Period Growth Rate Sales HPR Acct. % of sales
. Options 1 5% 1.05 CGS 60%
     An option is a contract between the seller, or the writer of the option, and the buyer of the option.  There are call and put options.  A call option gives the buyer the option to buy some pre-specified asset,  2 5% 1.05 Sales % 10%
1 investmentvaluationforyou.com like a share of stock, or as stock options that are traded on major exchanges, 100 shares is the usual contract amount.  The option contract allows the buyer to buy this asset at a pre-specified 3 5% 1.05
price, the exercise price, E, for a certain period of time, T.   It seems rather obvious to see that the lower the exercise price, the price at which the buyer of the option can buy the under-lying instrument, in this 4 5% 1.05 Taxes % 50%
case 100 shares of stock, the more the call option is worth.  When you buy an option, the most you can lose is the price of the option.  If you have an option to buy a stock at $60 per share, and the market price 5 5% 1.05
of the stock is at $50 per share, and the option is at expiration, then it is going to expire worthless, as most options do. However, when you buy an option, you can make unlimited profit.  Suppose the  6 5% 1.05
This site is a decision support system (DSS) for investment analysis and portfolio management, corporate finance, and commercial loan analysis.   market price of the stock goes to 68, and the option is at expiration, then the option would be worth $8 per share.  The longer time till expiration, the more the option is worth.  You can only lose the price 7 5% 1.05
A decision support system is a model based system for processing data  to assist in making decisions.  This site has several valuation models of the option, but you could make unlimited profit, if the price of the underlying instrument (100 shares of stock) moves in the right direction.  The more time you have for this to  happen, the more  8 5% 1.05
for different types of investments, and provides the fomulas and the necessary variable inputs for these models.  We will be estimating the  the option is worth. It is easy to see the price of an option at expiration.  The call option is worth S-E, the market price of the stock minus the exercise price.  A put option is worth E-S, the exercise price 9 5% 1.05 Long-term debt  $   400.00
"present value" of investments, and determining how this present value compares to the price of the investment.  If the  minus the stock price.   10 5% 1.05 APR 5.00%
present value that we have calculated is higher than the price, the investment is under priced.  This is what you are looking for, or at least Data as of 02/01/18      In the example below, if S=75 at expiration, then the call option will be worth $15x100=$1500).  If you had bought a call option when T=.241 (88 days till expiration) when the price of the call option 11 5% 1.05 Monthly % 0.41667%
an investment that is fairly priced.  A different approach to this problem is to estimate the return of an investment, and compare this estimated S&P 500 2762.13 was at 10.61, then you would have made a profit of (15-10.61)=4.61x100=$461.  This represents a return of 4.61/10.61=43% in 88 days(an annualized return of 180%).  If you owned the stock during this 12 5% 1.05 Monthly pmt.  $      1.67
return with thef the investment's "required return".  The meaning of an investment's required return will be explained later.                                                            D/P last 12 months 1.79% 154.39 period, you would have made the same profit, but your return would have been 7/68=10.3%, an annualized return of 48%.  What is illustrated here is that the option is a highly leveraged investment 13 5% 1.05 This year, only the interest on this $400 debt is being paid.  The principal ($400) remains the same throughout the year.
     Let me describe the layout of this web site.  I start off with some narrative about capital markets which explains how a fundamental analyst goes D/P projected 1.83% 18.28 17.89 instrument, much more than just owning the stock.  You get a big bang for your buck.  You can make a lot or loose a lot very quickly.  Besides being used for speculation, options can be used to 
about analyzing common stock.  This section can get rather complicated, especially when discussing the Capital Asset Pricing Model and the 2822.249 hedge your investments, thereby lowering your risk.  Let's say you own 100 shares of the stock referred to below.  If you could buy a put with an exercise price of 68, and you bought the stock for 60 today, Short-term debt
underlying mathematics.  It definitely requires some investment acumen, or at least some knowledge of the underlying mathematics involved you could lock in a profit of 8x100=$800.  The put guarantees that you can sell each share of stock at 68 in 88 days.  There are options and futures for a variety of commodities.  They are traded in  1 2 3 4 5 6 7 8 9 10 11 12 Total
(Probability and Statistics).   S&P 500 EPS Est. organized exchanges, mostly in Chicago.  Lets say you’re a farmer, and you have a crop of oranges that is going to be harvested in two months.  You are not sure what the price of oranges will be in two months, Short-term Debt  $   944.79 783.75 591.61 401.33 201.35 33.99
     I will have a section on options and futures, including a discussion of the Black-Scholes option pricing formula, as well as a look at some INDEX NAME Q1,'18 Q2,'18 Q3.'18 Q4,'18 EPS'18              Price          P/E and this presents some risk for you.  You know what the price is on a put option on oranges that matures in two months at an exercise price that you find acceptable, so you buy enough puts that will APR=10% 0.0083333  $      7.87  $      6.53  $      4.93  $      3.34  $      1.68  $      0.28
traditional option trading strategies.  We will look at theories affecting the pricing of interest rate instrument futures and the term structure of S&P 500 $35.66 $38.15 $39.61 $40.97 $154.39 2762.13 17.89 allow you to sell your oranges when they are harvested, at a price that you find acceptable. These puts you have bought you a guarantee of what price you will be able to sell your oranges for.  On the other side of   Monthly %=0.8333%
interest rates.  We will also look at bond valuation, and it's implications for portfolio management.  There will be a section about overall portfolio S&P 500 Consumer Discretionary $8.73 $10.08 $10.24 $10.50 $39.55 $848.74 21.46 the transaction, you have someone that wrote the puts who is speculating that orange prices will go up, the options will expire worthless, and he will profit by keeping the option premium that he took in Monthly HPR=1.008333
management, with recommendations for different asset allocations for different investment objectives.   S&P 500 Consumer Staples $6.82 $7.53 $7.95 $8.09 $30.39 $593.21 19.52 when he sold the puts.  The put options have functioned as an insurance policy for the farmer, locking in the price that he can sell his oranges for.  Call options can also act as insurance for someone Monthly pmt.(int. only) on LTD  $      1.67  $      1.67  $      1.67  $      1.67  $      1.67  $      1.67  $      1.67  $      1.67  $      1.67  $      1.67  $      1.67  $      1.67  $     20.00
     I also have a section that is designed more for the corporate user.  One section deals with extensive fundamental analysis of the corporation, S&P 500 Energy $5.70 $6.47 $6.58 $6.27 $25.02 $559.45 22.36 who needs to be able to buy a particular asset in the future at a guaranteed price, while the person selling the call option might just be a speculator, betting that the price of this asset will go down, Int. on short-term debt  $      7.87  $      6.53  $      4.93  $      3.34  $      1.68  $      0.28
analysis that might be suitable if you are not only a stock holder, but someone who is actually involved in the management of the company. S&P 500 Financials $8.27 $8.53 $8.77 $8.94 $34.51 $498.32 14.44 and he will profit by being able to keep the option premium (the option price is sometimes called the premium).  Futures can also act as a hedge to protect assets, and speculators can use futures Total int.  $      9.54  $      8.20  $      6.60  $      5.01  $      3.34  $      1.95  $      1.67  $      1.67  $      1.67  $      1.67  $      1.67  $      1.67
There is ratio analysis, Sources and Uses of Funds, break-even analysis, calculations and implications of degrees of operational and financial S&P 500 Health Care $14.55 $15.16 $15.37 $15.30 $60.38 $1,019.21 16.88 to attempt to profit from price changes in assets.  If you own a bunch of oranges, you could take a short position in the orange futures market, thus hedging your oranges against any price changes in the      
leverage. S&P 500 Industrials $7.21 $9.30 $9.46 $9.40 $35.37 $669.91 18.94 future.  Usually, a farmer is just trying to grow his crop and sell it for a known profit.  He does not want to speculate on price changes of his crop, that is not his business.  However, it is the business of 
     Finally, I have a section that is designed to produce pro-forma financial statements (projected financial statements).  The user will input some S&P 500 Information Technology $14.02 $14.12 $15.04 $18.14 $61.32 $1,189.61 19.40 some people, options and futures speculators who want to take on this risk.  Thus options and futures markets are born.  I have shown you how agricultural commodities are well-suited for these  Income Statement (000's)
beginning numbers, like projected sales figures, as well as assumptions about projected growth rates, as well as % of sales for certain items.  S&P 500 Materials $5.37 $5.89 $4.92 $5.14 $21.32 $389.09 18.25 markets, and so there are organized exchanges, where options and futures contracts on agricultural commodities are traded.  You can see why most of these exchanges are located in Chicago, the  1 % of Sales 2 % of Sales 3 % of Sales 4 % of Sales 5 % of Sales 6 % of Sales 7 % of Sales 8 % of Sales 9 % of Sales 10 % of Sales 11 % of Sales 12 % of Sales 13 % of Sales Yr.Tot.(1-12) % of Sales
Pro-forma statements can be useful for internal planning, as well as indicating to investors about the future prospects for this company.   S&P 500 Telecommunication Services $3.62 $3.77 $3.76 $3.43 $14.58 $169.13 11.60 place where the rail roads from all over the Midwest come together and meet.  Traditionally, agricultural commodities from all over the Midwest would be transported by rail thru Chicago, and on to the major Sales  $1,500.00 100%  $1,575.00 100%  $1,653.75 100%  $1,736.44 100%  $1,823.26 100%  $1,914.42 100%  $2,010.14 100.00%  $2,110.65 100.00%  $2,216.18 100.00%  $2,326.99 100.00%  $2,443.34 100.00%  $2,565.51 100.00%  $2,693.78 100.00% Sales  $23,875.69 100.00%
If you are starting a new business enterprise, and if you are seeking either debt or equity financing, any potential investor is going to want S&P 500 Utilities $3.95 $3.36 $5.09 $3.17 $15.57 $254.88 16.37 population centers in the east.  From Chicago, commodities could be shipped thru the Great Lakes, on thru the Erie Canal, and then by rail or trucks to the major cities in the East.   Cost of Goods Sold  $   900.00 60%  $   945.00 60%  $   992.25 60%  $1,041.86 60%  $1,093.96 60%  $1,148.65 60%  $1,206.09 60.00%  $1,266.39 60.00%  $1,329.71 60.00%  $1,396.20 60.00%  $1,466.01 60.00%  $1,539.31 60.00%  $1,616.27 60.00% Cost of Goods Sold  $14,325.41 60.00%
to see pro-forma statements.  Typically, if you had a business plan, you would go to an accountant to prepare pro-forma statements for any potential S&P 500 Real Estate $1.15 $1.25 $1.26 $1.39 $5.05 $196.14 38.84 Anyhow, you can see why agricultural commodity option and futures exchanges were in Chicago.  Exchanges for trading options and futures on stocks, bonds, stock indexes, and just about any Gross Profit  $   600.00 40%  $   630.00 40%  $   661.50 40%  $   694.58 40%  $   729.30 40%  $   765.77 40%  $   804.06 40.00%  $   844.26 40.00%  $   886.47 40.00%  $   930.80 40.00%  $   977.34 40.00%  $1,026.20 40.00%  $1,077.51 40.00% Gross Profit  $  9,550.28 40.00%
investors.  However, this can be very expensive.  With "investmentvaluationforyou.com" you can produce your own pro-forma statements.  INDEX NAME Q1,'19 Q2,'19 Q3,'19 Q4,'19 EPS'19                     P/E financial asset you can think of, as well as agricultural commodities, are mostly located in Chicago or New York.  Exchanges for options, futures, stocks (like the OTC market), don't have to be Selling Exp.(Variable)  $   150.00 10%  $   157.50 10%  $   165.38 10%  $   173.64 10%  $   182.33 10%  $   191.44 10%  $   201.01 10.00%  $   211.07 10.00%  $   221.62 10.00%  $   232.70 10.00%  $   244.33 10.00%  $   256.55 10.00%  $   269.38 10.00% Selling Exp.(Variable)  $  2,387.57 10.00%
     Before I describe what this DSS does, I would like to present some background information about capital markets.  S&P 500 $39.28 $41.86 $43.51 $45.21 $169.86 16.61 located in buildings, but they can exist as a computer network, with buyers and sellers linked together online.   Gen. and Ad. Exp.(Fixed)  $     20.00 1%  $     20.00 1%  $     20.00 1%  $     20.00 1%  $     20.00 1%  $     20.00 1%  $     20.00 0.99%  $     20.00 0.95%  $     20.00 0.90%  $     20.00 0.86%  $     20.00 0.82%  $     20.00 0.78%  $     20.00 0.74% Gen. and Ad. Exp.(Fixed)  $     240.00 1.01%
If you would like to skip this material and proceed to analysis of the aggregate stock market, page down about 11 times to row 505, and you  S&P 500 Consumer Discretionary $9.96 $11.48 $11.70 $11.90 $45.04 18.84 Pg. Dn. Depreciation  $     50.00 3%  $     50.00 3%  $     50.00 3%  $     50.00 3%  $     50.00 3%  $     50.00 3%  $     50.00 2.49%  $     50.00 2.37%  $     50.00 2.26%  $     50.00 2.15%  $     50.00 2.05%  $     50.00 1.95%  $     50.00 1.86% Depreciation  $     600.00 2.51%
will get into analysis of the aggregate market (S&P 500), as well as analysis of the individual economic sectors.  We will then get into S&P 500 Consumer Staples $7.35 $8.12 $8.59 $8.74 $32.80 18.07 E.B.I.T.  $   380.00 25%  $   402.50 26%  $   426.13 26%  $   450.93 26%  $   476.98 26%  $   504.33 26%  $   533.04 26.52%  $   563.20 26.68%  $   594.85 26.84%  $   628.10 26.99%  $   663.00 27.14%  $   699.65 27.27%  $   738.14 27.40% E.B.I.T.  $  6,322.71 26.48%
some analysis of individual stocks. S&P 500 Energy $6.00 $6.97 $7.29 $7.18 $27.44 20.40 Interest  $      9.54 1%  $      8.20 1%  $      6.60 0%  $      5.01 0%  $      3.34 0%  $      1.95 0%  $      1.67 0.08%  $      1.67 0.08%  $      1.67 0.08%  $      1.67 0.07%  $      1.67 0.07%  $      1.67 0.06% Interest  $      44.64 0.19%
