2. Measuring Returns and Risk

Question 1

The holding period return (HPR) relates the accrued profit on an investment directly to its beginning value.

Answer 1

True.

Question 2

An asset’s per-period return (PPR) is defined as the sum of that period’s income payments and price appreciation minus its beginning-of-period price.

Answer 2

False. An asset’s PPR is defined as the sum of that period’s income payments and price
appreciation divided by its beginning-of-period price, not minus.

Question 3

Accumulating returns over time and earning a return on the return of previous periods is called compounding.

Answer 3

True.

Question 4

If an investor deposits $10,000 in an account that yields 5 percent compounded annually, and has no withdrawals for 2 years, then he or she will end up with $11,000.

Answer 4

False. This $11,000 ignores the effects of compounding. The ending value will be
$11,025.

Question 5

Unless the PPRs are all identical, the geometric mean return will always be less than the arithmetic mean.

Answer 5

True.

Question 6

In investments, the term ‘risk’ refers to the dispersion of possible returns.

Answer 6

True.

Question 7

In insurance, one can insure for the expected profit, as well as compensation for loss.

Answer 7

False. In insurance, one can only insure for loss of value, not for loss of potential profits.

Question 8

Inflation risk is the risk that the general level of prices will grow, but at steadily slower rates.

Answer 8

False. Inflation risk is the risk of the loss of purchasing power. Disinflation is the
phenomenon where the general level of prices grows at a slower rate.

Question 9

‘Additional commitment risk’ is the risk that one might have to put additional monies into an investment.

Answer 9

True.

Question 10

‘Interest rate risk’ is the risk that the interest rate on a margin loan will exceed the rate of return on the investment.

Answer 10

False. Interest rate risk has two components: price risk and reinvestment rate risk. The
first is the change in value due to changes in interest rates, and the latter is the impact
due to changes in interest rates on the income produced by the reinvestment of cash
flows. The rate on margin loans has nothing to do with this risk.

Question 11

Market risk is the risk that a security cannot be sold quickly at the current market value.

Answer 11

False. The potential inability to quickly sell an asset with no price concession is liquidity
risk.

Question 12

‘Business risk’ is the risk from events that affect a particular company.

Answer 12

True.

Question 13

‘Political risk’ is the risk of loss in value due to fluctuations in the exchange rate.

Answer 13

False. Political risks are the risks that come from events while operating in a foreign
country, such as expropriation. Exchange rate risk is the risk of loss in value due to
fluctuations in the exchange rate.

Question 14

‘Tax risk’ is the risk that one might have to pay taxes on one’s income.

Answer 14

False. Tax risk is the risk of loss due to changes in the tax code.

Question 15

The standard deviation of returns is a measure of an asset’s riskiness.

Answer 15

True.

Question 16

The range of returns is among the simplest and most informative measures of the dispersion of returns.

Answer 16

False. Although the range of returns is among the simplest, it is not informative, as
the sum is always zero.

Question 17

Calculating the semivariance involves using only the prospective returns that are above the expected return.

Answer 17

False. Calculating the semivariance involves using only the prospective returns that
are below the expected return.

Question 18

Calculating the semivariance involves using only the prospective returns that are above the expected returns.

Answer 18

False. The variance must be computed first; then the standard deviation can be computed
by taking the square root of the result.

Question 19

The calculation of the expected return when all returns are equally weighted is the same process as calculating the arithmetic mean return when using historical data.

Answer 19

True.

Question 20

If the standard deviation of a security’s return is 10 percent, then its variance is 100 percent squared, or .100.

Answer 20

False. The variance is 100 percent-squared, but in decimal notation this is .0100 (.10 x
.10 = .01).

Question 21

The initial margin rate is the amount of money one can borrow to buy a security.

Answer 21

False. The initial margin rate is the amount of cash the investor must put up.

Question 22

The Federal Reserve Board has set the initial margin requirement on stocks at 25 percent.

Answer 22

False. The initial margin requirement on stocks is set at 50 percent.

Question 23

Buying on margin reduces the variability of an investor’s returns.

Answer 23

False. Buying on margin increases the variability of an investor’s returns.

