an introduction to risk and retur

Chapter 4

An Introduction to Risk and Return

Slide Contents

Learning Objectives

1. Calculate Realized and Expected Rates of Return.

2. Compute Geometric and Arithmetic Average Rates of Return.

3. Defining Risk and Calculating risk

Principles Used in This Chapter

Principle 1: There is a Risk-Return Tradeoff.

We will expect to receive higher returns for assuming more risk.

Principle 2: Market Prices Reflect Information.

Depending on the degree of efficiency of the market, security prices

may or may not fully reflect all information.

Investment Process

1. Market and Security Analysis

2. Formation of Optimal Portfolio

Return

Income received on an investment plus any change

in market price, usually expressed as a percent

of the beginning market price of the investment.

The total gain or loss experienced on an investment over a

given period of time.

Calculating the Realized Return from an

Investment

Realized return or cash return measures the gain or loss on an

investment.


Investment (cont.)

We can also calculate the rate of return as a percentage. It is

simply the cash return divided by the beginning stock price.


Investment (cont.)

Example 1: You invested in 1 share of Apple (AAPL) for $95 and

sold a year later for $200. The company did not pay any dividend

during that period. What will be the cash return on this

investment?


Investment (cont.)

Cash Return = $200 + 0 - $95

= $105


Investment (cont.)

Example 2: You invested in 1 share of share Apple (AAPL) for $95

and sold a year later for $200. The company did not pay any

dividend during that period. What will be the rate of return on

this investment?


Investment (cont.)

Rate of Return = ($200 + 0 - $95) ÷ 95

= 110.53%

Table 7-1 has additional examples on measuring an investor’s realized rate

of return from investing in common stock.


Investment (cont.)

Table 7-1 indicates that the returns from investing in common

stocks can be positive or negative.

Furthermore, past performance is not an indicator of future

performance.

However, in general, we expect to receive higher returns for

assuming more risk.

Calculating the Expected Return from an

Investment

Expected return is what you expect to earn from an

investment in the future.

It is estimated as the average of the possible returns, where each

possible return is weighted by the probability that it occurs.


Investment (cont.)


Investment (cont.)

Expected Return

= (-10%×0.2) + (12%×0.3) + (22%×0.5)

= 12.6%

Measuring Risk

In the example on Table 7-2, the expected return is 12.6%;

however, the return could range from -10% to +22%.

This variability in returns can be quantified by computing the

Variance or Standard Deviation in investment returns.

Geometric

vs. Arithmetic Average

Rates

of Return

Geometric vs. Arithmetic Average Rates of

Return

Arithmetic average may not always capture the true rate of return

realized on an investment.

In some cases, geometric or compound average may be a more

appropriate measure of return.


Return (cont.)

For example, suppose you bought a stock for $25. After one year,

the stock rises to $30 and in the second year, it falls to $15. What

was the average return on this investment?


Return (cont.)

The stock earned +20% in the first year and -50% in the second

year.

Simple average = (20%-50%) ÷ 2 = -15%

Computing Geometric Average Rate of

Return


Return (cont.)

However, over the 2 years, the $25 stock lost the equivalent of

22.54%

{(1+.20)(1-.50)}1/2 - 1 = -22.54%.

Here, -15% is the simple arithmetic average while -22.54% is the

geometric or compound average rate.

Which one is the correct indicator of return? It depends on the

question being asked.


Return (cont.)

The geometric average rate of return answers the question,

“What was the growth rate of your investment?”

The arithmetic average rate of return answers the question, “what

was the average of the yearly rates of return?


Return (cont.)

Compute the arithmetic and geometric average for the following

stock.

Year Annual Rate of Return

Value of the stock

0 $25

1 40% $35

2 -50% $17.50


Return (cont.)

Arithmetic Average = (40-50) ÷ 2 = -5%

Geometric Average

= [(1+Ryear1) × (1+Ryear 2)]1/2 - 1

= [(1.4) × (.5)] 1/2 - 1

= -16.33%

Choosing the Right “Average”

Both arithmetic average geometric average are important and correct. The following

grid provides some guidance as to which average is appropriate and when:

Question being addressed:

Appropriate Average Calculation:

What annual rate of return can we expect for next year?