     An important area of academic research in finance has been in the area of the efficiency of capital markets. S&P 500 Financials $9.33 $9.62 $9.81 $10.02 $38.78 12.85 E.B.T.  $   370.46 25%  $   394.30 25%  $   419.53 25%  $   445.92 26%  $   473.63 26%  $   502.38 26%  $   531.38 26.43%  $   561.53 26.60%  $   593.19 26.77%  $   626.43 26.92%  $   661.34 27.07%  $   697.99 27.21% E.B.T.  $  6,278.07 26.29%
"An efficient capital market is one in which security prices adjust rapidly to the infusion of new information, S&P 500 Health Care $15.82 $16.41 $16.63 $16.39 $65.25 15.62 Taxes  $   185.23 12%  $   197.15 13%  $   209.76 13%  $   222.96 13%  $   236.82 13%  $   251.19 13%  $   265.69 13.22%  $   280.76 13.30%  $   296.59 13.38%  $   313.22 13.46%  $   330.67 13.53%  $   348.99 13.60% Taxes  $  3,139.03 13.15%
and current market prices fully reflect all available information that is relevant."1   Many studies have been conducted concerning the efficiency S&P 500 Industrials $8.35 $10.42 $10.51 $10.43 $39.71 16.88 Net Income  $   185.23 12%  $   197.15 13%  $   209.76 13%  $   222.96 13%  $   236.82 13%  $   251.19 13%  $   265.69 13.22%  $   280.76 13.30%  $   296.59 13.38%  $   313.22 13.46%  $   330.67 13.53%  $   348.99 13.60% Net Income  $  3,139.03 13.15%
of capital markets, and the empirical results indicate that capital markets are more or less efficient for a vast majority of investors. S&P 500 Information Technology $15.09 $15.22 $16.38 $19.70 $66.39 17.91
The efficient market hypothesis (EMH) has important implications for technical and fundamental analysis.  Let me define both of these types of financial analyses. S&P 500 Materials $6.32 $6.41 $5.51 $5.59 $23.83 16.33 Balance Sheet (000's)
Technical analysts operate on the premise that stock price movements take place in trends that persist for a period of time. S&P 500 Telecommunication Services $3.66 $3.77 $3.83 $3.64 $14.90 11.35 Beg. 1 2 3 4 5 6 7 8 9 10 11 12
The trader has a system that, given a certain signal,  can detect the beginning  of a price movement and profit on the movement yet to come. S&P 500 Utilities $4.06 $3.56 $5.14 $3.64 $16.40 15.54 Assets
This signal can be found in the action of the market itself.  The belief that future prices can be predicted by examining past trading data S&P 500 Real Estate $1.25 $1.35 $1.42 $1.54 $5.56 35.20 Current Assets
is in direct contrast with the efficient market hypothesis , which states that the information dissemination process is extremely quick, INDEX NAME Q1,'16 Q2,'16 Q3,'16 Q4,16 EPS'16 Cash  $   400.00  $   100.00  $   100.00  $   100.00  $   100.00  $   100.00  $   108.69 328.66 508.91  $   699.97  $   902.38  $1,116.70  $1,343.52
and most investors receive the information at about the same time.  If current stock prices fully reflect all relevant information, as the EMH contends, S&P 500 $23.97 $25.70 $28.69 $27.90 $106.26                             "  The Black-Scholes Option Pricing Formula Accounts Rec.  $          -    $1,500.00  $1,575.00  $1,653.75  $1,736.44  $1,823.26  $1,914.42  $2,010.14  $2,110.65  $2,216.18  $2,326.99  $2,443.34  $2,565.51
then any technical trading system that depends on past trading data should be totally useless.  By the time the information is made public, the resulting price change S&P 500 Consumer Discretionary $7.70 $8.56 $8.54 $8.50 $33.30          C= SN(d)1-Ee-rtN(d2) Inventory  $   900.00  $   945.00  $   992.25  $1,041.86  $1,093.96  $1,148.65  $1,206.09  $1,266.39  $1,329.71  $1,396.20  $1,466.01  $1,539.31  $1,616.27
has already taken place. S&P 500 Consumer Staples $5.71 $6.30 $6.79 $6.53 $25.33 $2,821.37           T= 88 days or .241 of a year         d1= (ln(S/E)+(r+˝σ2)T)/σ√T Total Current Assets  $1,300.00  $2,545.00  $2,667.25  $2,795.61  $2,930.39  $3,071.91  $3,229.20  $3,605.19  $3,949.27  $4,312.35  $4,695.38  $5,099.35  $5,525.30
     Fundamental analysts contend that stocks have a basic, intrinsic value that is determined by the levels of underlying economic variables such as S&P 500 Energy ($2.71) ($2.06) $0.87 $0.41 ($3.49) $2,762.13         d2= d1-σ√T Net Fixed Assets  $   800.00  $   750.00  $   700.00  $   650.00  $   600.00  $   550.00  $   500.00  $   450.00  $   400.00  $   350.00  $   300.00  $   250.00  $   200.00
earnings, earnings growth, and levels of risk. If the analyst can do a superior job of estimating these variables, he can acquire undervalued stocks and S&P 500 Financials $5.47 $5.46 $7.00 $5.86 $23.79 0.1964 C=Call option price Total Assets  $2,100.00  $3,295.00  $3,367.25  $3,445.61  $3,530.39  $3,621.91  $3,729.20  $4,055.19  $4,349.27  $4,662.35  $4,995.38  $5,349.35  $5,725.30
earn above-average returns.  Let me stress that the analyst must project the future value of these variables in order to determine the present value of a particular stock. S&P 500 Health Care $10.49 $10.94 $10.69 $10.32 $42.44           S= 68 S=Stock price
Simply examining past data is unlikely to produce any above-average returns.  While advocates of fundamental analysis do believe   S&P 500 Industrials $5.88 $7.14 $7.50 $6.55 $27.07 ab           E= 60 E=Exercise price Liabilities
that capital markets are pretty efficient, they do realize that the price mechanism is not always perfect.  Over bought or over sold conditions S&P 500 Information Technology $8.03 $8.27 $9.23 $12.45 $37.98           T= 0.241             T= 0.490918 T=Time till expiration Current Liabilities
do occur, producing temporarily overvalued or undervalued stocks.  While advocates of fundamental analysis do believe that market price S&P 500 Materials $1.67 $4.91 $3.73 $2.71 $13.02 12 σ√T= 0.795271           r= 6% r=riskless interest rate Accounts Payable  $   900.00  $   945.00  $   992.25  $1,041.86  $1,093.96  $1,148.65  $1,206.09  $1,266.39  $1,329.71  $1,396.20  $1,466.01  $1,539.31  $1,616.27
and intrinsic value can differ, they also believe that the market will eventually recognize this discrepancy and correct it.  Fundamental analysis involves several levels S&P 500 Telecom. Services $2.80 $1.92 $2.60 $2.54 $9.86 56 0.632456              σ= 0.4               σ2= 0.16 σ=Standard deviation Accrued Exp.  $          -    $     20.00  $     20.00  $     20.00  $     20.00  $     20.00  $     20.00  $     20.00  $     20.00  $     20.00  $     20.00  $     20.00  $     20.00
of investigation, beginning with aggregate market analysis, and then moving on to sector and industry analysis, and then company analysis.  The analyst will attempt to estimate S&P 500 Utilities $3.66 $2.68 $4.93 $2.40 $13.67 0.316228          C= SN(d1)-Ee-rtN(d2) σ2=the variance Total Current Liabilities  $   900.00  $   965.00  $1,012.25  $1,061.86  $1,113.96  $1,168.65  $1,226.09  $1,286.39  $1,349.71  $1,416.20  $1,486.01  $1,559.31  $1,636.27
future aggregate stock market earnings per share, and  with the dividend yield and the projected growth rate, the value of the market.  The way to determine if S&P 500 Real Estate $2.55 $1.51 $1.48 $1.83 $7.37           σ2= 0.16          C= 68N(d1)-60e-.06(.241)N(d2) Long Term Debt $400.00 $400.00 $400.00 $400.00 $400.00 $400.00 $400.00 $400.00 $350.00 $300.00 $250.00 $200.00 $150.00
the aggregate market looks attractive is to try to determine if the current market is over or underpriced, or if it seems to be fairly priced.  If the aggregate INDEX NAME Q1'17 Q2'17 Q3'17 Q4'17 EPS'17         d1= (ln(S/E)+(r+˝σ2)T)/σ√T   ln68/60= 0.125163 Sub-total Liabilities  $1,300.00  $1,365.00  $1,412.25  $1,461.86  $1,513.96  $1,568.65  $1,626.09  $1,686.39  $1,699.71  $1,716.20  $1,736.01  $1,759.31  $1,786.27
market seems to be underpriced according to your analysis, then this would imply that an above-average return could be earned in the market. S&P 500 $28.82 $30.51 $31.33 $33.47 $124.13         d1=ln(68/60)+(.06+.5*.16).241/.4*.491   SN(d1)= 53.72 Short Term Debt  $          -    $   944.77 $772.62  $   591.60 $401.33  $   201.33  $          -    $          -    $          -    $          -    $          -    $          -    $          -  
This type of analysis should be done before industry or company analysis is attempted.  The technique we will use to estimate market values S&P 500 Consumer Discretionary $8.05 $8.71 $8.86 $8.93 $34.55 N(d1)=N(.808)=.790          d1= (.125+.034)/.196=.808       e-.rt= 0.98564 43.11207 Total Liabilities   $1,300.00  $2,309.77  $2,184.87  $2,053.46  $1,915.29  $1,769.98  $1,626.09  $1,686.39  $1,699.71  $1,716.20  $1,736.01  $1,759.31  $1,786.27
is the present value of dividends approach. Specifically,  we will apply the basic dividend valuation model (which I will describe below), S&P 500 Consumer Staples $5.99 $6.72 $7.17 $7.40 $27.28 N(d2)=N(.612)=.729          d2= d1-σ√T      Ee-rt= 59.1386 Equity
to the aggregate stock market.  This technique is referred to as a micro analysis of the aggregate market.  This formula is also used for individual stocks, portfolios of stocks like sectors or industries, S&P 500 Energy $3.88 $2.75 $3.73 $4.29 $14.65          d2= .808-.196 Ee-rtN(d2)= 43.1121 43.1121 Common Stock $800.00 $800.00 $800.00 $800.00 $800.00 $800.00 $800.00 $800.00 $800.00 $800.00 $800.00 $800.00 $800.00
and the aggregate market, or a proxy thereof, like the S&P 500.  What follows is the present value of dividends model.  S&P 500 Financials $6.83 $6.98 $6.14 $6.78 $26.73          d2= 0.612            C= 10.6079 10.60793 Retained Earnings  $          -    $   185.23  $   382.38  $   592.15  $   815.11  $1,051.92  $1,303.11  $1,568.80  $1,849.56  $2,146.16  $2,459.37  $2,790.04  $3,139.03
P=D1/(i-g) i=D1/P+g S&P 500 Health Care $10.52 $11.78 $11.37 $12.05 $45.72           C= 10.60793 Total Equity $800.00 $985.23 $1,182.38 $1,392.15 $1,615.11 $1,851.92 $2,103.11 $2,368.80 $2,649.56 $2,946.16 $3,259.37 $3,590.04 $3,939.03
P=price S&P 500 Industrials $6.37 $8.02 $8.05 $6.74 $29.18 Total Liabilities and Equity  $2,100.00  $3,295.00  $3,367.25  $3,445.61  $3,530.39  $3,621.91  $3,729.20  $4,055.19  $4,349.27  $4,662.35  $4,995.38  $5,349.35  $5,725.30
D1=dividend in year 1 S&P 500 Information Technology $10.30 $10.81 $12.20 $16.24 $49.55            Hedge ratio= h=-N(d1)
i= the estimated rate of return S&P 500 Materials $4.62 $4.89 $4.10 $4.05 $17.66 h=-.790 Sources and Uses of Funds
g=long term growth rate of earnings and dividends S&P 500 Telecom. Services $2.70 $2.80 $2.74 $1.92 $10.16 Sources 1 2 3 4 5 6 7 8 9 10 11 12
You can transform this formula into the pragmatic earnings multiplier model as follows:  S&P 500 Utilities $3.75 $3.06 $4.66 $3.31 $14.78 Net Income  $   185.23  $   197.15  $   209.76  $   222.96  $   236.82  $   251.19  $   265.69  $   280.76  $   296.59  $   313.22  $   330.67  $   348.99
     P/E1= D1/E1 S&P 500 Real Estate $1.37 $1.29 $1.56 $1.45 $5.67      The Black-Scholes formula gives the price of this call option at 10.61.  If the market price of the option is different than this, then significant profit opportunities exist for investors.  The strategies  Depreciation  $     50.00  $     50.00  $     50.00  $     50.00  $     50.00  $     50.00  $     50.00  $     50.00  $     50.00  $     50.00  $     50.00  $     50.00
(i-g) for reaping these profits will be discussed later. Increase in Liabilities  $     60.30  $     13.32  $     16.49  $     19.81  $     23.30  $     26.97
E1=earnings in year 1     Finding a mispriced option is just the start of the arbitrage strategy.  Arbitrage means making riskless profit.  You take a position in the option, and also a position in the underlying stock,  The  Decrease in Cash  $   300.00
The P/E ratio is the earnings multiplier.  It is determined by: investor now has a hedge against stock price movements.  The hedge ratio tells you how much the price of an option changes, given a 1 point change in the underlying stock.  The hedge ratio Increase in Short Term Debt  $   944.77
1.  The expected dividend payout ratio, D1/E1 is h=-N(d1).  N(d1) is part of the first term in the Black-Scholes formula, N(.) is the cumulative normal distribution function, whose values are given in the table below.   Increase in Accounts Pay.  $     45.00
2.  The estimated rate of return on the stock, i N(d1)=ΔC/ΔS, where Δ (delta) means change.  The negative sign means that the option and the stock are held in opposite positions.  Once a mispriced option is discovered, the investor  Increase in Accrued Exp.  $     20.00
3.  The expected rate of growth of earnings and dividends for the stock, g. can form a riskless hedge by buying N(d1) shares of stock for each option written.  The option and the stock are held in opposite positions, thus the minus sign in front of N(d1).   Total  $1,545.00  $   247.15  $   259.76  $   272.96  $   286.82  $   301.19  $   375.99  $   344.08  $   363.08  $   383.03  $   403.97  $   425.96
The difficult parameters to estimate are i and g, or more specifically, the spread between i and g.   However, it can be shown that D1/P=(i-g).  Therefore, If the option is over-priced, then a short position is taken in the option (it is written), and a long position is taken in the stock.  Long means buy, short  means sell.  In the above example Uses
you can use the indicated dividend yield to estimate (i-g).   h=-N(.808)=-.790 is the hedge ratio.  This means that if the stock changes by one point, the call option will change a little less than ,8 points.  However, the hedge ratio holds only for small Increase in Accounts Rec.  $1,500.00  $     75.00  $     78.75  $     82.69  $     86.82  $     91.16  $     95.72  $   100.51  $   105.53  $   110.81  $   116.35  $   122.17
    P/E1=         D1/E1= M, the multiplier changes in the price of the underlying instrument.  If the underlying instrument (Stock) changes by 5 or 10 points, the ratio will not assure the riskless hedge will be maintained.  The ratio Increase in Inventory  $     45.00  $     47.25  $     49.61  $     52.09  $     54.70  $     57.43  $     60.30  $     63.32  $     66.49  $     69.81  $     73.30  $     76.97
      (i-g) must be reevaluated and adjusted whenever the stock price changes significantly.  Also, as time to expiration goes down, the price of the option goes down, and the hedge must again Decrease in Liabilities  $   124.90  $   131.41  $   138.18  $   145.30  $   143.90
         P=       E1 x D1/E1 be adjusted.  The solution for N(d1) and N(d2), N(∙) is the cumulative normal distribution function, and the tables showing the N(d) for each d is shown 3 pages down. Increase in Cash  $          -    $          -    $      8.69  $   219.97  $   180.25  $   191.06  $   202.41  $   214.32  $   226.82