Question 24

All margin loans are at the call-loan rate.

Answer 24

False. The investor is charged the call-loan rate plus a mark-up that depends on the
size of the loan balance.

Question 25

Margin loans may remain outstanding as long as the borrower’s equity position does not fall below the maintenance margin percentage.

Answer 25

True.

Question 26

A margin call is issued whenever the market value of the account is less than the loan balance divided by one minus the maintenance margin rate.

Answer 26

True.

Question 27

The maintenance margin rate is lower than the initial margin rate.

Answer 27

True.

Question 28

The three ways to satisfy a margin call are to add cash to the account, add marginable securities, or to sell holdings and use the proceeds to pay down the loan balance.

Answer 28

True.

Question 29

Buying on margin is even riskier when a significant portion of the portfolio is concentrated in one holding.

Answer 29

True.

Question 30

As long as an account has borrowing power, one can withdraw cash by increasing the loan balance.

Answer 30

True.

Question 31

Compute the HPRR and the HPR for an investment in land purchased for $5,000 and sold 3 years later for $7,000.

Answer 31

HPRR = $7,000/$5,000 = 1.4

HPR = 1.4 – 1 = .4 = 40%

or HPR = ($7,000 – $5,000)/$5,000 = .4 = 40%

Question 32

Compute the HPRR and the HPR for a building that is held for 9 months, during which time it generates $3,500 in net cash inflows and is then sold for a $30,000 profit. Its original purchase price was $195,000.

Answer 32

HPRR = $228,500/$195,000 = 1.17

HPR = 1.17 – 1 = .17 = 17%

or HPR = ($30,000 + $3,500)/$195,000 = 17 = 17%

Question 33

Compute the PPRs and the PPRRs for an investment worth $10 at the
start of the first period, $11 at the end of the first period, $12 at the end of
the second period, and $11 at the end of the third period. Assume a $.50
dividend is paid at the end of each period.

Answer 33

For the arithmetic mean return: (–22% + 7% + 21%)/3 = 2%. For the geometric mean return, we must begin by computing the PPRRs as PPRR1 = 1 – .22 = .78, PPRR2 = 1 + .07 = 1.07, and PPRR3 = 1 + .21 = 1.21. Next, we compute the HPRR as: HPRR =(0.78)(1.07)(1.21) = 1.0099. Then we can compute the geometric mean return by taking the cube root of the HPRR (it is the cube root because there are three time periods), and subtracting one, as follows:

GMR = (HPRR)1/n – 1 = GMR = 1.00991/3 – 1= 1.0033 – 1 = .0033, or .33 percent.
HP-10BII keystrokes: SHIFT, C ALL, .78, x, 1.07, x, 1.21, =, SHIFT, yx, .333333, =,
–, 1, = (display: 0.0033)

Question 34

You are considering an investment that you believe has a 25 percent probability of a –15 percent rate of return, a 40 percent probability of a 5 percent rate of return, and a 35 percent probability of a 25 percent rate of return. What is the expected rate of return on this investment?

Answer 34

Expected return = .25 x (–15) + .40 x 5 + .35 x 25 = 7.00

Keystrokes: SHIFT, C ALL, .25, x, 15, =, +/–, M+, .40, x, 5, =, M+,.35, x, 25, =, M+,
RM (display: 7.0000)

Question 35

You have a holding period rate of return of 4 percent. What is your effective
annual rate of return if your holding period is
a. 3 months
b. 6 months
c. 9 months
d. 15 months

Answer 35

a. Rear = (1 + rHPR)# – 1 = (1 + .04)12/3 – 1 = .1699 or 16.99%

KEYSTROKES: SHIFT, C, ALL, 12, ÷, 3, =, →M, 1, +, .04, SHIFT, yx, RM, =, –, 1, =

b. Rear = (1 + rHPR)# – 1 = (1 + .04)12/6 – 1 = .0816 or 8.16%
c. Rear = (1 + rHPR)# – 1 = (1 + .04)12/9 – 1 = .0537 or 5.37%
d. Rear = (1 + rHPR)# – 1 = (1 + .04)12/15 – 1 = .0319 or 3.19%

Question 36

If an investor wants a 5 percent real rate of return, and the inflation rate is
4 percent, what nominal rate must he or she obtain?

