Unformatted text preview: Chapter 10 Investment Returns and Aggregate Measures of Stock Market By Dr. M. Metghalchi The capital market consists of a place (New York Stock Exchange) or a mechanism
(NASDAQ) or institutions and procedures that provide for transactions in longterm
financial instruments. (longterm = greater than a year).
Long‐term financial instruments are equity and bonds. Equity market includes common stocks and preferred stocks, in this chapter we mostly discuss the common stock and leave the bond discussion in later chapters. In this chapter we discuss the aggregate measures of the stock market and discuss historical returns of these aggregate stock market measures. There a few very well known stock indexes that measures aggregate stock market: Market Indexes:
Dow Jones Averages (priceweighted average). Each stock is weighted by its price. Changes in
high priced stocks have more impact on the index than changes in low priced stocks. A priceweighted index is similar to what you normally use in computing means.
Standard & Poor’s Index (market valueweighted index of large capitalization stocks). The
Standard & Poor’s 500 Stock Index is comprised of 371 industrials, 15 transportation firms, 49
utilities, and 65 financial firms.
Wilshire 5000 Index (market valueweighted index, the broadest U.S. index, consisting of
5,700 stocks, this number keeps changing as new stocks come into the market). The Wilshire
5000 Equity Index represents the total dollar value of 5,700 stocks. It includes all New York
Stock Exchange and American Stock Exchange stocks and most of the active NASDAQ issues.
Value Line Average (equally weighted index, smaller stocks are more representative)
Russell 1,000, 2000 and 3000 (Value weighted of stocks). Russell 1000 includes 1000 largest
companies, Russell 2000 include smallest 2000 companies and Russell 3000 is the sum of the
two indexes.
1 A valueweighted average of stock prices considers not only the price of the stock but also the total value
of all shares (i.e., the number of shares outstanding times the price of each share). If a firm has a large
number of shares outstanding, the value of these shares plays a larger role in the determination of a
valueweighted index.
Microsoft and ExxonMobil both have a large number of shares outstanding. Their impact on a
valueweighted index is larger than a firm with fewer shares outstanding.
The Dow Jones industrial average includes the prices of only thirty companies and is a simple average.
The S&P 500 stock index is more broadbased (500 stocks) and is a valueweighted average which gives
more weight to the stocks with the largest capitalization. The S&P 500 is expressed relative to a base
year, while the Dow Jones industrial average is an absolute number and not expressed relative to a base
year.
The Value Line Index is a broadbased index that includes the stock covered by the Value Line
Investment Survey. It is computed using a geometric average. Let’s see the differences in index construction, lets construct various indices of two stocks,
Victoria Inc. (VI) and Houston Inc. (HI), given the followings: Stock Beginning of
year price End of year
price Shares
outstanding Beginning of
year value End of year
value VI $50 $60 20 million $1000 M $1,200 M HI $200 $180 1 million $200 M $180 M $1,200 M $1,380 M Total Question: Construct a price weighted index, a value weighted index and an equally weighted
index and calculate the performance of each index over the one year period.