(i-g) The variables that are the inputs to the formula need to be estimated.  The stock price, the exercise price, and the time to maturity can be found in the option quotation.  The time to maturity is  Total  $1,545.00  $   247.15  $   259.77  $   272.96  $   286.82  $   301.18  $   376.00  $   344.08  $   363.08  $   383.03  $   403.97  $   425.95
         P= E1 x M calculated by taking the number of calendar days until maturity, and then dividing it by 365 to express it in annual terms.  We need to find the risk-free interest rate for borrowing and lending during 
You can think of the multiplier as the value you are assigning to the stream of earnings.  Changes in earnings or changes in the multiplier will affect the price. the period of the option.  Since these rates usually differ, an average of the two can used.  The duration of the interest rate used should be as close as possible to the time of expiration
     We will use the Standard & Poor's 500 as a proxy for the aggregate stock market.  The S&P 500 is a market-weighted index of 500 leading companies of the option.  If the option has 3 months till expiration, then a 3 month interest rate should be used.  T-bill rates are generally too low.  They are lower than the borrowing rate for anyone  
in leading industries of the U.S. economy.  Each company in the index is weighted by it's total market capitalization, which is the price x the number except the U.S. government (which guarantees re-payment).  Certificate if deposit rates are good to use, or rates for top quality commercial paper.  The price of the option does not Ratio Analysis
of shares issued, authorized, and outstanding.  The S&P 500 includes about 75% of U.S. equities.  Fortunately, Standard & Poor's provides per share change a whole lot when the interest rate changes.  The volatility of the stock price is the most difficult input to estimate.  A good first approximation of the volatility of the stock is to look 4 Liquidity Ratios 1 2 3 4 5 6 7 8 9 10 11 12 Total
earnings data for the S&P 500,  which works quite well for our micro analysis of the aggregate market.  We are going to estimate the earnings and dividends per share at past data.  Price observations over the past several months should be used.  If we go back too far, the returns will be less reflective of the future volatility of the stock.   Current Ratio Current Assets/Current Liabilities 2.64 2.63
for the S&P 500 for the upcoming year, as well as the growth of earnings and dividends for the upcoming year.  We will then be able to calculate the expected return      The measure of volatility that is used in the formula is the variance (σ2) of returns.  Let's assume we have end-of-week quotes for 20 weeks.  In order to change prices into rates-of-return Acid Test Cash+Marketable Securities+Accounts Receivable/Current Liabilities 1.66 1.65
for the S&P 500, and then the value of the S&P 500.  Standard & Poor's estimates future earnings per share for the S&P 500 to be as follows: we have Rk=Sk/Sk-1.  Sk=the stock price at the end of the kth week.  We get the natural log of the return to get the continuously compounded return.  Now we calculate the mean of  Profitability Ratios
Estimates of EPS for the S&P 500 the return, m=1/20 Σ ln Rk.  ln Rk is the natural log of Rk.  We calculate the variance by by summing (ln Rk-m)2 over all periods, then dividing by(N-1).  We divide by (N-1) to account for the  Return on Assets Net Income/Total Assets 0.06 0.059 0.061 0.063 0.065 0.067  $     0.066
12 Month Operating Earnings Per Share degrees of freedom. Return on Equity Net Income/Equity 0.19  $      0.06
2018 estimate=154.39      The hedge ratio = h = -N(d1).  The hedge ratio in our example is -,790.  If the market price of the call option is below the formula price, it is underpriced.  In this case you would Financial Leverage Ratios
2019 estimate=169.86 buy 1 option contract (which controls 100 shares of stock) , and short sell 79 shares of stock, using the remaining balance to invest at the NRFR.  If the option is overpriced, you would  After-tax % cost of interest           Interest exp.-Tax shield Created/Amount borrowed    0.01
The required return on common stocks is a function of the economy's" real" risk-free rate (RFR), the expected rate of inflation (I), and the risk premium for common  sell, or write1 option contract, buy 79 shares of the stock with the proceeds of the the option sale, and borrow the remaining amount at the NRFR.  In either case, an arbitrage profit % Total debt of total assets Total debt/Total assets 0.70
stocks.  The "real" RFR is combined with the expected inflation rate to arrive at the nominal risk-free rate (NRFR).  The relationship is actually multiplicative, but  is possible. Capitalization ratio Long-term debt/Long-term debt+Equity 0.29
just adding the two rates together serves as a good approximation.   Applications of the Option Pricing Formula Interest coverage E.B.I.T./Interest 39.83
     Nominal rates are the rates that you actually observe in the market place.  The current yield to maturity of a government bond equal to your investment planning      "We begin by estimating the variables that go into the formula.  The interest rate should be the risk-free borrowing and lending rate over the period of the option.  Since these rates Burden coverage Profit before fixed charges and taxes/Fixed charges+principal payments 13.54
horizon should serve as a good proxy for the current nominal risk-free rate.  The current risk premium of the expected return of the market E(Rm) over the NRFR typically differ, we can use an average of the two.  The treasury bill rate will typically underestimate the correct rate (it is below everyone's borrowing rate but the US government). Efficiency Ratios
needs to be estimated.  If ten years is your planning horizon, then the yield on 10 year treasury bonds on 02/02/18 was 2.87%.  (see section on The Capital Asset Pricing Model). The Certificate of Deposit rate or the rate of top quality commercial paper.  The interest rate used in the formula should be the same duration as the the option whose value you Return on Assets Net Income/Total assets=Net income/Sales*Sales/Total assets 0.06
We must now determine the risk premium for common stocks.  "A study by Ibbotson and Sinquefield estimated the equity risk premium as the difference are trying to determine.  The option price is not very sensitive to small changes in the interest rate.  Volatility in the formula is shown by the standard deviation of the underlying Accounts receivable turnover Sales/Accounts receivable 1.00
in the annual rates of return for common stocks and treasury bills.  They found that from 1926-1978, the geometric mean of this risk premium was 6.2%."2 instrument.  There are several ways to estimate the standard deviation of the underlying instrument.  You can use past data to estimate the current standard deviation. Days sales outstanding Accounts receivable/Sales for year/360 360.00
We could say that during this period, the average risk premium for common stocks over the 90 day T-bill rate was 6.2%.  If we assume that the current risk premium of the stock market First you convert prices to rates of return.  Rk=Sk/Sk-1.  Sk=stock price at the end of kth week.  We then get the natural log of the return to approximate the continuously  Inventory turnover Sales/ending inventory 1.59
is similar now, then we can estimate the investor's required return for the stock market. The 3 month T-bill rate now is about 1.59%, so 7,79% would be the expected compounded return.  Next we calculate the mean of the return, m=1/20Σln Rk.  The variance is then calculated by summing (ln Rk-m)2 over all the periods, then dividing by  Return on equity Net Income/equity=Sales/Total assets*Net income/Sales*Total assets/equity 0.19
return of the market E(Rm).  As I will show you, the E(Rm) for the upcoming year is more than twice  that amount. (N-1).  We divide by (N-1) to take account of the degrees of freedom in the calculation of the variance in order the get an unbiased estimate.  Also, we do not have to use
     "The intrinsic determinants of the aggregate market risk premium are the business risk (BR), financial risk (FR), and liquidity risk (LR) of the aggregate market.   52 weekly intervals.  Monthly, bi-weekly, or even daily data can be used.  Remember that we are using past data to predict future data.  The variance does change over time, Sources and Uses of Funds Statement
The market measure of this risk is the variance of returns, σ2,  for the stock market."3   Both of these methods are trying to determine the same thing, the  so the more recent the data, the more likely that it will be reflective of what the current data is.  The more entries that are used, the more precise the sample will be.  Sometimes Sources of funds
level of risk in the stock market, and thus the investors' required rate of return for the market, i. the time of day that the observations are taken may be different, and the shorter the interval the more this can matter.  This can make daily observations less dependable.  With Net Income
These intrinsic determinants cause volatility in the market, which is measured by the variance of returns for the market, σ2. weekly observations this problem becomes less significant.  Twenty weekly observation seems to be a good compromise amongst these considerations. Plus depreciation 4.1666667
The required return on common stocks can be summarized as follows: The use of time series analysis produces a forecast of the variance that is much more accurate than just using a direct application of the historical data.  If we also use Funds provided by operations
i=f(NRFR, BR, FR, LR) P/O ratio =Div./Op. Earn. 2/1/2018 firm, industry, and market data along with time series analysis, we obtain a forecast that is even more accurate than just using time series analysis alone. Increase in liabilities
or Ret. rate =1-PO ratio S&P close of: 2762.13 The forecasted data produces results that match the actual data quite closely, and thus produces a forecast of future results that are about as accurate as we are able Increase in common stock
i=f(NRFR), σ2) Dividend yield (last 12 months: Jan, '18) 1.79% to obtain.   Total
This reads "i is a function of NRFR, BR, FR, and LR or i is a function of NRFR and σ2".      Dividend yield (current indicated rate) 1.83%     Another method of producing a forecast of future volatility is the use the variance that is implied by plugging the option's price and other statistics into the option formula, Uses of funds
     The growth rate of dividends and earnings, g, is a function of the retention ratio, b, and the return on equity, ROE. and seeing what variance is implied by the option formula.  We are using the formula to supply an estimate of the variance, and the same Dividends
g=b x ROE.  ROE can be broken down as follows:  P/O ratio= 32.8% estimate of the variance is required as an input to the formula.  It seems that the reasoning here is kind of circular.  However, obtaining the implied variance does give us an idea                                                                                                                                                                                                                                                                                                       Increase in cash
ROE=Net Income/Equity=Net Income/Sales x Sales/Total Assets xTotal Assets/Equity             b= 67.2% of what the market sees as the variance of the underlying instrument.  It fits the actual data fairly well, and does not require data collection, as the other method does, so it may be easier. Increase in accounts receivable
ROE=(Profit Margin)(Total Asset Turnover)(Leverage) 2766.6667 Taking an average of the implicit volatilities of several options may be the best way to determine what the market's opinion of the variance is.  Also, differing tax rates for different investors Increase in inventory
b = the retention ratio, the % of earnings that are retained, or 1-(D1/E1) means there may not be one price that eliminates profit opportunities for all investors.  Uncertainty in interest rates and variance cause uncertainty in the correct option price and Net addition to total fixed assets
D1/E1 =payout ratio ROE can be increased by increasing profit margin, total asset turnover, or leverage.  However, increasing leverage will inrease financial risk. the correct hedge ratio.  
Currently, the retention ratio for the S&P 500 is about 67.3%, and the most recent ROE was 15.4%.  This makes for a projected growth rate of about 10.4%.      The arbitrage strategy involves maintaining a dynamic hedge.  The position of the stock relative to the option must be constantly adjusted as the time to maturity decreases and as A net increase in any asset account is a use of funds, a decrease is a source.  A net increase in any liability or equity account is a source, a decrease is
The expected dividend yield for the upcoming year for the S&P 500 is 1.83%.  We know from the present value of dividend model that investors' expected rate of return for the S&P 500 is as follows: the price of the stock changes.  Each adjustment will require transaction costs.  Empirical evidence by Galai (1975)  suggests that even a 1% transaction cost will eliminate any  a use.
P=D1/i-g Estimated return (i) with long term growth rate of 13.4% Note:This 13.4% long term growth rate is larger than the  profit potential from following the hedging strategy for a typical investor.  Even a floor trader whose transaction costs are very low, after considering the implicit costs involved (i.e. exchange
i=D1/P+g        P/E= 18.28 i=.0183+.134 10.4% growth rate calculated by using b x ROE. seat cost, etc.)  it may make the strategy impractical.  The questioni is can an investor profit from use of the option pricing formula?  The formula is widely known and used.  A majority  Break-even  Analysis
i=.0183+.104 10 yr. T-Bond=2.87% i=.1523         Risk Premium of E(Rm) less T-Bond yield = 0.1236 of market makers on the option exchanges subscribe to option services who use the option pricing model.  The price of an option calculated with the pricing model is well-known information. Break-even units=Total fixed expenses/selling price per unit-variable expense per unit.  FE/SPPU-VEPU.
i=.1223 Dividend yield=D/P D/P=(i-g)              i-g= 0.0183 An option that is mispriced will be acted on immediately by a large group of investors, and this will change the price to it's correct level.  In retrospect, it is the option model's popularity Break-even dollars of sales=FE/1-(VE/S).