Answer 36

(1 + .05) x (1 + .04) – 1 = .092 or 9.20%

Question 37

If an investor is receiving a nominal rate of return of 15 percent, and the
inflation rate is 10 percent, what is his or her real rate of return?

Answer 37

(1 + .15)/(1 + .10) – 1 = .0455 or 4.55%

Question 38

Assume XYZ stock has the following rates of return for the last 3 years:
–15 percent, 10 percent, and 35 percent. Determine XYZ’s mean return,
variance, standard deviation, and coefficient of variation based on this
historical data.

Answer 38

Mean return = (–15% + 10% + 35%)/3 = 10%

Note: If the problem is solved by plugging the number into equations 2-14 and 2-15,
then one must solve for the variance first, followed by the standard deviation. If this
problem is solved using the special keystrokes, then the standard deviation is computed first.

Variance = [(–15 – 10)2 + (10 – 10)2 + (35 – 10)2]/(3 – 1) = [625 + 0 + 625]/2 = 625
percent-squared
Standard deviation = (625 percent-squared).5 = 25%

KEYSTROKES: SHIFT, C ALL, 15, +/–,Σ+, 10,Σ+, 35, Σ+, SHIFT, SxSy

VARIANCE: 25, SHIFT, x2

Coefficient of variation = 25/10 = 2.5

Question 39

–5 percent rate of return, a 60 percent probability of a 15 percent rate of
return, a 15 percent probability of a 60 percent rate of return, is it skewed to
the right or left, and is it more or less attractive to an investor than a normal
distribution with the same expected rate of return and standard deviation?

Answer 39

This distribution is skewed to the right (i.e., the long part of the tail is to the right of
the largest part of the distribution. This would be more attractive to an investor
than a normal distribution with the same expected rate of return and standard
deviation, as investors love positive skewness.

Question 40

David Gordon buys 100 shares of Uhoh Corp. at $50 per share. The initial margin rate is 60 percent, the maintenance margin rate is 25 percent, and he buys it with minimum margin.
a. How much does he initially borrow?
b. If the stock rises to $80 per share shortly thereafter (ignore interest
charges), how much cash can he withdraw from his account?
c. If the stock rises to $80 per share shortly thereafter (ignore interest
charges), how many additional shares can he buy without depositing any
additional cash?
d. If the stock falls rather than rises, what is the lowest price to which it can
fall without receiving a margin call?
e. Suppose the stock falls to $20 per share. What is the size of the margin
call?

Answer 40

a. $50 x 100 shares = $5,000
$5,000 x .60 initial margin rate = $3,000 cash paid in $5,000 purchase price – $3,000 cash paid in = $2,000 loan

b. Maximum cash withdrawal = [MV x (1 – IMR)] – L = [(100 shares x $80) x (1 –
.60)] – $2,000 = [$8,000 x .40] – $2,000 = $3,200 – $2,000 = $1,200

c. Buying power = (E/IMR) – MV = [(MV – L)/IMR] – MV = [($8,000 – $2,000)/.60]
– $8,000 = $10,000 – $8,000 = $2,000. Number of new shares bought = buying
power/price per share = $2,000/$80 = 25 new shares

d. MV x (1 – MMR) ≥ Loan
MV x (1 – .25) ≥ $2,000
MV ≥ $2,000/.75 = $2,666.67
MV/number of shares = $2,666.67/100 = $26.67

e. Cash added = Loan – MV x (1 – MMR) = $2,000 – ($20 x 100) x (1 – .25) = $2,000
– $1,500 = $500

Question 41

Ralph Harrison buys 100 shares of Dynamite Corp. for $30 per share. He
purchases the stock with 70 percent margin (that is, he borrows 30 percent
of the purchase price). The stock jumps to $35 per share shortly thereafter.

What are his ROA and ROE? (Ignore interest charges.)

Answer 41

ROA = ($35 – $30)/$30 = .1667, or 16.67%

ROE = ($3,500 – $3,000)/($3,000 x .70) = .2381, or 23.81%