compute the simple rate of return, \(R_{UK}\), from the UK stock prices.
- Compute the simple rate of return, \(R_{US}\), from the prices in US dollars.
- What is the mathematical relationship between \(R_{US}\), \(R_{UK}\) and \(R_{e}\)?

**Exercise 1.5**Add R examples here. To be completed…