The S&P 500 was at 2821.37 on 02/02/18, with the estimated EPS for 2018 at $154.39 (the estimated dividend is $50.63).  The estimated dividend yield (D/P) for the S&P 500 is 1.83% that prevents it's profitable use.  When just a small group of investors has the informatiion that the model provides, then it can be used profitably by this small group.  Hovever, when                                                                                  
and the long term growth rate of earnings (and dividends) was estimated to be 13.4%.  These numbers indicate an expected return of 15.2%.. The risk premium over the 10 year bond is 12.4%. everyone knows about the formula, any mispricings quickly disappear.  The formula has become part of the market's information set.  Then why are there mispriced options according Fixed versus variable expenses
The 90 day T-Bill yield was about 1.59%, making the risk premium of the E(Rm) less the T-Bill yield about 13.7%, more than double what we decided the "normal" premium had ben (6.2%) between to the formula.  It is because the formula is not the only information in this set.  There are other firm, industry, or market indicators that investors consider in determining the market price Ex.1 Ex.2
1926 and 1978.  The P/E ratio (the multiplier) was 18.28%.  The P/E ratio for the S&P 500 has been going up for the past 5 years because prices have been increasing at a faster rate  of the option.  Investors, floor traders, or market makers combine the information in the set, one of which is the formula, and determine the market price  of the option. Sales 200000 200000
than EPS have. When some new information enters this set, the market price of the option changes very quickly, and the only people to profit from this change (if the change is positive) are the people who owned the option CGS 75% 150000 150000
Modern Portfolio Theory and the Capital Asset Pricing Model before this new information is released.  When there is a divergence between the market price of an option and it's formula price, this implies that the market believes that the Gross margin 50000 50000
      We will begin with an assumption that investors are risk averse.  This means that given a choice of two assets that have the same expected return, formula's price is not correct. Variable expense 20% 0 40000
the investor will select the asset with the lowest level of risk.  This means that there should be a positive relationship between expected return       The Black-Scholes option formula makes some assumptions that are unrealistic.  The original formulation does not allow for dividends.  This formulation needs to be adjusted  Fixed expense 46000 6000
and risk.  For a given level of risk, investors prefer higher returns to lower returns. Similarly, for a given level of return, investors prefer less because dividend payments usually make the price of the stock drop.  If the option is not payout protected, a drop in the price of the stock implies a drop in the option price.  The holder E.B.I.T. 4000 4000
risk to more risk.  Although the definitions of risk and volatility are different, for our purposes and in most financial literature, the terms are of a call option may want to exercise the option just before the stock goes ex-dividend so they can capture the value of the stock before the decline.  When evaluating an option, the  Level of sales where E.B.I.T.is the same under each alternative.
used interchangeably.  "The basic portfolio model, developed by Harry Markowitz, showed that the variance of the rate of return was a meaningful possibility of an early exercise must be taken into account.  If the dividend payout is known, you can replace the stock price with the stock price minus the dividend in the formula. Sales-.75Sales-46000=Sales-.95Sales-6000
measure of risk under a reasonable set of assumptions, and derived the formulas for computing the variance of the portfolio."4 If dividends are paid at a continuous rate, rather than lump sums, then the formula can be adjusted.  Let the dividend yield be ⱷ.  If the stock price is $100 and the dividend is $5,  Sales=200000
     While the expected rate of return for a portfolio of assets is simply the weighted average of the expected rates of return for each asset in the portfolio, the same then ⱷ=.05.  In order to modify the Black-Scholes formula to include this rate, we replace S in the first term of the formula with e-TS, and we replace the interest rate r in the d1 and
is not true for the variance of the rate of return for a portfolio of assets.  Let us look at the simplest case, a portfolio of two assets.  First we will  the d2 terms with r-.  These modificatios reduce the stock price S by the amount of the dividend, and adjust the return from the riskless hedging strategy accordingly."16 Degree of Operating Leverage
demonstrate the computation of the standard deviation (the square root of the variance) for an individual asset. The % change in net income brought about by a 1% change in sales.
      σ.=the standard deviation=  the of (Σ (Ri-E(Ri))2 x Pi) Traditional  Option Trading Strategies: Spreads and straddles  -1% 1% -1% 1%
      σ2=the variance I am goingto tell you about a couple of basic option trading stragies.  They hsve nothing to do with the option pricing model.  They are strategies concerning your outlook on the Sales 990 1000 1010 1485 1500 1515
      Ri=the rate or return for i underlying instrument of an option.  They could be options on stock, one of several stock index futures, many agricultural commodity futures, Treasury Bill Futures, Treasury Bond                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Var. exp. 742.5 750 757.5 1113.75 1125 1136.25
      E(Ri)=the expected rate of return of i (the arithmetic average of possible returns) Futures, precious metals futures, and others that I can not think of right now.  Let's say you are bullish on a particlar stock.  We will use IBM stock and some call options  Fixed exp. 100 100 100 100 100 100
      Σi=the summation of each i on it. We will now create a bull spread. Net Inc. 147.5 150 152.5 271.25 275 278.75
      Pi=the probability of each possible rate of return 7/30/2016 Strike Price D.O.L.= 1.67% 1.36%
     In order to discuss the derivation of the variance of a portfolio of assets, I need to present two basic concepts in statistics, covariance and correlation. IBM Stock $160.96 You can also calculate D.O.L.with the following formula.  D.O.L.=Q(SPPU-VEPU)/Q(SPPU-VEPU)-FE
Covariance is a measure of the degree to which two variables, in this case the returns of two assets, move together over time.  A positive covariance Options Call Q=Volume in units
means that the variables tend to move in the same direction at the same time, and a negative covariance means that they move in opposite directions. Exp. 1/15/2017 SPPU=Selling price per unit
Covariance is an absolute measure.  For two assets, i and j, the covariance of monthly returns is as follows: Strike 160 165 170 175 155 150 VEPU=Variable expense per unit
1/12 x Σ (Ri-E(Ri))(Rj-E(Rj)) Price  $    7.20  $    4.95  $    3.21  $    2.02  $   10.05  $   13.45 FE=Total fixed expenses
Ri=return of asset i   These optios have about 5˝ months till expiration. D.O.L. Sales=1000=1.67%
Rj=return of asset j I will go long (buy) the option with a strike price of 165, and write, or go short (sell) the option with a strike price of 175. D.O.L. Sales=1500=1.36%
The correlation coefficient is a relative measure of the degree to which two variables move together.  The correlation coefficient is defined as follows: Strike Price
r=Covijiσj 165  $    4.95 Degree of Financial Leverage
Covij=the covariance of assets i and j. 175  $   (2.02) The % change in earnings per share brought about by a 1% change in earnings before interest and taxes.
σi=the standard deviation of asset i Net  $    2.93 -1% 1% -1% 1%
σj=the standard deviation of asset j My total investment is $2.93.  On 1/15/17, IBM Stock is at 175.  The 165 strike finishes at $10, and the 175 strike expires worthless.  My net profit is as follows: E.B.I.T. 792000 800000 808000 990000 1000000 1010000
The correlation coefficient can vary between -1 and 1.  1 means there is a perfect, positive linear relationship between 2 variables, while -1  Strike Price E.P.S. 0.784 0.8 0.816 1.18 1.2 1.22
means that the variables move in completely opposite directions.  0 means that there is no linear relationship between the two variables.   165  $   10.00 D.O.F. 2.00% 1.67%
A correlation of -1 gives you the maximum benefits of diversification.  In a portfolio with 2 assets, perfect negative correlation produces a standard deviation 175  $        -   You can also calculate the D.O.F. with the following formula.  D.O.F.=E.B.I.T./(E.B.I.T.-F)
of 0, a risk-free portfolio. Net  $   10.00 F=Annual interest expense
The formula for the standard deviation of a portfolio is as follows: Spread  $   (2.93)
σp=the √σp2 = (Σ Wi2σi2 + 2 x ΣΣWiWjCovij) σp=the standard deviation of a portfolio Total  $    7.07 With IBM finishing at $175, my profit is $7.07.  If IBM finishes at 176, my profit is still $7.07. You currently own a company.  You own 10 shares of stock.  The tax rate is 50%.  E.B.I.T.is 200.  E.P.S. is 10.
Wi=the weight of each asset in the portfolio σp2=the variance of a portfolio Strike Price Now
WiWj=the weights of each pair of assets in the portfolio 165  $   11.00 E.B.I.T. 200
We see from the formula that not only do the individual variances affect the portfolio variance, but the covariance's between each individual asset also  175  $   (1.00) Taxes 100
affect the portfolio variance.  In fact, it can be shown that in a portfolio with a large number of assets, the variance of the portfolio becomes the Net  $   10.00 If IBM finishes at $167.93, my profit is at: If IBM finishes at $160, my profit is at: With this bull spread, you can make a maximum of E.A.T.C.S. 100
summation of the weighted covariances.  This means that when you add an asset to a portfolio with a large number of assets, it is not the asset's Spread  $   (2.93) Strike Price Strike Price $7.07 (the $10 difference in the strike prices less  E.P.S. 10
variance, but the asset's covariance with all the other assets in the portfolio that is important. Total  $    7.07 165 $2.93 165  $        -   the initial investment of $2.93).  You can lose a  Shares 10
     When assets with correlations of less than one are combined into portfolios, we get the benefits of diversification.  If all the possible If IBM finishes at $175 or higher, I will get the maximum profit on this spread of $7.07.  If IBM finishes at $174, my profit is as follows: 175  $        -   175  $        -   maximum of $2.93, (the $10 difference in the strike  You can expand E.B.I.T. by 160.  It will cost 500.  You can raise the money by issuing 10 new shares of stock, or by borrowing the money.
portfolios of assets are shown on a graph where the x-axis is the standard deviation and the y-axis is the expected return, we will get a curve on the 2008 EPS 2008 P/E 2009 EPS 2009 P/E 2010 EPS 2010 P/E 2011 EPS 2011 P/E 2012 EPS 2012 P/E 2013 EPS 2013 P/E 2014 EPS 2014 P/E Strike Price Net  $    2.93 Net  $        -   prices less the maximum profit of $7.07).   Stock Loan
edge of these groups of portfolios called the efficient frontier (see graphs 2,3).   All the portfolios on the efficient frontier are superior to all the other portfolios that S&P 500 $49.51 18.24 $56.86 19.61 $83.77 15.01 $96.44 13.04 $96.82 14.73 $107.31 17.22 $113.01 18.22 165 $9.00 Spread  $   (2.93) Spread  $   (2.93) $7.07+$2.93=$10, the difference in the strike prices. E.B.I.T. 360 360
are not on the efficient frontier.  When I say superior, I mean that every portfolio on the efficient frontier has a higher expected return for a given Con. Disc. $5.28 32.08 $10.96 21.45 $18.20 16.24 $20.81 14.83 $22.27 16.89 $25.12 21.1 $27.70 20.67 175  $      -   Total  $        -   Total  $        -   Interest 0 40
level of standard deviation, or a lower standard deviation for a given expected return, compared to every portfolio that is not on the efficient frontier. Con. Stap. $17.17 14.37 $18.36 14.94 $19.45 15.61 $21.38 15.69 $22.59 15.97 $23.95 18.48 $24.16 20.68 Net  $   9.00 Taxes 180 160
r=0 Energy $50.93 7.59 $17.26 24.91 $35.21 14.39 $47.94 10.86 $44.30 12.03 $42.35 15.39 $42.93 13.66 Spread  $  (2.93) If IBM finishes at  165 or lower,both options will expirre worthless, and I will lose my initial investment of $2.93.  With a bull spread you buy an option at a strike price, E.A.T.C.S. 180 180
       σ       Return Weight 1 Weight 2 Stock E(Ri) σ2 σ Financials ($21.24) -7.95 $4.40 44.04 $14.82 14.49 $16.23 10.79 $16.44 13.46 $21.07 13.99 $21.96 15.18 Total  $   6.93 and then write an option with a higher strike price, and thus at a lower price.  You get to lower your investment (if you just bought the IBM 165 you would have to pay E.P.S. 9 16
0.0812 0.18 0.2 0.8 1 0.1 0.0049 0.07 Health Care $24.47 12.64 $26.40 13.72 $28.90 12.62 $31.08 12.93 $31.53 14.68 $32.43 19.81 $35.79 22.13 $4.95), but you give up the potential of the possibility to make a lot of money (If IBM stock went to $200, you would make $35.00).  When you buy a call option, you Shares 20 10
0.0662 0.16 0.4 0.6 2 0.2 0.01 0.1 Industrials $21.18 9.78 $14.22 17.09 $18.41 16.36 $20.96 13.95 $22.28 14.76 $24.88 18.19 $26.83 18.13 Graph 1  may lose the money you paid for the option, but you can make the difference between  what the stock is trading for now, and what your exercise price is.  If the stock
0.061 0.15 0.5 0.5 Info. Tech. $16.12 14.38 $17.48 21.21 $26.25 15.41 $31.44 13.04 $32.69 14.19 $33.58 17.44 $36.99 18.7 Standard Deviation & Return when r=0 price goes way above the exercise price, you can make a lot of money.  The amount of money you could make is only limited by how high the stock price Stock (20 shares, you own 10) Loan
0.058 0.14 0.6 0.4 Materials $8.09 17.01 $7.09 28.18 $13.33 17.97 $16.20 13.07 $14.67 16.2 $14.04 20.78 $15.86 19.25 goes.  A put works the same way, but in reverse (with a put, you are betting that the stock will go down).  However, there is no limit on how high the E.B.I.T. E.A.T.C.S. E.P.S. E.B.I.T. Interest E.A.T.C.S. E.P.S.
