Unit06 Sample Homework Problems CHAPTER 10. INVESTMENT RETURNS AND AGGREGATE MEASURES OF STOCK MARKETS

Size: px

Start display at page:

Download "Unit06 Sample Homework Problems CHAPTER 10. INVESTMENT RETURNS AND AGGREGATE MEASURES OF STOCK MARKETS"

Lorin Scott
8 years ago
Views:

1 Unit06 Sample Homework Problems CHAPTER 10. INVESTMENT RETURNS AND AGGREGATE MEASURES OF STOCK MARKETS 1. You buy a stock for $40. After a year the price rises to $50 but falls back to $40 at the end of the second year. What was the average percentage return? Since the beginning and closing prices are the same, the return has to be 0%. The percentage changes for each year are 25% and 20%, so the average percentage change is 2.5%. 3. You make an investment and the annual return are as follows: Year Return 1 15% 2 13% 3-8% 4 0% 5 5% The average annual return is 3 percent. What is the geometric time-weighted return? The geometric time-weighted average return is [(1.15)(1.13)(0.92)(1)(1.05)]^(1/5)-1 = 4.65% 4. Given the following information concerning four stocks, --- Price Number of Shares Stock A $20 100,000 Stock B 34 50,000 Stock C ,000 Stock D ,000 a. Construct a simple price-weighted average, a value-weighted average, and a geometric average. 1

Since the beginning and closing prices are the same, the return has to be 0%. The percentage changes for each year are 25% and 20%, so the average percentage change is 2.5%. 3.

2 a. Simple price-weighted average: ($ )/4 = $30 Value-weighted average: Stock Price Number of Shares Total Value A $20 100,000 $2,000,000 B 34 50,000 1,700,000 C ,000 3,900,000 D ,000 8,000, ,000 $15,600,000 Average value per share: $15,600,000/500,000 = $31.20 Geometric average: [($20)(34)(26)(40)]^(1/4)=707200^(0.25)=$29 b. What is the percentage change in a value-weighted average if the stocks' prices become $10, $17, $13, $40? The new Value-weighted average: Stock Price Number of Shares Total Value A $10 100,000 $1,000,000 B 17 50, ,000 C ,000 1,950,000 D ,000 8,000, ,000 $11,800,000 Average value per share: $11,800,000/500,000 = $23.60 % Change =23.60/31.2-1=-24.36% 6. An investor buys a stock for $10 and sells it for $15 after five years. There is no dividend income. (a)what is the holding period return on an annual basis? (b) What is the true annual rate of return? a. The holding period return: 15/10-1 = 50%. On an annual basis: 50%/5 = 10% b. The true annual rate of return: $10(1 + r)^5 =15 Solving for r: r=(15/10)^(1/5)-1=8.45% 2

What is the percentage change in a value-weighted average if the stocks' prices become $10, $17, $13, $40?

3 13. You purchase a stock for $30 and sell it for $40 after holding it for five years. During this period you collected an annual dividend of $1. Did you earn more than 10 percent annual internal rate of return on your investment? What was the annual dollar-weighted rate of return? Price = div div + sales price (1 + r) (1 + r) n (1 + r) n In this case: 30=$1*(PVFS, r=?, n=5)+40 (PVF r=?, n=5) PVFS=present value factor sum PVF=present value factor 30=1*(1-(1/(1+r)^5))/r)+40/(1+r)^5 For a given rate to be the correct return, the two sides of the equation must be equal. At 10 percent, the right-hand side of the equation equals to $1( ) + 40( ) = $ ( is the PVFS when r=10% and n= is the PVF when r=10% and n=5.) The present value of the cash flows ($28.63) does not equal the cost of the investment ($40), so the rate of return is less than 10 percent. The actual dollar-weighted return is 8.92 percent. (PV = -40; N = 5; PMT = 1; FV = 40; I =? = 8.92%.) You can use Excel to solve this problem by trying different r s until the equation 30=1*(1-(1/(1+r)^5))/r)+40/(1+r)^5 holds. 18. You believe that QED stock may be a good investment and decide to buy 100 shares at $30. You subsequently buy an additional $3,000 worth of the stock every time the stock s price declines by an additional $3. (That means you bought $3000 when the price went down to $27, and another $3000 when the price went down to $24). If the stock s price declines to $22 and rebounds to $32, at which time you sell your holdings, what is your profit? (Assume that no fractional shares may be purchased so you may spend a little less than 3,000 each time.) 3

.. + div + sales price (1 + r) (1 + r) n (1 + r) n In this case: 30=$1*(PVFS, r=?, n=5)+40 (PVF r=?

