Risk and Return

Chapter Two

Risk and Return

The Concept of Risk Vs Return

Investment decisions are backed by various motives. Some people make investment to acquire control and enjoy prestige associated with it, some to display their wealth and some just for the sake of putting their excess money in some place or the other. But most of the people invest with an aim to get certain benefits in future. These future benefits are the returns you get on the investment. Return is the driving force behind investment. It represents rewards for making an investment.

In the case of a fixed income security like a debenture, the returns you get are in the form of periodic interest payments and repayment of principal at the end of the maturity period.

Similarly, in the case of an equity share, the returns are in the form of dividends and the price appreciation of the share.

Meaning of Risk

  • It is the possibility that actual future returns will deviate from expected returns
  • It is the variability of returns
  • It is a chance of unfavorable event to occur

From the perspective of financial analysis then, it is the possibility that the actual cash flow will be different from forecasted cash flows (returns). Therefore if an investment’s returns are known for certainty the security is called a risk free security. An example on this regard is Government treasury securities. This is basically because virtually there is no chance that the government will fail to redeem these securities at maturity or that the treasury will default on any interest payment owed.


When it comes to investments, there are always some levels of uncertainty associated with future holding period returns. Such uncertainty is commonly known as the risk of the investment. Then the question will be what causes the uncertainty (or volatility) of an investment’s returns? The answer depends on the nature of the investment, the performance of the economy, and other factors. In other words, when you “dissect” the uncertainty of an investment’s return, you will realize that it is made up of different components.


The purpose of this chapter is to help you understand how to choose among alternative investment assets. This selection process requires that you estimate and evaluate the expected risk-return trade-offs for the alternative investments available. Therefore, you must understand how to measure the rate of return and the risk involved in an investment accurately. To meet this need, in this section we examine ways to quantify return and risk. The presentation will consider how to measure both historical and expected rates of return and risk.

We consider historical measures of return and risk because this chapter and other publications provide numerous examples of historical average rates of return and risk measures for various assets, and understanding these presentations is important. In addition, these historical results are often used by investors when attempting to estimate the expected rates of return and risk for an asset class.

The first measure is the historical rate of return on an individual investment over the time period the investment is held (that is, its holding period). Next, we consider how to measure the average historical rate of return for an individual investment over a number of time periods. The third subsection considers the average rate of return for a portfolio of investments. Given the measures of historical rates of return, we will present the traditional measures of risk for a historical time series of returns (that is, the variance and standard deviation). Following the presentation of measures of historical rates of return and risk, we turn to estimating the expected rate of return for an investment. Obviously, such an estimate contains a great deal of uncertainty, and we present measures of this uncertainty or risk.

Measures of Historical Rates of Return

When you are evaluating alternative investments for inclusion in your portfolio, you will often be comparing investments with widely different prices or lives. As an example, you might want to compare a $10 stock that pays no dividends to a stock selling for $150 that pays dividends of $5 a year. To properly evaluate these two investments, you must accurately compare their historical rates of returns. A proper measurement of the rates of return is the purpose of this section.

When we invest, we defer current consumption in order to add to our wealth so that we can consume more in the future. Therefore, when we talk about a return on an investment, we are concerned with the change in wealth resulting from this investment. This change in wealth can be either due to cash inflows, such as interest or dividends, or caused by a change in the price of the asset (positive or negative).

If you commit $200 to an investment at the beginning of the year and you get back $220 at the end of the year, what is your return for the period? The period during which you own an investment is called its holding period, and the return for that period is the holding period return (HPR). In this example, the HPR is 1.10, calculated as follows:

                                            HPR = Ending Value of Investment / Beginning Value of Investment

                                                            $220/$200 = 1.10

This value will always be zero or greater—that is, it can never be a negative value. A value greater than 1.0 reflects an increase in your wealth, which means that you received a positive rate of return during the period. A value less than 1.0 means that you suffered a decline in wealth, which indicates that you had a negative return during the period.  An HPR of zero indicates that you lost all your money.

Although HPR helps us express the change in value of an investment, investors generally evaluate returns in percentage terms on an annual basis. This conversion to annual percentage rates makes it easier to directly compare alternative investments that have markedly different characteristics. The first step in converting an HPR to an annual percentage rate is to derive a percentage return, referred to as the holding period yield (HPY). The HPY is equal to the HPR minus 1.

