Answers to Questions
1.The purpose of market indexes is to provide a general indication of the aggregate
market changes or market movements. More specifically, the indicator series are used
to derive market returns for a period of interest and then used as a benchmark for
evaluating the performance of alternative portfolios. A second use is in examining the
factors that influence aggregate stock price movements by forming relationships
between market (series) movements and changes in the relevant variables in order to
illustrate how these variables influence market movements. A further use is by
technicians who use past aggregate market movements to predict future price
patterns. Finally, a very important use is in portfolio theory, where the systematic risk
of an individual security is determined by the relationship of the rates of return for the
individual security to rates of return for a market portfolio of risky assets. Here, a
representative market indicator series is used as a proxy for the market portfolio of
risky assets.
2. A characteristic that differentiates alternative market indexes is the sample--the size of
the sample (how representative of the total market it is) and the source (whether
securities are of a particular type or a given segment of the population (NYSE, TSE).
The weight given to each member plays a discriminatory role--with diverse members
in a sample, it would make a difference whether the series is price-weighted, value-
weighted, or unweighted. Finally, the computational procedure used for calculating
return--i.e., whether arithmetic mean, geometric mean, etc.
3. A price-weighted index is an unweighted arithmetic average of current prices of the
securities included in the sample--i.e., closing prices of all securities are summed and
divided by the number of securities in the sample.
A $100 security will have a greater influence on the series than a $25 security because
a 10 percent increase in the former increases the numerator by $10 while it takes a 40
percent increase in the price of the latter to have the same effect.
4. A value-weighted index begins by deriving the initial total market value of all stocks
used in the series (market value equals number of shares outstanding times current
market price). The initial value is typically established as the base value and assigned
an index value of 100. Subsequently, a new market value is computed for all
securities in the sample and this new value is compared to the initial value to derive
the percent change which is then applied to the beginning index value of 100.
5.Given a four security series and a 2-for-1 split for security A and a 3-for-1 split for
security B, the divisor would change from 4 to 2.8 for a price-weighted series.
Stock Before Split Price After Split Prices
B 30 10
20
C 20
D 30 30Total 100/4 = 25 70/x = 25
x = 2.8
The price-weighted series adjusts for a stock split by deriving a new divisor that will
ensure that the new value for the series is the same as it would have been without the
split. The adjustment for a value-weighted series due to a stock split is automatic.
The decrease in stock price is offset by an increase in the number of shares
outstanding.
Before Split
Stock P rice/Share # of Shares Market Value
A $20 1,000,000 $ 20,000,000
B 30 500,000 15,000,000
C 20 2,000,000 40,000,000
D 30 3,500,000 105,000,000
Total $180,000,000
The $180,000,000 base value is set equal to an index value of 100.
After Split
Stock Price/Share # of Shares Market Value
A $10 2,000,000 $ 20,000,000
B 10 1,500,000 15,000,000
C 20 2,000,000 40,000,000
D 30 3,500,000 105,000,000
Total $180,000,000
New Index Value = [(Current Market Value)/ Base Value] x Beginning Index Value
= [180,000,000/180,000,000] x 100 = 100
This is what one would expect since there has been no change in prices other than the
split.
6.In an unweighted price index, all stocks carry equal weight irrespective of their price
and/or their value. One way to visualize an unweighted index is to assume that equal
dollar amounts are invested in each stock in the portfolio, for example, an equal
amount of $1,000 is assumed to be invested in each stock. Therefore, the investor
would own 25 shares of GM ($40/share) and 40 shares of Coors Brewing ($25/share).
a $100 stock. An unweighted price index which consists of the above three stocks
would be constructed as follows:
Stock Price/Share # of Shares Market Value
GM $40 25 $1,000Coors 25 40 1,000
Total $2,000
A 20% price increase in GM:
Stock Price/Share # of Shares Market Value
Coors 25 40 1,000
Total $2,200
A 20% price increase in Coors:
Stock Price/Share # of Shares Market Value
Coors 30 40 1,200
Total $2,200
Therefore, a 20% increase in either stocks would have the same impact on the total
value of the index (i.e., in all cases the index increases by 10%. An alternative
treatment is to compute percentage changes for each stock and derive the average of
these percentage changes. In this case, the average would be 10% (20% - 0%)). So in
the case of an unweighted price-indicator series, a 20% price increase in GM would
have the same impact on the index as a 20% price increase of Coors Brewing.
7.Based upon the sample from which it is derived and the fact that is a value-weighted
index, the Wilshire 5000 Equity Index is a weighted composite of the NYSE
composite index, and the Nasdaq composite index. We would expect it to have the
highest correlation with the NYSE Composite Index because the NYSE has the
highest market value.
8.The high correlations between returns for alternative NYSE price indicator series can
be attributed to the source of the sample (i.e. stock traded on the NYSE). The four
series differ in sample size, that is, the DJIA has 30 securities, the S&P 400 has 400
securities, the S&P 500 has 500 securities, and the NYSE Composite about 2818
stocks. The DJIA differs in computation from the other series, that is, the DJIA is a
price-weighted series where the other three series are value-weighted. Even so, there
is strong correlation between the series because of similarity of types of companies.