Answer:
1. Price weighted Index:
Initial value of index (average price) = (50+200)/2 = 125
Final value of index = (60 + 180)/2 = 120
Performance of price weighted index = (120 – 125)/125 =  4 %
2. Value weighted index:
The final value of all stocks at the end of the year = $1,380 The initial value of all stocks = $1,200 2 The performance of value weighted index = (1380 – 1200)/1200 = 15 % Usually we set the beginning value of index to 100, [(1200/1200)*100] so the end of year value of index would be (1380/1200) * 100 = 115 or 15 % increase. 3. Equally weighted index: The same dollar amount in each stock. Therefore, assume we hold 1000 shares of HI with a total value o f $200,000 initially and 4000 shares of VI with the same initial value of $200,000. With a total value of portfolio = $400,000. End of the year, the value of portfolio will be: 1000* 180 + 4000* 60 = 180,000 + 240,000 = $420,000 Performance of equally weighted index = (420,000 ‐400,000)/400,000 = 5 % Note on equally weighted index: at the end of the year, your portfolio is no longer equally weighted, it is more heavily invested in VI, so in order to have equally weighted at the end of the year, you must increase HI or decrease VI or a combination in order to have again an equal weighted portfolio, so rebalancing is necessary for equally weighted index. As you can see we have constructed three different indexes and each index has shown a different performance over the one year period. Most analysts believe that the value‐weighted index is the best index that is why many indexes have been constructed using value weights. Holding Period Return or Realized Return versus Expected Return:
The expected return is the average rate of return over a security’s life. But you may not hold the
security forever, you may sell the security before its expected life, then the holding period return
(realized Return), HPR, is: HPR = IncomeEarnedduringth eperiod + capitalGai ns Purchae Pr ice 3 Example:
Let’s consider you buy a stock at a price of $1,000. And sell it next year at a price of $1272.07
and during the year you get $100 dividend from that stock. What is your holding period return?
Capital Gain =1272.07 –1,000 = $272.07
HPR = [100 (Dividend income) + 272.07 (capital gain)]/1000 = 372.07/1000=37.2 %
Example:
You buy Exxon stock at $50. You keep it for one year and sell it at $46 a year later. What is your HPR both in dollar and in percent? Assume Exxon pays $1.50 annual dividend. Capital Gain = 4650 = $4.0
Dividend = $1.50
Dollar return = ‐4 + 1.5 = ‐$2.5 Percent return = (1.5 – 4)/50 = ‐5 % Risk and Return: The Investors’ Experience Data have been compiled on the actual returns for various portfolios of securities from 19262006. Table 1 is taken from Fundamental of Investments (By J. Bradford and T. Miller, p.20) Table 1: Data from 1926‐2006 Average nominal Annual annual rate standard of return deviation 3.8 % 3.1 % Treasury Bonds 5.8 9.2 Corporate Bonds 6.2 8.5 Large firm stocks 12.3 20.1 Small firm stocks 17.4 32.7 Portfolio Treasury Bills 4 As you can see from Table 2, investors historically have received greater returns for greater
taking. (Standard deviation is a measure of risk). As we can see, the average return per year for
large capitalization stocks is 12.3 % over the past 80 years while it is 17.4% for small stocks.
Question: Looking at table 1, what is the historical market risk premium? Answer: Risk premium = return of risky asset – return of risk free asset Market risk premium = return of market (S&P500) – T Bill return Answer: 12.3 %  3.8 % = 8.5 % Market risk premium is a very important concept. This market risk premium is used with CAPM to estimate required rate of return for various stocks. Note: the answer in your textbook could be a touch different from other authors. The reason would be that other authors could use a different source of historical returns than your textbook or use another definition of risk free rate such as Treasury bond rate. My own preference is to use government bond rate as risk free rate, in this case, then the market risk premium is: Market risk premium = return of market (S&P500) – Treasury Bond Return Market risk premium = 12.3 – 5.8 = 6.8 % So please be careful when using market risk premium for various estimations. We will use this market risk premium to estimate required rate of return for any stock. According to Capital asset Pricing Model (CAPM) the required rate of return, K, for any stock is given by: K = Risk Free Rate + beta of the stock * market risk premium Question: which market risk premium should we use? 8.5 % or 6.8 % of the above estimations? Answer: If we use Treasury Bill rate for risk free rate in the CAPM, then we will use 8.5% for market risk premium. If we use Treasury Bond rate (10‐year TB rate) for risk free rate in the CAPM, then we will use 6.8% for market risk premium. I prefer the second method, using 10‐year Treasury Bond rate for risk free asset and use 6.8 % for market risk premium. 5 Note: since this historical market risk premium is an estimate and depends to source of data, and depends when the start date is and when the ending date is for taking the average, it is not written in the stone that it must be 6.8%. ANY NUMBER BETWEEN 6 to 7 PERCENT IS ACCEPTABLE. When using 10‐year Treasury bond yields for risk free rate. Different researchers use different numbers. EXAMPLE:
On May 22nd of 2009, estimate the required rate of return for General Electric Company (GE).
