Measuring Risk

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Investment Analysis & Portfolio Management (FIN630)

Lesson # 33

PORTFOLIO THEORY

Measuring Risk:

Risk is often associated with the dispersion in the likely outcomes. Dispersion refers to

variability. Risk is assumed to arise out of variability', which is consistent with our

definition of risk as the chance that the actual outcome of an investment will differ from the

expected outcome. If an asset's return has no variability, in effect it has no risk. Thus, a one-

year treasury bill purchased to yield 10 percent and held to maturity will, in fact, yield (a

nominal) 10 percent No other outcome is possible, barring default by the U.S. government,

which is not considered a reasonable possibility.

Consider an investor analyzing a series of returns (TRs) for the major types of financial

asset over some period of years. Knowing the mean of .this series is not enough; the

investor also needs to know something about the variability in the returns. Relative to the

other assets, common stocks show, the largest variability (dispersion) in returns, with small

common stocks showing f ten greater variability. Corporate bonds have a much smaller

variability and therefore a more compact distribution of returns. Of course, Treasury bills

are the least risky. The "dispersion of annual returns for bills is compact.

Standard Deviation:

The risk of distributions' can be measured with an absolute measure of dispersion, or

variability. The most commonly used measure of dispersion over some period of years is the

standard deviation, which measures the deviation of each observation from the arithmetic

mean of the observations and is a reliable measure of variability, because all the information

in a sample is used.

The standard deviation is a measure of the total risk of an asset or a portfolio. It captures the

total variability in the assets or portfolios return whatever the source of that variability. The

standard deviation can be calculated from the variance, which is calculated as:

σ = ∑ (X - X)

i=1

n-1

Where;

σ2 = the variance of a set of values

X = each value in the set

X = the mean of the observations

n = the number of returns in the sample

σ2 = (σ2) 1 / 2 = standard deviation

Knowing the returns from the sample, we can calculate the standard deviation quite easily.

Dealing with Uncertainty:

Realized returns are important for several reasons. For example, investors need to know

how their portfolios have performed. Realized returns, also can be particularly important in

helping investors to form expectations about future returns, because investors must concern

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Investment Analysis & Portfolio Management (FIN630)

themselves with their best estimate of return over the next year, or six months, or whatever.

How do we go about estimating returns, which is what investors must actually do in

managing their portfolios?

The total return measure, TR, is applicable whether one is measuring realized returns; or

estimating, future (expected) returns. Because it includes everything the investor can expect

to receive over any specified future period, the TR is useful in conceptualizing the estimated

returns from securities.

Similarly, the variance, or its square root, the standard deviation, is an accepted measure of

variability for both realized returns and expected returns. We will calculate both the

variance and the standard deviation below and use them interchangeably as the situation

dictates. Sometimes it is preferable to use one and sometimes the other.

Using Probability Distributions:

The return an investor will earn from investing is not known; it must be estimated. Future

return is an expected return and may or may not actually be-realized. An investor may

expect the TR on a particular security to be 0.10 for the coming year, but in truth this is

only a "point estimate." Risk, or the chance that some unfavorable event will occur, is

involved when investment decisions are made. Investors are often overly optimistic about

expected returns.

Probability Distributions:

To deal with the uncertainty of returns, investors need to think explicitly about a: security's

distribution of probable TRs. ln other words, investors need to keep in mind that, although

they may expect a security to return 10 percent, for example, this is only a one-point

estimate of the entire range of possibilities. Given that investors must deal with the

uncertain future, a number of possible returns can, and will, occur.

In the case of a Treasury bond paying fixed rate of interest, the interest payment will be

made with l00-percent certainty barring a financial collapse of the economy. The probability

of occurrence is 1.0; because no other outcome is possible.

With the possibility of two or more outcomes, which is the norm for common stocks, each

possible likely outcome must be considered and a probability of its occurrence assessed.

The probability for a particular outcome is simply the chance that the specified outcome

will occur. The result of considering these outcomes and their probabilities together is a

probability distribution consisting of the specification of the likely outcomes that may occur

and the probabilities associated with these likely outcomes.

Probabilities represent the likelihood of various outcomes and are typically expressed as a

decimal. The sum of the probabilities of all possible outcomes must be 1.0, because they

must completely describe all the (perceived) likely occurrences.

How are these probabilities and associated outcomes obtained? In the final analysis,

investing for some future period involves uncertainty, and therefore subjective estimates.

Although past occurrences (frequencies) may be relied on heavily to estimate the

probabilities the past must be modified for any changes expected in the future.

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Investment Analysis & Portfolio Management (FIN630)

Calculating Expected Return for a Security:

To describe the single most likely outcome from a particular probability distribution, it is

necessary to calculate its expected value. The expected value is the weighted average of'all

possible return outcomes, where each outcome is weighted by its respective probability of

occurrence. Since investors are interested in returns, we will-call this expected value the

expected rate of return, or simply expected-return, and for any security, it is calculated as;

E (R) = ∑ Ri pri

i=1

Where;

E (R) = the expected return on a security'

= the ith possible return

pri = the probability of the ith return Ri

m = the number of possible returns

Calculating Risk for a Security:

Investors must be able to quantify and measure risk. To calculate the total risk associated

with the expected return, the variance or standard deviation is used, the variance and, its

square root, standard deviation, are measures of the spread or dispersion in the probability

distribution; that is, they measure the dispersion of a random variable around its mean. The

larger this dispersion, the larger the variance or standard deviation.

To calculate the variance or standard deviation from the probability distribution, first

calculate the expected return of the distribution. Essentially, the same procedure used to

measure risk, but now the probabilities associated with the outcomes must be included,

The variance of returns = σ2 = ∑ - [Ri E (R)]2pri

i=1

And

The standard deviation of returns = σ = (σ2)1/2

Portfolio Expected Return:

The expected return on any portfolio is easily calculated as a weighted average of the

individual securities expected returns. The percentages of a portfolio's total value that are

invested in each portfolio asset are referred to as portfolio weights, which will denote by w.

The combined portfolio weights are assumed to sum to 100 percent of, total investable

funds, or 1.0, indicating that all portfolio funds are invested. That is,

w1 + w2 + ... + wn = ∑ wi = 1.0

i=1

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Investment Analysis & Portfolio Management (FIN630)

Portfolio Risk:

The remaining computation in investment analysis is that of the risk of the portfolio. Risk is

measured by the variance (or standard deviation) of the portfolio's return, exactly as in the

case of each individual security. Typically, portfolio risk is stated in terms of standard

deviation which is simply the square root of the variance.

It is at this point that the basis of modern portfolio theory emerges, which can be stated as

follows: Although the expected return of a portfolio is a weighted average of its expected

returns, portfolio risk (as measured by the variance or standard deviation) is not a weighted

average of the risk of the individual securities in the portfolio. Symbolically,

E (Rp) = ∑ wi E (Ri)

i=1

But

σ2p ≠ ∑ wi σ2i

i=1

Precisely, investors can reduce the risk of a portfolio beyond what it would be if risk were,

in fact, simply a weighted average of the individual securities' risk. In order to see how this

risk reduction can be accomplished, we must analyze portfolio risk in detail.

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