Overview of Risk and Return

1
Overview of Risk and Return

• Timothy R. Mayes, Ph.D.
• FIN 3600 Chapter 2

2
Risk and Return are Both Important

• It is important to consider both risk and return
  when making investment decisions. Over long
  periods of time (more than a year or two), risk
  and return tend be highly correlated as shown in
  the table below.

3
Sources of Returns

• Returns on investment can come from one of two
  sources or both
• Capital Gain This is the increase (or decrease)
  in the market value of the security
• Income This is the periodic cash flows that an
  investment may pay (e.g., cash dividends on
  stock, or interest payments on bonds)
• Note that your total return is the sum of your
  capital gains and income

4
Measuring Returns for One Period

• Investors look at returns in various ways, but
  the most basic (not necessarily the best) is the
  single period total return
• A period is defined as any appropriate period of
  time (year, quarter, month, week, day, etc.)
• This measure is known as the Holding Period
  Return (HPR)

5
HPR Example

• Suppose that you purchased 100 shares of XYZ
  stock for 50 per share five years ago.
  Recently, you sold the stock for 100. In
  addition, the company paid a dividend each year
  of 1.00 per share. What is your HPR?

6
Annualizing HPRs

• If a calculated HPR is for a non-annual holding
  period, we generally annualize it to make it
  comparable to other returns
• The general formula is
• Where m is the number of periods per year
• Note that m will be gt 1 for less than annual
  periods and lt 1 for greater than annual periods

7
Annualizing HPRs (cont.)

• As an example, suppose that you earned a return
  of 5 over a period of three months. There are 4
  three-month periods in a year, so your annualized
  HPR is
• Note that this calculation assumes that you can
  repeat this performance every three months for a
  year

8
Annualizing HPRs (cont.)

• For another example, suppose that you earned an
  HPR of 47 over a period of 5 years. In this
  case, your annualized HPR would be 8.01 per year
• Note that in this case, we use an exponent (m) of
  1/5 because a year is 1/5th of a five-year period

9
Multi-period Returns

• HPRs provide an interesting bit of data, but they
  suffer from some flaws
• The HPR ignores compounding
• The HPR is usually not comparable to other
  returns because it isnt an necessarily
  annualized return
• The solution to these problems is to calculate
  the IRR of the investment
• A security investments IRR is usually referred
  to as its Holding Period Yield (HPY)

10
Calculating the HPY

• Since the HPY is the same as the IRR, there is no
  general formula for finding the HPY
• Instead, we must use some iterative procedure (or
  a financial calculator or spreadsheet function)
• For the XYZ investment, the HPY is 16.421 per
  year

11
Problems with the HPY

• Generally, the HPY is superior to the HPR as a
  measure of return, but it also has problems
• The HPY assumes that cash flows are reinvested at
  the same rate as the HPY
• The HPY assumes that the cash flows are equally
  spaced in time (i.e., every year or every month)
• The HPY makes no provision for stock splits,
  stock dividends, or partial purchases or sales of
  holdings

12
The Reinvestment Assumption

• To see that the reinvestment assumption is
  implicit in the calculation of the HPY, lets try
  a few different reinvestment rates and see what
  the compound average annual rate of return is

13
The Timing Assumption

• In practice, investments often do not pay all
  cash flows at convenient equally spaced time
  periods
• This will cause most calculator and spreadsheet
  functions to not work properly unless adjustments
  are made
• The adjustment is to change to a common
  definition of a period, and to include cash flows
  of 0 for periods without a cash flow

14
An Example of a Timing Problem

• In this example, we simply change the timing of
  the dividends (Note that the period 3 dividend
  was omitted).

15
Handling Stock Splits, etc.

• Stock splits and stock dividends complicate the
  finding of the true HPY.
• For example, suppose that XYZ split 2 for 1
  immediately after period 3. In this case, your
  dividends would be only 0.50 per share in
  periods 4 and 5, and you would be selling the
  stock for 50 in period 5 (but you will have the
  same wealth).
• Your true HPY is the same, but if you dont
  adjust for the split you will get an incorrect
  HPY of 1.61 per year (and, your HPR would be
  8.00).

16
Arithmetic vs. Geometric Returns

• When they need to calculate a rate of return over
  a number of periods, people often use the
  arithmetic average. However, that is incorrect
  because it ignores compounding, and therefore
  tends to overstate the return.
• Suppose that you purchased shares in CDE
  two-years ago. During the first year, the stock
  doubled, but it fell by 50 in the second year.
  What is your average annual rate of return (it
  should be obvious)?
• Arithmetic
• Geometric

17
Returns on Foreign Investments

• Calculating the return on a foreign investment is
  very similar to domestic investments, except that
  we must take the change in the currency into
  account. So, we actually have two sources of
  return.
• For example, suppose that you purchased shares of
  Pohang Iron Steel (POSCO) on the Korean Stock
  Exchange (KSE) on Jan 3, 1997 and sold them on
  Dec 27, 1997. Here are the details