0.0595 0.12 0.8 0.2 Telecom. $8.21 13.61 $7.22 15.88 $7.36 17.5 $6.85 18.95 $3.38 43.26 $12.31 12.63 $7.21 21.16
price of a stock will go, but it can only go down to 0.  Therefore, if you have a call and a put both with the same parameters, the 100 50 2.5 100 40 30 3
Here we see how the standard deviation and the return are affected by changing the weights of stock 1 and stock 2.  The correlation coefficient in Utilities $12.25 12.08 $11.50 13.74 $12.34 12.92 $12.47 14.67 $11.97 14.85 $12.15 15.91 $13.22 18.16 call will be worth a little more than the put.  Anyway, back to our spread, you must realize that you have written an option, 200 100 5 200 40 80 8
this scenario is 0 (see Graph1). Real Estate $1.79 43.56 $0.99 95.13 $1.32 91.14 $2.49 52.33 $3.13 48.33 $3.31 44.95 $5.34 35.22 a 175 call as part of the spread.  If you have written an option and do not have it covered by a long position on an option or the stock, you 2000 1000 50 2000 40 980 98
     "Capital market theory builds on portfolio theory, resulting in a model that can be used to price all risky assets.  We end up with the 2015 EPS 2015 P/E 2016 EPS 2016 P/E 2017 EPS 2017 P/E 2018 EPS 2018 P/E 2019EPS 2019 P/E Prices will leave yourself open to unlimited loss if the stock goes through the roof.  However, you are long (bought) the 165 call  as 10000 5000 250 10000 40 4980 498
Capital Asset Pricing Model that will tell us how to determine the required rate of return for all risky assets."5  If we look at the graph of the       12'15     12'16     12'17 the other part of the spread, so you are covered (what ever you lose on the short  side, you will gain on the 165 call that you  200000 100000 5000 200000 40 99980 9998
efficient frontier, the risk-free asset is on the y-axis.  It has a 0 standard deviation.  The covariance between the risk-free asset and any risky asset S&P 500 $100.45 20.35 $106.26 21.07 $124.13 21.54 $154.39 18.28 $169.86 16.61 2043.94 2238.83 2673.61 are long.  With this spread, you are anticipating that the stock will finish somewhere in a range (165-175).  Actually, if the stock The growth rate of E.P.S. will always be greater than the growth rate of E.B.I.T. whenever financial leverage is used.  With common stock, they will be equal.
or portfolio of risky assets is zero.  Therefore, the variance of this portfolio is as follows: Con. Disc. $30.44 20.4 $33.30 19.46 $34.55 22.73 $39.55 21.46 $45.04 18.84 621.02 647.82 785.33 finishes at 175 or above, you will make the maximum you can make, $7.07.
σp2=(1-WRF)2σi2 WRF=the weight of the risk-free asset Con. Stap. $24.32 21.31 $25.33 21 $27.28 21.53 $30.38 19.52 $32.81 18.07 518.42 531.79 587.39      If you think that the underlying instrument of an option, in this case IBM stock, is going down (you are bearish on the
WRF=weight of the risk-free asset Energy ($13.71) -32.7 ($3.49) -158.91 $14.64 36.43 $25.02 22.36 $27.43 20.4 448.44 554.5 533.41 stock), then a bear spread might be approriate.  You can write the IBM 160 call for $7.20, and buy the 170 call for $3.21, for a net inflow
A straight line going from the risk-free asset on the y-axis to a point tangent to the efficient frontier and beyond is the capital market line (CML).  The point Financials $23.01 13.98 $23.78 16.25 $26.72 17.36 $34.52 14.44 $38.79 12.85 321.73 386.53 463.94 of $3.99. Remember that the stock is at $160.96 on 07/03/16.  These options have 5˝ months till expiration.
of tangency to the efficient frontier is the market portfolio, point M (See graph 5, point M is .05, .068).  Any portfolio along the line from RFR to M Health Care $38.72 21.52 $42.45 18.77 $45.72 20.92 $60.39 16.88 $65.24 15.62 833.23 796.91 956.32 Strike Price
represents some combination of investment in the risk-free asset and the market portfolio.  Point M means that 100% of your wealth is invested in the  Industrials $28.00 16.56 $27.07 19.88 $29.18 21.86 $35.38 18.94 $39.70 16.88 463.53 538.07 637.81 160  $    7.20
market portfolio.  Any portfolio on the CML that is above point M represents investing 100% of your wealth in the market portfolio, and then borrowing some % of Info. Tech. $37.97 19 $37.99 21.27 $49.55 22.32 $61.32 19.4 $66.40 17.91 721.48 807.95 1106.18 170  $   (3.21)
your wealth at the risk-free rate and also investing this in the market portfolio.  Suppose that the expected risk-free rate is 6%, and the expected return on the  Materials $8.49 32.23 $13.01 23.99 $17.66 21.46 $21.32 18.25 $23.82 16.33 273.64 312.16 378.94 Net  $    3.99
market is 12%.  The expected return and the expected standard deviation from a portfolio that is invested 50% in the risk-free asset and 50% in the  Telecom. $12.15 12.34 $9.86 17.9 $10.17 16.34 $14.58 11.6 $14.90 11.35 149.91 176.61 166.07 My net inflow is $3.99.  There is no initial investment like with the bull spread shown above, but an initial positive cash flow, and you hope 
market portfolio would be as follows: Utilities $11.25 19.55 $13.67 18.05 $14.78 18.09 $15.57 16.37 $16.41 15.54 220 246.83 267.37 that no negative cash flows occour later.
E(Rport)=WRF(RFR) + (1-WRF)(E(Rm)) Real Estate $5.45 34.89 $7.38 25.78 $5.67 35.95 $5.05 38.84 $5.57 35.2 190.22 190.23 203.86 On 1/15/17 IBM finishes at $160.  Both options expire worthless, and you keep your $3.99 inflow.
=.5(.06) + .5(.12) Stan. Dev. Return Strike Price
=.03 + .06 $2,044.16 $2,238.90 $2,673.76 $2,822.25 $2,821.37 0.0812 0.18 160  $        -  
=.09 In the row above, the  P/E ratios of the index and each sector of the index for historical data is based  on year end prices (2015 thru 2017).  The P/E ratios  0.0662 0.16 170  $        -  
E(Rport)=the expected return of the portfolio of estimated data are based on current prices (very small differences are based on rounding errors). 0.061 0.15 Net  $        -  
E(Rm)=the expected return of the market 0.058 0.14 Spread  $    3.99
E(σp)=(1-WRF)(σm) 0.0595 0.12 Total  $    3.99 160.96
=.5σm If IBM finishes at 150, you get the same result. 3.99
E(σp)=the expected standard deviation of the portfolio Strike Price 164.95
σm=the standard deviation of the market 160  $        -  
If you invested an amount equal to 100% of your wealth in the market portfolio, and then borrowed an amount equal to 50% of your wealth at the  170  $        -  
risk-free rate and also invested this in the market portfolio, the WRF would not be positive, but a negative 50%.  Your expected return and  Net  $        -  
expected standard deviation would be as follows: Spread  $    3.99
E(Rp)=-.5(.06) + 1.5(.12) Total  $    3.99
=-.03 + .18 If IBM finishes anywhere from $160 or below, you make your $3.99 profit on the spread.  If IBM finishes at $161, your profit will be as follows.
=.15       t 0 1 2 3 4 5 6 7 8 9 Strike Price
E(σp)=1.5(σm) -3 0.0013 160  $    1.00
Risk and return increase in a linear fashion along the CML, which is above and thus superior to the original efficient frontier.  The CML is the  -2.9 0.0019 0.0018 0.0018 0.0017 0.0017 0.0016 0.0015 0.0015 0.0014 0.0014 170  $        -  
"new" efficient frontier.  "The market portfolio, point M, is the tangent portfolio that gives the highest portfolio possibility line.  Therefore, everybody  -2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.002 0.0019 Net  $   (1.00)
will want to invest in this risky asset portfolio M, and borrow or lend to be somewhere on the CML.  Because all investors want this portfolio M -2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.003 0.0029 0.0028 0.0027 0.0026 Spread  $    3.99
as part of their total portfolio, all risky assets must be in the portfolio.  If a risky asset were not in this portfolio, it would have no demand, and -2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.004 0.0039 0.0038 0.0037 0.0036 Total  $    2.99
therefore no value.  When the market is in equilibrium, all assets that are included in this portfolio are in proportion to their market value. -2.5 0.0062 0.006 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 If IBM finishes at 163.99, then your profit is $0.
The market portfolio does not include only common stocks, but also bonds, options, real estate, coins, etc.  Because the market portfolio -2.4 0.0082 0.008 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064 Strike Price
contains all risky assets, it is a completely diversified portfolio."6 -2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 160  $    3.99
     Let me talk about systematic and unsystematic risk.  Systematic risk is the risk associated with the overall market.  This type of risk (i.e. volatility) -2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.011 170  $        -  
is caused by macroeconomic variables that affect all risky assets.  Unsystematic risk is the risk associated with each individual company.  In the  -2.1 0.0179 0.0174 0.017 0.0166 0.0162 0.0158 0.0154 0.015 0.0146 0.0143 Net  $   (3.99)
market portfolio, this unsystematic risk is diversified away, and only the systematic risk remains.  Since unsystematic risk can be eliminated through  -2 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 Spread  $    3.99
diversification, the investor can only expect to be compensated for the level of systematic risk that he assumes.  Every investor wants to invest  -1.9 0.0287 0.0281 0.0275 0.0268 0.0262 0.0256 0.025 0.0244 0.0239 0.0233 Total  $        -  
in portfolio M, the market portfolio.  The only decision is the financing decision, which depends on your risk preferences.  If you are relatively risk -1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.03 0.0294 If IBM finishes at $170,then your profit is $0.
averse, you will invest some of your wealth in the risk-free asset, and the remainder in the market portfolio.  If you are more of a risk taker, you will -1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 Strike Price
invest all of your wealth in the market portfolio.  If you want to take an even higher risk (and higher expected return) position, you could also borrow some -1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 160  $   10.00
portion of your wealth at the RFR and invest it all in portfolio M.  Because the only relevant portfolio is the market portfolio, the only important consideration -1.5 0.0668 0.0655 0.0643 0.063 0.0618 0.0606 0.0594 0.0582 0.0571 0.056 170  $        -  
is a stock's covariance with the market portfolio (Covim).  The ratio of a stock's covariance with the market over the variance of the market is equal to  -1.4 0.0808 0.0793 0.0778 0.0764 0.075 0.0735 0.0721 0.0708 0.0694 0.0681 Net  $  (10.00)
beta (Bi=Covim/σ2m), (Bi=beta of stock i).  A beta of 1 means that the covariance of a stock with the market is equal to the market variance.  A beta of  -1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 Spread  $    3.99
2 means that the covariance of a stock with the market is twice the market variance.  The Capital Asset Pricing Model (CAPM) is as follows: -1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.102 0.1003 0.0985 Total  $   (6.01)
E(Ri)=RFR + Bi(E(Rm) - RFR) -1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.123 0.121 0.119 0.117 If IBM finishes at 175, you lose:
E(Ri)=the expected return of stock i -1 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 Strike Price
"The expected rate of return for a stock is determined by the RFR plus a risk premium that is a function of the systematic risk of a stock (Bi), and  -0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.166 0.1635 0.1611 160  $   10.00
the prevailing market risk premium (Rm - RFR).  If we expect the economy's RFR to be .08 and the return of the market (Rm) to be .14, then the -0.8 0.2119 0.209 0.2061 0.2033 0.2005 0.1977 0.1949 0.1921 0.1894 0.1867 170  $    5.00
expected return for a stock with a beta of .7 would be as follows:  -0.7 0.242 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 Net  $  (10.00)
E(Ri)=.08 + .7(.14 - .08) -0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 Spread  $    3.99
=.122 -0.5 0.3085 0.305 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.281 0.2776 Total  $   (6.01)
Beta is a normalized measure of systematic risk."7  As I stated before, this systematic risk is the only risk that the investor can expect to be -0.4 0.3446 0.34 0.3372 0.3336 0.33 0.3264 0.3228 0.3192 0.3156 0.3121 If IBM finishes at $160 or below, you make a maximum profit of $3.99, (the $10 difference in the strike prices less the maximum loss of $6.01).
compensated for.  The relationship between the expected return of a stock and it's beta is shown in graph 4.  The diagonal line is the Security Market -0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.352 0.3483 If IBM finishes at $170 or above, you lose a maximum of $6.01, (the $10 difference in the strike prices less your initial inflow of $3.99).
Line (SML).  The RFR, which is 5%, is where the SML intersects the y-axis (point 0).  The M portfolio (point 1), with an expected return of 15% and a beta of 1,  -0.2 0.4207 0.4168 0.4129 0.409 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 $3.99+$6.01=$10, the difference in the strike prices.
