Risk and return is one of the important topics in finance. The economic foundation of this topic is based on the following assumptions:

Investor rationality: Investors are rational. This means they prefer more money to less and less risk to more, all else equal. The result of this assumption means that the ex-ante risk-return is positively related. Higher the risk higher return.

As risk-averse return-seekers, investors will take actions consistent with the rationality (above) assumptions. They will require higher returns to invest in riskier assets and are willing to accept lower returns on less risky assets.

Similarly, they will seek to reduce risk while attaining the desired level of return or increase return without exceeding the maximum acceptable level of risk.

In this topic, you will learn the following concepts.

Concept 1: Measures of return: Holding period return, Expected return

Concept 2: Measures of risk: variance, standard deviation

Concept 3: How investors use risk, return and Normal distribution to make decisions.

Concept 4: Relation between risk and return

Concept 5: Portfolio return and risk

Concept 1: Measures of return: Holding period return and Expected return.

Return

Return is the net dollar gain or loss of an investment over a specified time period. When you use the term rate of return, then it is expressed as a percentage of the investment’s initial cost. Rate of return is calculated by determining the percentage change from the $ value in the beginning of the period until the end.

Dollar returns

Total dollar return = income from investment + capital gain (loss) due to change in price

Example:

You bought a bond for $950 one year ago. You have received two coupons of $30 each. You can sell the bond for $975 today. What is your total dollar return?

• Income = $30 + $30 = $60
• Capital gain = $975 – $950 = $25
• Total dollar return = $60 + $25 = $85

Percentage returns

It is generally more intuitive to think in terms of percentage rather than dollar returns:

• Dividend yield = income / beginning price
• Capital gains yield = (ending price – beginning price)/ beginning price
• Total percentage return = dividend yield + capital gains yield

Note that the ‘dividend’ yield is just the yield on cash flows received from the security (other than the selling price).

You will learn two types of returns; holding period returns and expected returns.

Example: Calculating Holding Period Returns

You bought a share for $35 in 2015, and you received dividends of $1.25. The share is now selling for $40 in 2022.

2.1	Holding period return (Discrete compounding)	[latex]R_T=R_{CA}+R_I=\frac{P_1-P_0}{P_0}+\frac{{CF}_1}{P_0}[/latex]

What is your total dollar return?

• Dollar return = $1.25 + ($40 – $35) = $6.25

What is your percentage return?

• Dividend yield = $1.25 / $35 = 3.57%
• Capital gains yield = ($40 – $35) / $35 = 14.29%
• Total percentage return = 3.57% + 14.29% = 17.86%

If you notice carefully, the total percentage return is equal to total dollar return / beginning share price.

• Total percentage return = $6.25 / $35 = 17.86%

Expected returns

Expected returns are based on the probabilities of possible “future” outcomes. This is measured in the context of the future.

2.3

Expected return on an asset (Probability given)

[latex]{E(R}_{Asset})=\sum_{i=1}^{n}{(p_i\times R_i)}[/latex]

Suppose you have predicted the following returns for shares CCC and TTT in three possible states of the economy. What are the expected returns?

State	Probability	CCC	TTT
Boom	0.3	0.15	0.25
Normal	0.5	0.10	0.20
Recession	?	0.02	0.01

What is the probability of a recession? 1 – 0.3 – 0.5 = 0.2

What is the expected return of share CCC and TTT?

E(RCCC)=9.9%=0.3*0.15+0.5*0.10+0.2*0.02

E(RTTT)=17.7%=0.3*0.25+0.5*0.20+0.2*0.01

We also use ‘expected’ means ‘average’ if the process is repeated many times. The ‘expected’ return does not have to be a possible return.

2.4

Expected return on an asset (number of periods/equal probability is given)

[latex]{E(R}_{Asset})=\frac{\sum_{i=1}^{n}R_i}{n}[/latex]

Let’s say you have daily/monthly returns of a stock for the last 10 years. In this case, you can use formula 2.4 to calculate “expected”/”average” return. This can give you some idea about what the return is going to be in the future. The reason you apply formula 2.4 in this case because these returns have already occurred so you assign equal probabilities to each occurrence of return.

You need to annualise expected returns so that you can compare the performance of different securities available in the market. Annualise expected return so you can interpret it as return per annum.

Concept 2: Measures of risk: variance, standard deviation

Risk is there whenever investors are not certain about the outcome of an investment. Risk is measured by variance or standard deviation. Essentially, it means by how much a particular return deviates from an expected return (average return).

Variance

When probabilities are given use 2.5 to calculate variance:

Example: Variance of CCC and TTT share.