4 This problem illustrates dollar cost averaging. Price of the Number of Shares Cost of Stock Stock Purchased $ $3, , , $8,997 Average cost per share: $8,997/336 = $26.78 Proceeds of the sale: 336 X $32 = $10,752 Profit: $10,752 8,997 = $1, On January 31, 2001, you bought 10 shares of AVAYA (AV) for $7 a share. Subsequent prices of AV were: January 1, 2002, $5; January 1, 2003, $2; January 1, 2004, $15. You owned the stock for three years (2001 through 2004). What was your holding period return? What was the average percentage return? What was your true return on this investment? Holding period return: $15/$7-1 =114.29% Yearly percentage returns: First year: $5/7-1 = % Second year: 2/5-1 = -60% Third year: 15/2-1 = 650% Average percentage return for three years: [(-28.57%) + (-60%) + 650%]/3 = % True return: $7(1 + r) 3 = $15 r=(15/7)^(1/3)-1=28.92% Averaging percentage returns can produce very misleading results. 4

78 Proceeds of the sale: 336 X $32 = $10,752 Profit: $10,752 8,997 = $1,775 19. On January 31, 2001, you bought 10 shares of AVAYA (AV) for $7 a share.

5 Teaching Guides for the Financial Advisor s Case: The Calculation of Returns 1. Return on the market: (1.12)(1.17)(1.02)(0.97)(1.14).2 = = 8.13% Return on the stock: Annual returns Year 1: Amount invested $10 Yearly cash inflow $ Percentage return 8% Year 2: Amount invested $10.50 Yearly cash inflow $ Percentage return 23.8% Year 3: Amount invested $12 Yearly cash inflow $ Percentage return -2.1% Year 4 Amount invested $11.25 Yearly cash inflow $ Percentage return 20.0% Year 5 Amount invested $12.50 Yearly cash inflow $ Percentage return 22.2% (1.08)(1.238)(0.979)(1.20)(1.22).2 = = 13.89% 2. Questions 2 and 3 require considering the beta of the stock and the standard deviations of the returns. The standard deviation of the returns is 8.5%, and the standard deviation of the stock returns is 11.1%. The beta coefficient is with the coefficient of determination of Based on this information, the stock is less volatile than the market because the beta is less than 1.0. These investors bear less systematic risk, but the low coefficient of determination 5

50 Yearly cash inflow $12 + 1 Percentage return 23.8% Year 3: Amount invested $12 Yearly cash inflow $11.25 + 0.50 Percentage return -2.1% Year 4 Amount invested $11.25 Yearly cash inflow $12.

6 suggests there is considerable unsystematic risk associated with the security. 3. Since the standard deviation of the stock exceeds the standard deviation of the market returns, the stock returns are more variable than the market returns. 4. The answer to this question depends on the measure of risk used. The beta indicates that the stock is less responsive to changes in the market, but there is more variability in its returns. 5. The dollar-weighted return is $10 = $ $ $ $ $ (1 + r) (1 + r) 2 (1 + r) 3 (1 + r) 4 (1 + r) 5 r = 13.7% 6. If the investor purchased $1,000 (100 shares) and reinvested all dividends, the number of shares at the end of each year would be End of Year $30/(10.50) = shares $ /(12.00) = shares $55.714/(11.25) = shares $116.38/(12.50) = shares $ /(14.00) = shares The investor now holds shares worth $1, ( X $14). Based in the beginning and ending values of the shares, the annual rate of return is $1,000(1 + r) 5 = $1, r = ( ).2-1 = 13.9%. 7. Summary of the rates on the stock: 6

The answer to this question depends on the measure of risk used. The beta indicates that the stock is less responsive to changes in the market, but there is more variability in its returns. 5.

7 Time-weighted: 13.9% Dollar-weighted: 13.7% Dividend reinvested: 13.9% There is very little difference in the returns primarily because the cash flows and the stock prices do not dramatically change each year. The 22 percent return in year 5 favors the dividend-reinvested return. 8. If the individual invests $1,000 each year as well reinvesting the dividends, the total shares acquired are End of Shares Cash Received Total Year Purchased and Invested Shares /(10.50) = /10.50 = /(12.00) = /10.50 = /(11.25) = /11.25 = /(12.50) = /12.50 = /14.00 = The investor has shares worth $7, ( X 14 = 7,809.10). In this case the investor earns the highest return because the stock rose 20 and 22 percent during the last two years. Thus, the larger number of shares purchased during the first three years earned the highest return when the stock's price rose during the last year, which increased the overall return on the strategy. Since the investor is concerned with the return earned on each dollar invested, the dollar-weighted rate of return gives that individual the best indication of the investment's performance. 7

The 22 percent return in year 5 favors the dividend-reinvested return. 8.

Chapter 9. The Valuation of Common Stock. 1.The Expected Return (Copied from Unit02, slide 39)

Readings Chapters 9 and 10 Chapter 9. The Valuation of Common Stock 1. The investor s expected return 2. Valuation as the Present Value (PV) of dividends and the growth of dividends 3. The investor s required