18
Returns on Foreign Investments (cont)

• Now, if you were a Korean investor your return
  for the year would have been 23.06
• However, as a U.S. investor your return was a
  negative 30.88! Quite a difference, and it was
  entirely due to the loss in value of the won
  relative to the dollar during the Asian
  Contagion currency crisis that began in Thailand
  in June 1997

19
Returns on Foreign Investments (cont)

• To calculate this return, we first need to
  calculate your investment in dollar terms
• Where P0 is the cost in foreign currency, and FC0
  is the exchange rate (foreign currency
  unit/dollar). Your proceeds from the sale are
  calculated the same way
• Combining the equations into a rate of return,
  and rearranging we get the return in local
  currency (RLC)

20
Returns on Foreign Investments (cont)

• Now, we can see that your return in dollar terms
  was -30.88
• So, you made money on the stock, lost on the
  currency, and overall you lost a lot of money on
  this investment

21
Returns on Foreign Investments (cont)

• Heres another example. On Jan 27, 1999 Diageo
  PLC (LSE DGE) was selling for 630p. One year
  earlier it was selling for 542p, so a British
  investor would have earned a return of 16.24.
  However, an American investor would have made
  17.78
• The American made more because the British pound
  () appreciated against the dollar over that
  year. Note that the American originally paid
  8.87, but received 10.45 and the return is
  17.78.

22
Negative Returns

• All of the examples weve seen so far assume that
  your investment appreciates in value. However,
  its very likely that you will lose money
  occasionally.
• The formulas that weve seen work just as well
  for negative returns as for positive returns.
• For example, assume that you purchased a stock
  for 50 three months ago, and it is now worth
  40. What is your HPR and annualized HPR?
  Assume no dividends were paid.

23
Negative Returns (cont.)

• An often overlooked problem with losses is that
  you must earn a higher percentage return than you
  lost just to get even.
• Using our example, you lost 20. If the stock
  now rises by 20 you are not back to 50.
• To figure the gain to recover use the formula
  (L is the loss)
• So, you would need to earn a return of 25 to get
  back to 50

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What is Risk?

• A risky situation is one which has some
  probability of loss
• The higher the probability of loss, the greater
  the risk
• If there is no possibility of loss, there is no
  risk
• The riskiness of an investment can be judged by
  describing the probability distribution of its
  possible returns

• Types of Risk
• Default Risk
• Credit Risk
• Purchasing Power Risk
• Interest Rate Risk
• Systematic (Market) Risk
• Unsystematic Risk
• Event Risk
• Liquidity Risk
• Foreign Exchange (FX) Risk

25
Probability Distributions

• A probability distribution is simply a listing of
  the probabilities and their associated outcomes
• Probability distributions are often presented
  graphically as in these examples

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The Normal Distribution

• For many reasons, we usually assume that the
  underlying distribution of returns is normal
• The normal distribution is a bell-shaped curve
  with finite variance and mean

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The Expected Value

• The expected value of a distribution is the most
  likely outcome
• For the normal dist., the expected value is the
  same as the arithmetic mean
• All other things being equal, we assume that
  people prefer higher expected returns

E(R)
28
The Expected Return An Example

• Suppose that a particular investment has the
  following probability distribution
• 25 chance of 10 return
• 50 chance of 15 return
• 25 chance of 20 return
• This investment has an expected return of 15

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The Variance Standard Deviation

• The variance and standard deviation describe the
  dispersion (spread) of the potential outcomes
  around the expected value
• Greater dispersion generally (not always!) means
  greater uncertainty and therefore higher risk

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Calculating s 2 and s An Example

• Using the same example as for the expected
  return, we can calculate the variance and
  standard deviation

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The Scale Problem

• The variance and standard deviation suffer from a
  couple of problems
• The most tractable of these is the scale problem
• Scale problem - The magnitude of the returns used
  to calculate the variance impacts the size of the
  variance possibly giving an incorrect impression
  of the riskiness of an investment

32
The Scale Problem an Example
33
The Coefficient of Variation

• The coefficient of variation (CV) provides a
  scale-free measure of the riskiness of a security
• It removes the scaling by dividing the standard
  deviation by the expected return
• In the previous example, the CV for XYZ and ABC
  are identical, indicating that they have exactly
  the same degree of riskiness

34
Historical vs. Expected Returns Risk

• The equations just presented are for ex-ante
  (expected future) data.
• Generally, we dont know the probability
  distribution of future returns, so we estimate it
  based on ex-post (historical) data.
• When using ex-post data, the formulas are the
  same, but we assign equal (1/n) probabilities to
  each past observation.

35
Portfolio Risk and Return

• The preceding risk and return measures apply to
  individual securities. However, when we combine
  securities into a portfolio some things
  (particularly risk measures) change in, perhaps,
  unexpected ways.
• In this section, we will look at the methods for
  calculating the expected returns and risk of a
  portfolio.