SOLUTIONS:
K = Risk Free Rate + beta of the stock * market risk premium
K = 3.28 + 1.59 * 6.8% = 14.09 %
On 5/22/09, 10year Treasury Bond rate was 3.28%, and beta of GE was 1.59. Beta was taken from
Yahoo.com. Arithmetic versus Geometric Average Return: To calculate average returns we add up the annual returns and divide by the number of years.
Example:
Assume stock X had the following returns over the past 4 years: 10%, 25%,  20%, and 25%;
what is its arithmetic and geometric average return over the past 4 years:
ANSWER:
n Arithmetic average return= K = ∑ Ki
= (10 + 25 –20 +25)/4 = 10 %
n i =1
We use the following formula for geometric return: Geometric average return = [(1 + K1)* (1 + K2)* (1 + K3)*….*(1 + KN)]1/N – 1
Where, k1, K2, and so on are each year’s return
Using the above formula for the above example we estimate the geometric mean as:
KG = [(1+.10)*(1+.25)*(1.20)*(1+.25)]1/4 – 1 = .0829 0r 8.29 % 6 The geometric average return can easily be estimated by using a financial calculator or Excel.
The geometric average return is the rate that equalizes the Future Value (FV) with Present Value
(PV) over the period under consideration.
Example: Assume your parents invested $1 in the large stock market in 1926. Assume in 2002
the value of this investment was $1,775.34. What is the geometric average return of this
investment over 77 years (19262002)?
My calculator: HP12C: 1 PV 1775.34CHS FV 77 N , then push on i. Then answer is 10.20 %.
(Note: in my calculator CHS changes the number to negative)
Excel Solution:
Click on formula, then financial, then choose rate. Then enter 77 for NPR, 0 for payment, 1 for
PV, and 1775.34 for FV, and 0 for type. You get the answer of 10.20%
10.20 = RATE (77,0,1,1775.34,0)
Also we could use the following formula for Geometric return, KG:
KG = [FV/PV](1/N) – 1
KG = [1775.34/1] (1/77) – 1 = 1.1020 1.0 = .1020 or 10. 20 %
Some Exercises CAPITAL GAINS YIELD C 1. Capital gains yield is defined as: A. (Pt – Pt + 1) / Pt + 1 B. Dt+1 / Pt C. (Pt + 1 – Pt) / Pt D. (Pt – Pt + 1) / Pt 7 E. Pt / (Pt + 1 – Pt) GEOMETRIC RETURN C 2. A stock has had returns of –12 percent, 31 percent, 14 percent, 4 percent and 9 percent over the past five years. What is the geometric return for this stock? A. 7.10% B. 7.80% C. 8.30% D. 9.20% E. 10.02% Solutions to the end of chapter problems on page 370‐372. 6. a. The holding period return: ($56.38 ‐ 35)/$35 = 61%. On an annual basis: 61%/5 = 12.2% 5 b. The true annual rate of return: $35(1 + r) = $56.38 5 (1 + r) = 56.38/35 = 1.61 1.61 is the interest factor for the future value of $1 for five years, so the rate of growth (i.e., the rate of return) is 10 percent. (PV = ‐35; FV = 56.38; N = 5, PMT = 0; I = ? = 10.00.). You could also use Excel (Highly recommended). This problem illustrates the potential bias when using the holding period return instead of the true annualized return. 7. The holding period return is ($980‐795)/$795 = 23.3%. There are several methods to prove that the annualized return is not 23.3%. One possibility it to use an interest table The true annualized return is 5 $795(1 + r) = $980 8 5 (1 + r) = 980/795 = 1.233 1.233 is the interest factor for the future value of $1 for five years, The interest factor 1.233 produces a return of between 4 and 5 percent. An alternative method is to use a financial calculator: PV = ‐795; FV = 980; N = 5, PMT = 0; I = ? = 4.27. You could use Excel. 