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 > 1 for less than annual periods and < 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 isn’t an necessarily annualized return The solution to these problems is to calculate the IRR of the investment A security investment’s 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, let’s 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 don’t 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) Here’s 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 we’ve seen so far assume that your investment appreciates in value. However, its very likely that you will lose money occasionally. The formulas that we’ve 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:

24. 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 Types of Risk: Default Risk – This is the risk that a counterparty won’t make required payments (e.g., a bond issuer missing an interest payment) Credit Risk – This is the risk that a counterparty’s credit rating will decline, thereby leading to a loss of value to your investment Purchasing Power Risk – The risk that inflation will be higher than you expect, thus your real purchasing power will increase by a lesser amount than anticipated Interest Rate Risk – This is the risk that interest rates will move adversely to your position (e.g., and increase in rates causes your long bond position to lose value) Systematic (Market) Risk – The fluctuations in your portfolio value that can be explained by changes in the overall market Unsystematic (Company-specific) Risk - The fluctuations in your portfolio value that are not explained by changes in the overall market Event Risk – The risk that some unexpected event (e.g., a strike by union employees) will occur causing a decline in the value of your investment Liquidity Risk – This risk that you will be unable to convert your investment into cash quickly and with minimal discounts to market value (This happened in the 1987 market crash, and again to LTCM in 1998) Foreign Exchange (FX) Risk – The risk that the value of the dollar will fluctuate relative to a foreign currency to which you have exposure, causing an unanticipated decline in the value of your investment Note that there are many additional sources of risk that your investments will face. This list is only a summary of the major risks that investors may face. Many of these risks may also work in your favor depending on your position. For example, an issuer’s credit rating may actually increase instead of decrease, causing an increase in the value of your investment. Or, an increase in interest rates may increase the value of your investment (e.g., call options actually increase in value when interest rates rise, or your short bond position will also increase in value). Types of Risk: Default Risk – This is the risk that a counterparty won’t make required payments (e.g., a bond issuer missing an interest payment) Credit Risk – This is the risk that a counterparty’s credit rating will decline, thereby leading to a loss of value to your investment Purchasing Power Risk – The risk that inflation will be higher than you expect, thus your real purchasing power will increase by a lesser amount than anticipated Interest Rate Risk – This is the risk that interest rates will move adversely to your position (e.g., and increase in rates causes your long bond position to lose value) Systematic (Market) Risk – The fluctuations in your portfolio value that can be explained by changes in the overall market Unsystematic (Company-specific) Risk - The fluctuations in your portfolio value that are not explained by changes in the overall market Event Risk – The risk that some unexpected event (e.g., a strike by union employees) will occur causing a decline in the value of your investment Liquidity Risk – This risk that you will be unable to convert your investment into cash quickly and with minimal discounts to market value (This happened in the 1987 market crash, and again to LTCM in 1998) Foreign Exchange (FX) Risk – The risk that the value of the dollar will fluctuate relative to a foreign currency to which you have exposure, causing an unanticipated decline in the value of your investment Note that there are many additional sources of risk that your investments will face. This list is only a summary of the major risks that investors may face. Many of these risks may also work in your favor depending on your position. For example, an issuer’s credit rating may actually increase instead of decrease, causing an increase in the value of your investment. Or, an increase in interest rates may increase the value of your investment (e.g., call options actually increase in value when interest rates rise, or your short bond position will also increase in value).

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

26. 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

27. 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

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%

29. 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

30. Calculating s 2 and s : An Example Using the same example as for the expected return, we can calculate the variance and standard deviation:

31. 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 don’t 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 can’t just calculate a weighted average of the standard deviations of the individual securities because that ignores the fact that securities don’t 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 portfolio’s returns (i.e., reduce portfolio risk).

39. 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.

40. 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

45. 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