36
Portfolio Expected Return

• For a portfolio, the expected return calculation
  is straightforward. It is simply a weighted
  average of the expected returns of the individual
  securities
• Where wi is the proportion (weight) of security i
  in the portfolio.

37
Portfolio Expected Return (cont.)

• Suppose that we have three securities in the
  portfolio. Security 1 has an expected return of
  10 and a weight of 25. Security 2 has an
  expected return of 15 and a weight of 40.
  Security 3 has an expected return of 7 and a
  weight of 35. (Note that the weights add up to
  100.)
• The expected return of this portfolio is

38
Portfolio Risk

• Unlike the expected return, the riskiness
  (standard deviation) of a portfolio is more
  complex.
• We cant just calculate a weighted average of the
  standard deviations of the individual securities
  because that ignores the fact that securities
  dont always move in perfect synch with each
  other.
• For example, in a strong economy we would expect
  that stocks of grocery companies would be
  moderate performers while technology stocks would
  be great performers. However, in a weak economy,
  grocery stocks will probably do very well
  compared to technology stocks. Both are risky,
  but by owning both we can reduce the overall
  riskiness of our portfolio.
• By combining securities with less than perfect
  correlation, we can smooth out the portfolios
  returns (i.e., reduce portfolio risk).

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Portfolio Risk (cont.)

• The following chart shows what happens when we
  combine two risky securities into a portfolio.
  The line in the middle is the combined portfolio.
  Note how much less volatile it is than either of
  the two securities.

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Portfolio Risk (cont.)

• The key to the risk reduction shown on the
  previous chart is the correlation between the
  securities.
• Note how Stock A and Stock B always move in the
  opposite direction (when A has a good year, B has
  a not so good year and vice versa). This is
  called negative correlation and is great for
  diversification.
• Securities that are very highly (positively)
  correlated would result in little or no risk
  reduction.
• So, when constructing a portfolio, we should try
  to find securities which have a low correlation
  (i.e., spread your money around different types
  of securities, different industries, and even
  different countries).

41
Portfolio Risk Quantified

• The correlation coefficient (rxy) describes the
  degree to which two series tend to move together.
  It can range from 1.00 (they always move in
  perfect sync) to -1.00 (they always move in
  different directions). Note that rxy 0 means
  that there is no identifiable (linear)
  relationship.
• Our measure of portfolio risk (standard
  deviation) must take account of the riskiness of
  each security, the correlation between each pair
  of securities, and the weight of each security
  in the portfolio.
• For a two-security portfolio, the standard
  deviation is
• The equation gets longer as we add more
  securities, so we will concentrate on the
  two-security equation.

42
Portfolio Risk Quantified (cont.)

• Suppose that we are interested in two securities,
  but they are both very risky. Security 1 has a
  standard deviation of 30 and security 2 has a
  standard deviation of 40. Further, the
  correlation between the two is quite low at 20
  (r1,2 0.20).
• What is the standard deviation of a portfolio of
  these two securities if we weight them equally
  (i.e., 50 in each)?
• Note that the standard deviation of the portfolio
  is less than the standard deviation of either
  security. This is what diversification is all
  about.

43
Determining the Required Return

• The required rate of return for a particular
  investment depends on several factors, each of
  which depends on several other factors (i.e., it
  is pretty complex!)
• The two main factors for any investment are
• The perceived riskiness of the investment
• The required returns on alternative investments
  (which includes expected inflation)
• An alternative way to look at this is that the
  required return is the sum of the risk-free rate
  (RFR) and a risk premium

44
The Risk-free Rate of Return

• The risk-free rate is the rate of interest that
  is earned for simply delaying consumption and not
  taking on any risk
• It is also referred to as the pure time value of
  money
• The risk-free rate is determined by
• The time preferences of individuals for
  consumption
• Relative ease or tightness in money market
  (supply demand)
• Expected inflation
• The long-run growth rate of the economy
• Long-run growth of labor force
• Long-run growth of hours worked
• Long-run growth of productivity

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The Risk Premium

• The risk premium is the return required in excess
  of the risk-free rate
• Theoretically, a risk premium could be assigned
  to every risk factor, but in practice this is
  impossible
• Therefore, we can say that the risk premium is a
  function of several major sources of risk
• Business risk
• Financial leverage
• Liquidity risk
• Exchange rate risk

46
The MPT View of Required Returns

• Modern portfolio theory assumes that the required
  return is a function of the RFR, the market risk
  premium, and an index of systematic risk
• This model is known as the Capital Asset Pricing
  Model (CAPM).

47
Risk and Return Graphically
The Market Line
Rate of Return
RFR
Risk
f(Business, Financial, Liquidity, and Exchange
Rate Risk)
Or
b or s