9.The two stock price indexes (Tokyo SE and Nikkei) for the Tokyo Stock Exchange
show a high positive correlation (.82). However, the two indexes represent
substantially different sample sizes and weighting schemes. The Nikkei-Dow Jones
Average consists of 225 companies and is a price weighted series. Alternatively, the
Tokyo SE encompasses a much large set of 1800 companies and is a value-weighted
series.
The correlation between the Tokyo SE and S&P 500 is substantially lower at 0.328.
These results support the argument for diversification among countries.
10.Since the Wilshire 5000 equal weighted series implies that all stocks carry the same
weight, irrespective of price or value, the results indicate that on average all stocks in
the index increased by 23 percent. On the other hand, the percentage change in thevalue of a large company has a greater impact than the same percentage change for a
small company in the value weighted index like the Wilshire 5000 market value
weighted series. Therefore, the difference in results indicates that for this given
period, the smaller companies in the index outperformed the larger companies.
11.The bond-market series are more difficult to construct due to the wide diversity of
bonds available. Also bonds are hard to standardize because their maturities and
market yields are constantly changing. In order to better segment the market, you
could construct five possible sub indexes based on coupon, quality, industry, maturity,
and special features (such as call features, warrants, convertibility, etc.).
12.Since the Merrill Lynch-Wilshire Capital Markets index is composed of a distribution
of bonds as well as stocks, the fact that this index increased by 15 percent, compared
to a 5 percent gain in the Wilshire 5000 Index indicates that bonds outperformed
stocks over this period of time.
13.The Russell 1000, and Russell 2000 represent two different population sample of
stocks, segmented by size. The fact that the Russell 2000 (which is composed of the
smallest 2,000 stocks in the Russell 3000) increased more than the Russell 1000
(composed of the 1000 largest capitalization U.S. stocks) indicates that small stocks
performed better during this time period.
14.One would expect that the level of correlation between the various world indexes
should be relatively high. These indexes tend to include the same countries and the
largest capitalization stocks within each country.
15.High yield bonds (ML High Yield Bond Index) have definite equity characteristics.
Consequently, they are more highly correlated with the NYSE composite stock index
rather than the ML Aggregate Bond Index.
16.Indexes with the broadest representation of U.S. stocks include the Nasdaq Composite
(5575 stocks), Wilshire 500 Equity Value (5000 stocks), and the NSYE Composite
(2818 stocks). For portfolio managers wishing to construct a broadly diversified
portfolio of U.S. stocks these indexes would be appropriate benchmarks.CHAPTER 7
Answers to Problems
1
(a) Given a three security series and a price change from period T to T+1, the percentage
change in the series would be 42.85 percent.
Stock Period T Period T+1
A $60 $80
B 20 35
C 18 25
Divisor 3 3
Average 32.67 46.67
Percentage change = (46.67 - 32.67)/ 32.67 = 42.85%
(b) Period T
Stock Price/Share # of Shares Market Value
A $60 1,000,000 $ 60,000,000
B 20 10,000,000 200,000,000
C 18 30,000,000 540,000,000
$800,000,000
Period T+1
Stock Price/Share # of Shares Market Value
A $80 1,000,000 $80,000,000
B 35 10,000,000 350,000,000
C 25 30,000,000 750,000,000
$1,180,000,000 Percentage change = (1,180,000,0000 – (800,000,000)/800,000,000 = 47.50%
(c) The percentage change for the price-weighted series is a simple average of the
differences in price from one period to the next. Equal weights are applied to each
price change.
The percentage change for the value-weighted series is a weighted average of the
differences in price from one period T to T+1. These weights are the relative market
values for each stock. Thus, Stock C carries the greatest weight followed by B and
then A. Because Stock C had the greatest percentage increase and the largest
weight, it is easy to see that the percentage change would be larger for this series
than the price-weighted series.2.
(a) Period T
Stock Price/Share # of Shares Market Value
B 20 50.00 1,000.00
C 18 55.56 1,000.00
$3,000.00
Period T+1
Stock Price/Share # of Shares Market Value
A $80 16.67 $1,333.60
B 35 50.00 1,750.00
C 25 55.56 1,3.00
Total $4,472.60
Percentage change = (4,472.60 - 3,000)/(3,000) = 49.09%
(b) Stock A = (80 – 60)/60 = 33.33%
Stock B = (35 – 20)/20 = 75.00%
Stock C = (25 – 18)/18 = 38.%
Arithmetic average = (33.33% + 75.00% + 38.%)/3 = 49.07%
The answers are the same (slight difference due to rounding). This is what you
would expect since Part A represents the percentage change of an equal-weighted
series and Part B applies an equal weight to the separate stocks in calculating the
arithmetic average.
(c) Geometric average is the nth root of the product of n items.
