Chapter 4 - Risk and Return: The Basics

1. Investment Returns
   

   One way to express the return on an investment is how much of a dollar
   return you have made.

   
   

   Dollar Return = Amount received - Amount invested

   
   

   The problem with this is that It does not tell you how long the money
   was held nor how much money it took to get that return. In its place we
   can use a Rate of Return

   
   

   Rate of Return = (Amount received - Amount invested)/Amount
   Invested

   
   

   This resolves the problem by giving us a percentage of the original
   investment. If we then can express this across the years that the
   security is held then we can get the interest as an annual rate of
   return.

   
   


2. Stand Alone Risk
   

   Stand alone risk is defined as an exposure to loss or injury. In any
   security purchased one will be exposed to a chance that their will not be
   a payback of the money invested as well as any extra payback. This is the
   risk taken to invest in the security.

   
   

   One can look at these risks in two ways. The stand alone risk is where
   the security is the only one owned. If you had one security that was
   fairly secure in its return (a treasury bill for example) then it would
   be considered risk free. On the
   other hand, something moth return the same rate of return but have a
   chance of loosing your money, this is would be highly risky. How you
   invest would be a consideration of if you feel you can handle risk or
   not. In no case should you invest if your expected rate of return is not
   high enough to compensate for the perceived risk of the investment.

   
   

   One good/bad think about risky assets is they rarely return their
   expected rate of return. It is usually much higher (good) or much lower
   (bad).

   
   
   
   
   1. Probability Distributions
      

      This is defined as the chance that an event may occur. In
      investing we can say that a particular security could have a chance
      of returning a certain amount on its investment. This can even be
      broken down to the chance being strong, normal and weak, or even more
      shades as well as the percentages that they will happen with them.
      This becomes the probability distribution.

      
      
   
   
   2. Expected Rate of Return
      

      Multiply the possible outcomes by the probability that they will
      occur. Then take them and sum them up. This is the weighted average
      of outcomes. This is also know as the expected rate of return. It is
      named in formulas as r with a ^ symbol over it (called r-hat). Many
      securities can wind up with the same Expected Rate of Return even
      though they are widely varied in the chance they will succeed.
      Obviously we need another tool.

      
      
   
   
   3. Measuring Stand-Alone Risk: The Standard Deviation
      

      If we graph the probability distribution in a continuous curve we
      can see that some securities will have a tighter graph than others
      will. The tighter the graph, the smaller the risk is for the
      security. We measure this tightness by using a 'standard deviation',
      the symbol being σ and pronounced sigma. To find the standard
      deviation we do 4 steps:

      
      
      
      
      1. Calculate the expected rate of return
         

         Expected rate of return = r-hat = $$
         
         ∑
         
         i
         =
         1
         
         n
         
         P_ir_i

         
         
      
      
      2. Subtract the expected rate of return (r-hat) from each possible
         outcome (r_i) to get a set of deviations.
         

         Deviation = r_i - r-hat

         
         
      
      
      3. Square the deviation and multiple it by the chance that it
         might occur and then sum them to get the variance.
         

         Variance = σ² = $$
         
         ∑
         
         i
         =
         1
         
         n
         
         (r_i - r-hat)²P_i

         
         
      
      
      4. Finally do a square root of the Variance to find the standard
         deviation.
         

         The lower this standard deviation is, the tighter the graph it
         would produce and the less risk that it has. All things being
         normal, you can expect the actual return will be within one
         standard deviation of the expected rate of return.

         
         
      
      
      
      
   
   
   4. Using Historical Data to Measure Risk
      

      We have assumed to this point that we have a known probability
      distribution. If we have some sample return data form past periods we
      can figure out standard deviation as well.

      
      

      Estimated σ = S =$$
      
      
      
      
      ∑
      
      r
      =
      1
      
      n
      
      
      (
      
      rbar
      t
      
      
      −
      
      rbar
      Avg
      
      
      )
      
      
      
      2
      
      
      
      
      n
      −
      1
      
      
      
      

      
      

      Historic sigma is often an indicator of future sigma.

      
      
   
   
   5. Measuring Stand -Alone Risk: The Coefficient of Variation
      

      Given a choice, we will tend to choose the investment with the
      less risk, so will choose between two investments with the same
      expected returns the one with the lowest standard deviation. If two
      had the same standard deviation but one a higher expected return we
      would go for it. What do you do if neither has one that is the same.
      The coefficient of variation (CV devides the standard deviation by
      the expected return.

      
      

      CV = $$
      
      σ
      rhat
      
      

      
      

      This shows the risk per a unit of return so that they can be
      compared better. The lower this number is, the better the chance that
      it will bring a good return.

      
      
   
   
   6. Risk Aversion and Required Returns
      

      Most people will choose the less risky return on investment and
      therefore we could say that they have risk aversion. While this is
      not bad in itself, it can have influence on things that get invested
      in. If you had two stocks, one was less riskier, that sold for the
      same price, most would go for the less riskier one. Since there would
      be more demand for it, the price would go up and the return would go
      down. Likewise those who own the risker one would sell causing its
      price to drop, changing its risk and return. The differences in the
      start and finish price is known as the risk premium (RP).