The arithmetic average calculated using annual rates of return.

What annual rate of return can we expect over a multi-year horizon?

The geometric average calculated over a similar past period.

Checkpoint 7.2

Computing the Arithmetic and Geometric Average Rates of Return

Five years ago Mary’s grandmother gave her $10,000 worth of stock in the shares of a publicly traded company

founded by Mary’s grandfather. Mary is now considering whether she should continue to hold the shares, or

perhaps sell some of them. Her first step in analyzing the investment is to evaluate the rate of return she has

earned over the past five years.

The following table contains the beginning value of Mary’s stock five years ago as well as the values at the end of

each year up until today (the end of year 5):

What rate of return did Mary earn on her investment in the stock given to her by her grandmother?

Checkpoint 7.2

Checkpoint 7.2: Check Yourself

• Mary has decided to keep the stock given to her by her grandmother. However, now she wants to consider the prospect of selling another gift made to her five years ago by her grandmother. What are the arithmetic and geometric average rates of return for the following investment?

• See table on the next slide.

Problem (cont.)

Year Annual Rate of Return

Value of the Stock

0 $10,000.00

1 -15.0% $8,500.00

2 15.0% $9,775.00

3 25.0% $12,218.75

4 30.0% $15,884.38

5 -10.0% $14,295.94

Step 2: Decide on a Solution Strategy

We need to calculate the arithmetic and geometric average.

The arithmetic average fails to capture the effect of compound

interest, which can be measured by geometric average.

Step 3: Solve

Calculate the Arithmetic Average

Arithmetic Average

= Sum of the annual rates of return ÷ Number of years

= 45% ÷ 5 = 9%

Based on past performance of the stock, Mary should expect that

it would earn 9% next year.

Step 3: Solve (cont.)

Calculate the Geometric Average

Geometric Average = [(1+Ryear1) × (1+Ryear 2 ) × (1+Ryear3) × (1+Ryear4)

× (1+Ryear5) ]1/5 - 1

= [(.85) × (1.15) × (1.25) × (1.30) × (.90)] 1/5 - 1

= 7.41%

Step 4: Analyze

The arithmetic average is 9% while the geometric average is

7.41%. The geometric average is lower as it incorporates

compounding of interest.

Both of these averages are useful and meaningful but in answering

two very different questions.

Step 4: Analyze (cont.)

The arithmetic average answers the question, what rate of return

Mary can expect from her investment next year assuming all else

remains the same as in the past?

The geometric average answers the question, what rate of return

Mary can expect over a five-year period?

Defining Risk

The variability of returns from those that are expected.

The chance of financial loss or more formally the variability

of returns associated with a given asset.

Market Risk

The chance that the value of an investment will affect because

of market factors such as economic, political &social events.

In general, the more a given investment’s value responds to

the market, the greater its risk & vice versa.

Interest rate risk

The chance that the changes in interest rates will adversely

affect the value of an investment.

Most investments lose value when the interest rate rises &

increases in value when it falls

Inflation Risk

Refers to uncertainty of purchasing power of cash flows to be

received out of investment.

Investment involves a postponement of current

consumption. If during this period prices of goods and

services go up, Investor loses in terms of purchasing power.

Business Risk

Variability in the income of the firm and the expected

dividend.

Some Industries have higher business risk than others.

Security of firm with higher business risk are risky.

Financial Risk

Degree of debt financing used by the firm in the capital

structure.

Higher the debt financing, the greater the degree of financial

risk.

Debt Financing increases the risk of equity shares by

- Increasing the variability of returns of equity shares

- -increasing the risk of non-receipt of capital in case of

winding up of company.

Types of Risk

Out of the five sources of risk in the investments, the first

three are external to the firm and are uncontrollable

Last two risks i.e business risk and the financial risk are

internal to the firm and are controllable.

