Lecture 1- Part 2

1
Lecture 1- Part 2

• Risk Management and Derivative by Stulz, Ch2
• Expected Return and Volatility

2
Knowing risk and return of securities and
portfolios

• A key measure of investors success is the rate
  at which their funds have grown
• Holding-period return (HPR) of shares is composed
  of capital gain and dividend
• HPR (P1-Po Cash Dividend)/Po
• This definition assumes end of period returns and
  ignores re-investment of income

3
Return Distributions

• If the return on a stock is fixed, there will be
  100 probability (Certaininty)that the return
  will be realized, like in bonds and T-bills
• In stocks, return is not fixed so the probability
  of all likely outcomes should be assessed
• Probability is the chance that the specified
  outcome will occur

4

• Probability distribution is the specification of
  likely outcomes and the probability associated
  with each outcome
• Suppose we expect that PPL can give either 10,
  20 or -5 return. So we have three possible
  outcomes, if we associate chances of occurrence
  with each return, then it becomes probability
  distribution

5
Expected Return

• Expected return is the single most likely outcome
  from a PD
• It is calculated by taking a weighted average of
  all possible return outcomes
• E(R) SRiPi

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An Example
Oil Prices Return Probability RiPi
Flat 0.10 0.30 0.03
Rise 0.2 0.50 .1
Fall 0 0.20 0
Sum .13
7
Variance of Returns (Risk)

• Variance of a random variable is a statistical
  tools that measures how the realization of the
  random variable are distribute around their
  expected values
• In other words it measures risk
• Variance SRi-E(R)2 Pi

8
Variance of Returns (Risk)

• Standard Deviation taking square root of
  .00600, we get value of 0.077 or 7.7

Return Probability RiPi Ri-E(R) SRi-E(R)2 SRi-E(R)2 Pi
0.10 0.30 0.03 -.03 .0009 .00027
0.2 0.50 .1 .07 .0049 .00245
0 0.20 0 -.13 .0169 .00338
Sum .13 .00600
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Cumulative Distribution Function

• The cumulative distribution function of a random
  variable y specifies, for any number y, the
  probability that the realization of the random
  variable will be no greater than y
• For POL, a reasonable estimate of the stock
  return volatility is 9.2 with expected return of
  13, the following table show cumulative
  distributions functions for different levels of
  returns

10
Cumulative Distribution Function
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How to Calculate CDF

• CDF can easily be calculated with MS Excel
• Put the equal sign in a cell
• Open parenthesis and give x value (x means the
  level of return for which you want cumulative
  probability)
• Then give the mean return value,
• Then the standard deviation value
• And finally write TRUE and close parenthesis

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Interpretation

• Taking values from the table in the previous
  slide, CDF is .59 with the 20 return level
• It means that there is 59 probability that
  return on POL will be less than 20
• An investor has Rs.100,000 investment in POL and
  he wants that he does not lose more than Rs.30000
  of his investment, what is the probability of
  this occurrence

13
Return of a Portfolio

• To calculate an average rate on a combination of
  stocks, we simply take the weighted average
  return of all stocks
• E(Rp) Swi E(Ri)
• Wi Weight of the security in the poftfolio
• E(Rp) The expected return on the portfolio

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Calculating portfolio return

• Calculating weights PPL 20000/60000 .33
• FFC 30000/6000 .5 Lucky 10000/60000
  .16

Stock Value Return Weight Swi E(Ri
PPL 20000 15 .33 4.95
FFC 30000 12 .5 6
Lucky 10000 10 .16 1.6
Sum 12.55
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Calculating Portfolio Risk

• Risk of the porftolio is not the weighted average
  risk of the individual securities
• Rather it is determined by three factors
• 1.the SD of each security
• 2. the covariance between the securities
• The weights of securities in the portfolio

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Diversification

• By combining negatively correlated stocks, we can
  remove the individual risks of the stocks
• Example Pol face the risk of falling oil prices
• PIA face the risk of rising oil prices
• By combining these two stocks, reduction in
  return in one stock due to change in oil price is
  compensated by increase in return of the other
  stock
• However, all of market risk cannot be eliminated
  through diversification

17
Efficient Frontier

• Investors should select portfolios on the basis
  of expected return and risk
• A portfolio is efficient if
• 1 it has the smallest level of risk for a given
  return or
• 2. largest return for a given level of risk
• To select efficient portfolios, investors should
  find out all portfolios opportunities set
• i.e find out risk and return set for all
  portfolios

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Efficient Frontier

• Example given in the Excel File
• Steps
• 1. Calculate securities return
• 2. calculate portfolio returns
• 3. Find portfolio risk
• 4. Make different portfolios by changing weights
  of the securities
• 5. Find risk and return of each portfolio
  developed in step 4
• 6. Plot the risk and return of these portfolios
• 7. Find the minimum variance portfolio
• 8. Portfolios above the minimum variance
  portfolios are efficient