8. If returns of 10% for stock and 6% for corporate bonds are achieved, the $1,000 invested in stock grows to $2,689.62. 10 $1,000(1 + .10) = $2,594 (PV = ‐1000; I = 10; N = 10, PMT = 0; FV = 2594.) The $1,000 invested in corporate bonds grows to $1,774.02. 10 $1,000(1 + .06) = $1,791 (PV = ‐1000; I = 6; N = 10, PMT = 0; FV = 1791.) If these returns are earned for twenty years, the $1,000 invested in stock grows to $6,727. 20 $1,000(1 + .10) = $6,727 (PV = ‐1000; I = 10; N = 20, PMT = 0; FV = 6727.) The $1,000 invested in corporate bonds grows to $3,207. 20 $1,000(1 + .06) = $3,207 (PV = ‐1000; I = 6; N = 20, PMT = 0; FV = 3207.) The difference exceeds $3,500 and strongly argues for including stocks in a portfolio whose objective is to finance spending (e.g., retirement or a child’s education) in the distant future. 9. This problem is another version of the previous problem. If the range in the Dow Jones Industrial Average grew annually at 10.4 percent for nine years, the low value would be 9 9,796(1 + .104) = 23,865. (PV = ‐9796; I = 10.4; N = 6, PMT = 0; FV = 23865.) 9 The high value would be 9 11,723(1 + .104) = 28,560. (PV = ‐11723; I = 10.4; N = 20, PMT = 0; FV = 28560.) If the historic returns continued for the nine years 2001‐2009, the range in the DJIA would have been 28,560‐23,865. For 2009, the range was 10606‐6440! From 1973 through 1982, the DJIA was virtually unchanged, so there are similarities between 1973‐1982 and 2000‐2009. 11. a. $80 = $4 + $4 + $4 + $100 2 3 3 (1 + r) (1 + r) (1 + r) (1 + r) At 12 percent: $4(2.402) + $100(.712) = $80.81 The expected rate of return is approximately 12 percent. (PV = ‐80; FV = 100; N = 3, PMT = 4; I = ? = 12.38.) b. $60 = $4 + $4 + $4 + $100 2 3 3 (1 + r) (1 + r) (1 + r) (1 + r) At 24 percent: $4(1.981) + $100(.524) = $60.32 The expected rate of return is approximately 24 percent. (PV = ‐60; FV = 100; N = 3, PMT = 4; I = ? = 24.27.) c. $80 = $1 + $1 + $1 + $100 2 3 3 (1 + r) (1 + r) (1 + r) (1 + r) At 9 percent: $1(2.531) + $100(.772) = $79.73 The expected rate of return is 9 percent. (PV = ‐80; FV = 100; N = 3, PMT = 1; I = ? = 8.88.) 12. The cash flows are Present: $100 outflow (purchase of one share) End of year one: $5.50 inflow (the dividend on one share) and $130 outflow (the second purchase) 10 End of year two: $11 (dividend on two shares) $280 (sale of two shares) The equation for determining dollar‐weighted rate of return: $100 = $5.50 ‐ $130 + $11 + 280 2 (1 + r) (1 + r) ( 1 + r) Select a rate (e.g., 24%); substitute into the equation: ? $100 = $5.50 ‐ $130 + $11 + 280 2 (1 + .24) (1 + .24) ( 1 + .24) At 24%: $5.50(.806) ‐ $130(.806) + $291(0.650) = $4.43 ‐ $104.78 + $189.15 = $88.80 Since $88.80 is less than $100, the selected rate was too high, and the process must be repeated with a lower rate. At 18%: $5.50(.847) ‐ $130(.847) + $291(0.718) = $4.66 ‐ $110.11 + $208.94 = $103.49 Since $103.49 is greater than $100, the selected rate was too low, and the process must be repeated with a higher rate. At 20%: $5.50(.833) ‐ $130(.833) + $291(0.694) = $4.58 ‐ $108.29 + $201.95 = $98.24 The dollar‐weighted rate of return is between 18 and 20 percent. (If the students are limited to the use of the interest tables in the text, this is as close an answer they may derive without interpolating. This problem may also be solved using a financial calculator that accepts uneven cash inflows or Excel, which yields a return of 19.3 percent.) To determine the time‐weighted rate of