Systematic risk

Variability in return due to general factors in the market such

as money supply, inflation, economic recession , interest rate

policy of the government, political factors, credit policy, tax

reforms.

No investor can avoid this risk whatever precaution or

diversification may be resorted to.

It is also called as non-diversifiable risk

Unsystematic Risk

Fluctuations in return from an investment due to factors that

are specific to the particular firm not the market as a whole.

These are unique to the firm, these must be examined

separately for each firm and the industry

For example, the risk that airline industry employees will go

on strike, and airline stock prices will suffer as a result, is

considered to be unsystematic risk

It is also called as diversifiable risk.

Measuring Risk

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Number of Securities

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Market risk

Unique

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Risk Illustrated

Possible Returns on the Stock

Probability

-30% -20% -10% 0% 10% 20% 30% 40%

Outcomes that produce harm

The range of total possible returns on the stock runs from -30% to more than +40%. If the required return on the stock is 10%, then those outcomes less than 10% represent risk to the investor.

A

CHAPTER 8 – Risk, Return and Portfolio Theory

8 - 49

Range

The difference between the maximum and minimum values is

called the range.

As a rough measure of risk, range tells us that common stock is

more risky than treasury bills.

Differences in Levels of Risk Illustrated

Possible Returns on the Stock

Probability

-30% -20% -10% 0% 10% 20% 30% 40%

Outcomes that produce harm The wider the range of probable outcomes the greater the risk of the investment.

A is a much riskier investment than B B

A

Measuring Risk

Variance - Average value of squared deviations from mean.

A measure of volatility.

Standard Deviation - Average value of squared

deviations from mean. A measure of volatility.

Measuring Risk

Problem 1

Estimate the standard deviation of the historical returns on

investment A that were:

Time Return

1 10%

2 24%

3 -12%

4 8%

5 10%

Measuring Risk

Step 1 – Calculate the Historical Average Return

Step 2 – Calculate the Standard Deviation

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Measures of Risk

54

Given an asset's expected return, its variance can be calculated

using the following equation:

N

Var(r) = 2 = S pi(ri – E[r])2

i=1

Where:

N = the number of states

pi = the probability of state i

ri = the return on the stock in state i

E[r] = the expected return on the stock

Measures of Risk

55

The standard deviation is calculated as the positive square root of

the variance:

SD(R) = = 2 = (2)1/2 = (2)0.5

Measures of Risk

56

Probability Distribution:

State Probability Return On Return On

Stock A Stock B

1 20% 5% 50%

2 30% 10% 30%

3 30% 15% 10%

4 20% 20% -10%

E[r]A = 12.5%

E[r]B = 20%

Expected Return

57

In this example, the expected return for stock A would be

calculated as follows:

E[r]A = .2(5%) + .3(10%) + .3(15%) + .2(20%) = 12.5%

Now you try calculating the expected return for stock B!

Expected Return

58

Did you get 20%? If so, you are correct.

If not, here is how to get the correct answer:

E[r]B = .2(50%) + .3(30%) + .3(10%) + .2(-10%) = 20%

So we see that Stock B offers a higher expected return than Stock A.

However, that is only part of the story; we haven't considered risk.

Measures of Risk

59

The variance and standard deviation for stock A is calculated as follows:

2A = .2(.05 -.125)2 + .3(.1 -.125)2 + .3(.15 -.125)2 + .2(.2 -.125)2 = .002625

A (.002625)0.5 .0512 5.12%

Now you try the variance and standard deviation for stock B!

Measures of Risk

60

If you didn’t get the correct answer, here is how to get it:

2

B = .2(.50 -.20)2 + .3(.30 -.20)2 + .3(.10 -.20)2 + .2(-.10 - .20)2 = .042

B (.042)0.5 .2049 20.49%

Although Stock B offers a higher expected return than Stock A, it also is riskier since its variance and standard deviation are greater than Stock A's.

This, however, is still only part of the picture because most investors choose to hold securities as part of a diversified portfolio.

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