2.5

Variance of return on an asset (Probability Given)

[latex]Var(R)=\sigma_R^2=\sum_{i=1}^{n}\left\{(p_i\times\left[R_i-E(R)\right]^2\right\}[/latex]

Variance of CCC = 0.3*(0.15 – 0.099)² + 0.5*(0.10 – 0.099)² + 0.2*(0.02 – 0.099)² = 0.00209

Variance of TTT = 0.3*(0.25 – 0.177)² + 0.5*(0.20 – 0.177)² + 0.2*(0.01 – 0.177)² = 0.007441

		Population Variance:  [latex]Var\left(R\right)=\sigma_R^2=\frac{\sum_{i=1}^{n}\left[R_i-E\left(R\right)\right]^2}{n}[/latex]
2.6	Variance of return on an asset
		Sample Variance:  [latex]Var\left(R\right)=\sigma_R^2=\frac{\sum_{i=1}^{n}\left[R_i-E\left(R\right)\right]^2}{n-1}[/latex]

Let’s say you have daily/monthly returns of a stock for the last 10 years. In this case you can use formula 2.6 to calculate variance of return. This can give you some idea about what the variance of return is going to be in the future. The reason you apply formula 2.6 in this case because these returns have already occurred, so you assign equal probabilities to each occurrence of return.

Standard Deviation

This is another measure of risk. It is just the square root of variance. It is easier to interpret than variance because it is measured in the same unit as the return (%).

Use formula 2.7 to compute standard deviation of returns.

2.7

Standard Deviation

[latex]StDev\left(R\right)=\sqrt{Var\left(R\right)}=\sqrt{\sigma_R^2}[/latex]

Annualise the standard deviation of returns so you can interpret it as risk per annum.

Risk Premium

Risk premium is how much compensation you have to pay to investors to get them to invest in equity as a class of asset. Therefore, it is measured as the difference between average equity return and the risk-free rate (government bond rate).

If the average return of CCC and TTT are 9.9% and 17.7%. Risk-free rate is 4.15%, what is the risk premium of CCC and TTT?

Share CCC: 9.9 – 4.15 = 5.75%

Share TTT: 17.7 – 4.15 = 13.55%

Concept 3: How investors use risk, return and Normal distribution to make decisions.

Investors use mean, standard deviation of returns and normal distribution to make investment decisions. Let’s first see what is normal distribution.

Normal distribution

The normal distribution is a symmetric, bell-shaped frequency distribution. It is completely defined by its mean and standard deviation. We will assume that returns are normally distributed. The probability on the left hand side is 50% and on the right hand side is 50%. The total area is 100% or 1.

Image source: https://commons.wikimedia.org/wiki/File:Standard_deviation_diagram_micro.svg

Once you know the normal distribution rule you can apply these rules, mean and standard deviation to measure the probability if return is going to be positive, negative or range of returns.

The normal distribution is just the picture of all possible return outcomes since we have assumed returns are distributed normally. The mean return is the central point of the distribution. The standard deviation is the average deviation from the mean. Assuming investors are rational, the mean is a proxy for expected return and the standard deviation is a proxy for total risk.

Concept 4: Relation between risk and return

There is also a trade-off between risk and return. Higher the risk, higher the return. To give you an example. When you buy a government bond, you get a return of about 2.5% (I am talking about the short term government bond rate here) but if you invest in a real estate stock then you may get a return of 6% or 7% but there is a high risk involved in it. If the market for real estate stock does not do well, you may lose all your money. However, in case of an Australian government bond there is no such risk. Note that there is a positive relation between risk and return.

There are three types of investors. They are:

Risk-neutral investor:

One whose utility is not affected by risk; when chooses to invest, investor focuses only on expected return

Risk-averse investor:

One who demands compensation in the form of higher expected returns in order to be induced into taking on more risk.

Risk-seeking investor:

One who derives utility from being exposed to high risk. The investor may be willing to give up some expected return in order to be exposed to additional risk.

The standard assumption in finance theory is all investors are risk averse.

This does not mean an investor will refuse to bear any risk at all.

Rather, investors consider risk as something undesirable, but may take if compensated with sufficient return; it’s a trade-off between risk and return.

The risk of assets

The risk of an individual asset is summarised by the standard deviation (or variance) of returns.

Investors usually invest in a number of assets (a portfolio) and will be concerned about the risk of their overall portfolio.

You need to consider how individual asset risks will interact to provide you with overall portfolio risk.

Concept 5: Portfolio return and risk

A portfolio is a collection of assets. An asset’s risk and return are important in how they affect the risk and return of the portfolio. The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation. The concept is similar to how you measure risk and return of individual assets.

