How To Measure the Risk of Expected Rates of Return?

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'll demonstrate how variance and standard deviation measure the dispersion of possible rates of return around the expected rate of return. The formula for variance is as follows:

Variance The larger the variance for an expected rate 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 (risk-free) 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 second example would be:

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

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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How To Calculate Expected Rates of Return?

Risk is the uncertainty that an investment will earn its expected rate of return. An investor who is evaluating a future investment alternative expects or anticipates a certain rate of return. 

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:

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:

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How To Compute Arithmetic Mean & Geometric Mean

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 is the geometric mean return. To find the arithmetic mean (AM), the sum (Σ) of annual HPYs is divided by the number of years (n) as follows:

Where:

ΣHPY = 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 minus one.

Where:

π = the product of the annual holding period returns as follows:

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

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How To Measure Holding Period Yield & Annual Holding Period Yield

Holding Period Return (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.

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

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

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What Is Historical Rates of Return?

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).

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 $15 stock that pays no dividends to a stock selling for $250 that pays dividends of $10 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.

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:

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Definition Of Investment

Investment can be defined as the current commitment of dollars for a period of time in order to derive future payments that will compensate the investor for (i) the time the funds are committed, (ii) the expected rate of inflation during this time period, and (iii) the uncertainty of the future payments. The “investor” can be an individual, a government, a pension fund, or a corporation. Similarly, this definition includes all types of investments, including investments by corporations in plant and equipment and investments by individuals in stocks, bonds, commodities, or real estate. This text emphasizes investments by individual investors. In all cases, the investor is trading a known dollar amount today for some expected future stream of payments that will be greater than the current dollar amount today.

Hence, the question is why people invest and what they want from their investments. They invest to earn a return from savings due to their deferred consumption. They want a rate of return that compensates them for the time period of the investment, the expected rate of inflation, and the uncertainty of the future cash flows. This return, the investor’s required rate of return, is discussed throughout this book. A central question of this book is how investors select investments that will give them their required rates of return.

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