return, the first calculation is Year one: ($130 + $5.50)/$100 = 1.355 Year two: ($140 + $5.50)/$130 = 1.119 11 Next compute the geometric average to determine the time‐weighted rate of return: .5 [(1.355)(1.119)] ‐ 1 = 1.231 ‐1 = 23.1%. In this example the dollar‐weighted return is lower because a smaller return was earned in the second year when more dollars were invested. In the second part of the problem the cash flows are Present: $100 outflow (purchase of one share) End of year one: $5.50 inflow (the dividend on one share) and $110 outflow (the second purchase) End of year two: $11 (dividend on two shares) $280 (sale of two shares) The equation for determining dollar‐weighted rate of return: $100 = $5.50 ‐ $110 + $11 + 280 2 (1 + r) (1 + r) ( 1 + r) Select a rate (e.g., 24%); substitute into the equation: ? $100 = $5.50 ‐ $110 + $11 + 280 2 (1 + .24) (1 + .24) ( 1 + .24) At 24%: $5.50(.806) ‐ $110(.806) + $291(0.650) = $4.43 ‐ $88.66 + $189.15 = $104.92 Since $104.92 exceeds $100, the selected rate was too low, and the process must be repeated with a higher rate. At 28%: $5.50(.781) ‐ $110(.781) + $291(0.610) = 12 $4.30 ‐ $85.91 + $177.51 = $95.90 The dollar‐weighted rate of return is between 24 and 28 percent (26.2 percent using a calculator). To determine the time‐weighted rate of return, first determine the return for each period: Year one: ($110 + $5.50)/$100 = 1.155 Year two: ($140 + $5.50)/$110 = 1.323 Next compute the geometric average to determine the time‐weighted rate of return: .5 [(1.155)(1.323)] ‐ 1 = 1.236 ‐ 1 = 23.6%. In the second part of this problem, the dollar‐weighted return is higher because a larger return was earned in the second year when more dollars were invested. 13. An alternative means to test whether the student understands the computation of rates of return is to ask if an investment yields a particular rate of return. Price = div + ... + div + sales price n n (1 + r) (1 + r) (1 + r) For a given rate to be the correct return, the two sides of the equation must be equal. At 12 percent: $2(3.605) + 50(.567) = $35.56. (3.605 and .567 are the interest factors for the present value of an annuity and of a $1 at 12 percent for five years.) The present value of the cash flows ($35.56) does not equal the cost of the investment ($40), so the rate of return is not 12 percent. The actual dollar‐weighted return is 9.16 percent. (PV = ‐40; N = 5; PMT = 2; FV = 50; I = ? = 9.16%.) 14. Once again the student is asked to determine a rate of return for uneven cash flows. Unless some type of computer program or a financial calculator that accepts uneven cash inflow is used, the task is tedious. The equation to be solved for r is $35 = $1.00 + $3.15 + $2.09 + $1.71 + 41.00 2 3 3 (1 + r) (1 + r) (1 + r) (1 + r) At 10%: $1(.909) + 3.15(.826) + 2.09(.751) + 42.71(.683) = $34.33 At 9%: $1(.917) + 3.15(.842) + 2.09(.772) + 42.71(.708) = $35.32 13 The return is between 9 and 10 percent, and the answer is 9.4 percent using a financial calculator. (Be certain to point out that this return is the dollar‐weighted rate of return. The time‐weighted rate of return cannot be calculated because the problem does not specify the fund's net asset value at the end of each year.) Copy right 2011, Dr. Massoud Metghalchi, all right reserved. 14 ...
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 Stock market index, Poor’s 500 Stock Index

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