Each individual has their own level of risk tolerance. Some people are naturally more inclined to take risk, and they will not require the same level of compensation as others for doing so. Individual risk preferences also change through time. You may be willing to take more risk when you are young and without a family. But, once you start a family, your risk tolerance may drop.

Portfolio Weights

Suppose you have $15 000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security?

• $2000 of CCC
• $3000 of KKK
• $4000 of NNN
• $6000 of BBB

Weight of CCC = $2000 / $15000 = 13.33%

Weight of KKK = $3000 / $15000 = 20.00%

Weight of NNN = $4000 / $15000 = 26.67%

Weight of BBB = $6000 / $15000 = 40.00%

Remember the sum of weights of a portfolio is always 100% or 1.

13.33% +20.00% + 26.67% + 40.00% = 100%

Portfolio expected returns

The expected return of a portfolio is the weighted average of the expected returns of the individual assets in the portfolio.

If portfolio return and probabilities are given for different state of the economy, you can find the expected return by finding the portfolio return in each possible state and computing the expected value as you did with individual securities.

Let’s say the expected return of the assets are given as follows:

CCC 19.69%

KKK: 5.25%

NNN: 16.65%

BBB: 18.24%

What is the expected return of the portfolio.

2.11

Expected return for a portfolio

[latex]E(R_p)=\sum_{j=1}^{n}{w_jE\left(R_j\right)}[/latex]

E(RP) = 0.133(19.69%) + 0.2(5.25%) + 0.267(16.65%) + 0.4(18.24%) = 15.41%

Portfolio variance

It is a bit complicated to compute portfolio variance of a 4-asset portfolio. Therefore, for MAF203 we will focus on only on a two-asset portfolio.

Portfolio (comprising two assets) risk depends on:

• the proportion of funds invested in each asset (w)
• the riskiness of the individual assets (σ2)
• the relationship between each asset in the portfolio with respect to risk, correlation (ρij)
• for a two-asset portfolio the variance is:

2.12

Two-asset portfolio variance

[latex]\sigma \:_p^2=\:w_1^2\:\sigma \:_1^2+\:w_2^2\:\sigma \:_2^2+2w_1w_2\rho \:_{1,2}\:\sigma _1\sigma _2\:[/latex] Or [latex]\sigma _p^2=\:w_1^2\:\sigma _1^2+\:w_2^2\:\sigma _2^2+2w_1w_2Cov\left(R_1,R_2\right)[/latex]

Relationship measures

• Covariance
  • This describes the relationship between two variables
  • It is positive if both variables move together in the same directions.
  • is negative if both variables move in the opposite direction.
• Correlation coefficient
  • It is another measure of the strength of a relationship between two variables. It is very similar to covariance.
  • It is equal to the covariance divided by the product of the asset’s standard deviations.
  • It is simply a standardisation of the covariance, and for this reason is bounded by the range +1 to –1.
  • Correlations are measured in percentage. Therefore, it is easy to interpret than covariance.

Example:

Suppose you invest in a portfolio that consists of two stocks. Stock CCC is worth $50,000 and has a standard deviation of 20%. Stock BBB is worth $100,000 and has a standard deviation of 10%. The correlation between the two stocks is 0.85.

What are the portfolio weights?

What are the portfolio risk?

Total $ value of the portfolio = $100,000 + $50,000 = $150,000

Given this, the portfolio weight of Stock CCC is 33.3% = ($50,000 / $150,000)

Portfolio weight of Stock BBB is 66.7% = ($1000,000 / $150,000)

If you plug in this information into the formula, the variance is calculated to be:

Variance = (33.3%^2 x 20%^2) + (66.7%^2 x 10%^2) + (2 x 33.3% x 20% x 66.7% x 10% x 0.85) = 1.64%

Since Variance is not a very intuitive statistic to interpret on its own, you calculate the standard deviation, which is simply the square root of variance. In this example, the square root of 1.64% is 12.81%.

The portfolio risk is 12.81%

References:

• Peirson, G., Brown, R., Easton, S. A., Howard, P., & Pinder, S. (2015). Business finance (Twelfth edition). McGraw-Hill Education.
• Ross, S. A., Trayler, R., Hambusch, G., Koh, C., Glover, K., Westerfield, R., & Jordan, B. (2021). Fundamentals of corporate finance (Eighth edition.). McGraw-Hill Education (Australia) Pty Limited.
• Parrino, R., Au Yong, H. H., Dempsey, M. J., Ekanayake, S., Kidwell, D. S., Kofoed, J., Morkel-Kingsbury, N., & Murray, J. (2014). Fundamentals of corporate finance (Second edition.). John Wiley and Sons Australia, Ltd.