topic1[1]

Topic 1: Relationship Between Risk and Return

BUS 442 Investment Theory and Portfolio Management

One of the most important concepts in investment theory is the relationship between risk and return. It is this relationship that drives the theoretical foundation of many investment models (such as the Capital Asset Pricing Model). Before we begin our discussion on the development of theoretical models that attempt to “capture” the relationship between risk and return, we need to first understand how these two variables (or measurements) are determined.

1. (Rate of) Return

In your own words, explain what rate of return represents and why it is so important to an investor?

One of the measurements of return is the holding period return (HPR), which represents the return an investor received for holding an investment for a certain amount of time. The formula for determining the HPR is as follows:

price

Purchase income

Current gain/loss

Capital price

Purchase income

Current price) Purchase price (Sale +=

+-=

HPR

From the above formula, it is clear that the HPR is dependent on two components:

(a) Capital gain/loss

(b) Current income

Refer to In-class E xample 1

It is important to understand that the HPR is an ex-post return, i.e. a return that has already taken place. It is sometimes known as the historical return. Another thing that you should be aware of is that the HPR is a measurement for return over a single period (i.e. 4 months, 5 years, etc.)

What happen if you needed to determine the investment returns over multiple periods? In other words, what if you are interested in the average returns of an investment over a number of quarters or years? There are three different measures for average returns: (a) arithmetic average, (b) geometric average, and (c) dollar-weighted average return. It is important for you to understand the advantages and disadvantages for each of the three return measurements.

(a)Arithmetic average

(i)Advantage

(ii)Disadvantage

(b)Geometric average

(i)Advantage

(ii)Disadvantage

(c)Dollar-weighted average return

(i)Advantage

(ii)Disadvantage

Refer to In-class E xample 2

In order to compare the performances among different investments, it is important to make sure that you are doing so on equal terms. In other words, you need to “compare apples with apples”. One way to do this is to annualize all your returns before making the comparisons. There are two common annualized returns: (a) annual percentage rates (APR) and (b) effective annual rate (EA R).

(a) Annual percentage rate (APR)

Explain how the APR of an investment is determined. What does thi s measuremen t ignored?

(b) Effective annual rate (EA R)

Explain the difference between APR and EAR.

The relationship between the EAR and the APR of an investment can be expressed with the formula as follows:

n

n APR EAR ??? ?

?

+=+11

where n = number of compounding period per year.

Keep in mind that the formula has to be “modified” when you are dealing with continuous compounding. In this case, the relationship between EAR and APR is determined by the formula as follows:

1-=APR

e EAR

Refer to In-class E xample 3

2. Uncertainty and its Impact on Return

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.

What cause s 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 real ize that it is made up of different components. The following are some of the components:

(a) Business risk : This is the uncertainty regarding the earnings (or profitability) of a firm as a result of

changes in demand, input prices, and technological obsolescence.

(b) Default risk : This is the uncertainty regarding an issuing firm’s ability to pay interest, principal, etc. on its

debt instruments.

(c) Inflation risk : This is the uncertainty over future rates of inflation. If the return from an investment is barely

kee ping up with the rate of inflation, an investor’s purchasing power will be eroded as time goes on. In other words, the investor will receive a lesser amount of purchasing power than what was originally invested because the cost of buying everything has gone up. Inflation risk is also known as purchasing power risk .

(d) Market risk : This represents the changes in an investment’s price (or market value) as a result of an event

that affects the entire market. An example is the impact of a market correction or a market crash on an investment’s return.

(e) Interest rate risk : This represents the fluctuation in the value of an investment when market interest rate

changes. This has a big impact on interest-paying investments because as market interest rate rises (falls), a n investor’s money is tied up in a bond that pay less (more) than the going rate, and hence the value of the investor’s bond decreases (increases).

(f) Liquidity risk : This is the risk of not being able to sell an investment immediately with a reasonable price. (g) Political risk : This is caused by changes in the political environment that affect an investment’s market

value. Political risk can be classified as either domestic or foreign political risk. An example of domestic political risk is a change in the tax laws, and an example of foreign political risk is a change in a foreign government’s policy regarding capital outflow.

(h) Callability risk : This is the risk that an investment is recalled (or retired) prior to the original stated date.

This type of risk is most applicable to long-term bonds and preferred stocks. This usually happens when the issuing firms find the market conditions favorable in “refinancing” such investments.

(i) Exchange rate risk : This is the uncertainty regarding the changes in exchange rates th at might affect the

value of an investment. Exchange rate uncertainty has an impact on both domestic and foreign investments. Why is this the case?

It is important to understand that the components of risk present and the “size” of each component differ f rom one investment to another investment. For example, certain investments have no liquidity risk while other investments have extremely high level of liquidity risk.

Now that you knew more about the concept of risk, how do we measure it? We can determine the risk of an investment using the scenario analysis approach. This approach is based on an investment’s expected return rather than its historical return.

In your own words, explain what the expected return represents (or measures).

The expected return of an investment can be determined using the following formula:

)(...)()(][)(2211n n i i p r p r p r p r r E ?++?+?=?=∑

where i p = probability of a given scenario and n = the number of scenarios.

The above formula should look familiar to you since you would have encountered it in statistics and introductory finance courses. Now that you are once again comfortable with the formula to calculate the expected return, lets take a look at a slightly more complex formula for calculating the standard deviation, which is a common measurement for risk.

[]

[]2

2)()(r E p r

i i

-?=

∑σ

It is easier to understand the concept of standard deviation when we look at the distribution curve of the returns of an asset. The following graph depicts the distribution curves of the returns of two assets, A and B, which have the same expected return.

F requency

Return

M ean

A sset A

A sset B

We know the standard deviation measures how far the returns deviate from the asset’s average return. In the preceding graph, we know that the standard deviation of Asset B is greater than that of Asset A because we can visually verify that the returns of Asset B deviate more from its mean return. Another way to compare the volatility (or standard deviation) of two assets is to look at the shapes of the distribution curves. The flatter (or more spread out) the curve, the more volatile the returns; and the narrower the curve, the less volatile the returns.

Refer to In-class E xample 4

It is important to understand that the formula above is an ex-ante formula. In other words, it tries to “predict” the volatility of a particular investment’s returns in the future. However, it is also crucial to analyze how the investment had behaved in the past. In order to accomplish this, we will need to analyze the volatility of the investment’s historical return.

The concepts behind standard deviations based on expected returns and historical returns are very similar. They both look at the average deviation of the returns from the investment’s average return. The only differ ence is that one uses the expected return for the average return while the other uses average historical return for the average return (which is denoted by x ). The following is the formula for calculating the standard deviation based on historical returns:

[]1

2

2

--=

∑n x

n x s

It is important to note that most investors use a small sample of an investment’s historical returns to determine its volatility. As a result, we will be determining the standard deviation of an investment’s return based on a sample rather than the population. In other words, we are solving for the sample standard deviation, and thi s is usually denoted by s rather than σ (as indicated by the formula above).

Refer to In-class E xample 5