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3.7. Other concepts of return

3.7.1. Introduction

In the previous section we outlined the main return measures that are applied in the financial markets. In this section we cover the return concepts that are found in financial literature and have practical and theoretical significance:

• Risk-free rate.

• Expected rate of return.

• Required rate of return.

3.7.2. Risk-free rate

The risk-free rate (rfr) is a concept that has a centre-stage place in finance. It is a concept that some scholars have difficulty in defining (some have even said that it does not exist).

In our view the rfr is the rate on government securities (treasury bills - TBs - and government bonds) of the applicable term to maturity. For example, the applicable rfr in a 3-month option is the 3-month TB rate. If the intention is to hold shares for 5 years, then the relevant rfr (in the CAPM and the CGDDM13) is the 5-year government bond ytm, and so on...

The rfr is the lowest rate that can be earned for the relevant period with certainty. It is certain that the rfr rate will be earned because governments don't default14 (because they have the authority to borrow and tax).

3.7.3. Expected rate of return

The expected rate of return (ER) applies to the future in the case of risky assets (particularly shares). An investment now in a 6-month treasury bill will provide a certain return, whereas an investment in a single share or a portfolio of shares will provide a return in the future that is unknown. It may be expressed as:

ER = [(P, - P0) + I] / P0

where:

P0 = purchase price of share

P1 = expected selling price of share

e x income amount expected(dividend).

expected return and probability of return on a risk-free security

Figure 2: expected return and probability of return on a risk-free security

For example if the present price of the share is LCC10 and it is expected to be LCC12 in 6-months' time, and the expected dividend is LCC0.6, the ER is:

ER = [(EP - P0) + EI] / P0

= [(LCC12 - LCC10) + LCC0.6] / LCC10 = LCC2.6 / LCC10

= 0.26

= 26%.

The concept expected rate of return introduces the probability of return because the return is not certain. Any person in the investment game expects a particular return from an investment but can never be certain about the return - except in the case of a risk-free asset. The latter case may be portrayed as in Figure 2. An investor buys a 3-month TB at a rate 10% pa and intends to hold it to maturity; the probability of receiving the return = 1.0.

Market conditions

Probability

Rate of return

Boom economic conditions ahead; inflation rising

0.25

15%

Moderate economic conditions; little inflation

0.60

10%

Weak economic conditions; falling inflation

0.15

2%

Total

Table 3: Example of expected returns and associated probabilities

With the purchase of a share, no such certainty exists. The investor intuitively assigns or consciously is obliged to assign probabilities to the possible outcomes of the investment in equities. S/he may decide that the expected returns and associated probabilities are as presented in Table 3 and portrayed in Figure 3.

The ER on this investment (call it investment A) is the product of the weighted (by the probabilities -P) returns (R):

ERA = P1R1 + P2R2 + P3R3

= (0.25 x 15%) + (0.60 x 10%) + (0.15 x 2%) = 3.75 + 6.0 + 0.30

= 10.05%.

Another (ridiculous) example is presented in Table 4 (investment B).

possible returns and probability distribution of returns on a risky security

Figure 3: possible returns and probability distribution of returns on a risky security

Outcome number

Probability (P)

Rate of return (R)

Product (P x R)

1

0.1

-75%

-7.5

0.1

-60%

-6.0

0.1

-30%

-3.0

0.1

-15%

-1.5

0.1

0%

0.0

0.1

25%

2.5

0.1

45%

4.5

0.1

60%

6.0

0.1

70%

7.0

0.1

80%

8.0

10.0 = ER

Table 4: Example of expected returns and associated probabilities

In this example the ER is the same as the ER in the case of investment A, i.e. 10%. However, it will be apparent that investment B is substantially more risky than in the case of investment A because the outcome is highly uncertain.

If the investor has a choice between the risk-free asset, investment A and investment B, s/he will no doubt choose the risk-free asset. Second choice is investment A, and last is investment B. This is because investors are risk averse, meaning that they will choose assets that offer greater certainty of return, or least uncertainty of return. To these concepts we shall return.

3.7.4. Required rate of return

The required rate of return (rrr) is a return concept that has pride of place in the CAPM, developed by Nobel Economics Laureate Prof William Sharpe. The essence of the model is that buyers of equity require a particular rate of return that is above the risk-free rate (rfr) and compensates them for the risk inherent in equity investments. Thus the rrr is made up of two elements: the rfr and a premium for risk.

What is the size of the premium? Is it a gut-feel approximation or is it based on something that can be measured? The answer is that, generally, the thinking investor would most like to base it on something, and that something is what happened to the particular market of which the share is a part (obviously the share market).

The accepted measure is the volatility of return relative to the market of which the share is a part. This "risk" is measured by the so-called beta coefficient. It measures the tendency of a share's return to fluctuate relative to fluctuations in the market (in practice a market index). We will return to this issue a little later.

 
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