          HPY = HPR - 1

In our example: 

          HPY = 1.10 - 1 = 0.1 = 10% 

To derive an annual HPY, you compute an annual HPR and subtract 1. Annual HPR is found by:

        Annual HPR = HPR 1/n                                                         


        n = number of years the investment is held

Consider an investment that cost $250 and is worth $350 after being held for two years:

        HPR = Ending Value of Investment / Begining Value of Investmnent = $350/$250 = 1.40

       Anual HPR = 1.40 1/n = 1.40 1/2 = 1.1832

      Anual HPY = 1.1832 - 1 = 0.1832 = 18.32%                          

If you experience a decline in your wealth value, the computation is as follows:


A multiple year loss over two years would be computed as follows:



In contrast, consider an investment of $100 held for only six months that earned a return of $112:



Note that we made some implicit assumptions when converting the HPY to an annual basis. This annualized holding period yield computation assumes a constant annual yield for each year. In the two-year investment, we assumed an 18.32 percent rate of return each year, compounded. In the partial year HPR that was annualized, we assumed that the return is compounded for the whole year.

That is, we assumed that the rate of return earned during the first part of the year is likewise earned on the value at the end of the first six months. The 12 percent rate of return for the initial six months compounds to 25.44 percent for the full year. Because of the uncertainty of being able to earn the same return in the future six months, institutions will typically not compound partial year results.


Remember one final point: The ending value of the investment can be the result of a positive or negative change in price for the investment alone (for example, a stock going from $20 a share to $22 a share), income from the investment alone, or a combination of price change and income.Ending value includes the value of everything related to the investment.

Computing Mean Historical Returns

Now that we have calculated the HPY for a single investment for a single year, we want to consider mean rates of return for a single investment and for a portfolio of investments. Over a number of years, a single investment will likely give high rates of return during some years and low rates of return, or possibly negative rates of return, during others. Your analysis should consider each of these returns, but you also want a summary figure that indicates this investment’s typical experience, or the rate of return you should expect to receive if you owned this investment over an extended period of time. You can derive such a summary figure by computing the mean annual rate of return for this investment over some period of time.

Alternatively, you might want to evaluate a portfolio of investments that might include similar investments (for example, all stocks or all bonds) or a combination of investments (for example, stocks, bonds, and real estate). In this instance, you would calculate the mean rate of return for this portfolio of investments for an individual year or for a number of years. Single Investment Given a set of annual rates of return (HPYs) for an individual investment, there are two summary measures of return performance. The first is the arithmetic mean return, the second the geometric mean return. To find the arithmetic mean (AM), the sum (∑) of annual HPYs is divided by the number of years (n) as follows:

                                     AM =∑HPY/n


∑HPY = the sum of annual holding period yields

An alternative computation, the geometric mean (GM), is the nth root of the product of the HPRs for n years.



  • the product of the annual holding period returns as follows:

       ( HPR1) × (HPR2) ... (HPRn)

To illustrate these alternatives, consider an investment with the following data:



Beginning value

Ending value



















  • Compute AM and GM          

Investors are typically concerned with long-term performance when comparing alternative investments. GM is considered a superior measure of the long-term mean rate of return because it indicates the compound annual rate of return based on the ending value of the investment versus its beginning value.


Although the arithmetic average provides a good indication of the expected rate of return for an investment during a future individual year, it is biased upward if you are attempting to measure an asset’s long-term performance.


When rates of return are the same for all years, the GM will be equal to the AM. If the rates of return vary over the years, the GM will always be lower than the AM. The difference between the two mean values will depend on the year-to-year changes in the rates of return. Larger annual changes in the rates of return—that is, more volatility—will result in a greater difference between the alternative mean values.

An awareness of both methods of computing mean rates of return is important because published accounts of investment performance or descriptions of financial research will use both the AM and the GM as measures of average historical returns

Measuring Expected Rates of Return

Risk is the uncertainty that an investment will earn its expected rate of return. In the examples in the prior section, we examined realized historical rates of return. In contrast, an investor who is evaluating a future investment alternative expects or anticipates a certain rate of return. The investor might say that he or she expects the investment will provide a rate of return of 10 percent, but this is actually the investor’s most likely estimate, also referred to as a point estimate.