Geometric average = [(1.3333)(1.75)(1.38)]1/3 - 1
= [3.2407]1/3 - 1
= 1.4798 - 1 = .4798 or 47.98%
The geometric average is less than the arithmetic average. This is because
variability of return has a greater affect on the arithmetic average than the geometric
average.
3. Student Exercise
4
(a) DJIA = ∑=30
1
/i adjusted it Divisor P
Day 1
Company Price/Share
A
12
B 23
C 52
DJIA = (12 + 23 + 52)/3 = 29
Day 2
(Before Split)
(After Split) Company Price/Share Price/Share A 10 10 B 22
44 C 55
55 DJIA = (10 + 22 + 55)/3 = 29
DJIA = (10 + 44 + 55)/X = 29 X
= 3.7586 (new divisor) Day 3
(Before Split) (After Split) Company Price/Share Price/Share A
14
14 B 46 46 C 52
26 DJIA = (14 + 46 + 52) /3.7586
DJIA = (14 + 46 + 26)/Y = 29.798 = 29.798 Y
= 2.8861 (new divisor)
Day 4
Company Price/Share
A 13
B 47
C 25
DJIA = (13 + 47 + 25)/ 2.8861= 29.452
Company Price/Share
B 45
C 26
DJIA = (12 + 45 + 26)/2.8861 = 28.759
(b) Since the index is a price-weighted average, the higher priced stocks carry more
weight. But when a split occurs, the new divisor ensures that the new value for the
series is the same as it would have been without the split. Hence, the main effect of
a split is just a repositioning of the relative weight that a particular stock carries in
determining the index. For example, a 10% price change for company B would
carry more weight in determining the percent change in the index in Day 3 after the
reverse split that increased its price, than its weight on Day 2.
(c) Student Exercise
5
(a) Base = ($12 x 500) + ($23 x 350) + ($52 x 250)
= $6,000 + $8,050 + $13,000
= $27,050
Day 1 = ($12 x 500) + ($23 x 350) + ($52 x 250)
= $6,000 + $8,050 + $13,000
= $27,050
Index1 = ($27,050/$27,050) x 10 = 10
Day 2 = ($10 x 500) + ($22 x 350) + ($55 x 250)
= $5,000 + $7,700 + $13,750
= $26,450
Index2 = ($26,450/$27,050) x 10 = 9.778
Day 3 = ($14 x 500) + ($46 x 175) + ($52 x 250)
= $7,000 + $8,050 + $13,000
= $28,050
Index3 = ($28,050/$27,050) x 10 = 10.370
Day 4 = ($13 x 500) + ($47 x 175) + ($25 x 500)
= $6,500 + $8,225 + $12,500
= $27,225Index4 = ($27,225/$27,050) x 10 = 10.065
Day 5 = ($12 x 500) + ($45 x 175) + ($26 x 500)
= $6,000 + $7,875 + $13,000
= $26,875
Index5 = ($26,875/$27,050) x 10 = 9.935
(b) The market values are unchanged due to splits and thus stock splits have no effect.
The index, however, is weighted by the relative market values.
6. Price-weighted index(PWI)2004 = (20 + 80+ 40)/3 = 46.67
To account for stock split, a new divisor must be calculated:
(20 + 40 + 40)/X = 46.67
X = 2.143 (new divisor after stock split)
Price-weighted index(PWI)2005 = (32 + 45 + 42)/2.143 = 55.53
= 20(100,000,000) + 80(2,000,000) + 40(25,000,000)
VWI
2004
= 2,000,000,000 + 160,000,000 + 1,000,000,000
= 3,160,000,000
assuming a base value of 100 and 2004 as the base period, then
3,160,000,000/3,160,000,000 x 100 = 100
VWI
= 32(100,000,000) + 45(4,000,000) + 42(25,000,000)
2005
= 3,200,000,000 + 180,000,000 + 1,050,000,000
= 4,430,000,000
assuming a base value of 100 and 2004 as the base period, then
4,430,000,000/3,160,000,000 x 100= 1.4019 x 100 = 140.19
6.
(a) Percentage change in PWI = (55.53 - 46.67)/46.67 = 18.99%
Percentage change in VWI = (140.19 - 100)/100 = 40.19%(b) The percentage change in VWI was much greater than the change in the PWI
because the stock with the largest market value (K) had the grater percentage gain in
price (60% increase).
(c) December 31, 2002
Stock Price/Share # of Shares Market Value
K $20 50.0 $1,000.00
M 80 12.5 1,000.00
R 40 25.0 1,000.00
Total $3,000.00
December 31, 2003
Stock Price/Share # of Shares Market Value
A $32 50.0 $1,600.00
B 45 25.0* 1,125.00
C 42 25.0 1,050.00
Total $3,775.00
(*Stock-split two-for-one during the year)
Percentage change = (3,775.00 - 3,000)/3,000 = 25.83%
(As a geometric average = [(1.60)(1.125)(1.05)]1/3 - 1
= [1.]1/3 - 1
= 1.23 - 1
= 0.23 or 23.%
Unweighted averages are not impacted by large changes in stocks prices (i.e. price-
weighted series) or in market values (i.e. value-weighted series).下载本文