is where the SML intersects the horizontal and vertical lines.  When markets are efficient and in equilibrium, all assets or portfolios of assets should plot on the -0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
SML.  Any security that is above the SML is underpriced because it's estimated return would be above what is required from a security with this level           -.0 0.5 0.496 0.492 0.488 0.484 0.4801 0.4761 0.4721 0.4681 0.4641
of systematic risk.  In contrast, any security that is below the SML would be overpriced, because its estimated return would be below what is required from a 
security with this level of systematic risk.  In the example above, the required return is 12.2%.  If the estimated return for the upcoming period             .0 0.5 0.504 0.508 0.512 0.516 0.5199 0.5239 0.5279 0.5319 0.5359
is higher than this, then the stock is underpriced.  Securities' markets are pretty much efficient, but not completely efficient.  There are times when 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
not everyone is aware of all the relevant informationabout a security.  If an analyst can derive estimates of returns that are consistently superior to the 0.2 0.5793 0.5832 0.5871 0.591 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
aggregate market's estimates, then he will be able to earn above average returns on a risk-adjusted basis. 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.648 0.6517
     The systematic risk of an asset can be calculated with a statistical methodology called linear regression, which allows you to derive an asset's characteristic  0.4 0.6554 0.6592 0.6628 0.6664 0.67 0.6736 0.6772 0.6808 0.6844 0.688
line with the market portfolio of the following form: 0.5 0.6915 0.695 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.719 0.7224
E(Rit)=αi + BiRmt + e 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
E(Rit)=the expected rate of return for asset i during period t 0.7 0.758 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
Rmt=the rate of return for the market portfolio during period t 0.8 0.7881 0.791 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
Bi=beta of asset i 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.834 0.8365 0.8389
αi=constant term for asset i 1 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
e=random error term 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.877 0.879 0.881 0.883
The characteristic line is the line of best fit through a scatter plot of rates of return for asset i, coinciding with market rates of return over some past 1.2 0.8849 0.887 0.8888 0.8907 0.8925 0.8944 0.8962 0.898 0.8997 0.9015
time period.  The characteristic line is positioned in such a way that the summation of (yi - Yi)^2 is minimized.  It is located in such a way as to  1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
minimize the sum of the squares of the errors, thus it is called the least-squares method. 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
yi=value of y for observation i 1.5 0.9332 0.9345 0.9357 0.937 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
Yi=predicted value of y for observation i 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
Consider the following example of the computation of the beta coefficient for IBM in 1979 relative to the S&P 500  E(RIBM)=-1.1% 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
B=CovIBM,mm2 σ2IBM=14.73 σm2=13.786 B=11.111/13.785 E(Rm)=1.04% 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.97 0.9706
B=beta σIBM=3.838 σm=3.713 B=.806 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.975 0.9756 0.9761 0.9767
CovIBM,m=the covariance or the returns for IBM and the S&P 500 r=CovIBM,M/σIBMσm                  α= -1.93824 2 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
=11.111                  r= 0.78 α=E(RIBM)-B*E(Rm) 2.1 0.9821 0.9826 0.983 0.9834 0.9838 0.9842 0.9846 0.985 0.9854 0.9857
σm2=the variance of the returns for the market α=-1.1-(.806*1.04) 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.989
=13.785 σm=3.713 Characteristic Line= RIBM=-1.94 + .806(Rm) 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
This is the simple linear regression model, with the market returns as the independent variable x, and the stock returns (IBM) as the dependent 2.4 0.9918 0.992 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
variable y.  This model measures the type and extent of the relationship between variables.  Excel and Lotus both have a linear regression function 2.5 0.9938 0.994 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
that automatically calculates the above values.  There are many services that calculate betas for stocks.  Yahoo, for instance, gives you a beta for each stock. 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.996 0.9961 0.9962 0.9963 0.9964
     The number of observations and the time interval between them used to calculate the characteristic line varies amongst services.  Value Line 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.997 0.9971 0.9972 0.9973 0.9974
Investment Services uses weekly rates of return over a 5 year period.  Other services use monthly rates of return over a 5 year period. 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.998 0.9981
There is no set limit on how many observations to use or the interval between them.. You want to use enough observations to get an accurate picture of the 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9987
characteristic line, but not so many that you are going so far back in time that the company may have changed a lot.  Typically, the S&P 500 is used 3 0.9987
as a proxy for the market portfolio when determining the characteristic line.  We must remember that we are using historical data to estimate the future
beta for an asset. "Studies have shown that betas for individual stocks over short periods are rather unstable, but the stability of portfolio betas over longer
periods increased dramatically."8 N(d1)=N(.808)=.790
     If the estimated return of a stock is equal to the expected return, then the stock will plot on the SML (Graph 4), and it is properly valued.  If the  N(d2)=N(.612)=.729
estimated return is greater than the expected return, then the stock will plot above the SML and it would be undervalued.  If the estimated return is 
less than the expected return, then the stock would plot below the SML and it would be overvalued.  
The expected return is synonymous with the required return.  It is the return that the investor requires from a stock, a portfolio, or the  
market as a whole. The CAPM shows how the systematic risk of an asset, Beta (B), determines the investor's required rate of return and thus it's 
value.  The separation theorem implies that everyone will want to invest in this mythical market portfolio, the only difference between investors is how
they will finance this investment.  First of all, there is no portfolio that contains all risky assets.  Also, if you can not borrow at the RFR, how can you attain a 
high-risk portfolio?  Although the mythical "market" portfolio containing all risky assets (stocks, bonds, real estate, coins, stamps, antiques, etc.) does not
exist, an investor can acquire a diversified portfolio of some of these assets that would be highly correlated with  the market portfolio.  It is possible 
to acquire well-diversified portfolios of stock through large, balanced mutual funds.  These funds are highly correlated the S&P 500 index.  You can also 
acquire well-diversified portfolios of municipal, government, or corporate bonds through fixed-income mutual funds.  A real estate investment trust (REIT)
can provide a diversified portfolio of real estate.  If you can not borrow and leverage your portfolio, and you want to have a portfolio with a beta of 1.3, 
you can simply build a well-diversified portfolio of stocks with a weighted average beta of 1.3.  
     The question is whether or not the CAPM can explain the returns of risky assets.  Is there a positive relationship between systematic risk (beta) and return?
"A study by Sharpe and Cooper generally supports a positive relationship, although it was not completely linear.  In the highest risk classes, there was 
a tendency for returns to level off and even decline slightly.  This study also shows that betas for portfolios were stable."9
     An investor wants to analyze the performance of their investment portfolio,  whether they do their own analysis or have it done by a professional money
manager.  If you do their own analysis, you want to see if the time and effort spent was worth it in terms of results.  If you paying a professional 
money manager (through a mutual fund or an investment counselor), you need to evaluate his performance and determine if the results are worth
what he is charging you.  We will now take a look at how we evaluate portfolio performance.  We will examine three composite performance measures
that determine risk and return.  When evaluating the performance of portfolio managers, we will consider two major factors.  First, we will look at the
ability to derive above average-returns for a given risk-class.  Also, we will look at the ability to diversify, and thus eliminate all unsystematic risk
from the portfolio.  Superior risk-adjusted returns can be achieved through superior timing or superior stock selection.  The "superior analyst" can
consistently select stocks that plot above the SML (see graph 4).  The level of diversification in a portfolio can be determined by a portfolio's  Stan. Dev. Return
correlation with the market portfolio, which is completely diversified.   0.12 0.22
     We will now take a look at different methods of ranking portfolio managers' risk-adjusted return performance.  The first composite performance measure 0.12 0.21
we will look at is the Treynor method, developed by Jack Treynor in an article in the "Harvard Business Review" 10 0.12 0.2
The Treynor performance measure (T) is as follows: 0.12 0.19
         Ti= Ri-RFR 0.12 0.18
   Bi 0.11 0.21
Ti=the Treynor performance measure for portfolio i during a given time  period 0.11 0.2
Ri=the average rate of return for portfolio i during a given time period 0.11 0.19
RFR=the average risk-free rate during a given time period 0.11 0.18 Graph 2
Bi=the beta of the portfolio i, which indicates relative volatility and systematic risk 0.11 0.17 The Efficient Frontier
" Another measure of portfolio performance is the Sharpe measure, developed by William F. Sharpe that was designed to evaluate the performance of  0.1 0.19
mutual funds.  The Sharpe measure is as follows: 0.1 0.18
        Si= Ri-RFR 0.1 0.17
σi 0.1 0.16
Si=the Sharpe portfolio performance measure for portfolio i during a given time period 0.1 0.15
Ri=the average rate of return for portfolio i during a given time period P 0.09 0.17
RFR=the average risk-free rate during a given time period 0.09 0.16
σi=the standard deviation of portfolio i 0.09 0.15
The difference between the two measures is that the Treynor measure uses beta, the systematic risk, as the denominator, while the Sharpe measure uses the 0.09 0.14
standard deviation, the total risk, as the denominator.  "The Sharpe measure not only evaluates a portfolio manager on the basis of return performance, but also 0.09 0.13
measures how well diversified the portfolio is.  If a portfolio is perfectly diversified, (i.e. does not contain any unsystematic risk), the two measures would 0.08 0.14
give identical rankings, the total variance would equal the systematic variance.  In a well-diversified portfolio such as a mutual fund, the two measures  0.08 0.13
will give very similar rankings.  The Jensen measure is another composite performance measure, and like the others it is based on the Capital Asset 0.08 0.12
Pricing Model.  If this model is empirically valid, we can express the expectations formula in terms of realized rates of return.   0.08 0.11
Rjt=RFRt + Bj(Rmt - RFRt) + Ujt 0.08 0.1
Rjt=the realized rate of return on portfolio j during time period t 0.07 0.11
RFRt=the risk-free rate during time period t 0.07 0.1
Bj=the beta of portfolio j 0.07 0.09
Ujt=random error term 0.07 0.08
Subtract the RFRt from both sides. 0.07 0.07
Rjt - RFRt=Bj(Rmt - RFRt) + Ujt 0.06 0.07
If a portfolio manager is superior because he can forecast market turns or consistently select undervalued securities, the risk premium he experiences 0.06 0.06
will exceed the premium implied by the model.  The superior portfolio manager will have consistently positive random error terms.  His actual returns will 0.06 0.05
consistently be above the expected returns.  In order to measure this superior performance,  we will have to allow for an intercept that measures these 0.06 0.04
positive residuals.  This intercept is alpha, and if it is positive, it measures how much the actual returns are above the expected returns. 0.06 0.03
Rjt - RFRtj + Bj(Rmt - RFRt) + U 0.05 0.06
αj=alpha of portfolio j  0.05 0.05
If a regression of the portfolio j risk premium (Rjt - RFRt) and the market risk premium (Rmt - RFRt) indicates a positive, statistically significant intercept, 0.05 0.04
then this intercept shows how much the portfolio's risk-adjusted return exceeds the aggregate market's return.  However, if these residuals 0.05 0.03
are consistently negative, the alpha will be negative.  This indicates an inferior portfolio manager.  His actual returns will consistently 0.05 0.02
be below the expected returns.  An alpha of zero indicates that the portfolio manager has essentially matched the expected return.
The correlation coefficient, R squared, is a good measure of diversification.  A correlation coefficient of 1 means the portfolio is perfectly
correlated with the market.  This portfolio is just as diversified as the market is.  
     The above mentioned portfolio performance measures can be used to evaluate open-end mutual funds.  The return for each year for these funds  Graph 3
can be calculated as follows. Eff. Fron. Stan. Dev. Return
         Rit= EPit + Divit + Cap Distit - Bpit 0.12 0.22
Bpit
0.11 0.2
Rit=the return for fund i during year t 0.1 0.19
Epit=the ending price for fund i during year t 0.09 0.17
Divit=the dividend payment made by fund i during year t 0.08 0.14
Cap Distit=capital gain distributions made by fund i during year t 0.07 0.11
Bpit=the beginning price for fund i during year t 0.06 0.07
Mutual funds provide a convenient way to achieve instant diversification in a certain area.   There are many different types of mutual funds that invest  0.05 0.06
in different types of stocks and bonds.  Mutual funds may be very appealing to investors with neither the time nor the inclination to do fundamental analysis
themselves.  They offer a wide variety of investments in terms of risk and return.  There are two main types of mutual funds, closed-end and 
open-end funds.  Both of these types of mutual funds begin like any other publicly held company, with an initial offering of stock to investors. 
The difference between these two types of funds is what they do after this.  The closed-end mutual fund, again like any other publicly held company, 
has stock that trades on a secondary market, and the price of it's shares depends on supply and demand.  When buying or selling shares of a 
closed-end fund, the investor pays a regular trading commission.  There are no additional shares offered after the initial offering, and the company
does not repurchase shares on demand.  The net asset value (NAV) is the total market value of all the securities in the fund divided by
the number of shares of the fund outstanding.  Surprisingly, the NAV and the market price of the fund's shares are almost never the same.
Typically, the market price of the shares is 5 to 20 percent below the NAV.  An open-end mutual fund continues to sell shares after the initial
offering is made.  They sell shares at the NAV of the fund, with or without a sales charge.  Also, they stand ready to buy back shares at the NAV
at any time, with or without a sales charge.  Open-end mutual funds are either load or no-load.  A load fund charges a sales fee when the
fund is initially offered.  This sales fee is usually 7.5 - 8 percent of the NAV.  If we assume an 8 percent load, then an investor with $1,000 will
only receive $920 worth of securities.  A load fund typically will not charge a redemption fee, which means they redeem their shares at the NAV.
A load fund is listed with a bid and an ask price.  The bid price is the redemption price, which is the NAV.  The asking price is the NAV divided
by .92, assuming an 8 percent load.  A no-load mutual fund does not charge a sales fee with the initial offering.  There might be a small
redemption fee with these funds of one-half of one percent.  The bid price of a no-load fund is the NAV, which is the redemption price.
The ask price is also the NAV.  Mutual funds fall into several categories, depending on the type of securities they invest in.  There are 
funds that invest only in common stock.  Some funds emphasize "growth" stocks.  These are stocks of companies that retain a lot of 
their earnings (as opposed to paying a high dividend), thus creating capital gains for the investor.  Some funds concentrate on stocks of companies
that payout a large percentage of their earnings as dividends.  There are index funds that invest in a cross section of stocks in a particular
series, such as the S&P 500.  Their intent is to match the return of the series.  There are mutual funds that concentrate on bonds, in order to
produce current income.  Some invest solely in high-grade corporate bonds, while others may hold a combination of different grades of
bonds.  There are funds that concentrate on municipal bonds, which are not subject to federal taxes.  There are balanced funds, that invest 
in both stocks and bonds.  Finally, there are money market funds, which invest in a variety of short-term securities such as treasury bills, bank
certificates of deposit, bank acceptances, and commercial paper.  As you can see, there are mutual funds to suit almost any need.  
There have been a lot of studies that have examined the performance of mutual funds, and the results have been very consistent.
If you do not consider the expenses of running the fund (gross returns), about half the funds performed better than the market, and the
other half performed worse. This is what you would expect with random selection.  If you consider the expenses of running the fund (net returns),
only about one-third of funds do better than the market, and two-thirds do worse.  Also, funds were not consistent in their performance.