Pressed further, the investor would probably acknowledge the uncertainty of this point estimate return and admit the possibility that, under certain conditions, the annual rate of return on this investment might go as low as –10 percent or as high as 25 percent. The point is, the specification of a larger range of possible returns from an investment reflects the investor’s uncertainty regarding what the actual return will be. Therefore, a larger range of expected returns makes the investment riskier.

An investor determines how certain the expected rate of return on an investment is by analyzing estimates of expected returns. To do this, the investor assigns probability values to all possible returns. These probability values range from zero, which means no chance of the return, to one, which indicates complete certainty that the investment will provide the specified rate of return. These probabilities are typically subjective estimates based on the historical performance of the investment or similar investments modified by the investor’s expectations for the future.

As an example, an investor may know that about 30 percent of the time the rate of return on this particular investment was 10 percent. Using this information along with future expectations regarding the economy, one can derive an estimate of what might happen in the future. The expected return from an investment is defined as:


Let us begin our analysis of the effect of risk with an example of perfect certainty wherein the investor is absolutely certain of a return of 5 percent. Exhibit 1.2 illustrates this situation.

Perfect certainty allows only one possible return, and the probability of receiving that return is 1.0. Few investments provide certain returns. In the case of perfect certainty, there is only one value for PiRi:

                                                    E(Ri) = (1.0)(0.05) = 0.05

In an alternative scenario, suppose an investor believed an investment could provide several different rates of return depending on different possible economic conditions. As an example, in a strong economic environment with high corporate profits and little or no inflation, the investor might expect the rate of return on common stocks during the next year to reach as high as 20 percent. In contrast, if there is an economic decline with a higher-than-average rate of inflation, the investor might expect the rate of return on common stocks during the next year to be –20 percent. Finally, with no major change in the economic environment, the rate of return during the next year would probably approach the long-run average of 10 percent.

The investor might estimate probabilities for each of these economic scenarios based on past experience and the current outlook as follows:


The computation of the expected rate of return [E(Ri)] is as follows:


Measuring the expected Risk of Expected Rates of Return

We have shown that we can calculate the expected rate of return and evaluate the uncertainty, or risk, of an investment by identifying the range of possible returns from that investment and assigning each possible return a weight based on the probability that it will occur. Although the graphs help us visualize the dispersion of possible returns, most investors want to quantify this dispersion using statistical techniques. These statistical measures allow you to compare the return and risk measures for alternative investments directly. Two possible measures of risk (uncertainty) have received support in theoretical work on portfolio theory: the variance and the standard deviation of the estimated distribution of expected returns.

In this section, we demonstrate how variance and standard deviation measure the dispersion of possible rates of return around the expected rate of return. We will work with the examples discussed earlier. The formula for variance is as follows:


Variance The larger the variance for an expected rates of return, the greater the dispersion of expected returns and the greater the uncertainty, or risk, of the investment. The variance for the perfect-certainty example would be:


Note that,  in perfect certainty, there is no variance of return because there is no deviation from expectations and, therefore, no risk or uncertainty. The variance for the secon                                        

Standard Deviation The standard deviation is the square root of the variance:


For the second example, the standard deviation would be:



Therefore, when describing this example, you would contend that you expect a return of 7 percent, but the standard deviation of your expectations is 11.87 percent.

A Relative Measure of Risk In some cases, an unadjusted variance or standard deviation can be misleading. If conditions for two or more investment alternatives are not similar—that is, if there are major differences in the expected rates of return—it is necessary to use a measure of relative variability to indicate risk per unit of expected return. A widely used relative measure of risk is the coefficient of variation (CV), calculated as follows:


The CV for the preceding example would be:


This measure of relative variability and risk is used by financial analysts to compare alternative investments with widely different rates of return and standard deviations of returns. As an illustration, consider the following two investments:

Comparing absolute measures of risk, investment B appears to be riskier because it has a standard deviation of 7 percent versus 5 percent for investment A. In contrast, the CV figures show that investment B has less relative variability or lower risk per unit of expected return because it has a substantially higher expected rate of return:


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