X$402$ Math for Management: Problem Set $10$

Problems $1-2$ use the following table:

Table $1$: Selected Performance Statistics, $1926-1988$

Series Annual Return $($Geometric Mean Rate of Return$)$ Highest Annual Return Lowest Annual Return Standard Deviation of Annual Returns

Common Stock $10\mathrm{.}0\%54\mathrm{.}0\%-43\mathrm{.}3\%20\mathrm{.}9\%$

Small Company Stocks $12\mathrm{.}3\%142\mathrm{.}9\%-49\mathrm{.}8\%35\mathrm{.}6\%$

Long$-$term Corporate Bonds $5\mathrm{.}0\%43\mathrm{.}8\%-8\mathrm{.}1\%8\mathrm{.}4\%$

US Treasury Bills $3\mathrm{.}5\%14\mathrm{.}7\%0\mathrm{.}0\%3\mathrm{.}3\%$

Source: Malkiel, Burton Gordon. A Random Walk Down Wall Street. New York: Norton, $1973.$ Page $64.$

$1.($Multiple choice. Choose one answer.$)$ The average returns in the above table provide evidence that:

a$.$ There is a negative relationship between risk and return

b$.$ Stocks are more risky than bonds, on average

c$.$ Years with positive returns are followed by years with negative returns

d$.$ The excess return from Small Company Stocks does not adequately compensate investors for the additional volatility of their returns

$2.$ Assuming a risk$-$free rate of $3\mathrm{.}5\%,$ calculate the Sharpe Ratio for each of the asset types above. Show your calculations. Which asset class offers the greatest return per unit of risk?

$3.$ Table $3($below$)$ shows annual returns for the S&P $500$ for the years $2000-2016$:

Table $3$: Annual Returns

Year Returns

$2000-9\mathrm{.}0\%$

$2001-11\mathrm{.}9\%$

$2002-22\mathrm{.}0\%$

$200328\mathrm{.}4\%$

$200410\mathrm{.}7\%$

$20054\mathrm{.}8\%$

$200615\mathrm{.}6\%$

$20075\mathrm{.}5\%$

$2008-36\mathrm{.}6\%$

$200925\mathrm{.}9\%$

$201014\mathrm{.}8\%$

$20112\mathrm{.}1\%$

$201215\mathrm{.}9\%$

$201332\mathrm{.}2\%$

$201413\mathrm{.}5\%$

$20151\mathrm{.}4\%$

$201611\mathrm{.}7\%$

Calculate:

a$.$ The cumulative return over the $17$ years;

b$.$ The average annual return;

c$.$ The standard deviation;

d$.$ The Sharpe Ratio $($assuming a risk free rate of $2\mathrm{.}3\%$ on average$)$

$4.$ A Price$-$to$-$Earnings $($P$/$E$)$ ratio is a very popular and simple method to compare stock values.

a$.$ Briefly describe how P$/$E is used to value stocks

b$.$ Describe $2$ limitations or pitfalls of using P$/$E ratios to value stocks

$$

For problems $5-7,$ refer to the following table:

Table $4$: Problem $5-7$ Data

Stock Price Shares Outstanding Market Capitalization Earnings EPS P$/$E

$$25\mathrm{.}901,500,000$ $$450,000$

$20\mathrm{.}50260,000218,000$

$4\mathrm{.}005,200,0001,398,000$

$6\mathrm{.}50820,000(95,000)$

$108\mathrm{.}30520,000640,000$

$4\mathrm{.}106,300,000(8,210)$

$7\mathrm{.}00890,000129,000$

$9\mathrm{.}207,800,0005,158,000$

$21\mathrm{.}80650,00065,300$

$41\mathrm{.}20250,000561,000$

$5.$ Complete the column $$Market Capitalization.$$

$6.$ Complete the column $$EPS$.$

$7.$ Complete the column $$P$/$E$.$

$8.$ You calculate P$/$Es for $3$ stocks in an industry and get $217\mathrm{.}0,5\mathrm{.}2$ and $8\mathrm{.}6.$ What could be possible reasons for the outlier $(217\mathrm{.}0)?$

a$.$ The company$$s earnings suffered a one$-$time dip for a change in accounting method on its inventory.

b$.$ The company is relatively young, and its earnings are close to zero.

c$.$ The company$$s product is considered $$the next big thing$$ and was named $$hottest stock$$ by Valuation magazine.

d$.$ Any or all of the above.

$9.$

a$.$ When you want to know the volatility of a portfolio, you calculate the $\_\_\_\_\_\_\_\_\_\_\_\_\_.$

b$.$ When you want to know how stocks move relative to one another, you calculate the $\_\_\_\_\_\_\_\_\_\_\_\_\_.$

$10.$ Consider the portfolios presented in the following table:

Table $6$: Returns

Stock Portfolio $1$ Portfolio $2$ Portfolio $3$

A $8\mathrm{.}1\%-2\mathrm{.}1\%-2\mathrm{.}1\%$

B $-6\mathrm{.}8\%1\mathrm{.}1\%6\mathrm{.}0\%$

C $3\mathrm{.}0\%3\mathrm{.}0\%-2\mathrm{.}8\%$

D $-5\mathrm{.}2\%2\mathrm{.}0\%4\mathrm{.}8\%$

E $4\mathrm{.}8\%-0\mathrm{.}1\%-0\mathrm{.}1\%$

a$.$ Assuming the stocks are equally weighted in the portfolio, calculate the average return for each of the $3$ portfolios.

Consider portfolios $1$ and $2.$

b$.$ How are they the same?

c$.$ How are they different?

d$.$ If you were to invest money in one of the portfolios $(1$ or $2)$ going forward $($for the next period$),$ which would you choose? Why?

Using a spreadsheet program, you calculate the standard deviation of each portfolio and find:

Table $7$: Standard Deviation

Portfolio $1$ Portfolio $2$ Portfolio $3$

$6\mathrm{.}5\%2\mathrm{.}0\%4\mathrm{.}0\%$

e$.$ Do these results support or change your choice in part d above?

f$.$ Calculate the Sharpe Ratio for each of the portfolios $($assume a risk$-$free rate of $0\%).$

g$.$ What does it tell you?

Compare portfolios $2$ and $3.$

h$.$ Which would you choose based on the Mean Return alone?

i$.$ Based on the Sharpe Ratio?