     If you had your own portfolio manager, these are some of the functions you would want him to perform.  First, you would want him to S&P 500 Rm RIBM Rm-E(Rm) RIBM-E(RIBM) Points on characteristic line (tab 1 page over for graph of characteristic line)
determine your risk-return preferences and develop a portfolio that is consistent with what you want.  You would want him to diversify your 12 96.11 x y
portfolio to eliminate unsystematic risk, and monitor your portfolio to maintain diversification, and make sure you remain in your desired risk class. 1 99.93 3.97 3.4 2.93 4.5 13.185 8.5849 20.25 -10 -10
You would want him to attempt to achieve a risk-adjusted performance that is superior to the aggregate market (a positive, statistically significant 2 96.28 -3.65 -2.8 -4.69 -1.7 7.973 21.9961 2.89 -9 -9.194
alpha, using the Jensen portfolio performance measure).  Also, the portfolio manager needs to know what your investment objectives are.  He 3 101.59 5.52 5.2 4.48 6.3 28.224 20.0704 39.69 -8 -8.388
needs to know what your cash flow objectives are for your portfolio, and the implications this may have on your tax situation.  If you are a small 4 101.76 0.17 -0.4 -0.87 0.7 -0.609 0.7569 0.49 -2 -3.552
investor, you may not have the resources to buy 100 shares of 10 or 12 different issues in order to achieve diversification.  In such cases, 5 99.08 -2.63 -3.3 -3.67 -2.2 8.074 13.4689 4.84 0 -1.94
mutual funds can provide diversification for an investment as low as $1000.  The investor can acquire a portfolio that is .9 correlated with the 6 102.91 3.87 -2.8 2.83 -1.7 -4.811 8.0089 2.89 2 -0.328
market portfolio, and thus it is 90 percent diversified. 7 103.81 0.87 -5.6 -0.17 -4.5 0.765 0.0289 20.25 8 4.508
     The CAPM sounds good, but the test of any theory should be how well it explains relationships that exist in the real world.  Two questions  8 109.32 5.31 0.4 4.27 1.5 6.405 18.2329 2.25 10 6.12
arise when considering the CAPM.  Is there are positive relationship between beta and the rate of return on risky assets.  Also, how stable is  9 109.32 0 -3.2 -1.04 -2.1 2.184 1.0816 4.41
beta, the measure of systematic risk.  As I stated before, we are using past betas to estimate future risk. " A study by Sharpe and Cooper  10 101.82 -6.86 -7.9 -7.9 -6.8 53.72 62.41 46.24
generally showed that there is a positive relationship between beta and rates of return, although it might not be completely linear."11  Specifically, 11 106.16 4.26 4.6 3.22 5.7 18.354 10.3684 32.49
rates of return tend to level off for the highest risk classes.  "This study, like others by Blume and Levy, show stability in beta over time, 12 107.94 1.68 -1.3 0.64 -0.2 -0.128 0.4096 0.04
especially with larger portfolios (25 or 50 stocks).  Blume  found that in portfolios of 20 or more stocks, the correlation with the market, R squared, 1.04 -1.1 133.336 165.4175 176.73
ranged from .93 to .98."12  In general, most studies show that portfolio betas are very stable over short and long runs.  These results are 
encouraging for investors who want to use beta as a measure of future risk of a portfolio. CovIBM,m=133.336/12=11.111 r=CovIBM,m/σIBM*σm 14.7275 13.78479 y=-1.94 + .806x -1.94
     Many portfolio managers use the S&P 500 as a market proxy.  However, all of the above mentioned performance measures are σ2m=∑((Rmt-E(Rm)2/N 0.77969     σ2IBM = 14.7275 0.806
essentially based on the CAPM which is based on the idea of a "market portfolio", that is the point of tangency between the 13.78479 α=E(RIBM)-BE(Rm)      σIBM = 3.837643 2.09
Markowitz efficient frontier and the SML (see graph 6).  This  theoretical "market portfolio" should contain all risky assets in the economy, 13.785 -1.1 α=-1.10-.806(1.04)
stocks as well as bonds, gold, real estate, coins, stamps, etc.  The fact that this "market portfolio" does not in reality exist does not B=11.111/13.785 -0.83824 -1.93824
negate the usefulness or the CAPM.  The analyst must realize that the true SML could have a higher slope because the true efficient frontier 0.806021 -1.93824 x y 3.82
could be higher than the one that was derived by using the S&P 500 as a market proxy.  The portfolio manager might want to adjust his performance σm=13.785.5 -2 -3.552 0.039746
targets to account for this. 3.712816 -1 -2.746
     I think that I have done enough explaining, so lets do some actual analysis.  We will start with an analysis of the aggregate market.  We will use  E(RIBM)=-1.94 + .806E(Rm) 0 -1.94
the S&P 500 as a proxy for the market.  First, let's take a look at the last several quarters' pay-out ratio, return on equity, and the growth rate for -1.10176 1 -1.134
dividends and earnings for the S&P 500, as well as some earnings estimates for the S&P 500.  It was demonstrated earlier that with an ROE of 2 -0.328 Note: To see the scatter plot of estimated returns for IBM and the S&P 500, page down.
around 15%, and a retention ratio of 70%, the current annual growth rate of earnings and dividends should be 10.5%.  On 12/31/14, the expected 3 0.478 The characteristic line that was fitted from the scatter plot with the
dividend yield on the S&P 500 was 2.00%, and thus the expected return per share on the S&P 500 was 12.5%.  4 1.284 least-squares method is a page over.
5 2.09
2/1/2018 Note:  The annual growth rate in earnings of the S&P 500 that is shown by the formula g=b x ROE 
S&P 500 2763.13 of 10.38%. The growth rate for the S&P 500 shown 1 page dowm of 13.4% is the 5 yr. prog. annual
OP EARN      P/E Div. D/P D/E growth rate.  This is the rate that is used in the valuation model P=Div1//(i-g).
2018 EST 154.39 17.90 50.630 1.83% 32.79%
2019 EST 169.86 16.27
The 5 year average annual growth rate in earnings from 2018 to 2019 forecasted by S&P (13.4%) is higher than the growth rate (10.4%)forecasted using b x ROE.  This is because
this formula only accounts for earnings growth that results from capital raised by retaining earnings.  It does not account for growth that comes
from capital that comes from other sources, like taking on more debt.  You can raise EPS by simply increasing the % of total assets that are
financed by debt.  However, this additional debt increases the degree of financial leverage that the market has, and consequently increases the
level of risk for the market.
b=retention ratio
ROE=return on equity
"Let me point out that the earnings estimates that we are using are bottom up estimates.  These are estimates that are built from the Capital IQ consensus
estimates for specific issues, building from the bottom up to the index level estimate.  There are also estimates known as top down estimates,  which  0.1523
incorporate economic and financial models to produce estimates, and do not come down from the issue level.  The earnings estimates that I have been
using in "investmentvaluationforyou.com have been bottom up estimates.  Actual numbers are always bottom up, while estimates are bottom up or
top down.  The current expected dividend yield for the S&P 500 is 1.83%, and the current expected earnings and dividend growth is approximately
13.4%"13.  With i=D/p+g, we have i=15.23%, as I have shown.
Note:  The above narrative begins 10 pages up and continues about 4 more pages down.  It is a column beginning in the upper left hand corner of this spreadsheet, and if you page down about
15 pages you will get to the end of this first section in this web site.  If you have read to this point, continue paging down to the end of this first section.  If you go to the top of this column of pages,
you will find S&P earnings estimates  for 2015 to the right.  There are earnings estimates for the index, as well as estimates for each sector of the S&P 500.  Immediately to the right of this section you
will find the section about options.  If you page down from there you will get to the section about the "Black Scholes Option Pricing Model.  If you page up 6 pages and over 1 page you will find a table
showing EPS and P/E for the S&P 500 for the years 2008-2015.  If you page over again you will find Graph 1, showing the risk and return for a particular investment.  If you page down 3 pages from there, you
will find several graphs that are referred to in the opening narrative (the first column of pages).  If you page down 1 page from Graph 3, you will find the results of  a simple linear regression of the
returns of IBM and the S&P 500.  1 page down from there is a scatter diagram of these returns, and 1 page over is the resulting characteristic line from this regression. If you page down 2 times from
Risk-adjusted performance measurement methods there you will find Graph 4, and page down 3 times to find Graph 5.
Treynor Sharpe Jensen
          Ti= Ri-RFR         Si= Ri-RFR E(Rj)=NRFR+[E(Rm)-NRFR]
    Βi      σi
Review of formulas
α=E(Ri)-B*E(Rm)
Modern Portfolio Theory NRFR=nominal risk-free rate
     Covij= ∑(Ri-E(Ri)) (Rj-E(Rj))/n i=estimated return
         r= Covim
him
Standard Deviation of a two asset portfolio
σport=√∑Wi2σi2+2∑∑WiWjCovij
a b
Capital Asset Pricing Model
E(Ri)=RFR+Bi(Rm-RFR) NRFR=RFR      NRFR= 2.87% α=E(Ri)-Bi(E(Rm))
          B= Covim RP=E(Rm)-NRFR      E(Rm)= 15.23% α=alpha
    σm          RP= 12.36% E(Ri)=expected return of security i
Beta is a normalized measure of systematic risk E(Rm)=expected return of the market
S&P 500 
Present Value of Dividends Model 2/1/2018 2/1/2015
           P= Div1           i= D/P+g      Price 2762.13 P=Div1/(i-g) 2762.13    Price 2762.13 Beta Return
(i-g)      D/P= (i-g)      Div1 50.63 i=D/P+g 15.23%     P/E 17.89 0 0.15 Horizontal Line
           g= bxROE      D/P 1.83%      EPS1= 154.39 0.2 0.15
     (i-g) 1.83% 0.4 0.15
            P= 2762.13      ROE 15.4% g=b x ROE 10.32% However, this g is the growth generated from b X ROE. 0.6 0.15
     PO 33% As I stated earlier, growth can come from other sources, like 0.8 0.15
     b 67% taking on more debt. S&P has the 5 year expected average annual growth   Market 1 0.15 Intersection of Diagonal, Vertical, and Horizontal Lines
     g  13.40% rate for the index over the next 5 years to be 13.4% as of 02/02/18.  1 0.13 Vertical Line
     I chose to use the Jensen measure (α) and the Treynor measure because they both look at systematic risk.  Remember that since non-systematic risk 1 0.11
can be diversified away, the market will only compensate you for the systematic risk that you assume.  There is some discrepancy in the rankings of 1 0.09
the two methods, however both methods agree on the top five sectors, and these are the only sectors that have positive alphas.  That does not mean 1 0.07
that you can't find stocks with forecasted positive alphas in the lower five sectors, but you will find more stocks with forecasted positive alphas in the 1 0.05
top five sectors.  We are now at the stage where we can start looking at individual stocks.  We want to determine the beta of a stock, and then estimate 1 0.03
the return on the stock.  We then determine what the market's required return on this stock should be, and then select stocks that have an estimated  1 0.01
risk adjusted return that is greater than their required return. SML Market 1 0.15 Diagonal Line 
     The investor must now decide what his investment objectives are.  Is the investor looking for capital gains, or is he looking for stocks that pay-out 0.8 0.13
most of their earnings in dividends?  The type of stock selected is a function of the investor's over-all goals for his portfolio.  The investor might be  0.6 0.11
quite wealthy, and he does not need current cash flow from his investments.  This is the type of investor who wants stocks that retain most of their 0.4 0.09
earnings, producing capital gains.  Other types of investors may need current income from their investments, and they look for stocks that pay-out 0.2 0.07
a large portion of their earnings in dividends.  This type of investor might prefer to invest in bonds, which produce interest payments (usually twice  NRFR 0 0.05
a year).  Remember, dividends and interest payments are taxable when they are received (a small amount of dividends are deductible, but the 
rest are taxed).  Capital gains are not taxable until you sell the stock and realize the gain.  Also, the tax rate on capital gains is lower than it is
on regular income.  If you already have enough current income to live on, then you can just hang on to your stocks and the capital gains will not
be taxed until you sell them.  If you never sell them, then you will never pay taxes on the capital gains.  If you do finally sell the stock, you will
have to pay taxes on the capital gains, but chances are that if you wait long enough your tax rate will decline.
     The individual investments in an investor's portfolio are a function of the investor's overall wealth position, his tax situation, income requirements,
and age.  All these factors go into determining the investor's risk tolerance.  Age alone is a big factor.  If you are young, have a job that provides you
income to cover your living expenses, you have a fairly high risk tolerance.  You can withstand some volatility in your portfolio's return.  However,
if you are older, maybe retired, then your risk tolerance goes down.  You can withstand less volatility, and you may need to draw some income from
your portfolio.  Bonds may be suitable.  However, interest rates are so low these days that you might want to try some stocks that have a high
dividend yield.  S&P has a web site (it is free).  Select S&P 500, and then click on additional information.  Take a look at the Dividend Aristocrat
file.  This file contains stocks that have 25 years of increasing dividends, with liquidity and market value restrictions.  The Index Earnings file
is where I get all my S&P 500 and individual sector earnings data.
12/29/2017
INDEX NAME Price EPS1 P/E Div1 PO  b D/P S&P 500 5 yr. Proj. Annual Growth %
S&P 500 2762.13 $154.39 17.89  $     50.53 33% 67% 1.83% 13.4%
S&P 500 Consumer Discretionary 785.33 $39.55 19.86  $     13.05 33% 67% 1.66% 19.88% S&P 500 S&P 500 5YR
S&P 500 Consumer Staples 587.39 $30.38 19.33  $     10.03 33% 67% 1.71% 9.16% 2018 EST PROJ ANNUAL
S&P 500 Energy 533.41 $25.02 21.32  $      8.26 33% 67% 1.55% 30.47% OPER P/E GROWTH %
S&P 500 Financials 463.94 $34.52 13.44  $     11.39 33% 67% 2.46% 10.64% 18.28 13.4
S&P 500 Health Care 956.32 $60.39 15.84  $     19.93 33% 67% 2.08% 10.88% 21.46 19.88
S&P 500 Industrials 637.81 $35.38 18.03  $     11.68 33% 67% 1.83% 11.23% 19.52 9.16
S&P 500 Information Technology 1106.18 $61.32 18.04  $     20.24 33% 67% 1.83% 13.92% 22.36 30.47
S&P 500 Materials 378.94 $21.32 17.77  $      7.04 33% 67% 1.86% 11.65% 14.44 10.64
S&P 500 Telecommunication Services 166.07 $14.58 11.39  $      4.81 33% 67% 2.90% 9.10% 16.88 10.88
S&P 500 Utilities 267.37 $15.57 17.17  $      5.14 33% 67% 1.92% 4.15% 18.94 11.23
S&P 500 Real estate 203.86 $5.05 40.37  $      1.67 33% 67% 0.82% 5.64% 19.4 13.92
Ri=NRFR+Β(Rm-NRFR)         NRFR= 2.87% Note: There was no dividend info. for the sectors, so I assumed the same PO as index, and Div1, b, and D/P could be wrong 18.25 11.65
i=D/P+g RP=Rm-NRFR 0.1523 i=D/P+g P=Div1/(i-g)       B*RP= 0.1483 R=Required return                                     for the sectors. 11.6 9.1
15.230%      E(Rm)= 15.23%     D/P= 2.03% D/P=(i-g) α=E(Ri)-B(E(Rm))      NRFR= 0.0287 RP=Risk premium 16.37 4.15
          RP= 12.36%  NRFR= 2.87% g=ROE x b E(Rm)=NRFR + RP        E(Ri)= 0.1770 NRFR=Nominal risk-free rate 38.84 5.64 Graph 4
i=Estimated return E(Ri)=NRFR + (B x RP) E(Ri)=..0287+(1.2 x .1236) PO=Pay out ratio 1492.59% SML
Β=Beta   E(Rm)= 15.23%                 i= 15.23% E(Rm)=.0287.+.1236 E(Ri)=15.54% b=Retention ratio (1-PO)
P=Price E(Rm)=Expected return of the market 15.23%             B= 1.2 g=growth rate of earnings and dividends
α=Alpha Note:  g=ROExb if there is no new debt or any other factor Note:  Expected return and required return are 
σ=Standard deviation that has an affect on ROE and thus g.  The index has g=15%x70%=10.5%,  synonymous in this context.