      
      
   
   
   
   


3. Risk in a Portfolio Context
   

   By adding stocks together in a portfolio, the risks of one stock can
   offset the risks of other stocks. In fact many stocks can be up while
   others are down and this can balance out the portfolio.

   
   
   
   
   1. Portfolio Returns
      

      To get the expected return on a portfolio add together the
      weighted averages of all the members of the portfolios.

      
      

      rbar_p = $$
      
      ∑
      
      i
      =
      1
      
      n
      
      
      w
      i
      
      
      rbar
      i
      
      

      
      
   
   
   2. Portfolio Risk
      

      The risk of the portfolio will almost always be smaller than the
      weighted average of the asset's σ. One thing should be noted
      about stocks that are up when others are down. If we had ones that
      has a perfect correlation (one was at the exact opposite point of th
      other), they would cancel each other out and have no risk. In truth
      it is not possible to get stocks in perfect alignment like this so we
      measure the correlation coefficient (noted as ρ (pronounced
      rho)). ρ can range from -1 (if exact opposites) to +1 (if exactly
      the same). For that reason, in order to diversify we must find stocks
      that have ρ that cancel each other out. In general you would want
      to have investments in tow or more separate industries instead of
      just all in one. If they are all in one industry type, the problem is
      that when that industry goes into a slump so will all the investments
      that you own in it.

      
      
   
   
   3. Diversifiable Risk versus Market Risk
      

      It is not impossible to find stocks that are negatively correlated
      as they tend to work with the economy as a whole. So there is risk in
      any investment but not as much if all is held in one stock. A market
      portfolio, all the stocks combined, should have a standard deviation
      of about 20.1 %. By research it is found that 40 or more stocks in
      diversified industries should diversify out most risk involved. The
      risk involved in a stock that moves with the market itself is called
      Market Risk, the part that deals with the stock itself and how
      lawsuits, strikes, etc. can affect it is called diversifable risk.
      Market risk can not be diversified out, diversifiable risks can.
      Capital Asset Pricing Model (CAPM), is used to analyze the
      relationship between risk and rate of return.

      
      
   
   
   4. The Concept of Beta
      

      The relevant risk of an individual stock is called its beta
      coefficient. and is defined under CPAM as the amount of risk that the
      stock contributes to a portfolio.

      
      

      $$
      
      b
      i
      
      =
      
      (
      
      
      σ
      i
      
      
      σ
      M
      
      
      )
      
      
      ρ
      iM
      
      

      
      

      A stock with a high standard deviation ($$
      
      σ
      i
      
      ) will have a high beta. It is possible to use a calculator or
      a spreadsheet to do the job. You can also take a graph and plot the
      stock as its return on the y and the market portfolio as the x, we
      could plot the graph of the graph of expectations by setting another
      point by using the slope with the various beta possibilities (2.0
      high, 1.0 average, .5 low) then we can figure out the , then we could
      see volatility possibilities.

      
      
   
   
   
   


4. Calculating Beta Coefficients
   

   Different organizations calculate Betas in different way so other than
   sticking with a beta from one organization it is a good idea to calculate
   your own. The first step is to compile the data for the company you want
   plus a standard to go by (say the S & P 500 Index). Second is to
   convert the data to rates of return (change from previous month/this
   month value) for both the stock and the standard. Plot on a graph the
   returns of the company against the standard and run a line through them
   to show the regression (Spreadsheets may make this easier). The slope of
   the line would be the beta.

   
   


5. The Relationship Between Risk and Rates of Returns
   

   The Market Risk Premium ($$
   
   RP
   M
   
   ) is the premium that people want for bearing the risk of the
   average stock. It would be the current market risk minus the risk free
   premium. We can use this to calculate our required return

   
   

   Required return = Riskfree return + premium for risk .

   
   


6. This leads to the Security Market Line (SML)
   

   $$
   
   r
   i
   
   =
   
   r
   RF
   
   +
   
   (
   
   r
   M
   
   −
   
   r
   RF
   
   )
   
   b
   i
   
   
   

   
   
   
   
   1. The Impact of Inflation
   
   
   2. Changes in Risk Aversion
      

      The slope of the SML reflects the averseness to risk of the
      investor.

      
      
   
   
   3. Changes in a Stock's Beta Coefficient
      

      A firm can influence its own beta by the assets it has and the use
      of it's debt. Other external factors can influence it as well.

      
      
   
   
   
   


7. Projects versus Securities
   

   Only by analyzing these situations can we begin to understand
   comparing projects in a business environment

   
   


8. Some Concerns About Beta and the CAPM
   

   There are some problems with CAPM. The size of a firm and it's
   market/book ratio can affect the CAPM but have no real effect on the
   beta.

   
   


9. Volatility versus Risk
   

   Volatility and risk are not the same thing. A company can have wild
   fluctuation and still be very profitable. Rule to follow, earnings
   volatility does not necessarily mean risk but stock price volatility
   does.