R=Required return as compared to g=11.94% that S&P has calculated. NRFR=5%
INDEX NAME i=D/P+g Β E(Rs) P α σ Rm=15%
S&P 500 15.23% 1 15.23% 2762.13 0.00% 8.8
S&P 500 Consumer Discretionary 21.54% 1.1 16.47% 785.33 4.79% 1.2
S&P 500 Consumer Staples 10.87% 0.7 11.52% 587.39 0.21% 10.3
S&P 500 Energy 32.02% 1.1 16.47% 533.41 15.27% 18.8
S&P 500 Financials 13.10% 1.1 16.47% 463.94 -3.66% 13.6
S&P 500 Health Care 12.96% 1 15.23% 956.32 -2.27% 11.7
S&P 500 Industrials 13.06% 1.1 16.47% 637.81 -3.69% 11.2
S&P 500 Information Technology 15.75% 1.1 16.47% 1106.18 -1.00% 12.7
S&P 500 Materials 13.51% 1.2 17.70% 378.94 -4.77% 14.4
S&P 500 Telecommunication Services 12.00% 0.6 10.29% 166.07 2.86% 14.4
S&P 500 Utilities 6.07% 0.3 6.58% 267.37 1.50% 13.8
S&P 500 Real estate 6.46% 0.8 12.76% 203.86 -5.73% 13.2
Note:  The stock valuation model (P=Div1/(i-g) or P/E= D/E           P=Ex D/E D/E This is the multiplier (M)
(i-g) (i-g) (i-g) P=ExM
A word about the stock valuation model.  As I stated earlier, g=ROExb.  If i=ROE, then the dividend payout ratio does not affect the price.  If ROE is Point M, the market portfolio, is (1,.15) is where all 3 lines intersect.
greater than i, then the higher the retention ratio (b), the higher the price.  If ROE is less than i, then the higher the dividend payout ratio is (i.e. the lower The point where the SML intersects the y-axis is 5%, which is the NRFR.
the retention ratio is), the higher the price is.  The reverse is also true.  
In a lot of the analysis in this site, I will use the shorter version of the valuation model, simply to save space and time.
INDEX NAME Jenson α Rank Ts Rank Ss Rank
S&P 500 12.36% 1.40% The Treynor method uses Beta as the denominator,
S&P 500 Consumer Discretionary 4.79% 2 16.97% 2 15.56% 1 while the Sharpe method uses the standard 
S&P 500 Consumer Staples 0.21% 5 11.42% 5 0.78% 6 deviation (σ) as the denominator.  Beta is a measure
S&P 500 Energy 15.26% 1 26.50% 1 1.55% 2 of the systematic risk of a portfolio, while σ is a
S&P 500 Financials -3.66% 8 9.30% 8 0.75% 7 measure of the total risk.  If the two methods
S&P 500 Health Care -2.27% 7 10.09% 7 0.86% 5 give the same rankings, it shows that the
S&P 500 Industrials -3.69% 9 9.26% 9 0.91% 4 portfolios are perfectly diversified.  If the rankings 0.005 0.035 0.005 0.035
S&P 500 Information Technology -1.00% 6 11.71% 4 1.01% 3 are close, then the portfolios are fairly diversified. 0.029 0.045 0.029 0.045
S&P 500 Materials -4.77% 10 8.86% 10 0.74% 8 This is the case here.  These are portfolios of  0.051 0.055 0.051 0.055
S&P 500 Telecommunication Services 2.86% 3 15.21% 3 0.63% 9 stocks in the same sector, which would make 0.065 0.065 0.065 0.065
S&P 500 Utilities 1.50% 4 10.67% 6 0.23% 11 the returns of some of these stocks correlated. 0.071 0.075 0.071 0.075
S&P 500 Real estate -5.73% 11 4.48% 11 0.27% 10 0.075 0.085 0.075 0.085
These performance measures are usually used to rank the past performance of portfolios such as mutual funds.  Remember that we are using these methods 0.079 0.095 0.079 0.095
to rank the projected returns of different sectors.  The formula for determining the return for a portfolio is the capital gains plus any dividends earned.
           Rit= EPit+Divit+Cap.Dist.it-BPit
                  Bpit
Rit=return on fund i during year t
Epit=ending price for fund i during year t.
Cap.Dist.it=capital gain distributions for fund i during year t.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
Bpit=beginning price for fund i during year t.
The Jensen performance measure is based on the Capital Asset Pricing Model as follows.
E(Rj)=NRFR+Bj[E(Rm)-NRFR]
     NRFR=  2.87%     E(Rj)=  17.70%
         RP= 12.36%         Bj= 1.2
     E(Rm)= 15.23% E(Rm)=the expected return of the market
RP=E(Rm)-NRFR E(Rj)=the expected return of stock or portfolio j
    NRFR= Nominal risk-free rate Bj=Beta of stock or portfolio j
        RP= Risk premium
         Bj= The Beta of security or portfolio j
"We can express the expectations formula in terms of realized rates of return. 0.035 0.005
Rjt=NRFR+Bj[Rmt-NRFR]+Ujt 0.045 0.029
Ujt=Random error term 0.055 0.051
Subtracting the NRFR from both sides gives us 0.065 0.065
Rjt-NRFR=Bj[Rmt-NRFR]+Ujt 0.075 0.071
The risk premium for the jth security or portfolio equals Bj times the market risk premium plus a random error term.  If some portfolio managers 0.085 0.075
are superior, they will consistently achieve positive random error terms.  They are consistently achieving returns that are superior to the returns that 0.095 0.079
would be expected with the CAPM.  It is necessary to allow for an intercept (a nonzero constant) in this equation that will take account of the
positive residuals.  If we allow for a nonzero constant alpha (α), the equation becomes 
Rjt-NRFR=α+Bj[Rmt-NRFR]+Ujt
With this equation, a positive and statistically significant α indicates the ability to generate superior, risk-adjusted returns.
A superior analyst is one that can achieve consistently positive alphas (α) in the portfolios they manage.  They are able to generate superior
returns because they can forecast market turns or consistently select undervalued securities.  On the other hand, consistently negative, statistically
significant alphas indicate inferior returns on a risk-adjusted basis.
    It should be realized that these performance measures are meant to be used as methods to analyze return per unit of risk after the fact.
We are using them to analyze the estimated return of the aggregate market (S&P 500), as well as the different sectors of the market.
Once we have decided on some sectors that look attractive, we can then start looking at individual stocks in these sectors.
     Analyzing individual stocks is the same as analyzing the aggregate market or sectors of the market.  You need to find out what the EPS for the upcoming year is, as well as the
dividends, the expected growth rate of earnings and dividends, and some measure of the risk of the stock.  Let's look at a couple of stocks to get a feel for this.
     IBM
Price  $   157.37  $   157.38             PO= 36%
Div1  $      5.76 next 12 months             i= 8.17%     E(RIBM)= 12.65%
EPS1  $     16.01        P/E= 9.83      E(Rm)= 15.23%
D/P 3.66%  RP=risk premium               B= 0.91
(i-g) 0.0366 RP=E(Rm)-NRFR       NRFR=   2.87% 10 yr. T-Bond=NRFR
g 4.51% RP=.1523-.0287          RP= 12.36% 5.95%
P=D/(i-g)  $   157.38 RP=12.36%
i=D/P+g 8.17% last 12 months Required rate of return
EPS  $     11.90       P/E= 13.22             P/OIBM= 37% E(RIBM)=NRFR+B(Rm-NRFR)
Div.  $      4.40              RPm 12.36% E(RIBM)=.0287+.91(.1523-.0287)
α=i-B(Rm)                    α= -5.69%     E(RIBM)= 14.12%
IBM is way over-priced.  The required return of a stock with B=.91 in this environment is 14.12%.  The estimated return for IBM is only 8.17%.
0.1236
      GM next 12 months
Price  $     33.92       P/E= 7.73
Div1  $      1.20          B= 0.97             PO= 27%
EPS1  $      4.39
D/P 3.54% Required rate of return
(i-g) 3.54% E(RGM)=NRFR+B(Rm-NRFR)
g 14.06% E(RGM)=.0287+.97(.1523-.0287)      E(RGM)= 14.86%
P=D/(i-g)  $     33.90
i=D/P+g 17.60% α=.1760-(.97*,1523)
                 α= 2.83%
 $    33.90
GM is extremely underpriced.  The required return is 14.86%, while the estimated return is 17.60%.  Referring to graph 4, this stock would plot above the Security Market Line (SML).
Price 125.66
Div1 2.08
EPS1 9.74
D/P 1.66%
(i-g) 1.66%
g
P=D/(i-g) 126.06061
i=D/P+g
P/E 12.9
0 0.018
0.01 0.028
0.02 0.02 0.038
0.03 0.038 0.048
0.04 0.054 0.058
point M 0.05 0.068 0.068
0.06 0.074 0.078
0.07 0.078 0.088
0.08 0.08 0.098
Note:  This website is a work in progress.  It is in no way completed.  In fact, I will be adding new material once a week, sometimes more.  
On the first page of this site I have outlined the different topics that will be discussed in the different sections of this site.  There is still a lot
more to come.  
1 Frank K. Reilly, "Investments" (1982): 152.
2 Roger G. Ibbotson and Rex A. Sinquefield,  "Stocks, Bonds, Bills, and Inflation:  Historical Return" {1976-1978) (Charlottesville, VA: Financial 
Analysts Research Foundation 1979).
3 Frank K. Reilly, "Investments" (1982): 262.
4 Harry Markowitz, "Portfolio Selection,"  "Journal of Finance 7" (1952): 77-91; and idem, "Portfolio Selection-Efficient Diversification of Investments"
(New York:  John Wiley & Sons, 1959).
5 William F. Sharpe, "Capital Asset Prices:  A Theory of Market Equilibrium Under Conditions of Risk," "Journal of Finance 19 (1964):  425-442;
John Lintner, "Security Prices, Risk and Maximum Gains from Diversification,"  "Journal of Finance 20 (1965): 587-615; and J. Mossin, 
"Equilibrium in a Capital Asset Market,"  "Econometrica 34" (1966): 768-783.
6 Frank K. Reilly, "Investments" (1982):  591.
7 IBID  595 graph 5
8 Robert A. Levy, "On the Short-term Stationarity of Beta Coefficients," Financial Analysts Journal 27(1971):  55-62. CML
9 William F. Sharpe and Guy M. Cooper, "Risk-Return Classes of New York Stock Exchange Common Stocks:  1931-1977," "Financial
Analysts Journal 28 (1972):  46-54.
10 Jack L. Treynor, "How to Rate Management of Investment Funds," "Harvard Business Review" (1965):  63-75.
11 William F. Sharpe and Guy M. Cooper, "Risk-Return Classes of New York Stock Exchange Common Stocks:  1931-1977," "Financial
Analysts Journal 28 (1972):  46-54.
12 Marshall E. Blume, "On the Assessment of Risk," "Journal of finance NRFR=1.8%
13 Howard Silverblatt, S&P Senior Index Analyst, "S&P 500 EPS EST" Excel Spreadsheet, Standard & Poor's Dow Jones indices web-site Rm=6.8%
14 "The 2013 Economic Report of the President" Figure 2-14, 71
15 Howard Silverblatt, S&P Senior Index Analyst, "S&P 500 EPS EST" Excel Spreadsheet, Standard & Poor's Dow Jones indices web-site
16 Richard M. Bookstaber, Chapter 4, "Option Pricing and Strategies in Investing" (1981):  40-73. 
References
Frank K. Reilly, "Investments" (1982).
Roger G. Ibbotson and Rex A. Sinquefield, "Stocks, Bonds, Bills, and Inflation: Historical Return" (1976-1978). 
Harry Markowitz, "Portfolio Selection," "Journal of finance 7" (1952).
William F. Sharpe, "Capital Asset Prices:  A Theory of Market Equilibrium Under Conditions of Risk," "Journal of finance 19" (1964).
John Lintner, "Security Prices, Risk and Maximum Gains from Diversification, "Journal of Finance 20" (1965).
J. Mossin, "Equilibrium in a Capital Asset Market," "Econometrica 34" (1966).
Robert A. Levy, "On the short-term stationarity of Beta Coefficients," Financial Analysts Journal 27 (1971).
William F. Sharpe and Guy M. Cooper, "Risk-Return Classes of New York Stock Exchange Common Stocks:  1931-1977, Financial Analysts
Journal 28 (1972).
Jack L. Treynor, "How to Rate Management of Investment Funds, "Harvard Business Review" (1965)
Standard & Poor's Dow Jones Indices web-site, specifically the S&P 500 Earnings and Estimate Report spreadsheet prepared by  Point M (the market portfolio) is where the straight line (the CML) is tangent to the curved line (the old efficient frontier),
Howard Silverblatt, S&P Senior Index Analyst M=.05,.068, the CML is the new efficient frontier.
"The 2013 Economic Report of the President"  Transmitted to the Congress March 2013.
This site was created by Lawrence I. Kurtz, BBA and MS, both with majors in finance, Georgia State University, Atlanta, GA
My e-mail is bark23@att.net
Note:  This website is a work in progress.  It is in no way completed.  In fact, I will be adding new material once a week, sometimes more.  
On the first page of this site I have outlined the different topics that will be discussed in the different sections of this site.  There is still a lot
of material to come.